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Introduction to Mathematical Philosophy | Bertrand Russell — Transcript

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0:01preface of introduction to mathematical
0:03philosophy this is a LibriVox recording
0:06all LibriVox recordings are in the
0:08public domain for more information or to
0:12volunteer please visit librivox.org
0:15recording by Landon DC alind at the
0:18University of Iowa in Iowa City Iowa
0:22introduction to mathematical Philosophy
0:25by berand Russell
0:27preface this book is intended
0:29essentially as an introduction and does
0:32not aim at giving an exhaustive
0:34discussion of the problems with which it
0:36deals it seemed desirable to set forth
0:39certain results hitherto only available
0:42to those who have mastered logical
0:44symbolism in a form offering the minimum
0:47of difficulty to the
0:49beginner the utmost Endeavor has been
0:51made to avoid dogmatism on such
0:54questions as are still open to Serious
0:56doubt and this endeavor has to some
0:59extent dominated the choice of topics
1:02considered the beginnings of
1:04mathematical logic are less definitely
1:06known than its later portions but are of
1:10at least equal philosophical interest
1:13much of what is set forth in the
1:15following chapters is not properly to be
1:17called philosophy though the matters
1:20concerned were included in philosophy so
1:23long as no satisfactory science of them
1:26existed the nature of infinity and
1:29continuity for example belonged in
1:31former days to philosophy but belongs
1:34now to mathematics mathematical
1:37philosophy in the strictest sense cannot
1:40perhaps be held to include such definite
1:43scientific results as have been obtained
1:45in this region the philosophy of
1:48mathematics will naturally be expected
1:50to deal with questions on the frontier
1:52of knowledge as to which comparative
1:54certainty is not yet attained but
1:57speculation on such questions is hard ly
2:00likely to be fruitful unless the more
2:02scientific parts of the principles of
2:04mathematics are known a book dealing
2:07with those parts may therefore claim to
2:10be an introduction to mathematical
2:12philosophy though it can hardly claim
2:15except where it steps outside its
2:17Province to be actually dealing with a
2:20part of
2:21philosophy it does deal however with a
2:25body of knowledge which to those who
2:27accept it appears to invalidate much
2:29traditional philosophy and even a good
2:32deal of what is current in the present
2:34day in this way as well as by its
2:38bearing on still unsolved problems
2:41mathematical logic is relevant to
2:43Philosophy for this reason as well as on
2:46account of the intrinsic importance of
2:48the subject some purpose may be served
2:51by a succinct account of the main
2:53results of mathematical logic in a form
2:56requiring neither a knowledge of
2:58mathematics nor an aptitude for
3:00mathematical
3:02symbolism here however as elsewhere the
3:05method is more important than the
3:07results from the point of view of
3:09further research and the method cannot
3:11well be explained within the framework
3:13of such a book as the following it is to
3:16be hoped that some readers may be
3:18sufficiently interested to advance to a
3:20study of the method by which
3:22mathematical logic can be made helpful
3:25in investigating the traditional
3:27problems of philosophy but that is a
3:30topic with which the following Pages
3:32have not attempted to deal Bertrand
3:36Russell end of
3:43preface chapter one of introduction to
3:46mathematical Philosophy by burand
3:48Russell this LibriVox recording is in
3:51the public
3:52domain the series of natural
3:55numbers mathematics is a study which
3:58when we start from its most familiar
4:00portions may be pursued in either of two
4:03opposite directions the more familiar
4:06direction is constructive towards
4:08gradually increasing complexity from
4:10integers to fractions real numbers
4:13complex numbers from addition and
4:15multiplication to differentiation and
4:18integration and on to higher
4:21mathematics the other direction Which is
4:23less familiar proceeds by analyzing to
4:27greater and greater abstractness and
4:29iCal Simplicity instead of asking what
4:32can be defined and deduced from what is
4:35assumed to begin with we ask instead
4:38what more General ideas and principles
4:40can be found in terms of which what was
4:44our starting point can be defined or
4:47deduced it is the fact of pursuing this
4:50opposite direction that characterizes
4:52mathematical philosophy as opposed to
4:55ordinary
4:56mathematics but it should be understood
4:59that the distinction is one not in
5:02subject matter but in the State of Mind
5:04of the
5:05investigator early Greek geometers
5:08passing from the empirical rules of
5:10Egyptian land surveying to the general
5:12propositions by which those rules were
5:14found to be justifiable and then to
5:17ukids axioms and postulates were engaged
5:20in mathematical philosophy according to
5:22the above
5:23definition but when Once the axim and
5:26postulates had been reached their
5:27deductive employment as we find it in
5:30uid belonged to mathematics in the
5:32ordinary
5:33sense the distinction between
5:35mathematics and mathematical philosophy
5:38is one which depends upon the interest
5:40inspiring the research and upon the
5:43stage which the research has reached not
5:45upon the propositions with which the
5:47research is
5:49concerned we may State the same
5:51distinction in another way the most
5:54obvious and easy things in mathematics
5:56are not those that come logically at the
5:58beginning there are things that from the
6:00point of view of logical deduction come
6:03somewhere in the middle just as the
6:06easiest bodies to see are those that are
6:08neither very near nor very far neither
6:10very small nor very great so the easiest
6:13conceptions to grasp are those that are
6:16neither very complex nor very simple
6:19using simple in a logical
6:22sense and as we need two sorts of
6:24instruments the telescope and the
6:26microscope for the enlargement of our
6:28visual power
6:30so we need two sorts of instruments for
6:32the enlargement of our logical Powers
6:35one to take us forward to the higher
6:36mathematics the other to take us
6:38backward to The Logical foundations of
6:41the things that we are inclined to take
6:43for granted in
6:45mathematics we shall find that by
6:47analyzing our ordinary mathematical
6:49Notions we acquire fresh Insight new
6:53powers and the means of reaching whole
6:55new mathematical subjects by adopting
6:57fresh lines of advance after our
6:59backward Journey it is the purpose of
7:02this book to explain mathematical
7:05philosophy simply and un technically
7:08without enlarging upon Those portions
7:11which are so doubtful or difficult that
7:14an elementary treatment is scarcely
7:16possible a full treatment will be found
7:19in principia Mathematica footnote one
7:22Cambridge University press volume 1 1910
7:26Volume 2 1911 volume 3
7:291913 by Whitehead and Russell end of
7:33footnote 1 the treatment in the present
7:36volume is intended merely as an
7:39introduction to the average educated
7:42person of the present day the obvious
7:44starting point of mathematics would be
7:46the series of whole numbers 1 2 3 4
7:51Etc probably only a person with some
7:54mathematical knowledge would think of
7:56beginning with zero instead of one but
7:58we will presume this degree of knowledge
8:01we will take as our starting point the
8:03series 0 1 2 3 n n+ one and so on and it
8:10is this series that we shall mean when
8:13we speak up the series of natural
8:16numbers it is only at a high stage of
8:19civilization that we could take this
8:21series as our starting point it must
8:23have required many ages to discover that
8:26a brace of pheasants and a couple of
8:27days were both in instances of the
8:30number two the degree of abstraction
8:32involved is far from
8:34easy and the discovery that one is a
8:37number must have been difficult as for
8:40zero it is a very recent addition the
8:42Greeks and Romans had no such
8:45digit if we had been embarking upon
8:47mathematical philosophy in early days we
8:51should have had to start with something
8:52less abstract than the series of natural
8:54numbers which we should reach as a stage
8:57on our backward Journey when The Logical
8:59foundations of mathematics have grown
9:01more familiar we shall be able to start
9:04further back at what is now a late stage
9:06in our analysis but for the moment the
9:09natural numbers seem to represent what
9:11is easiest and most familiar in
9:15mathematics but though familiar they are
9:17not understood very few people are
9:20prepared with a definition of what is
9:22meant by number or zero or one it is not
9:26very difficult to see that starting from
9:28zero any other of the natural numbers
9:31can be reached by repeated additions of
9:33one but we shall have to Define what we
9:35mean by adding one and what we mean by
9:39repeated these questions are by no means
9:41easy it was believed until recently that
9:45some at least of these first Notions of
9:48arithmetic must be accepted as too
9:50simple and primitive to be defined since
9:53all terms that are defined are defined
9:55by means of other terms it is clear that
9:58human knowledge must always be content
10:00to accept some terms as intelligible
10:03without definition in order to have a
10:05starting point for its
10:07definitions it is not clear that there
10:09must be terms which are incapable of
10:12definition it is possible that however
10:14far back we go in defining we always
10:17might go further still on the other hand
10:21it is also possible that when analysis
10:23has been pushed far enough we can reach
10:26terms that really are simple and
10:28therefore for logically incapable of the
10:30sort of definition that consists in
10:33analyzing this is a question which it is
10:36not necessary for us to decide for our
10:39purposes it is sufficient to observe
10:41that since human powers are finite the
10:44definitions known to us must always
10:47begin somewhere with terms undefined for
10:50the moment though perhaps not
10:53permanently all traditional pure
10:55mathematics including analytical
10:57geometry may be regarded as consisting
11:00wholly of propositions about the natural
11:03numbers that is to say the terms which
11:06occur can be defined by means of the
11:08natural numbers and the propositions can
11:11be deduced from the properties of the
11:13natural numbers with the addition in
11:15each case of the ideas and propositions
11:18of pure
11:19logic that all traditional pure
11:22mathematics can be derived from the
11:23natural numbers is a fairly recent
11:26discovery though it had long been
11:28suspected
11:30Pythagoras who believed that not only
11:32mathematics but everything else could be
11:34deduced from numbers was the Discover of
11:37the most serious obstacle in this way of
11:39what is called the
11:57arithmetization two which appeared not
12:00to be a number at all the problem thus
12:03raised was solved only in our day and
12:06was only solved completely by the help
12:08of the reduction of arithmetic to logic
12:11which will be explained in following
12:13chapters for the present we shall take
12:16for granted the arithmetization of
12:18mathematics though this was a feat of
12:20the very greatest
12:23importance having reduced all
12:25traditional pure mathematics to the
12:27theory of the natural numbers the next
12:29step in logical analysis was to reduce
12:32this Theory itself to the smallest set
12:34of premises and undefined terms from
12:37which it could be derived this work was
12:39accomplished by piano he showed that the
12:42entire theory of the natural numbers
12:45could be derived from three primitive
12:47ideas and five primitive propositions in
12:50addition to those of pure logic these
12:53three ideas and five propositions thus
12:56became as it were hostages for the whole
12:59of traditional pure mathematics if they
13:01could be defined and proved in terms of
13:03others so could all pure mathematics
13:07their logical weight if one may use such
13:09an expression is equal to that of the
13:12whole series of Sciences that have been
13:15deduced from the theory of the natural
13:17numbers the truth of this whole series
13:20is assured if the truth of the five
13:23primitive propositions is guaranteed
13:25provided of course that there is nothing
13:28erroneous
13:29in the purely logical apparatus which is
13:32also involved the work of analyzing
13:35mathematics is extraordinarily
13:36facilitated by this work of
13:39panas the three primitive ideas in
13:42piana's arithmetic are zero number
13:46successor by successor he means the next
13:49number in the natural order that is to
13:52say the successor of zero is one the
13:55successor of one is two and so on by
13:59number he means in this connection the
14:01class of the natural numbers footnote
14:04one we shall use number in this sense in
14:07the present chapter afterwards the word
14:10will be used in a more General sense end
14:12of footnote
14:14one he is not assuming that we know all
14:17the members of this class but only that
14:20we know what we mean when we say that
14:22this or that is a number just as we know
14:25what we mean when we say Jones is a man
14:28though we do not know all men
14:31individually the five primitive
14:33propositions which Pano assumes are one
14:37zero is a number two the successor of
14:41any number is a number three no two
14:44numbers have the same successor four
14:48zero is not the successor of any
14:51number five any property which belongs
14:54to zero and also to the successor of
14:57every number which has the prop property
14:59belongs to all
15:01numbers the last of these is the
15:03principle of mathematical induction we
15:06shall have much to say concerning
15:08mathematical induction in the sequel for
15:11the present we are concerned with it
15:12only as it occurs in piano's analysis of
15:17arithmetic let us consider briefly the
15:19kind of way in which the theory of the
15:22natural numbers results from these three
15:24ideas and five
15:25propositions to begin with we Define one
15:28one as the successor of zero two as the
15:32successor of one and so on we can
15:35obviously go on as long as we like with
15:38these definitions since in virtue of two
15:41every number that we reach will have a
15:43successor and in virtue of three this
15:47cannot be any of the numbers already
15:49defined because if it were two different
15:51numbers would have the same successor
15:54and in virtue of four none of the
15:56numbers we reach in the series of
15:58success successors can be
16:00zero thus the series of successors gives
16:03us an endless series of continually new
16:06numbers in virtue of five all numbers
16:10come in this series which begins with
16:12zero and travels on through successive
16:15successors for a zero belongs to this
16:18series and B if a number n belongs to it
16:22so does its successor whence by
16:24mathematical induction every number
16:27belongs to the series
16:29suppose we wish to define the sum of two
16:32numbers taking any number M we Define M
16:36plus 0 as M and M + n + 1 as the
16:41successor of m +
16:44n in virtue of five this gives a
16:46definition of the sum of M and N
16:49whatever number n may be similarly we
16:53can Define the product of any two
16:55numbers the reader can easily convince
16:58himself
16:59that any Ordinary Elementary proposition
17:01of arithmetic can be proved by means of
17:03our five premises and if he has any
17:06difficulty he can find the proof in
17:10piano it is time now to turn to the
17:12considerations which make it necessary
17:14to advance beyond the standpoint of
17:16piano who represents the last Perfection
17:20of the arithmetization of mathematics to
17:23that of Fraga who first succeeded in
17:27logician I.E in reducing to Logic the
17:31arithmetical Notions which his
17:33predecessors had shown to be sufficient
17:35for
17:36mathematics we shall not in this chapter
17:39actually give fraus definition of number
17:41and of particular numbers but we shall
17:43give some of the reasons why piano's
17:46treatment is less final than it appears
17:49to
17:49be in the first place piana's three
17:53primitive ideas namely zero number and
17:56successor are cap capable of an infinite
17:59number of different
18:01interpretations all of which will
18:03satisfy the five primitive propositions
18:06we will give some
18:08examples one let zero be taken to mean
18:12100 and let number be taken to mean the
18:15numbers from 100 onward in the series of
18:18natural
18:19numbers then all of our primitive
18:21propositions are satisfied even the
18:23fourth for though 100 is the successor
18:26of 99 99 is not a number in the sense
18:30which we are now giving to the word
18:33number it is obvious that any number may
18:36be substituted for 100 in this
18:39example two let Zero have its usual
18:42meaning but let number mean what we
18:44usually call even numbers and let the
18:47successor of a number be what results
18:50from adding two to it then one will
18:53stand for the number two two will stand
18:55for the number four and so on the series
18:58of numbers now will be 0 2 4 6 8 and so
19:04on all pianos five premises are
19:07satisfied
19:09still three let zero mean the number one
19:12and let number mean the set 1 1/2 1/4
19:161/8 116th and so on and let successor
19:20mean half then all piano's five aums
19:24will be true of this set it is clear
19:27that such examples might be multiplied
19:30indefinitely in fact given any series X
19:34Sub 0 x sub1 x sub2 x sub3 x subn and so
19:40on which is endless contains no
19:44repetitions has a beginning and has no
19:47terms that cannot be reached from the
19:49beginning in a finite number of steps we
19:52have a set of terms verifying piano's
19:55axioms this is easily seen though the
19:57formal proof is somewhat long let zero
20:01mean X subz let number mean the whole
20:04set of terms and let the successor of x
20:07subn mean x subn + 1 then 1 zero is a
20:13number that is X subz is a member of the
20:18set two the successor of any number is a
20:21number that is taking any term X subn in
20:25the set x subn + 1
20:28is also in the
20:30set three no two numbers have the same
20:33successor that is if x subm and x subn
20:39are two different members of the set x
20:42subn + 1 and x subn + one are different
20:47this results from the fact that by
20:50hypothesis there are no repetitions in
20:52the set four zero is not the successor
20:56of any number that that is no term in
20:59the set comes before x
21:03subz five this becomes any property
21:07which belongs to X Sub 0 and belongs to
21:11x subn +1 provided it belongs to X subn
21:15belongs to all the
21:17X's this follows from the corresponding
21:20property for
21:22numbers a series of the form X Sub 0 x
21:26sub1 x sub 2 X subn and so on in which
21:31there is a first term a successor to
21:33each term so that there is no last term
21:36no repetitions and every term can be
21:39reached from the start in a finite
21:41number of steps is called a
21:44progression progressions are of great
21:47importance in the principles of
21:49mathematics as we have just seen every
21:52progression verifies piano's five
21:54axioms it can be proved conversely that
21:58every series which verifies Pano's five
22:00aums is a progression hence these five
22:04axium may be used to define the class of
22:07progressions progressions are those
22:09series which verify these five axioms
22:12any progressions may be taken as the
22:14basis of pure mathematics we may give
22:17the name zero to its first term the name
22:20number to the whole set of its terms and
22:23the name successor to the next in the
22:26progression the progression need not be
22:28composed of numbers it may be composed
22:31of points in space or moments in time or
22:34any other terms of which there is an
22:36infinite
22:37Supply each different progression will
22:40give rise to a different interpretation
22:43of all the propositions of traditional
22:45pure mathematics all these possible
22:47interpretations will be equally
22:50true in Pano's system there is nothing
22:53to enable us to distinguish between
22:55these different interpretations of his
22:57primitive
22:58ideas it is assumed that we know what is
23:01meant by zero and that we shall not
23:03suppose that this symbol means 100 or
23:06Cleopatra's Needle or any of the other
23:09things that it might
23:11mean this point that zero and number and
23:16successor cannot be defined by means of
23:18piano's five axioms but must be
23:21independently understood is important we
23:24want our numbers not merely to verify
23:27mathematical formula but to apply in the
23:30right way to Common objects we want to
23:33have 10 fingers and two eyes and one
23:36nose a system in which one meant 100 and
23:40two meant 101 and so on might be all
23:43right for pure mathematics but would not
23:46suit daily
23:47life we want zero and number and
23:50successor to have meanings which will
23:52give us the right allowance of fingers
23:55and eyes and
23:56noses we have have already some
23:58knowledge though not sufficiently
24:00articulate or analytic of what we mean
24:03by one and two and so on and our use of
24:07numbers in arithmetic must conform to
24:09this knowledge we cannot secure that
24:12this shall be the case by piano's method
24:15all that we can do if we adopt his
24:17method is to say we know what we mean by
24:21zero and number and successor though we
24:24cannot explain what we mean in terms of
24:26other simpler
24:28Concepts it is quite legitimate to say
24:31this when we must and at some point we
24:33all must but it is the object of
24:36mathematical philosophy to put off
24:38saying it as long as
24:40possible by The Logical theory of
24:42arithmetic we are able to put it off for
24:45a very long
24:47time it might be suggested that instead
24:50of setting up zero and number and
24:52successor as terms of which we know the
24:54meaning although we cannot Define them
24:57we might might let them stand for any
24:58three terms that verify piano's axioms
25:02they will then no longer be terms which
25:04have a meaning that is definite though
25:06undefined they will be variables terms
25:09concerning which we make certain
25:11hypotheses namely those stated in the
25:13five aums but which are otherwise
25:16undetermined if we adopt this plan our
25:19theorems will not be proved concerning
25:21an astain set of terms called the
25:23natural numbers but concerning all sets
25:26of terms having certain properties
25:28such a procedure is not vicious indeed
25:31for certain purposes it represents a
25:34valuable
25:35generalization but from the two points
25:37of view it fails to give an adequate
25:40basis for arithmetic in the first place
25:43it does not enable us to know whether
25:45there are any sets of terms verifying
25:47piano's axioms it does not even give the
25:50faintest suggestion of any way of
25:53discovering whether there are such sets
25:56in the first place as already observed
25:59we want our numbers to be such as can be
26:01used for counting common objects and
26:04this requires that our numbers should
26:06have a definite meaning not merely that
26:09they should have certain formal
26:12properties this definite meaning is
26:14defined by The Logical theory of
26:18arithmetic end of chapter
26:261
26:28chapter two of introduction to
26:30mathematical Philosophy by Bertrand
26:32Russell this LibriVox recording is in
26:35the public
26:36domain definition of Number the question
26:40what is a number is one which has been
26:42often asked but has only been correctly
26:45answered in our own time the answer was
26:48given by Fraga in
26:501884 in his gr log and de arithmetic
26:53footnote one the same answer is given
26:56more fully and with more development in
26:58his grun gazetta de arithmetic volume 1
27:031893 end of footnote
27:061 although this book is quite short not
27:10difficult and of the very highest
27:11importance it attracted almost no
27:14attention and the definition of number
27:16which it contains remained practically
27:19unknown until it was rediscovered by the
27:21present author in
27:241901 in Seeking a definition of number
27:28the first thing to be clear about is
27:30what we may call the grammar of our
27:32inquiry many philosophers when
27:34attempting to Define number are really
27:37setting to work to Define plurality
27:39which is quite a different thing number
27:42is what is characteristic of numbers as
27:45man is what is characteristic of men a
27:48plurality is not an instance of number
27:51but of some particular number a trio of
27:54men for example is an instance of the
27:56number three
27:58and the number three is an instance of
28:00number but the trio is not an instance
28:03of
28:04number this point may seem Elementary
28:07and scarcely worth mentioning yet it has
28:09proved too subtle for the philosophers
28:12with few
28:13exceptions a particular number is not
28:16identical with any collection of terms
28:18having that number the number three is
28:20not identical with the trio consisting
28:23of brown Jones and
28:25Robinson the number three is something
28:28which all trios have in common and which
28:30distinguishes them from other
28:33collections a number is something that
28:35characterizes certain collections namely
28:38those that have that
28:39number instead of speaking of a
28:42collection we shall as a rule speak of a
28:44class or sometimes a set other words
28:47used in mathematics for the same thing
28:49are Aggregate and
28:52manifold we shall have much to say later
28:54on about classes for the present we
28:57shall say as little as
29:00possible but there are some remarks that
29:03must be made
29:04immediately a class or collection may be
29:08defined in two ways that at first sight
29:10seem quite distinct we may enumerate its
29:13members as when we say the collection I
29:16mean is Brown Jones and
29:18Robinson or we may mention a defining
29:21property as when we speak of mankind or
29:24the inhabitants of London the definition
29:27would
29:27numerates is called a definition by
29:30extension and the one which mentions a
29:32defining property is called a definition
29:35by
29:36intention of these two kinds of
29:38definition the one by intention is
29:41logically more
29:42fundamental this is shown by two
29:45considerations one that the extensional
29:47definition can always be reduced to an
29:49intentional one two that the intentional
29:53one often cannot even theoretically be
29:55reduced to the extension
29:58one each of these points needs a word of
30:02explanation One Brown Jones and Robinson
30:06all of them possess a certain property
30:08which is possessed by nothing else in
30:10the whole universe namely the property
30:13of being either brown or Jones or
30:16Robinson this property can be used to
30:19give a definition by intention of the
30:21class consisting of brown and Jones and
30:24Robinson consider such a formula as X is
30:28brown or X is Jones or X is
30:31Robinson this formula will be true for
30:34just three X's namely Brown and Jones
30:37and Robinson in this respect it
30:40resembles a cubic equation with its
30:42three Roots it may be taken as assigning
30:45a property common to the members of the
30:47class consisting of these three men and
30:50peculiar to
30:52them a similar treatment can obviously
30:55be applied to any other class given
30:57given in
30:58extension two it is obvious that in
31:02practice we can often know a great deal
31:05about a class without being able to
31:07enumerate its members no one man could
31:10actually enumerate all men or even all
31:12the inhabitants of London yet a great
31:15deal is known about each of these
31:17classes this is enough to show that
31:20definition by extension is not necessary
31:23to knowledge about a class but when we
31:26come to consider infinite classes we
31:29find that enumeration is not even
31:30theoretically possible for beings who
31:33only live for a finite time we cannot
31:36enumerate all the natural numbers they
31:39are 0 1 2 3 and so
31:43on at some point we must content
31:46ourselves with and so on we cannot
31:49enumerate all fractions or all your
31:51rational numbers or all of any other
31:54infinite
31:55collection thus our knowledge and in
31:57regard to all such collections can only
31:59be derived from a definition by
32:03intention these remarks are relevant
32:06when we are seeking the definition of
32:07number in three different ways in the
32:10first place numbers themselves form an
32:13infinite collection and cannot therefore
32:15be defined by
32:17enumeration in the second place the
32:20collections having a given number of
32:21terms themselves presumably form an
32:24infinite
32:24collection it is to be presumed for
32:27example that there are an infinite
32:29collection of trios in the world for if
32:32this were not the case the total number
32:34of things in the world would be finite
32:37which though possible seems
32:39unlikely in the third place we wish to
32:42Define number in such a way that
32:44infinite numbers may be possible thus we
32:47must be able to speak of the number of
32:49terms in an infinite collection and such
32:52a collection must be defined by
32:55intention that is by by a property
32:57common to all its members and peculiar
32:59to
33:01them for many purposes a class and a
33:04defining characteristic of it are
33:06practically
33:07interchangeable the vital difference
33:09between the two consists in the fact
33:12that there is only one class having a
33:14given set of members whereas there are
33:17always many different characteristics by
33:19which a given class may be defined men
33:23may be defined as featherless bipeds or
33:25as rational animals or more correctly by
33:28the traits by which Swift delineates the
33:31yahoos it is this fact that a defining
33:34characteristic is never unique which
33:36makes classes useful otherwise we could
33:39be content with the properties common
33:42and peculiar to their members footnote
33:44one as will be explained later classes
33:48may be regarded as logical fictions
33:51manufactured out of defining
33:53characteristics but for the present it
33:55will simplify our exposition to treat
33:58classes as if they were real end of
34:01footnote
34:02one any one of these properties can be
34:05used in place of the class whenever
34:07uniqueness is not
34:10important returning now to the
34:12definition of number it is clear that
34:14number is a way of bringing together
34:17certain collections namely those that
34:20have a given number of terms we can
34:23suppose all couples in one bundle all
34:26trios in another and so on in this way
34:30we obtain various bundles of collections
34:33each bundle consisting of all the
34:36collections that have a certain number
34:38of
34:38terms each bundle is a class whose
34:42members are collections that is
34:45classes thus each is a class of
34:48classes the bundle consisting of all
34:51couples for example is a class of
34:54classes each couple is a class of two
34:56members
34:57and the whole bundle of couples is a
34:59class with an infinite number of members
35:03Each of which is a class of two
35:06members how shall we decide whether two
35:09collections are to belong to the same
35:11bundle the answer that suggests itself
35:13is find out how many members each has
35:17and put them in the same bundle if they
35:19have the same number of
35:21members but this presupposes that we
35:23have to find numbers and that we know
35:26how to discover how many terms a
35:28collection
35:30has we are so used to the operation of
35:32counting that such a presupposition
35:35might easily pass unnoticed in fact
35:38however counting though familiar is
35:41logically a very complex operation
35:43moreover it is only available as a means
35:46of discovering how many terms of a
35:48collection has when the collection is
35:51finite our definition of number must not
35:54assume in advance that all numbers are
35:56finite
35:57and we cannot in any case without a
35:59vicious circle use counting to Define
36:02numbers because numbers are used in
36:05counting we need therefore some other
36:08method of deciding when two collections
36:10have the same number of
36:12terms in actual fact it is simpler
36:15logically to find out whether two
36:17collections have the same number of
36:18terms than it is to Define what that
36:21number is an illustration will make this
36:25clear if there were no polygamy or
36:27polyandry anywhere in the world it is
36:30clear that the number of husbands living
36:32at any moment would be exactly the same
36:35as the number of
36:36wives we do not need a census to assure
36:39us of this nor do we need to know what
36:42is the actual number of husbands and
36:44wives we know that the number must be
36:46the same in both collections because
36:49each husband has one wife and each wife
36:52has one husband the relation of husband
36:54and wife is what is called one
36:57one a relation is said to be one one
37:01when if x has the relation in question
37:03to Y no other term X Prime has the same
37:07relation to Y and X does not have the
37:10same relation to any term y Prime other
37:13than
37:14y when only the first of these two
37:17conditions is fulfilled the relation is
37:19called one many when only the second is
37:22fulfilled it is called many
37:24one it should be observed that the
37:26number number one is not used in these
37:29definitions in Christian countries the
37:32relation of husband to wife is one one
37:35in mohamedan countries it is one many in
37:39Tibet it is many one the relation of
37:42father to son is one many that of son to
37:46father is many one but that of eldest
37:49son to father is 1
37:52one if n is any number the relation of n
37:55to n+1 1 is 1 one so is the relation of
38:00n to 2 N or to 3
38:02n when we are considering only positive
38:05numbers the relation of n to n s is 1
38:09one but when negative numbers are
38:11admitted it becomes 2 one since n and
38:15negative n have the same
38:17square these instances should suffice to
38:20make clear the Notions of one one one
38:22many and many one relations which play a
38:25great part in the principles of
38:27mathematics not only in relation to the
38:29definition of numbers but in many other
38:33connections two classes are said to be
38:36similar when there is a one- one
38:38relation which correlates the terms of
38:40the one class each with one term of the
38:42other classes in the same manner in
38:45which the relation of marriage
38:46correlates husbands with
38:49wives a few preliminary definitions will
38:51help us to State this definition more
38:54precisely the class of those terms ter
38:56that have a given relation to something
38:58or other is called the domain of that
39:02relation thus fathers are the domain of
39:04the relation of father to child husbands
39:07are the domain of the relation of
39:09husband to wife wives are the domain of
39:12the relation of wife to husband and
39:15husbands and wives together are the
39:17domain of the relation of
39:19marriage the relation of wife to husband
39:22is called the converse of the relation
39:25of husband to wife
39:27similarly less is the converse of
39:29greater later is the converse of earlier
39:32and so on generally the converse of a
39:36given relation is that relation which
39:38holds between Y and X whenever the given
39:42relation holds between X and Y the
39:45converse domain of a relation is the
39:48domain of its
39:49Converse thus the class of wives is the
39:52converse domain of the relation of
39:54husband to wife we may now State our
39:57definition of similarity as
40:00follows one class is said to be similar
40:03to another when there is a one one
40:06relation of which the one class is the
40:08domain while the other is the converse
40:11domain it is easy to prove one that
40:14every class is similar to itself two
40:18that if a class Alpha is similar to a
40:21class beta then beta is similar to Alpha
40:25three that if Alpha Al is similar to
40:27Beta And beta to gamma then Alpha is
40:30similar to
40:31gamma a relation is said to be reflexive
40:35when it possesses the first of these
40:37properties symmetrical when it possesses
40:39the second and transitive when it
40:41possesses the
40:43third it is obvious that a relation
40:46which is symmetrical and transitive must
40:48be reflexive throughout its
40:51domain relations which possess these
40:53properties are an important kind and it
40:56is worthwhile to note that similarity is
41:00one of this kind of
41:02relations it is obvious to common sense
41:06that two finite classes have the same
41:08number of terms if they are similar but
41:10not
41:11otherwise the act of counting consists
41:13in establishing a one- one correlation
41:16between the set of objects counted and
41:19the natural numbers excluding zero that
41:22are used up in the
41:24process accordingly Common Sense
41:27concludes that there are as many objects
41:29in the set to be counted as there are
41:31numbers up to the last number used in
41:33the
41:34counting and we also know that so long
41:37as we confine ourselves to finite
41:39numbers there are just n numbers from
41:42one up to
41:44n hence it follows that the last number
41:47used in counting a collection is the
41:50number of terms in the collection
41:52provided the collection is
41:54finite but this result besides being
41:57only applicable to finite collections
41:59depends upon and assumes the fact that
42:03two classes which are similar have the
42:05same number of terms for what we do when
42:08we count say 10 objects is to show that
42:11the set of these objects is similar to
42:13the set of numbers from 1 to 10 the
42:17notion of similarity is logically
42:19presupposed in the operation of counting
42:22and is logically simpler though less
42:24familiar in in counting it is necessary
42:28to take the objects counted in a certain
42:30order as first second third and the rest
42:35but order is not of the essence of
42:37number it is an irrelevant addition an
42:40unnecessary complication from The
42:43Logical point of view the notion of
42:45similarity does not demand an order for
42:49example we saw that the number of
42:51husbands is the same as the number of
42:53wives without having to establish an
42:56order of precedents among
42:58them the notion of similarity also does
43:01not require that the classes which are
43:04similar should be
43:06finite take for example the natural
43:08numbers excluding zero on the one hand
43:11and the fractions which have one for
43:13their numerator on the other hand it is
43:16obvious that we can correlate two with
43:181/2 3 with 1/3 and so on thus proving
43:23that the two classes are
43:25similar we may thus use the notion of
43:28similarity to decide when two
43:30collections are to belong to the same
43:32bundle in the sense in which we were
43:35asking this question earlier in this
43:38chapter we want to make one bundle
43:40containing the class that has no members
43:43this will be for the number zero then we
43:46want a bundle of all the classes that
43:49have one member this will be for the
43:51number
43:52one then for the number two we want a
43:55Bund
43:56consisting of all couples then one of
43:59all trios and so on given any collection
44:03we can Define the bundle it is to belong
44:05to as being the class of all those
44:08collections that are similar to
44:11it it is very easy to see that if for
44:15example a collection has three members
44:18the class of all those collections that
44:20are similar to it will be the class of
44:23trios and whatever number of terms a
44:26collection may have those collections
44:28that are similar to it will have the
44:31same number of terms we may take this as
44:34a definition of having the same number
44:37of
44:38terms it is obvious that it gives
44:40results conformable to usage so long as
44:43we can find ourselves to finite
44:46collections so far we have not suggested
44:49anything in the slightest degree
44:51paradoxical but when we come to the
44:53actual definition of numbers we cannot
44:56avoid what must at First Sight seem a
44:59paradox though this impression will soon
45:02wear off we naturally think that the
45:05class of couples for example is
45:07something different from the number two
45:10but there is no doubt about the class of
45:12couples it is indubitable and not
45:15difficult to Define whereas the number
45:18two in any other sense is a metaphysical
45:21entity about which we can never feel
45:23sure that it exists or that we have
45:26tracked it down it is therefore more
45:29prudent to content ourselves with the
45:31class of couples which we are sure of
45:34than to hunt for a problematical number
45:36two which must always remain
45:39elusive accordingly we set up the
45:42following
45:43definition the number of a class is the
45:46class of all those classes that are
45:48similar to it thus the number of a
45:52couple will be the class of all couples
45:55in fact the class of all couples will be
45:58the number two according to our
46:00definition at the expense of a little
46:03Oddity this definition secures
46:06definiteness and
46:08indubitable and it is not difficult to
46:10prove that numbers so defined have all
46:13the properties that we expect numbers to
46:17have we may now go on to Define numbers
46:20in general as any one of the bundles
46:23into which similarity collects
46:26classes a number will be a set of
46:29classes such as that any two are similar
46:33to each other and none outside the set
46:36are similar to any inside the set in
46:39other words a number in general is any
46:42collection which is the number of one of
46:44its members or more simply still a
46:48number is anything which is the number
46:51of some class such a definition has a
46:54verbal appearance of being being
46:56circular but in fact it is not we Define
47:00the number of a given class without
47:02using the notion of number in
47:04general therefore we may Define number
47:07in general in terms of the number of a
47:10given class without committing any
47:12logical
47:14error definitions of this sort are in
47:17fact very common the class of fathers
47:20for example would have to be defined by
47:22first defining what it is to be the
47:24father of somebody then the class of
47:26fathers will be those who are somebody's
47:29father similarly if we want to Define
47:32square numbers say we must first Define
47:35what we mean by saying that one number
47:37is the square of another and then Define
47:40square numbers as those that are the
47:43squares of other numbers this kind of
47:46procedure is very common and it is
47:48important to realize that it is
47:51legitimate and even often
47:54necessary we have we have now given a
47:56definition of numbers which will serve
47:58for finite collections it remains to be
48:01seen how it will serve for infinite
48:04collections but first we must decide
48:06what we mean by finite and infinite
48:09which cannot be done within the limits
48:11of the present
48:13chapter end of chapter
48:212 chapter 3 of introduction to
48:24mathematical philosophy by berand
48:26Russell this LibriVox recording is in
48:29the public
48:30domain finitude and mathematical
48:33induction the series of natural numbers
48:36as we saw in chapter one can all be
48:38defined if we know what we mean by the
48:41three terms zero number and successor
48:44but we may go a step farther we can
48:47Define all the natural numbers if we
48:49know what we mean by zero and
48:51successor it will help us to understand
48:54the difference between finite and
48:56infinite to see how this can be done and
48:59why the method by which it is done
49:01cannot be extended beyond the
49:03finite we will not yet consider how zero
49:06and successor are to be
49:09defined we will for the moment assume
49:11that we know what these terms mean and
49:14show how then all other natural numbers
49:17can be
49:18obtained it is easy to see that we can
49:21reach any assigned number say
49:2430,000 we first Define one as the
49:26successor of zero then we Define two as
49:29the successor of one and so on in the
49:32case of an assigned number such as
49:3430,000 the proof that we can reach it by
49:37proceeding step by step in this fashion
49:39may be made if we have the patience by
49:42actual
49:43experiment we can go on until we
49:45actually arrive at
49:4730,000 but although the method of
49:49experiment is available for each
49:51particular number it is not available
49:54for proving The General proposition that
49:56all such numbers can be reached in this
49:58way that is by proceeding from zero step
50:01by step from each number to its
50:04successor is there any other way by
50:07which this can be proved let us consider
50:10the question the other way round what
50:12are the numbers that can be reached
50:14given the terms zero and successor is
50:17there any way by which we can Define the
50:20whole class of such numbers we reach one
50:23as the successor of zero two two as the
50:26successor of one three as the successor
50:28of two and so on it is this and so on
50:33that we wish to replace by something
50:35less vague and indefinite we might be
50:38tempted to say that and so on means that
50:41the process of proceeding to the
50:43successor may be repeated any finite
50:46number of
50:47times but the problem upon which we are
50:49engaged is the problem of defining
50:52finite number and therefore we must not
50:55use use this notion in our
50:57definition our definition must not
50:59assume that we know what a finite number
51:02is the key to our problem lies in
51:05mathematical induction it will be
51:07remembered that in chapter 1 this was
51:10the fifth of the five primitive
51:12propositions which we laid down about
51:14the natural numbers it stated that any
51:17property which belongs to zero and the
51:20successor of any property which has the
51:22property belongs to all the natural
51:24numbers
51:26this was then presented as a principle
51:29but we shall now adopt it as a
51:32definition it is not difficult to see
51:34that the terms obeying it are the same
51:36as the numbers that can be reached from
51:38zero by successive steps from next to
51:42next but as the point is important we
51:45will set forth the matter in some
51:48detail we shall do well to begin with
51:50some definitions which will be useful in
51:52other connections also a property is
51:56said to be hereditary in the natural
51:58number series if whenever it belongs to
52:01a number n it also belongs to n + one
52:05the successor of
52:06n similarly a class is said to be
52:09hereditary if whenever N is a member of
52:12the class so is n + one it is easy to
52:16see though we are not yet supposed to
52:18know that to say a property is
52:20hereditary is equivalent to saying that
52:23it belongs to all the natural numers
52:25numbers not less than some one of them
52:27say it must belong to all that are not
52:30less than 100 or all that are less than
52:331,000 or it may be that it belongs to
52:36all that are not less than zero that is
52:38to all without
52:40exception a property is said to be
52:42inductive when it is a hereditary
52:44property which belongs to zero similarly
52:48a class is inductive when it is a
52:50hereditary class of which zero is a
52:54member given a hereditary class of which
52:57zero is a member it follows that one is
52:59a member of it because a hereditary
53:01class contains the successor of its
53:03members and one is the successor of zero
53:06similarly given a hereditary class of
53:09which one is a member it follows that
53:11two is a member of it and so on thus we
53:15can prove by a step-by-step procedure
53:18that any assigned natural number say
53:2030,000 is a member of every inductive
53:24class
53:25we will Define the posterity of a given
53:27natural number with respect to the
53:29relation immediate predecessor which is
53:32the converse of
53:34successor as all those terms that belong
53:37to every hereditary class to which the
53:39given number belongs it is again easy to
53:42see that the posterity of a natural
53:44number consists of itself and all
53:47greater natural numbers but this also we
53:50do not yet officially
53:52know by the above definitions the
53:55posterity of zero will consist of those
53:57terms which belong to every inductive
54:01class it is now not difficult to make it
54:04obvious that the posterity of zero is
54:06the same set as those terms that can be
54:09reached from zero by successive steps
54:12from next to next for in the first place
54:16zero belongs to both these sets in the
54:18sense in which we have to find our
54:20terms in the second place if n belongs
54:23to both sets so does
54:26n+1 it is to be observed that we are
54:29dealing here with the kind of matter
54:31that does not admit of precise proof
54:33namely the comparison of a relatively
54:36vague idea with a precise one the notion
54:39of those terms that can be reached from
54:41zero by successive steps from next to
54:44next is vague though it seems as if it
54:47conveyed a definite meaning on the other
54:50hand the posterity of zero is precise
54:53and explicit just where the other idea
54:56is hazy it may be taken as giving what
54:59we meant to mean when we spoke of the
55:02terms that can be reached from zero by
55:05successive
55:06steps we now lay down the following
55:09definition the natural numbers are the
55:11posterity of zero with respect to the
55:14relation immediate predecessor which is
55:16the converse of
55:19successor we have thus arrived at a
55:21definition of one of piano's three
55:23primitive ideas in terms of the other
55:26two as a result of this definition two
55:29of his primitive propositions namely the
55:31one asserting that zero is a number and
55:33the one asserting mathematical induction
55:36become unnecessary since they result
55:39from the
55:40definition the one asserting that the
55:42successor of a natural number is a
55:44natural number is only needed in the
55:46weakened form every natural number has a
55:50successor we can of course easily Define
55:53zero and successor by means of the
55:55definition of number in general which we
55:58arrived at in Chapter 2 the number zero
56:01is the number of terms in a class which
56:03has no members that is in the class
56:06which is called the null class by the
56:10general definition of Number the number
56:12of terms in the null class is the set of
56:15all classes similar to the null class
56:17that is as is easily proved the set
56:20consisting of the null class all alone
56:24that is the the class whose only member
56:26is the null class this is not identical
56:29with the null class it has one member
56:32namely the null class whereas the null
56:34class itself has no members a class
56:38which has one member is never identical
56:40with that one member as we shall explain
56:43when we come to the theory of
56:46classes thus we have the following
56:48purely logical
56:50definition zero is the class whose only
56:53member is the null class
56:56it remains to define successor given any
56:59number n let Alpha be a class which has
57:02n members and let X be a term which is
57:05not a member of alpha then the class
57:08consisting of alpha with X added on will
57:11have n+ one members thus we have the
57:14following
57:15definition the successor of the number
57:18of terms in the class Alpha is the
57:21number of terms in the class consisting
57:23of alpha together with X where X is any
57:27term not belonging to the
57:29class certain niceties are required to
57:32make this definition perfect but they
57:35need not concern us footnote one C
57:38principia Mathematica Volume 2 Star 110
57:42end of footnote
57:431 it will be remembered that we have
57:47already given in chapter 2 a logical
57:50definition of the number of terms in a
57:52class namely we defined it as as the set
57:55of all classes that are similar to the
57:57given
57:58class we have thus reduced piana's three
58:01primitive ideas to ideas of logic we
58:04have given definitions of them which
58:06make them definite no longer capable of
58:09an Infinity of different
58:11meanings as they were when they were
58:14only determinant to the extent of
58:16obeying piano's five axioms we have
58:19removed them from the fundamental
58:20apparatus of terms that must be merely
58:23apprehended
58:25and have thus increased the deductive
58:27articulation of
58:29mathematics as regards the five
58:31primitive propositions we have already
58:34succeeded in making two of them
58:35demonstrable by our definition of
58:37natural number how stands it with the
58:40remaining three it is very easy to prove
58:43that zero is not the successor of any
58:46number and that the successor of any
58:48number is a
58:50number but there is a difficulty about
58:53the remaining primitive proposition I
58:55namely no two numbers have the same
58:58successor the difficulty does not arise
59:01unless the total number of individuals
59:03in the universe is
59:05finite for given two numbers M and N
59:08neither which is the total number of
59:10individuals in the universe it is easy
59:13to prove that we cannot have n + 1 = n +
59:171 unless we have m is equal to
59:21n but let us suppose that the total
59:24number of individuals in the universe
59:26were say 10 then there would be no class
59:30of 11 individuals and the number 11
59:33would be the null class so with the
59:36number 12 thus we should have 11 = 12
59:40therefore the successor of 10 would be
59:43the same as the successor of 11 although
59:4610 would not be the same as
59:4811 thus we should have two different
59:50numbers with the same successor this
59:53failure of the Third axium however
59:55cannot arise if the number of
59:57individuals in the world is not finite
1:00:01we shall return to this topic at a later
1:00:03stage footnote 1 C chapter 13 end of
1:00:07footnote
1:00:081 assuming that the number of
1:00:10individuals in the universe is not
1:00:12finite we have now succeeded not only in
1:00:15defining piano's three primitive ideas
1:00:18but in seeing how to prove his five
1:00:20primitive propositions by means of
1:00:22primitive ideas and propositions
1:00:24belonging to logic it follows that all
1:00:28pure mathematics in so far as it is
1:00:30deducible from the theory of the natural
1:00:32numbers is only a prolongation of logic
1:00:36the extension of this result to those
1:00:38modern branches of mathematics which are
1:00:40not deducible from the theory of the
1:00:42natural numbers offers no difficulty of
1:00:45Principle as we have shown elsewhere
1:00:48footnote one for geometry in so far as
1:00:51it is not purely analytical see
1:00:54principles of mathematics part six for
1:00:57rational dynamics that same book part
1:01:00seven end of footnote
1:01:02one the process of mathematical
1:01:05induction by means of which we defined
1:01:07the natural numbers is capable of
1:01:10generalization we Define the natural
1:01:12numbers as the posterity of zero with
1:01:15respect to the relation of a number to
1:01:17its immediate successor if we call this
1:01:19relation n any number M will have this
1:01:23relation to m + 1 a property is
1:01:27hereditary with respect to n or simply n
1:01:30hereditary if whenever the property
1:01:32belongs to a number M it also belongs to
1:01:35m + one that is to the number to which M
1:01:39has the relation n and a number n will
1:01:43be said to belong to the posterity of M
1:01:47with respect to the relation n if n has
1:01:50every n hereditary property belonging to
1:01:53m these definitions can all be applied
1:01:57to any other relation just as well as to
1:02:00n thus if R is any relation whatever we
1:02:04can lay down the following definitions
1:02:06footnote two these definitions and the
1:02:09generalized theory of induction are due
1:02:11to fragga and were published so long ago
1:02:13as 1879 in his begriff shrift in spite
1:02:17of the great value of this work I was I
1:02:20believe the first person who ever read
1:02:22it more than 20 years after its public
1:02:25end of footnote
1:02:272 a property is called R hereditary when
1:02:31if it belongs to a term x and x has the
1:02:33relation R to Y then it belongs to y a
1:02:37class is R hereditary when its defining
1:02:40property is R
1:02:42hereditary a term X is said to be an R
1:02:45ancestor of the term y if y has every R
1:02:49hereditary property that X has provided
1:02:52X is a term which has the relation R to
1:02:55something or to which something has the
1:02:58relation R this is only to exclude
1:03:00trivial
1:03:02cases the r posterity of X is all the
1:03:06terms of which X is an R
1:03:09ancestor we have framed the above
1:03:11definition so that if a term is the
1:03:14ancestor of anything it is its own
1:03:16ancestor and belongs to its own
1:03:18posterity this is merely for
1:03:21convenience it will be observed that if
1:03:23we take are the relation parent ancestor
1:03:27and posterity will have the usual
1:03:29meanings except that a person will be
1:03:31included among his own ancestors and
1:03:35posterity it is of course obvious at
1:03:37once that ancestor must be capable of
1:03:40definition in terms of parent but until
1:03:43fragga developed his generalized theory
1:03:44of induction no one could have defined
1:03:47ancestor precisely in terms of parent a
1:03:50brief consideration of this point will
1:03:53serve to show the importance of the
1:03:55theory a person confronted for the first
1:03:57time with the problem of defining
1:03:59ancestor in terms of parent would
1:04:01naturally say that a is an ancestor of Z
1:04:05if between a and z there are a certain
1:04:08number of people b c and so on of whom B
1:04:12is a child of a each is a parent of the
1:04:15next until the last who is a parent of
1:04:18Z but this definition is not adequate
1:04:22unless we add that the number of inter
1:04:24intermediate terms is to be
1:04:26finite take for example such a series as
1:04:29the following -1
1:04:32-2 -
1:04:331/4
1:04:351/8 continuing to 1/8 1/4 1/2
1:04:41one here we have first a series of
1:04:44negative fractions with no end and then
1:04:46a series of positive fractions with no
1:04:48beginning shall we say that in this
1:04:50series /8 is an ancestor of 1/8 it will
1:04:54be so according to The Beginner's
1:04:56definition suggested above but it will
1:04:59not be so according to any definition
1:05:02which will give the kind of idea that we
1:05:04wish to Define for this purpose it is
1:05:07essential that the number of
1:05:08intermediaries should be finite but as
1:05:12we saw finite is to be defined by means
1:05:14of mathematical induction and it is
1:05:17simpler to define the ancestral relation
1:05:19generally at once than to Define it
1:05:22first only for the case of the relation
1:05:25of n to n + one and then extend it to
1:05:28other
1:05:29cases here as constantly elsewhere
1:05:32generality from the first though it may
1:05:34require more thought at the start will
1:05:36be found in the long run to economize
1:05:39thought and increase logical power the
1:05:43use of mathematical induction in
1:05:45demonstrations was in the past something
1:05:48of a mystery there seemed no reasonable
1:05:51doubt that it was a valid method of
1:05:52proof but no no one quite knew why it
1:05:55was valid some believed it to be really
1:05:58a case of induction in the sense in
1:06:00which the word is used in logic po
1:06:03footnote 1 science and Method chapter 4
1:06:07end of footnote 1 considered it to be a
1:06:09principle of the utmost importance by
1:06:12means of which an infinite number of
1:06:14syllogisms could be condensed into one
1:06:17argument we now know that all such views
1:06:19are mistaken and that mathematical
1:06:21induction is a definition not a princip
1:06:25there are some numbers to which it can
1:06:27be applied and there are others as we
1:06:30shall see in chapter 8 to which it
1:06:32cannot be
1:06:34applied we Define the natural numbers as
1:06:37those to which proofs by mathematical
1:06:39induction can be applied that is as
1:06:42those that possess all inductive
1:06:45properties it follows that such proofs
1:06:47can be applied to the natural numbers
1:06:50not in virtue of any mysterious
1:06:51Intuition or axium or principle
1:06:54but as a purely verbal
1:06:56proposition if quadraped are defined as
1:06:59animals having four legs it will follow
1:07:02that animals that have four legs are
1:07:04quadrip heads and the case of numbers
1:07:06that obey mathematical induction is
1:07:09exactly
1:07:11similar we shall use the phrase
1:07:13inductive numbers to mean the same set
1:07:15as we have hither to spoken of as the
1:07:18natural numbers the phrase inductive
1:07:21numbers is preferable as affording a
1:07:23reminder
1:07:24that the definition of this set of
1:07:26numbers is obtained for mathematical
1:07:29induction mathematical induction affords
1:07:32more than anything else the essential
1:07:34characteristic by which the finite is
1:07:36distinguished from the infinite the
1:07:38principle of mathematical induction
1:07:40might be stated popularly in some such
1:07:43form as what can be inferred from next
1:07:45to next can be inferred From First to
1:07:48Last this is true when the number of
1:07:50intermediate steps between the first and
1:07:52last is finite but not
1:07:55otherwise anyone who has ever watched a
1:07:57Goods train beginning to move will have
1:07:59noticed how the impulse is communicated
1:08:01with the jerk from each truck to the
1:08:04next until at last even the hindmost
1:08:07truck is in motion when the train is
1:08:10very long it is a very long time before
1:08:13the last truck moves if the train were
1:08:15infinitely long there would be an
1:08:17infinite succession of
1:08:19jerks and the time would never come when
1:08:22the whole train would be in motion
1:08:24nevertheless if there were a series of
1:08:26trucks no longer than the series of
1:08:28inductive numbers which as we shall see
1:08:31is an instance of the smallest of
1:08:33Infinities every truck would begin to
1:08:35move sooner or later if the engine
1:08:38persevered though there would always be
1:08:40other trucks further back which had not
1:08:42yet begun to
1:08:44move this image will help to elucidate
1:08:46the argument from next to next and its
1:08:49connection with finitude when we come to
1:08:52infinite numbers where are Arguments for
1:08:54mathematical induction will be no longer
1:08:56valid the properties of such numbers
1:08:58will help to make clear by contrast the
1:09:01almost unconscious use that is made of
1:09:04mathematical induction where finite
1:09:06numbers are
1:09:08concerned end of chapter
1:09:17three chapter four of introduction to
1:09:20mathematical Philosophy by berand
1:09:22Russell
1:09:24this LibriVox recording is in the public
1:09:27domain the definition of
1:09:29order we have now carried our analysis
1:09:32of the series of natural numbers to the
1:09:34point where we have obtained logical
1:09:36definitions of the members of this
1:09:38series of the whole class of its members
1:09:41and of the relation of a number to its
1:09:43immediate
1:09:44successor we must now consider the
1:09:46serial character of the natural numbers
1:09:49in the order 0 1 2 3 and so on
1:09:54we ordinarily think of the numbers as in
1:09:57this order and it is an essential part
1:10:00of the work of analyzing our data to
1:10:03seek a definition of order or Series in
1:10:06logical
1:10:07terms the notion of order is one which
1:10:10has enormous importance in mathematics
1:10:13not only the integers but also rational
1:10:15fractions and all real numbers have an
1:10:18order of magnitude and this is essential
1:10:21to most of their mathematical properties
1:10:24the order of points on a line is
1:10:26essential to Geometry so is the slightly
1:10:29more complicated order of lines through
1:10:31a point in a plane or of planes through
1:10:33a line dimensions in Geometry are a
1:10:36development of order the conception of a
1:10:39limit which underlies all higher
1:10:42mathematics is a Serial
1:10:45conception there are parts of
1:10:46mathematics which do not depend upon the
1:10:48notion of order but they are very few in
1:10:51comparison with the parts in which this
1:10:53notion is
1:10:54involved in Seeking a definition of
1:10:57order the first thing to realize is that
1:10:59no set of terms has just one order to
1:11:02the exclusion of others a set of terms
1:11:05has all the orders of which it is
1:11:07capable sometimes one order is so much
1:11:10more familiar and natural to our
1:11:12thoughts that we are inclined to regard
1:11:14it as the order of that set of terms but
1:11:18this is a
1:11:19mistake the natural numbers or the
1:11:22inductive numbers as we shall also call
1:11:24them occur to us most readily in order
1:11:26of magnitude but they are capable of an
1:11:29infinite number of other arrangements we
1:11:32might for example consider first all the
1:11:35odd numbers and then all the even
1:11:36numbers or first one then all the even
1:11:40numbers then the odd multiples of three
1:11:42then all the multiples of five but not
1:11:45of two or three then all the multiples
1:11:48of seven but not of two or three or five
1:11:51and so on through the whole series of
1:11:52primes
1:11:54when we say that we arrange the numbers
1:11:57in these various orders that is an
1:11:59inaccurate
1:12:00expression what we really do is to turn
1:12:03our attention to certain relationships
1:12:06between the natural numbers which
1:12:08themselves generate such in such an
1:12:10arrangement we can no more arrange the
1:12:13natural numbers than we can the starry
1:12:15Heavens but just as we may notice among
1:12:18the fixed Stars either their order of
1:12:20brightness or their distribution in the
1:12:22sky so there are various relations among
1:12:25numbers which may be observed and which
1:12:27give rise to various different orders
1:12:29among numbers all equally
1:12:32legitimate and what is true of numbers
1:12:34is equally true of points on a line or
1:12:36of the moments of time one order is more
1:12:40familiar but others are equally valid we
1:12:43might for example take first on a line
1:12:46all the points that have integral
1:12:48coordinates then all those that have
1:12:50non- integral rational coordinates then
1:12:52all those that have algebraic
1:12:54non-rational coordinates and so on
1:12:57through any set of complications we
1:12:59please the resulting order will be one
1:13:01which the points of the line certainly
1:13:03have whether we choose to notice it or
1:13:06not the only thing that is arbitrary
1:13:08about the various orders of a set of
1:13:10terms is our attention for the terms
1:13:13themselves have always all the orders of
1:13:15which they are
1:13:17capable one important result of this
1:13:20consideration is that we must not look
1:13:22for the definition of order in the
1:13:24nature of the set of terms to be ordered
1:13:27since one set of terms has many orders
1:13:30the order lies not in the class of terms
1:13:33but in a relation among the members of
1:13:35the class in respect of which some
1:13:37appear as earlier and some as later the
1:13:40fact that a class may have many orders
1:13:43is due to the fact that there can be
1:13:44many relations holding among the members
1:13:47of one single class what properties must
1:13:50a relation have in order to give rise to
1:13:52an order
1:13:54the essential characteristic of a
1:13:55relation which is to give rise to order
1:13:58may be discovered by considering that in
1:14:00respect of such a relation we must be
1:14:03able to say of any two terms in the
1:14:05class which is to be ordered that one
1:14:08precedes and the other
1:14:10follows now in order that we may be able
1:14:13to use these words in the way in which
1:14:15we should naturally understand them we
1:14:18require that the ordering relation
1:14:19should have three
1:14:21properties one if x proceeds y y must
1:14:25not also preced X this is an obvious
1:14:28characteristic of the kind of relations
1:14:30that lead to series if x is less than y
1:14:34y is not also less than
1:14:36x if x is earlier in time than y y is
1:14:40not Also earlier than x if x is to the
1:14:44left of y y is not to the left of X on
1:14:48the other hand relations which do not
1:14:50give rise to series often do not have
1:14:52this property
1:14:54if x is a brother or sister of y y is a
1:14:57brother or sister of x if x is of the
1:15:00same height as y y is of the same height
1:15:02as x if x is of a different height from
1:15:05y y is of a different height from X in
1:15:08all these cases when the relation holds
1:15:11between X and Y it also holds between Y
1:15:14and X but with serial relations such a
1:15:17thing cannot happen a relation having
1:15:19this first property is called
1:15:22asymmetrical
1:15:23two if x precedes Y and Y precedes z x
1:15:28must precede Z this may be illustrated
1:15:31by the same instances as before less
1:15:34earlier left of but as instances of
1:15:37relations which do not have this
1:15:39property only two of our previous three
1:15:42instances will serve if x is brother or
1:15:45sister of Y and Y of z x may not be
1:15:49brother or sister of Z since x and z may
1:15:53be the same person the same applies to
1:15:56difference of height but not to sameness
1:15:58of height which has our second property
1:16:00but not our first the relation father on
1:16:03the other hand has our first property
1:16:05but not our second a relation having our
1:16:08second property is called
1:16:11transitive three given any two terms of
1:16:15the class which is to be ordered there
1:16:17must be one which precedes and the other
1:16:19which follows for example of any two
1:16:22integers or fractions or real numbers
1:16:25one is smaller and the other greater but
1:16:28of any two complex numbers this is not
1:16:31true of any two moments in time one must
1:16:34be earlier than the other but of events
1:16:37which may be simultaneous this cannot be
1:16:40said of two points on a line one must be
1:16:44to the left of the other a relation
1:16:46having this third property is called
1:16:50connected when a relation possesses
1:16:52these three prop properties it is of the
1:16:54sort to give rise to an order among the
1:16:57terms between which it holds and
1:17:00wherever an order exists some relation
1:17:03having these three properties can be
1:17:05found generating
1:17:07it before illustrating this thesis we
1:17:10will introduce a few
1:17:12definitions one a relation is said to be
1:17:15a Leo relative footnote 1 this term is
1:17:18due to CS purse end of footnote one or
1:17:22to be contain Ed in or imply diversity
1:17:25if no term has this relation to itself
1:17:29thus for example greater different in
1:17:32size brother husband father are Alo
1:17:36relatives but equal born of the same
1:17:39parents dear friend or not two the
1:17:43square of a relation is that relation
1:17:45which holds between two terms x and z
1:17:49when there is an intermediate term y
1:17:52such that the given relation holds
1:17:54between X and Y and between Y and
1:17:58Z thus paternal grandfather is the
1:18:01square of Father Greater by two is the
1:18:04square of Greater by one and so
1:18:07on
1:18:09three the domain of a relation consists
1:18:12of all those terms that have the
1:18:14relation to something or other and the
1:18:16converse domain consists of all those
1:18:19terms to which something or other has
1:18:21the relation
1:18:23these words have been already defined
1:18:25but are recalled here for the sake of
1:18:28the following definition four the field
1:18:31of a relation consists of its domain and
1:18:34Converse domain
1:18:36together five one relation is said to
1:18:39contain or be implied by another if it
1:18:42holds whenever the other holds it will
1:18:46be seen that an asymmetrical relation is
1:18:48the same thing as a relation whose
1:18:51square is an Alo relative
1:18:53it often happens that a relation is an
1:18:56AO relative without being
1:18:59asymmetrical though an asymmetrical
1:19:01relation is always an Alo relative for
1:19:05example spouse is an Alo relative but is
1:19:09symmetrical since if x is the spouse of
1:19:11y y is the spouse of X but among
1:19:15transitive relations all Alo relatives
1:19:17are asymmetrical as well as vice
1:19:21versa from the definition
1:19:23it will be seen that a transitive
1:19:25relation is one which is implied by its
1:19:28square or as we also say contains its
1:19:32Square thus ancestor is transitive
1:19:35because an ancestor's ancestor is an
1:19:38ancestor but father is not transitive
1:19:41because a father's father is not a
1:19:44father a transitive Alo relative is one
1:19:47which contains its square and is
1:19:50contained in diversity or what comes to
1:19:53the same thing one whose square implies
1:19:56both it and
1:19:58diversity because when a relation is
1:20:00transitive asymmetry is equivalent to
1:20:03being an Alo
1:20:05relative a relation is connected when
1:20:09given any two different terms of its
1:20:11field the relation holds between the
1:20:14first and the second or between the
1:20:16second and the first not excluding the
1:20:19possibility that both may happen though
1:20:22both cannot happen happen if the
1:20:23relation is
1:20:25asymmetrical it will be seen that the
1:20:27relation ancestor for example is UN Alo
1:20:31relative and transitive but not
1:20:34connected it is because it is not
1:20:37connected that it does not suffice to
1:20:39arrange the human race in a
1:20:42series the relation less than or equal
1:20:45to among numbers is transitive and
1:20:48connected but not asymmetrical or an Alo
1:20:52relative
1:20:53the relation greater or less among
1:20:56numbers is an Alo relative and is
1:20:58connected but is not transitive for if x
1:21:02is greater or less than Y and Y is
1:21:05greater or less than Z it may happen
1:21:07that x and z are the same
1:21:10number thus the three properties of
1:21:13being one an AO relative two transitive
1:21:17and three connected are mutually
1:21:20independent since a relation may have
1:21:22any two without having the
1:21:24third we may now lay down the following
1:21:28definition a relation is serial when it
1:21:32is an Alo relative transitive and
1:21:34connected or what is equivalent when it
1:21:37is asymmetrical transitive and
1:21:41connected a series is the same thing as
1:21:45a serial relation it might have been
1:21:47thought that a series should be the
1:21:49field of a serial relation not the
1:21:52serial Rel
1:21:53itself but this would be an error for
1:21:56example 1 2 3 1 32 23 1
1:22:03213
1:22:05312 3 2 1 are six different series which
1:22:10all have the same field if the field
1:22:13were the series there could only be one
1:22:16series with a given field what
1:22:18distinguishes the above six series is
1:22:21simply the different order ordering
1:22:22relations in the six cases given the
1:22:26ordering relation the field and the
1:22:28order are both
1:22:30determinant thus the ordering relation
1:22:32may be taken to be the series but the
1:22:35field cannot be so
1:22:37taken given any serial relation say p we
1:22:41shall say that in respect of this
1:22:43relation X precedes y if x has the
1:22:47relation P to Y which we shall write X
1:22:51py for short
1:22:53the three characteristics which P must
1:22:55have in order to be serial are one we
1:22:59must never have X PX that is no term
1:23:03must precede itself two p^ 2 must imply
1:23:07P that is if x precedes Y and Y precedes
1:23:12z x must precede Z three if X and Y are
1:23:18two different terms in the field of P we
1:23:21shall have x p y y or
1:23:24ypx that is one of the two must precede
1:23:28the other the reader can easily convince
1:23:31himself that where these three
1:23:33properties are found in an ordering
1:23:36relation the characteristics we expect
1:23:38of series will also be found and vice
1:23:42versa we are therefore justified in
1:23:45taking the above as a definition of
1:23:47order or series and it will be observed
1:23:51that the definition is effective in
1:23:53purely logical
1:23:54terms although a transitive asymmetrical
1:23:57connected relation always exists
1:24:00wherever there is a series it is not
1:24:03always the relation which would most
1:24:05naturally be regarded as generating the
1:24:08series The Natural number series may
1:24:10serve as an
1:24:12illustration the relation we assumed in
1:24:14considering the natural numbers was the
1:24:17relation of immediate succession that is
1:24:20the relation between consecutive
1:24:22integers this relation is asymmetrical
1:24:26but not transitive or connected we can
1:24:29however derive from it by the method of
1:24:32mathematical induction The ancestral
1:24:35relation which we considered in the
1:24:37preceding
1:24:38chapter this relation will be the same
1:24:41as less than or equal to among inductive
1:24:45integers for purposes of generating the
1:24:48series of natural numbers we want the
1:24:50relation less than excluding equal to
1:24:55this is the relation of M to n when m is
1:24:58an ancestor of n but not identical with
1:25:01n or what comes to the same thing when
1:25:05the successor of M is an ancestor of n
1:25:08in the sense in which a number is its
1:25:11own ancestor that is to say we shall lay
1:25:14down the following
1:25:16definition an inductive number m is said
1:25:20to be less than another number n when n
1:25:24possesses every hereditary property
1:25:27possessed by the successor of
1:25:29M it is easy to see and not difficult to
1:25:33prove that the relation less than so
1:25:36defined is a symmetrical transitive and
1:25:40connected and has the inductive numbers
1:25:43for its field thus by means of this
1:25:46relation the inductive numbers acquire
1:25:49an order in the sense in which we
1:25:51defined the term order order and this
1:25:53order is the so-called natural order or
1:25:56order of
1:25:58magnitude the generation of series by
1:26:01means of relations more or less
1:26:03resembling that of n to n +1 is very
1:26:07common the series of the Kings of
1:26:10England for example is generated by
1:26:12relations of each to his successor this
1:26:15is probably the easiest way where it is
1:26:18applicable of conceiving the generation
1:26:21of a series in this method we pass on
1:26:24from each term to the next as long as
1:26:26there is a next or back to the one
1:26:28before as long as there is one
1:26:31before this method always requires the
1:26:34generalized form of mathematical
1:26:36induction in order to enable us to
1:26:39Define earlier and later in a series so
1:26:42generated on the analogy of proper
1:26:45fractions let us give the name proper
1:26:48posterity of X with respect to R to the
1:26:52the class of those terms that belong to
1:26:54the r posterity of some term to which X
1:26:57has the relation R in the sense which we
1:27:00gave before to posterity which includes
1:27:03a term in its own
1:27:05posterity reverting to the fundamental
1:27:07definitions we find that the proper
1:27:10posterity may be defined as follows the
1:27:14proper posterity of X with respect to R
1:27:18consists of all terms that possess every
1:27:21r R hereditary property possessed by
1:27:24every term to which X has the relation
1:27:28R it is to be observed that this
1:27:31definition has to be so framed as to be
1:27:34applicable not only when there is only
1:27:37one term to which X has the relation R
1:27:40but also in cases as say that of father
1:27:44and child where there may be many terms
1:27:47to which X has the relation R we Define
1:27:51further
1:27:52a term X is a proper ancestor of Y with
1:27:56respect to R if y belongs to the proper
1:28:00posterity of X with respect to R we
1:28:04shall speak for short of R posterity and
1:28:07R ancestors when these terms seem more
1:28:11convenient reverting now to the
1:28:14generation of series by the relation R
1:28:17between consecutive terms we see that if
1:28:20this method is to be POS possible the
1:28:23relation proper R ancestor must be an
1:28:26Alo relative transitive and connected
1:28:30under what circumstances will this occur
1:28:33it will always be transitive no matter
1:28:36what sort of relation R may be our
1:28:38ancestor and proper our ancestor are
1:28:40always both
1:28:42transitive but it is only under certain
1:28:45circumstances that it will be an Alo
1:28:47relative or connected consider for
1:28:50example the relation to one one's
1:28:52left-and neighbor at a round dinner
1:28:54table at which there are 12 people if we
1:28:57call this relation R the proper R
1:29:00posterity of a person consists of all
1:29:03who can be reached by going round the
1:29:05table from right to left this includes
1:29:08everybody at the table including the
1:29:11person himself since 12 steps brings us
1:29:14back to our starting
1:29:16point thus in such a case though the
1:29:19relation proper our ancestor is
1:29:21connected Ed and though R itself is an
1:29:24Alo relative we do not get a series
1:29:28because proper R ancestor is not an Leo
1:29:32relative it is for this reason that we
1:29:34cannot say that one person comes before
1:29:37another with respect to the relation
1:29:39right of or to its ancestral
1:29:43derivative the above was an instance in
1:29:46which the ancestral relation was
1:29:49connected but not contained in diversity
1:29:53an instance where it is contained in
1:29:55diversity but not connected is derived
1:29:58from the ordinary sense of the word
1:30:01ancestor if x is a proper ancestor of Y
1:30:05X and Y cannot be the same person but it
1:30:08is not true that of any two persons one
1:30:12must be an ancestor of the
1:30:14other the question of the circumstances
1:30:18under which series can be generated by
1:30:20ancestral relations derived from
1:30:22relations of consecutiveness is often
1:30:25important some of the most important
1:30:28cases are the following let R be a many
1:30:31one relation and let us confine our
1:30:33attention to the posterity of some term
1:30:36X when so confined the relation proper
1:30:39our ancestor must be connected therefore
1:30:43All That Remains to ensure it's being
1:30:45serial is that it shall be contained in
1:30:48diversity this is a generalization of
1:30:51the instant of the dinner table another
1:30:54generalization consists in taking R to
1:30:56be a one one relation and including the
1:30:59ancestry of X as well as the
1:31:02posterity here again the one condition
1:31:04required to secure the generation of a
1:31:06series is that the relation proper our
1:31:10ancestor shall be contained in
1:31:13diversity the generation of order by
1:31:15means of relations of consecutiveness
1:31:17though important in its own sphere is
1:31:21less gener
1:31:22than the method which uses a transitive
1:31:24relation to define the order it often
1:31:27happens in a series that there are an
1:31:30infinite number of intermediate terms
1:31:33between any two that may be selected
1:31:36however near together these may
1:31:39be take for instance fractions in order
1:31:42of magnitude between any two fractions
1:31:45there are others for example the
1:31:47arithmetic mean of the two consequently
1:31:51there is no no such thing as a pair of
1:31:53consecutive fractions if we depend upon
1:31:56consecutiveness for defining order we
1:31:59should not be able to define the order
1:32:01of magnitude among fractions but in fact
1:32:04the relations of greater and less among
1:32:07fractions do not demand generation from
1:32:10relations of consecutiveness and the
1:32:13relations of greater and less among
1:32:15fractions have the three characteristics
1:32:18which we need for defining serial
1:32:20relations in all such cases the order
1:32:23must be defined by means of a transitive
1:32:26relation since only such a relation is
1:32:29able to leap over an infinite number of
1:32:31intermediate terms the method of
1:32:34consecutiveness like that of counting
1:32:36for discovering the number of a
1:32:37collection is appropriate to the finite
1:32:41it may even be extended to certain
1:32:43infinite series namely those in which
1:32:46though the total number of terms is
1:32:48infinite the number of terms between any
1:32:50two is always finite but it must not be
1:32:53regarded as general not only so but care
1:32:57must be taken to eradicate from the
1:33:00imagination all habits of thought
1:33:02resulting from supposing it General if
1:33:05this is not done Series in which there
1:33:08are no consecutive terms will remain
1:33:11difficult and puzzling and such series
1:33:14are of vital importance for the
1:33:16understanding of continuity space time
1:33:19and
1:33:20motion there are many ways in which
1:33:23series may be generated but all depend
1:33:26upon the finding or construction of an
1:33:29asymmetrical transitive connected
1:33:32relation some of these ways have
1:33:34considerable importance we may take as
1:33:37illustrative the generation of series by
1:33:40means of a three-term relation which we
1:33:42may call
1:33:44between this method is very useful in
1:33:47geometry and may serve as an
1:33:49introduction to relations having more
1:33:51than two two terms it is best introduced
1:33:54in connection with Elementary
1:33:57geometry given any three points on a
1:34:00straight line in ordinary space there
1:34:02must be one of them which is between the
1:34:05other two this will not be the case with
1:34:08the points on a circle or any other
1:34:10closed curve because given any three
1:34:13points on a circle we can travel from
1:34:16any one to any other without passing
1:34:18through the third in fact the notion
1:34:21between is characteristic of open series
1:34:25or Series in the strict sense as opposed
1:34:28to what may be called cyclic Series
1:34:31where as with people at the dinner table
1:34:33a sufficient Journey brings us back to
1:34:35our starting point this notion of
1:34:38between may be chosen as the fundamental
1:34:41notion of ordinary
1:34:43geometry but for the present we will
1:34:45only consider its application to a
1:34:47straight line and to The Ordering of the
1:34:50points on a straight line
1:34:52footnote one confer Rista
1:34:55Mathematica for Pages 55 and following
1:34:59principles of Mathematics page 394
1:35:02section 375 end of footnote
1:35:061 taking any two points a b the line AB
1:35:12consists of three parts besides A and B
1:35:15themselves one points between A and B
1:35:19two points X such that a is between X
1:35:23and B three points y such that b is
1:35:27between Y and a thus the line AB can be
1:35:33defined in terms of the relation
1:35:36between in order that this relation
1:35:38between May arrange the points of the
1:35:41line in an order from left to right we
1:35:43need certain assumptions namely the
1:35:46following one if anything is between A
1:35:49and B A and B are not identical two
1:35:54anything between A and B is also between
1:35:57b and
1:35:58a three anything between A and B is not
1:36:02identical with a nor consequently with B
1:36:06in virtue of
1:36:07two four if x is between A and B
1:36:11anything between a and X is also between
1:36:15A and
1:36:16B five if x is between A and B and B is
1:36:21between X and Y then B is between a and
1:36:26Y six If X and Y are between A and B
1:36:30then either X and Y are
1:36:33identical or X is between a and Y or X
1:36:37is between Y and
1:36:38b seven if B is between a and x and also
1:36:43between a and Y then either X and Y are
1:36:47identical or X is between B and Y or Y
1:36:51is between B and
1:36:53X these seven properties are obviously
1:36:57verified in the case of points on a
1:36:58straight line in ordinary space any
1:37:01three term relation which verifies them
1:37:04gives rise to series as may be seen from
1:37:07the following definitions for the sake
1:37:10of definiteness let us assume that a is
1:37:12to the left of B then the points of the
1:37:15line a b are one those between which and
1:37:19B A lies
1:37:21these we will call to the left of a two
1:37:25a itself three those between A and B
1:37:30Four B itself five those between which
1:37:34and a lies B these we will call to the
1:37:38right of B we may now Define generally
1:37:41that of two points X Y on the line
1:37:45a we shall say that X is to the left of
1:37:49Y in any of the following foll in cases
1:37:52one when X and Y are both to the left of
1:37:55a and Y is between X and a two when X is
1:38:01to the left of a and Y is a or b or
1:38:05between a and b or to the right of
1:38:09B three when X is a and Y is between a
1:38:14and b or is b or is to the right of B
1:38:19four when X and Y are both between a and
1:38:22b and Y is between X and
1:38:26B five when X is between a and b and Y
1:38:31is b or to the right of
1:38:34B six when X is B and Y is to the right
1:38:39of B seven when X and Y are both to the
1:38:43right of B and X is between B and Y it
1:38:47will be found that from the seven
1:38:49properties which we have assigned to the
1:38:51relation between it can be deduced that
1:38:54the relation to the left of as above
1:38:57defined is a Serial relation as we
1:39:00defined that term it is important to
1:39:03notice that nothing in the definitions
1:39:05or the argument depends upon our meaning
1:39:08by Between the actual relation of that
1:39:11name which occurs in empirical space any
1:39:15three- term relation having the above
1:39:18seven purely formal properties will
1:39:21serve the purpose of the argument
1:39:23equally
1:39:24well cyclic order such as that of the
1:39:28points on a circle cannot be generated
1:39:31by means of three term relations of
1:39:33between we need a relation of four terms
1:39:36which may be called separation of
1:39:38couples the point may be illustrated by
1:39:41considering a journey around the world
1:39:44one may go from England to New Zealand
1:39:46by way of Suez or by way of San
1:39:49Francisco we cannot say definitely that
1:39:52either of these two places is between
1:39:54England and New Zealand but if a man
1:39:57chooses that route to go around the
1:39:59world whichever way round he goes his
1:40:02times in England and New Zealand are
1:40:05separated from each other by his times
1:40:07in Suz and San Francisco and
1:40:11conversely generalizing if we take any
1:40:14four points on a circle we can separate
1:40:16them into two couples say A and B and X
1:40:20and Y y such that in order to get from A
1:40:24to B one must pass through either X or Y
1:40:28and in order to get from X to Y one must
1:40:32pass through either A or
1:40:34B under these circumstances we shall say
1:40:38that the couple a are separated by the
1:40:41couple
1:40:42XY out of this relation a cyclic order
1:40:46can be generated in a way resembling
1:40:49that in which we generated an open order
1:40:52from between but somewhat more
1:40:54complicated footnote one confer
1:40:57principles of Mathematics page 205
1:41:00section 194 and references there given
1:41:04end of footnote
1:41:061 the purpose of the latter half of this
1:41:09chapter has been to suggest the subject
1:41:12which one may call generation of Serial
1:41:15relations when such relations have been
1:41:17defined the generation of them from
1:41:20other relations
1:41:21possessing only some of the properties
1:41:23required for series becomes very
1:41:26important especially in the philosophy
1:41:28of geometry and physics but we cannot
1:41:31within the limits of the present volume
1:41:33do more than make the reader aware that
1:41:36such a subject
1:41:38exists end of chapter
1:41:494
1:41:52chapter five of introduction to
1:41:54mathematical Philosophy by burand
1:41:57Russell this LibriVox recording is in
1:41:59the public
1:42:01domain kinds of
1:42:03relations a great part of the philosophy
1:42:06of mathematics is concerned with
1:42:08relations and many different kinds of
1:42:10relations have different kinds of uses
1:42:13it often happens that a property which
1:42:15belongs to all relations is only
1:42:18important as regards relations of
1:42:21certain sorts in these cases the reader
1:42:24will not see the bearing of the
1:42:25proposition asserting such a property
1:42:28unless he has in mind the sorts of
1:42:30relations for which it is useful for
1:42:33reasons of this description as well as
1:42:35from the intrinsic interest of the
1:42:37subject It is Well to have in our minds
1:42:40a rough list of the more mathematically
1:42:43serviceable varieties of
1:42:45relations we dealt in the preceding
1:42:48chapter with a supremely important class
1:42:51namely serial relations each of the
1:42:54three properties which we combined in
1:42:56defining series namely asymmetry
1:42:58transitives and
1:43:01connexity has its own importance we will
1:43:04Begin by saying something on each of
1:43:06these three asymmetry that is the
1:43:10property of being incompatible with the
1:43:12converse is a characteristic of the very
1:43:15greatest interest and importance in
1:43:18order to develop its functions we will
1:43:20consider consider various examples the
1:43:23relation husband is asymmetrical and so
1:43:26is the relation wife that is if a is
1:43:30husband of b b cannot be husband of a
1:43:33and similarly in the case of wife on the
1:43:37other hand the relation spouse is
1:43:40symmetrical if a is spouse of B then B
1:43:44is spouse of
1:43:46a suppose now we are given the relation
1:43:49spouse and we wish wish to derive the
1:43:51relation husband husband is the same as
1:43:55male spouse or spouse of a female thus
1:43:59the relation husband can be derived from
1:44:02spouse either by limiting the domain to
1:44:04males or by limiting the converse to
1:44:07females we see from this instance that
1:44:10when a symmetrical relation is given it
1:44:12is sometimes possible without the help
1:44:15of any further relation to separate it
1:44:18into two asymmetrical relations but the
1:44:21cases where this is possible are rare
1:44:23and exceptional they are cases where
1:44:26there are two mutually exclusive classes
1:44:29say Alpha and beta such that whenever
1:44:32the relation holds between two terms one
1:44:36of the terms is a member of Alpha and
1:44:38the other is a member of beta as in the
1:44:40case of spouse one term of the relation
1:44:43belongs to the class of males and one to
1:44:46the class of females in such a case the
1:44:50relation with its domain confined to
1:44:52Alpha will be asymmetrical and so will
1:44:55the relation with its domain confined to
1:44:59Beta but such cases are not of the sort
1:45:02that occur when we are dealing with
1:45:04series of more than two terms for in a
1:45:07series all terms except the first and
1:45:11last if these exist belong both to The
1:45:14Domain and to the converse domain of the
1:45:17generating relation so that a relation
1:45:20like husband where the domain and
1:45:22Converse domain do not overlap is
1:45:26excluded the question of how to
1:45:28construct relations having some useful
1:45:30property by means of operations upon
1:45:33relations which only have rudiments of
1:45:36the property is one of considerable
1:45:38importance transitives and connexity are
1:45:42easily constructed in many cases where
1:45:45the originally given relation does not
1:45:48possess them for example if R is any
1:45:51relation whatever the ancestral relation
1:45:54derived from R by generalized induction
1:45:57is
1:45:58transitive and if R is a many one
1:46:00relation The ancestral relation will be
1:46:03connected if confined to the posterity
1:46:06of a given term but asymmetry is a much
1:46:09more difficult property to secure by
1:46:12construction the method by which we
1:46:14derived husband from spouse is as we
1:46:17have seen not available in the most
1:46:19important cases such as greater before
1:46:23to the right of where domain and
1:46:25Converse domain overlap in all these
1:46:28cases we can of course obtain a
1:46:30symmetrical relation by adding together
1:46:33the given relation and its Converse but
1:46:35we cannot pass back from this
1:46:37symmetrical relation to the original
1:46:39asymmetrical relation except by the help
1:46:42of some asymmetrical
1:46:44relation take for example the relation
1:46:47greater the relation greater or less
1:46:50that is is unequal is symmetrical but
1:46:53there is nothing in this relation to
1:46:55show that it is the sum of two
1:46:57asymmetrical relations take such a
1:47:00relation as differing in shape this is
1:47:03not the sum of an asymmetrical relation
1:47:05and its Converse since shapes do not
1:47:07form a single series but there is
1:47:10nothing to show that it differs from
1:47:12differing in magnitude if we did not
1:47:15already know that magnitudes have
1:47:17relations of greater and
1:47:19less this illustrates the fundamental
1:47:21character of asymmetry as a property of
1:47:26relations from the point of view of the
1:47:28classification of relations being
1:47:30asymmetrical is a much more important
1:47:33characteristic than implying diversity
1:47:36asymmetrical relations imply diversity
1:47:38but the converse is not the case unequal
1:47:41for example implies diversity but is
1:47:45symmetrical broadly speaking we may say
1:47:47that if we wished as far as possible to
1:47:50dispense with relational propositions
1:47:52and replace them by such as ascribed
1:47:55predicates to subjects we could succeed
1:47:57in this so long as we can find ourselves
1:48:00to symmetrical
1:48:02relations those that do not imply
1:48:04diversity if they are transitive may be
1:48:07regarded as asserting a common predicate
1:48:10while those that do imply diversity may
1:48:12be regarded as asserting in compatible
1:48:15predicates for example consider the
1:48:18relation of similarity between classes
1:48:21by means of which we defined numbers
1:48:24this relation is symmetrical and
1:48:26transitive and does not imply
1:48:29diversity it would be possible though
1:48:31less simple than the procedure we
1:48:33adopted to regard the number of a
1:48:35collection as a predicate of the
1:48:38collection then two similar classes will
1:48:40be two that have the same numerical
1:48:43predicate while two that are not similar
1:48:46will be two that have different
1:48:47numerical predicates such such a method
1:48:50of replacing relations by predicates is
1:48:53formally possible though often very
1:48:55inconvenient so long as the relations
1:48:57concerned are
1:48:59symmetrical but it is formally
1:49:01impossible when the relations are
1:49:03asymmetrical because both sameness and
1:49:05difference of predicates are
1:49:08symmetrical asymmetrical relations are
1:49:11we may say the most characteristically
1:49:13relational of relations and the most
1:49:16important to the philosopher who wishes
1:49:18to study the old ultimate logical nature
1:49:22of
1:49:24relations another class of relations
1:49:26that is of the greatest use is the class
1:49:29of one many relations that is relations
1:49:33which at most one term can have to a
1:49:35given term such are father mother
1:49:38husband except in tobet square of sign
1:49:41of and so on but parent square root and
1:49:45so on are not one many it is possible
1:49:49formally to to replace all relations by
1:49:52one many relations by means of a
1:49:54device take say the relation less among
1:49:58the inductive numbers given any number n
1:50:02greater than one there will not be only
1:50:04one number having the relation less to n
1:50:08but we can form the whole class of
1:50:09numbers that are less than
1:50:12n this is one class and its relation to
1:50:15n is not shared by any other class we
1:50:19may call the class of numbers that are
1:50:21less than n the proper ancestry of n in
1:50:24the sense in which we spoke of ancestry
1:50:26and posterity in connection with
1:50:29mathematical
1:50:30induction then proper ancestry is a on-
1:50:33many relation one many will always be
1:50:36used so as to include one one since each
1:50:39number determines a single class of
1:50:41numbers as constituting its proper
1:50:44ancestry thus the relation less than can
1:50:47be replaced by being a member of the
1:50:50proper ancestry of in this way a one
1:50:53many relation in which the one is a
1:50:55class together with membership of this
1:50:58class can always formally replace a
1:51:00relation which is not one many Pano who
1:51:04for some reason always instinctively
1:51:06conceives of a relation as one many
1:51:09deals in this way with those that are
1:51:11naturally not
1:51:13so reduction to one many relations by
1:51:16this method however though possible as a
1:51:19matter of form does not represent a
1:51:22technical simplification and there is
1:51:24every reason to think that it does not
1:51:26represent a philosophical analysis if
1:51:30only because classes must be regarded as
1:51:33logical
1:51:34fictions we shall therefore continue to
1:51:36regard one many relations as a special
1:51:39kind of
1:51:40relation one many relations are involved
1:51:43in all phrases of the form the so and so
1:51:46of such and such the king of England the
1:51:49wife of Socrates the father of John
1:51:51Stewart Mill and so on all describe some
1:51:54person by means of a one many relation
1:51:57to a given term a person cannot have
1:52:00more than one father therefore the
1:52:02father of John Stewart Mill described
1:52:05some one person even if we did not know
1:52:07whom there is much to say on the subject
1:52:10of descriptions but for the present it
1:52:13is relations that we are concerned with
1:52:15and descriptions are only relevant as
1:52:18exemplifying the uses of one many
1:52:22relations it should be observed that all
1:52:25mathematical functions result from one
1:52:27many relations the logarithm of X the
1:52:30cosine of x and so on are like the
1:52:33father of X terms described by means of
1:52:36a one many relation logarithm cosine and
1:52:40so on to a given term
1:52:43X the notion of function need not be
1:52:46confined to numbers or to the uses to
1:52:49which mathematicians have accustomed us
1:52:52it can be extended to all cases of one
1:52:55many relations and the father of X is
1:52:59just as legitimately a function of which
1:53:01X is the argument as is the logarithm of
1:53:05X functions in this sense are
1:53:08descriptive functions as we shall see
1:53:11later there are functions of a still
1:53:13more General and more fundamental sort
1:53:16namely propositional functions but for
1:53:19the present we shall confine our
1:53:21attention to descriptive functions that
1:53:23is the term having the relation R tox or
1:53:27for short the r ofx where R is any one
1:53:31many
1:53:32relation it will be observed that if the
1:53:35r ofx is to describe a definite term X
1:53:39must be a term to which something has
1:53:41the relation R and there must not be
1:53:44more than one term having the relation R
1:53:46tox since the correctly used must imply
1:53:52uniqueness thus we may speak of the
1:53:54father of x if x is any human being
1:53:57except Adam and Eve but we cannot speak
1:53:59of the father of x if x is a table or a
1:54:03chair or anything else that does not
1:54:05have a father we shall say that the r
1:54:08ofx exists when there is just one term
1:54:11and no more having the relation R
1:54:14tox thus if R is a one many relation the
1:54:18r of X exist exists whenever X belongs
1:54:21to the converse domain of R and not
1:54:24otherwise regarding the r ofx as a
1:54:27function in the mathematical sense we
1:54:29say that X is the argument of the
1:54:32function and if Y is the term which has
1:54:34the relation
1:54:35r2x that is if Y is the r ofx then Y is
1:54:40the value of the function for the
1:54:43argument
1:54:44X if R is a one many relation the range
1:54:47of possible arguments to the function
1:54:50is the converse domain of R and the
1:54:52range of values is the domain thus the
1:54:55range of possible arguments to the
1:54:57function the father of X is all who have
1:55:00fathers that is the convers domain of
1:55:03the relation father while the range of
1:55:05possible values for the function is All
1:55:08fathers that is the domain of the
1:55:12relation many of the most important
1:55:14Notions in the logic of relations are
1:55:16descriptive functions for example
1:55:19Converse domain Converse domain field
1:55:23other examples will occur as we
1:55:26proceed among one many relations one one
1:55:29relations are a specially important
1:55:32class we've already had occasion to
1:55:34speak of one one relations in connection
1:55:37with the definition of number but it is
1:55:40necessary to be familiar with them and
1:55:42not merely to know their formal
1:55:45definition their formal definition may
1:55:47be derived from that of one many
1:55:50relations they may be defined as one
1:55:53many relations which are also the
1:55:55Converses of one many relations that is
1:55:58as relations which are both one many and
1:56:01many
1:56:02one one many relations may be defined as
1:56:06relations such that if x has the
1:56:09relation in question to Y there is no
1:56:11other term X Prime which also has the
1:56:15relation to y or again they may be
1:56:18defined as follows given two terms x and
1:56:21x Prime the terms to which X has the
1:56:24given relation and those to which X
1:56:26Prime has it have no member in
1:56:29common or again they may be defined as
1:56:33relations such that the relative product
1:56:35of one of them and its Converse implies
1:56:38identity where the relative product of
1:56:41two relations R and S is that relation
1:56:45which holds between x and z when there
1:56:48is an intermediate term y such that X
1:56:51has the relation R to Y and Y has the
1:56:54relation s to
1:56:56Z thus for example if R is the relation
1:57:00of father to son the relative product of
1:57:02R in its Converse will be the relation
1:57:05which holds between X and a man Z when
1:57:08there is a person y such that X is the
1:57:11father of Y and Y is the son of Z it is
1:57:15obvious that x and z must be the same
1:57:18person if on the other hand we take the
1:57:21relation of parent and child which is
1:57:24not one many we can no longer argue that
1:57:27if x is a parent of Y and Y is a child
1:57:30of z x and z must be the same person
1:57:34because one may be the father of why and
1:57:36the other the
1:57:38mother this illustrates that it is
1:57:40characteristic of one many relations
1:57:43when the relative product of a relation
1:57:45and its Converse implies
1:57:47identity in the case of of one one
1:57:50relations this happens and also the
1:57:52relative product of the converse and the
1:57:55relation implies
1:57:57identity given a relation R it is
1:57:59convenient if x has the relation R to Y
1:58:03to think of Y as being reached from X by
1:58:06an R step or an R Vector in the same
1:58:10case X will be reached from y by a
1:58:13backward R step thus we may State the
1:58:16characteristic of one many relations
1:58:18with which we have been dealing by
1:58:20saying that an R step followed by a
1:58:23backward R step must bring us back to
1:58:25our starting
1:58:26point with other relations this is by no
1:58:29means the case for example if R is the
1:58:33relation of child to parent the relative
1:58:35product of R and its Converse is the
1:58:38relation self or brother or sister and
1:58:41if R is the relation of grandchild to
1:58:44grandparent the relative product of R
1:58:46and its Converse is self or brother
1:58:49brother or sister or first
1:58:51cousin it will be observed that the
1:58:54relative product of two relations is not
1:58:56in general commutative that is the
1:58:59relative product of R and S is not in
1:59:02general the same relation as the
1:59:04relative product of
1:59:05SNR for example the relative product of
1:59:09parent and brother is uncle but the
1:59:11relative product of brother and parent
1:59:14is
1:59:15parent one one relations give a
1:59:18correlation of two classes term for term
1:59:21so that each term in either class has
1:59:24its correlate in the other such
1:59:26correlations are simplest to grasp when
1:59:29the two classes have no members in
1:59:31common like the class of husbands and
1:59:34the class of wives for in that case we
1:59:37know at once whether a term is to be
1:59:39considered as one from which the
1:59:41correlating relation R goes or as one to
1:59:45which it
1:59:46goes it is convenient to use the word
1:59:48reference
1:59:50for the term from which the relation
1:59:51goes and the term relatum for the term
1:59:55to which it
1:59:56goes thus If X and Y are husband and
1:59:59wife then with respect to the relation
2:00:02husband X is the reference and Y rotum
2:00:05but with respect to the relation wife Y
2:00:08is referent and X
2:00:11relatum we say that a relation and its
2:00:14Converse have opposite senses thus the
2:00:17sense of a relation that goes from X to
2:00:20Y is the opposite of that of the
2:00:22corresponding relation from y to X the
2:00:26fact that a relation has a sense is
2:00:28fundamental and is part of the reason
2:00:31why order can be generated by suitable
2:00:33relations it will be observed that the
2:00:36class of all possible reference to a
2:00:38given relation is its domain and the
2:00:41class of all possible relat is its
2:00:43Converse
2:00:45domain but it very often happens that
2:00:47the domain and Converse domain of a one
2:00:49one relation
2:00:51overlap take for example the first 10
2:00:54integers excluding zero and add one to
2:00:58each thus instead of the first 10
2:01:00integers we now have the integers 2 3 4
2:01:045 6 7 8 9 10
2:01:0911 these are the same as those we had
2:01:11before except that one has been cut off
2:01:14at the beginning and 11 has been joined
2:01:17on at the end there are still 10
2:01:19integers they are correlated with the
2:01:22previous 10 by the relation of n to n +
2:01:251 which is a 1 one
2:01:28relation or again instead of adding one
2:01:31to each of our original 10 integers we
2:01:34could have doubled each of them thus
2:01:35obtaining the integers 2 4 6 8 10 12 14
2:01:4116 18 20 here we still have five of our
2:01:46previous set of integers namely 2 4 6 8
2:01:5010 the correlating relation in this case
2:01:53is the relation of a number to its
2:01:55double which is again a one one relation
2:01:58or we might have replaced each number by
2:02:00its Square thus obtaining the set 1 49
2:02:0516 25 36 49 64 81
2:02:11100 on this occasion only three of our
2:02:14original set are left namely
2:02:17149 such processes of correlation may be
2:02:21varied
2:02:23endlessly the most interesting case of
2:02:25the above kind is the case where our one
2:02:28one relation has a convers domain which
2:02:30is part but not the whole of the
2:02:33domain If instead of confining the
2:02:36domain to the first 10 integers we had
2:02:38considered the whole of the inductive
2:02:40numbers the above instances would have
2:02:42Illustrated this case we may place the
2:02:45numbers concerned in two rows putting
2:02:47the correlate directly under the number
2:02:49whose correlate it is thus when the
2:02:52correlator is the relation of n to n +
2:02:55one we have the two rows 1 2 3 4 5 and
2:03:01so on to n and so on 2 3 4 5 6 and so on
2:03:072 n + 1 and so on when the correlator is
2:03:11the relation of a number to its double
2:03:14we have the two rows 1 2 3 4 5 and so on
2:03:18to n and so on 2 4 6 8 10 and so on to 2
2:03:25N and so on when the correlator is the
2:03:29relation of a number to its Square the
2:03:31rows are 1 2 3 4 5 and so on to n and so
2:03:37on 1 4 9 16 25 and so on to N squared
2:03:43and so on in all these cases all
2:03:46inductive numbers occur in the top row
2:03:48row and only some in the bottom
2:03:51row cases of this sort where the
2:03:53converse domain is a proper part of the
2:03:56domain that is a part not the whole will
2:03:59occupy us again when we come to deal
2:04:01with Infinity for the present we wish
2:04:03only to note that they exist and demand
2:04:07consideration another class of
2:04:09correlations which are often important
2:04:11is the class called permutations where
2:04:14the domain and Converse domain are
2:04:16identical consider for example the six
2:04:19possible Arrangements of three letterss
2:04:22ABC ACB BCA b a c cab
2:04:30CBA each of these can be obtained from
2:04:33any one of the others by means of a
2:04:35correlation take for example the first
2:04:38and last ABC and
2:04:40CBA here a is correlated with c b with
2:04:44itself and C with a it is obvious that
2:04:47the combination of two permutations is
2:04:50again a permutation that is the
2:04:53permutations of a given class form what
2:04:55is called a
2:04:57group these various kinds of
2:04:59correlations have importance in various
2:05:01connections some for one purpose some
2:05:04for another the general notion of one
2:05:06one correlations has boundless
2:05:09importance in the philosophy of
2:05:11mathematics as we have partly seen
2:05:13already but shall see much more fully as
2:05:16we proceed one of its uses will occupy
2:05:19us in our next
2:05:21chapter end of chapter
2:05:295 chapter six of introduction to
2:05:33mathematical Philosophy by berand
2:05:35Russell this LibriVox recording is in
2:05:38the public
2:05:39domain similarity of
2:05:42relations we saw in Chapter 2 that two
2:05:45classes have the same number of terms
2:05:47when they are similar
2:05:49that is when there is a one- one
2:05:51relation whose domain is the one class
2:05:54and whose Converse domain is the other
2:05:57in such a case we say that there is a
2:06:00one one correlation between the two
2:06:04classes in the present chapter we have
2:06:06to define a relation between relations
2:06:10which will play the same part for them
2:06:12that similarity of classes plays for
2:06:14classes we will call this relation
2:06:17similarity of relation
2:06:19or likeness when it seems to use a
2:06:22different word from that which we use
2:06:24for classes how is likeness to be
2:06:28defined we shall employ still the notion
2:06:31of correlation we shall assume that the
2:06:34domain of the one relation can be
2:06:36correlated with the domain of the other
2:06:39and the converse domain with the
2:06:40converse domain but that is not enough
2:06:43for the sort of resemblance which we
2:06:45desire to have between our two relations
2:06:48what we desire is that whenever either
2:06:51relation holds between two terms the
2:06:53other relation shall hold between the
2:06:55correlates of these two terms the
2:06:58easiest example of the sort of thing we
2:07:00desire is a map when one place is north
2:07:04of another the place on the map
2:07:06corresponding to the one is above the
2:07:08place on the map corresponding to the
2:07:10other when one place is west of another
2:07:14the place on the map corresponding to
2:07:16the one is to the left of the place on
2:07:19the map corresponding to the other and
2:07:22so on the structure of the map
2:07:25corresponds with that of the country of
2:07:27which it is a map the space relations in
2:07:30the map have likeness to the space
2:07:32relations in the country map it is this
2:07:36kind of connection between relations
2:07:38that we wish to
2:07:40Define we may in the first place
2:07:42profitably introduce a certain
2:07:45restriction we will confine ourselves in
2:07:47defining likeness
2:07:49to such relations as have fields that is
2:07:52to such as permit of the formation of a
2:07:55single class out of the domain and the
2:07:58converse domain this is not always the
2:08:00case take for example the relation
2:08:03domain that is the relation which the
2:08:06domain of a relation has to the
2:08:09relation this relation has all classes
2:08:12for its domain since every class is the
2:08:14domain of some relation and it has all
2:08:17relations for its compers domain since
2:08:19every relation has a
2:08:21domain but classes and relations cannot
2:08:24be added together to form a new single
2:08:27class because they are of different
2:08:29logical types we do not need to enter
2:08:32upon the difficult doctrine of types but
2:08:34it is well to know when we are
2:08:36abstaining from entering upon it we may
2:08:39say without entering upon the grounds
2:08:41for the assertion that a relation only
2:08:43has a field when it is what we call
2:08:46homogeneous that is when it's domain and
2:08:49Converse domain are of the same logical
2:08:51type and as a Rough and Ready indication
2:08:55of what we mean by a type we may say
2:08:57that individuals classes of individuals
2:08:59relations between individuals relations
2:09:02between classes relations of classes to
2:09:04individuals and so on are different
2:09:07types now the notion of likeness is not
2:09:10very useful as applied to relations that
2:09:12are not homogeneous we shall therefore
2:09:15in defining likeness simplify our
2:09:17problem by speaking of the field of one
2:09:20of the relations
2:09:21concerned this somewhat limits the
2:09:24generality of our definition but the
2:09:26limitation is not of any practical
2:09:29importance and having been stated it
2:09:31need no longer be remembered we may
2:09:34Define two relations p and Q as similar
2:09:38or as having likeness when there is a
2:09:40one one relation s whose domain is the
2:09:43field of p and whose Converse domain is
2:09:46the field of Q and which is such that if
2:09:49one term has the relation P to another
2:09:52the correlate of the one has the
2:09:53relation Q to the correlate of the other
2:09:56and vice versa a figure will make this
2:09:59clearer let X and Y be two terms having
2:10:02the relation P then there are to be two
2:10:05terms z w such that X has the relation s
2:10:09to z y has the relation s to w and z has
2:10:14the relation Q to
2:10:15W if this happens with every pair of
2:10:18terms such as X and Y and if the
2:10:21converse happens with every pair of
2:10:22terms such as Z and W it is clear that
2:10:26for every instance in which the relation
2:10:28P holds there is a corresponding
2:10:30instance in which the relation Q holds
2:10:33and vice
2:10:34versa and this is what we desire to
2:10:37secure by our
2:10:38definition we can eliminate some
2:10:41redundancies in the above sketch of a
2:10:42definition by observing that when the
2:10:45above conditions are
2:10:47realized the the relation p is the same
2:10:50as the relative product of s and Q and
2:10:53the converse of s that is the pep from X
2:10:56to Y may be replaced by the succession
2:11:00of the S step from X to Z the Q step
2:11:04from Z to W and the backward st- step
2:11:07from W to Y thus we may set up the
2:11:10following definitions a relation s is
2:11:14said to be a correlator or an ordinal
2:11:17correlator of two relations p and Q if s
2:11:21is one one has the field of Q for its
2:11:24Converse domain and is such that P is
2:11:26the relative product of s and Q and the
2:11:29converse of
2:11:30s two relations p and Q are said to be
2:11:33similar or to have likeness when there
2:11:36is at least one correlator of p and Q
2:11:39these definitions will be found to yield
2:11:42what we above decided to be
2:11:44necessary it will be found that when two
2:11:46relations are similar they share all
2:11:49properties which do not depend upon the
2:11:51actual terms in their fields for
2:11:53instance if one implies diversity so
2:11:56does the other if one is transitive so
2:11:59is the other if one is connected so is
2:12:02the other hence if one is serial so is
2:12:05the other again if one is one many or
2:12:08one one the other is one many or one one
2:12:12and so on through all the general
2:12:14properties of
2:12:15relations even statements involving the
2:12:18actual terms of the field of a relation
2:12:20though they may not be true as they
2:12:22stand when applied to a similar relation
2:12:24will always be capable of translation
2:12:26into statements that are analogous we
2:12:29are led by such considerations to a
2:12:31problem which has in mathematical
2:12:33philosophy an importance by no means
2:12:36adequately recognized hither to our
2:12:39problem may be stated as
2:12:41follows given some statement in a
2:12:43language of which we know the grammar
2:12:45and the syntax but not the vocabulary
2:12:48what are the possible meanings of such a
2:12:49statement and what are the meanings of
2:12:51the unknown words that would make it
2:12:53true the reason that this question is
2:12:56important is that it represents much
2:12:58more nearly than might be supposed the
2:13:00state of our knowledge of nature we know
2:13:03that certain scientific propositions
2:13:05which in the most advanced Sciences are
2:13:08expressed in mathematical symbols are
2:13:10more or less true of the world but we
2:13:13are very much at Sea as to the
2:13:15interpretations to be put upon the terms
2:13:17which occur in these propositions we
2:13:20know much more to use for a moment an
2:13:23oldfashioned pair of terms about the
2:13:25form of nature than about the matter
2:13:28accordingly what we really know when we
2:13:30enunciate a law of nature is only that
2:13:33there is probably some interpretation of
2:13:35our terms which will make the law
2:13:37approximately true thus great importance
2:13:41attaches to the question what are the
2:13:43possible meanings of a law expressed in
2:13:45terms of which we do not know the
2:13:47substantive meaning but only the grammar
2:13:49and syntax and this question is the one
2:13:52suggested above for the present we will
2:13:56ignore the general question which will
2:13:58occupy us again at a later stage the
2:14:01subject of likeness itself must first be
2:14:03further
2:14:05investigated owing to the fact that when
2:14:07two relations are similar their
2:14:09properties are the same except when they
2:14:11depend upon the fields being composed of
2:14:14just the terms of which they are
2:14:15composed it is desirable to have a n
2:14:18clature which collects together all the
2:14:20relations that are similar to a given
2:14:23relation just as we have called the set
2:14:25of those classes that are similar to a
2:14:27given class the number of that class so
2:14:30we may call the set of all those
2:14:32relations that are similar to a given
2:14:34relation the number of that relation but
2:14:38in order to avoid confusion with the
2:14:40numbers appropriate to classes we will
2:14:42speak in this case of a relation number
2:14:46thus we have the following definition
2:14:48the relation number of a given relation
2:14:51is the class of all those relations that
2:14:54are similar to the given
2:14:56relation relation numbers are the set of
2:14:59all those classes of relations that are
2:15:02relation numbers of various relations or
2:15:05what comes to the same thing a relation
2:15:07number is a class of relations
2:15:10consisting of all those relations that
2:15:12are similar to one member of the
2:15:15class when it is necessary to speak
2:15:18of the numbers of classes in a way which
2:15:20makes it impossible to confuse them with
2:15:23relation numbers we shall call them
2:15:25cardinal numbers thus cardinal numbers
2:15:28are the numbers appropriate to classes
2:15:31these include the ordinary integers of
2:15:33daily life and also certain infinite
2:15:36numbers of which we shall speak later
2:15:39when we speak of numbers without
2:15:41qualification we are to be understood as
2:15:43meaning cardinal numbers the definition
2:15:46of a cardinal number if it will be
2:15:48remembered is as follows the Cardinal
2:15:51number of a given class is the set of
2:15:54all those classes that are similar to
2:15:56the given class the most obvious
2:15:59application of relation numbers is to
2:16:01series two series may be regarded as
2:16:04equally long when they have the same
2:16:06relation number two finite series will
2:16:09have the same relation number when their
2:16:11fields have the same Cardinal number of
2:16:14terms and only then that is a series of
2:16:18say 15 terms will have the same relation
2:16:21number as any other series of 15 terms
2:16:25but will not have the same relation
2:16:27number as a series of 14 or 16 terms nor
2:16:30of course the same relation number as a
2:16:32relation which is not serial thus in the
2:16:35quite special case of finite series
2:16:38there is parallelism between Cardinal
2:16:40and relation numbers the relation
2:16:42numbers applicable to series may be
2:16:44called serial numbers what are commonly
2:16:46called ordinal numbers are a subass of
2:16:49these thus a finite serial number is
2:16:52determinant when we know the Cardinal
2:16:54number of terms in the field of a series
2:16:57having the serial number in question if
2:17:00N is a finite Cardinal number the
2:17:02relation number of a series which has n
2:17:05terms is called the ordinal number n
2:17:08there are also infinite ordinal numbers
2:17:10but of them we shall speak in a later
2:17:13chapter when the Cardinal number of
2:17:15terms in the field of a series is in
2:17:17infinite the relation number of the
2:17:19series is not determined merely by the
2:17:22Cardinal number indeed an infinite
2:17:24number of relation numbers exist for one
2:17:27infinite Cardinal number as we shall see
2:17:30when we come to consider infinite series
2:17:33when a series is infinite what we may
2:17:36call its length that is its relation
2:17:38number May Vary without change in the
2:17:41Cardinal number but when a series is
2:17:43finite this cannot
2:17:45happen we can define a addition and
2:17:48multiplication for relation numbers as
2:17:50well as for cardinal numbers and a whole
2:17:53arithmetic of relation numbers can be
2:17:56developed the manner in which this is to
2:17:58be done is easily seen by considering
2:18:00the case of series suppose for example
2:18:04that we wish to define the sum of two
2:18:06non-overlapping Series in such a way
2:18:09that the relation number of the sum
2:18:11shall be capable of being defined as the
2:18:14sum of the relation numbers of the two
2:18:16series in the first place it is clear
2:18:20that there is an order involved as
2:18:22between the two series one of them must
2:18:24be placed before the other thus if p and
2:18:27Q are the generating relations of the
2:18:29two Series in the series which is their
2:18:32sum with P put before Q every member of
2:18:35the field of P will precede every member
2:18:39of the field of
2:18:40Q thus the serial relation which is to
2:18:43be defined as the sum of p and Q is not
2:18:47P or Q simply but P or Q or the relation
2:18:51of any member of the field of P to any
2:18:54member of the field of Q assuming that P
2:18:57and Q do not overlap this relation is
2:19:00serial but P or Q is not serial being
2:19:04not connected since it does not hold
2:19:06between a member of the field of p and a
2:19:09member of the field of Q thus the sum of
2:19:12p and Q as above defined is what we need
2:19:15in order to define the sum of two
2:19:17relation
2:19:18numbers similar modifications are needed
2:19:21for products and Powers the resulting
2:19:24arithmetic does not obey the communative
2:19:26law the sum or product of two relation
2:19:28numbers generally depends upon the order
2:19:31in which they are taken but it obeys the
2:19:34associative law one form of the
2:19:36distributive law and two of the formal
2:19:39laws for Powers not only as applied to
2:19:41serial numbers but as applied to
2:19:43relation numbers
2:19:45generally relation arithmetic in fact
2:19:48though recent is a thoroughly
2:19:50respectable branch of
2:19:53mathematics it must not be supposed
2:19:55merely because series afford the most
2:19:58obvious application of the idea of
2:20:00likeness that there are no other
2:20:02applications that are important we have
2:20:05already mentioned maps and we might
2:20:07extend our thoughts from this
2:20:08illustration to Geometry
2:20:11generally if the system of relations by
2:20:13which a geometry is applied to a certain
2:20:16set of terms can be brought fully into
2:20:19relations of likeness with a system
2:20:21applying to another set of terms then
2:20:24the geometry of the two sets is
2:20:26indistinguishable from the mathematical
2:20:28point of view that is all the
2:20:31propositions are the same except for the
2:20:33fact that they are applied in one case
2:20:36to one set of terms and in the other to
2:20:39another we may illustrate this by the
2:20:42relations of the sort that may be called
2:20:44between which we considered in chapter
2:20:47four we there saw that provided a three-
2:20:51term relation has certain formal logical
2:20:54properties it will give rise to series
2:20:57it may be called a between
2:20:59relation given any two points we can use
2:21:02the between relation to define the
2:21:04straight line determined by those two
2:21:07points it consists of A and B together
2:21:10with all points X such that the between
2:21:13relation holds between the three points
2:21:16a b x X in some order or
2:21:19other it has been shown by oblin that we
2:21:23may regard our whole space as the field
2:21:26of a three term between relation and
2:21:28Define our geometry by the properties we
2:21:32assign to our between relation footnote
2:21:35one this does not apply to elliptic
2:21:38space but only to spaces in which the
2:21:40straight line is an open series modern
2:21:44mathematics edited by jwa Young Pages 3
2:21:48to 51 monograph by oblin on the
2:21:52foundations of geometry end of footnote
2:21:56one now likeness is just as easily
2:21:59definable between three term relations
2:22:01as between two term relations if B and B
2:22:05Prime are two between relations so that
2:22:08X has B to the ordered pair Y and Z
2:22:12means X is between Y and Z with respect
2:22:16to B
2:22:17we shall call S A correlator of B and B
2:22:20Prime if it has the field of B Prime for
2:22:23its Converse domain and is such that the
2:22:26relation B holds between three terms
2:22:29when B Prime holds between their s
2:22:31correlates and only then and we shall
2:22:35say that b is like B Prime when there is
2:22:38at least one correlator of B with B
2:22:40Prime the reader can easily convince
2:22:43himself that if B is like B Prime in
2:22:45this sense there can be no difference
2:22:47between the geometry generated by B and
2:22:50that generated by B
2:22:52prime it follows from this that the
2:22:55mathematician need not concern himself
2:22:58with the particular being or intrinsic
2:23:00nature of his points lines and planes
2:23:03even when he is speculating as an
2:23:05applied mathematician we may say that
2:23:08there is empirical evidence of the
2:23:10approximate truth of such parts of
2:23:12geometry as are not matters of
2:23:15definition but there is no empirical
2:23:17evidence as to what a point is to be it
2:23:20has to be something that as nearly as
2:23:22possible satisfies our axioms but it
2:23:25does not have to be very small or
2:23:27without Parts whether or not it is those
2:23:30things is a matter of indifference so
2:23:32long as it satisfies the
2:23:34axioms if we can out of empirical
2:23:37material construct a logical structure
2:23:40no matter how complicated which will
2:23:42satisfy our geometrical axioms that
2:23:45structure May legitimately be called a
2:23:47point we must not say that there is
2:23:50nothing else that could legitimately be
2:23:52called a point we must only say this
2:23:55object we have constructed is sufficient
2:23:57for the geometer it may be one of many
2:24:00objects any of which would be sufficient
2:24:03but that is no concern of ours since
2:24:05this object is enough to vindicate the
2:24:08empirical truth of geometry in so far as
2:24:11geometry is not a matter of
2:24:13definition this is only an illustration
2:24:16of the general princip principle that
2:24:18what matters in mathematics and to a
2:24:20very great extent in physical science is
2:24:22not the intrinsic nature of our terms
2:24:25but the logical nature of their inter
2:24:28relations we may say of two similar
2:24:31relations that they have the same
2:24:33structure for mathematical purposes
2:24:37though not for those of pure philosophy
2:24:39the only thing of importance about a
2:24:41relation is the cases in which it holds
2:24:44not its intrinsic nature just as a class
2:24:47may be defined by various different but
2:24:50coextensive concepts for example man and
2:24:54featherless biped so to relations which
2:24:57are conceptually different May hold in
2:25:00the same set of instances an instance in
2:25:03which a relation holds is to be
2:25:05conceived as a couple of terms with an
2:25:08order so that one of the terms comes
2:25:10first and the other second the couple is
2:25:13to be of course such that its first term
2:25:16has the Rel in question to its second
2:25:19take say the relation father we can
2:25:22Define what we may call the extension of
2:25:25this relation as the class of all
2:25:27ordered couples X Y which are such that
2:25:31X is the father of Y from the
2:25:33mathematical point of view the only
2:25:36thing of importance about the relation
2:25:38father is that it defines this set of
2:25:40ordered
2:25:42couples speaking generally we say the
2:25:45extension of a relation is the class of
2:25:47those ordered couples X Y which are such
2:25:51that X has the relation in question to
2:25:54Y we can now go a step further in the
2:25:57process of abstraction and consider what
2:26:00we mean by structure given any relation
2:26:03we can if it is a sufficiently simple
2:26:06one construct a map of it for the sake
2:26:09of definiteness let us take a relation
2:26:12of which the extension is the following
2:26:14couples AB AC a d b c c e d c d e where
2:26:21a b c d e are five terms no matter what
2:26:26we may make a map of this relation by
2:26:29taking five points on a plane and
2:26:31connecting them by arrows as in the
2:26:33accompanying figure what is revealed by
2:26:36the map is what we call the structure of
2:26:38the
2:26:40relation it is clear that the structure
2:26:42of the relation does not depend upon the
2:26:45particular terms that make up up the
2:26:47field of the relation the field may be
2:26:49changed without changing the structure
2:26:52and the structure may be changed without
2:26:54changing the field for example if we
2:26:57were to add the couple AE in the above
2:27:00illustration we should alter the
2:27:02structure but not the field two
2:27:04relations have the same structure we
2:27:06shall say when the same map will do for
2:27:09both or what comes to the same thing
2:27:12when either can be a map for the other
2:27:15since every relation can be its own map
2:27:19and that as a moment's reflection shows
2:27:21is the very same thing as what we have
2:27:23called likeness that is to say two
2:27:26relations have the same structure when
2:27:28they have likeness that is when they
2:27:31have the same relation number thus what
2:27:34we defined as the relation number is the
2:27:37very same thing as is obscurely intended
2:27:40by the word structure a word which
2:27:43important as it is is never so far as We
2:27:46Know defined in precise terms by those
2:27:49who use
2:27:50it there has been a great deal of
2:27:53speculation in traditional philosophy
2:27:56which might have been avoided if the
2:27:57importance of structure and the
2:27:59difficulty of getting behind it had been
2:28:02realized for example it is often said
2:28:05that space and time are subjective but
2:28:08they have objective counterparts or that
2:28:11phenomena are subjective but are caused
2:28:14by things in themselves which must have
2:28:16differences inters say corresponding
2:28:19with the differences in the phenomena to
2:28:21which they give rise where such
2:28:23hypotheses are made it is generally
2:28:26supposed that we can know very little
2:28:28about the objective
2:28:29counterparts in actual fact however if
2:28:33the hypothesis as stated were correct
2:28:35the objective counterparts would form a
2:28:37world having the same structure as the
2:28:40phenomenal world and allowing us to
2:28:42infer from phenomena the truth of all
2:28:45propositions that can be stated in
2:28:48abstract terms and are known to be true
2:28:50of phenomena if the phenomenal world has
2:28:53three dimensions so must the World
2:28:55Behind phenomena if the phenomenal world
2:28:58is ukian so must the other be and so
2:29:02on in short every proposition having a
2:29:06communicable significance must be true
2:29:09of Both Worlds or of neither the only
2:29:12difference must lie in just that essence
2:29:15of individuality which always eludes
2:29:17words and baffles description but which
2:29:20for that very reason is irrelevant to
2:29:22science now the only purpose that
2:29:24philosophers have in view in condemning
2:29:26phenomena is in order to persuade
2:29:28themselves and others that the real
2:29:31world is very different from the world
2:29:33of appearance we can all sympathize with
2:29:35their wish to prove such a very
2:29:37desirable proposition but we cannot
2:29:40congratulate them on their
2:29:42success it is true that many of them do
2:29:45not assert objective counterpart parts
2:29:46to phenomena and these escape from the
2:29:48above argument those who do assert
2:29:51counterparts are as a rule very reticent
2:29:54on the subject probably because they
2:29:57feel instinctively that if pursued it
2:30:00will bring about too much of a reproach
2:30:02me between the real and the phenomenal
2:30:04world if they were to pursue the topic
2:30:08they could hardly avoid the conclusions
2:30:10which we have been suggesting in such
2:30:12ways as well as in many others the
2:30:15notion of structure or ation number is
2:30:18important end of chapter
2:30:256 chapter S of introduction to
2:30:29mathematical Philosophy by Bertrand
2:30:32Russell this LibriVox recording is in
2:30:35the public
2:30:36domain rational real and complex
2:30:40numbers we have now seen how to define
2:30:43cardinal numbers and also relation
2:30:46numbers num of which what are commonly
2:30:48called ordinal numbers are a particular
2:30:51species it will be found that each of
2:30:54these kinds of number may be infinite
2:30:56just as well as finite but neither is
2:31:00capable as it stands of the more
2:31:03familiar extensions of the idea of
2:31:05number namely the extensions to negative
2:31:08fractional irrational and complex
2:31:11numbers in the present chapter we shall
2:31:14briefly Supply logical definition
2:31:16of these various
2:31:19extensions one of the mistakes that have
2:31:21delayed the discovery of correct
2:31:23definitions in this region is the common
2:31:26idea that each extension of number
2:31:29included the previous sorts as special
2:31:32cases it was thought that in dealing
2:31:35with positive and negative integers the
2:31:38positive integers might be identified
2:31:41with the original signless integers
2:31:44again it was thought that a fraction
2:31:46whose denominator is one may be
2:31:49identified with a natural number which
2:31:51is its numerator and the irrational
2:31:54numbers such as the square root of two
2:31:57were supposed to find their place among
2:32:00rational fractions as being greater than
2:32:02some of them and less than the others so
2:32:05that rational and irrational numbers
2:32:08could be taken together as one class
2:32:10called real numbers and when the idea of
2:32:13number was further extended so as to
2:32:15include complex numbers that is numbers
2:32:19involving the square Ro T
2:32:20of1 it was thought that real numbers
2:32:23could be regarded as those among complex
2:32:26numbers in which the imaginary part that
2:32:29is the part which was a multiple of the
2:32:32< TK of1 was
2:32:34Zero all these suppositions were
2:32:37erroneous and must be discarded as we
2:32:40shall find if correct definitions are to
2:32:42be
2:32:43given let us begin with positive and and
2:32:46negative integers it is obvious on a
2:32:49moment's consideration that + one and
2:32:53minus1 must both be relations and in
2:32:56fact must be each other's
2:32:59Converses the obvious and sufficient
2:33:01definition is that + one is the relation
2:33:04of n + 1 to n and minus one is the
2:33:07relation of n to n + 1 generally if m is
2:33:12any inductive number plus M will be the
2:33:15relation of n plus M to n for any n and
2:33:19minus M will be the relation of n to n+
2:33:24M according to this definition plus m is
2:33:28a relation which is 1 one so long as N
2:33:32is a cardinal number finite or infinite
2:33:35and M is an inductive Cardinal number
2:33:38but plus m is under no circumstances
2:33:41capable of being identified with M which
2:33:44is not a relation but a class class of
2:33:46classes indeed plus m is every bit as
2:33:50distinct from M as minus m
2:33:54is fractions are more interesting than
2:33:56positive or negative integers we need
2:33:59fractions for many purposes but perhaps
2:34:02most obviously for purposes of
2:34:04measurement my friend and collaborator
2:34:07Dr a an Whitehead has developed a theory
2:34:10of fractions specially adapted for their
2:34:13application to measurement which is set
2:34:16forth in principia Mathematica footnote
2:34:191 volume 3 star 300 and following
2:34:23especially 33 end of footnote 1 but if
2:34:27all that is needed is to Define objects
2:34:29having the required purely mathematical
2:34:32properties this purpose can be achieved
2:34:34by a simpler method which we shall here
2:34:37adopt we shall Define the fraction m / n
2:34:42as being that relation which holds
2:34:44between two inductive numbers X
2:34:46y when xn equal
2:34:50ym this definition enables us to prove
2:34:54that m / N is a 1 one relation provided
2:34:58neither M nor n is zero and of course n
2:35:02/ m is the converse relation to m / n
2:35:08from the above definition it is clear
2:35:10that the fraction m / 1 is that relation
2:35:13between two integers X and Y which
2:35:16consists in the fact that xal my
2:35:20y this relation like the relation plus m
2:35:24is by no means capable of being
2:35:26identified with the inductive Cardinal M
2:35:29because a relation and a class of
2:35:31classes are objects of utterly different
2:35:33kinds footnote one of course in practice
2:35:37we shall continue to speak of a fraction
2:35:39as say greater or less than one meaning
2:35:43greater or less than the ratio 1 /
2:35:461 so long as it is understood that the
2:35:49ratio 1 / 1 and the cardal number one
2:35:52are different it is not necessary to be
2:35:55always panic in emphasizing the
2:35:58difference end of footnote one it will
2:36:01be seen that 0 / n is always the same
2:36:05relation whatever inductive number n may
2:36:07be it is in short the relation of zero
2:36:11to any other inductive Cardinal we may
2:36:14call this the zero of r numbers it is
2:36:17not of course identical with the
2:36:19Cardinal number zero conversely the
2:36:22relation m /0 is always the same
2:36:25whatever inductive number M may be there
2:36:29is not any inductive Cardinal
2:36:30corresponding to M ID Z we may call it
2:36:34the Infinity of rationals it is an
2:36:37instance of the sort of infinite that is
2:36:39traditional in mathematics and that is
2:36:41represented by a
2:36:43lncape this is a totally different sort
2:36:46from the true cantoran infinite which we
2:36:48shall consider in our next chapter the
2:36:51Infinity of rationals does not demand
2:36:53for its definition or use any infinite
2:36:56classes or infinite integers it is not
2:36:59an actual fact a very important notion
2:37:02and we could dispense with it alt
2:37:04together if there were any object in
2:37:06doing so the cantoran infinite on the
2:37:09other hand is of the greatest and most
2:37:12fundamental importance the understanding
2:37:14of it opens the way to whole new Realms
2:37:17of mathematics and
2:37:21philosophy it will be observed that Zer
2:37:23and infinity alone among ratios are not
2:37:27one one zero is one many and infinity is
2:37:31many
2:37:32one there is not any difficulty in
2:37:35defining greater or less among ratios or
2:37:38fractions given two ratios m / n and p /
2:37:43Q we shall say that m / by n is less
2:37:47than P / Q If the product of M and Q is
2:37:52less than the product of p and N there
2:37:55is no difficulty in proving that the
2:37:57relation less than so defined is serial
2:38:01so that the ratios form a series in
2:38:03order of magnitude in this series zero
2:38:07is the smallest term and infinity is the
2:38:10largest if we omit zero and infinity
2:38:13from our series there is no longer any
2:38:15small or largest ratio it is obvious
2:38:19that if m / n is any ratio other than
2:38:23zero and infinity m / the product of 2
2:38:27and N is smaller and the product of 2
2:38:31and N / n is larger though neither is
2:38:35zero or Infinity so that m / n is
2:38:40neither the smallest nor the largest
2:38:42ratio and therefore when zero and
2:38:44infinity are om
2:38:46there is no largest or smallest since m
2:38:49/ n was chosen
2:38:51arbitrarily in like manner we can prove
2:38:54that however nearly equal two fractions
2:38:56may be there are always other fractions
2:38:59between them for let m / n and p / Q be
2:39:04two fractions of which P / Q is the
2:39:08greater then it is easy to see or to
2:39:12prove that the sum of M and P divided by
2:39:16the sum of N and Q will be greater than
2:39:19m / n and less than P / Q thus the
2:39:25series of ratios is one in which no two
2:39:27terms are consecutive but there are
2:39:30always other terms between any two since
2:39:33there are other terms between these
2:39:35others and so on at infinum it is
2:39:38obvious that there are an infinite
2:39:40number of ratios between any two however
2:39:43nearly equal these two may be footnote
2:39:46one strictly speaking this statement as
2:39:50well as those following to the end of
2:39:52the paragraph involves what is called
2:39:54the Axiom of infinity which will be
2:39:57discussed in a later chapter end of
2:40:00footnote
2:40:01one a series having the property that
2:40:04there are always other terms between any
2:40:06two so that no two are consecutive is
2:40:10called
2:40:11compact thus the ratios in order of
2:40:14magnitude form a compact
2:40:17series such series have many important
2:40:20properties and it is important to
2:40:22observe that ratios afford an instance
2:40:25of a compact series generated purely
2:40:28logically without any appeal to space or
2:40:32time or any other empirical
2:40:36datum positive and negative ratios can
2:40:38be defined in a way analogous to that in
2:40:42which we defined positive and negative
2:40:44integers
2:40:46having first defined the sum of two
2:40:49ratios m / n and p / Q as the sum of the
2:40:56product of M and Q and the product of p
2:40:59and N divided by the product of n and q
2:41:03we
2:41:05Define summing by P / Q as the relation
2:41:10of m / n summed with P / Q to m / n
2:41:18where m / n is any
2:41:21ratio and subtracting by P / Q is of
2:41:26course the converse of summing by P / Q
2:41:31this is not the only possible way of
2:41:32defining positive and negative ratios
2:41:35but it is a way which for our purpose
2:41:38has the Merit of being an obvious
2:41:40adaptation of the way we adopted in the
2:41:42case of
2:41:44integers we come now to a more
2:41:46interesting extension of the idea of
2:41:49number that is the extension to what are
2:41:52called real numbers which are the kind
2:41:55that Embrace irrationals in chapter one
2:41:58we had occasion to mention
2:42:00incommensurables and their Discovery by
2:42:03Pythagoras it was through them that is
2:42:06through geometry that irrational numbers
2:42:08were first thought of a square of which
2:42:12the side is one inch long will have a
2:42:14diagonal of which the length is the
2:42:17square root of 2
2:42:18in but as the Ancients discovered there
2:42:21is no fraction of which the square is
2:42:24two this proposition is proved in the
2:42:2710th book of uclid which is one of those
2:42:29books that school boys supposed to be
2:42:32fortunately lost in the days when uclid
2:42:35was still used as a textbook the proof
2:42:38is extraordinarily simple if possible
2:42:42let m / n be theare root of 2 so that
2:42:47m^2 / n^2 is equal to 2 that is m^2 is
2:42:54equal to the product of 2 and
2:42:57n^2 thus m^2 is an even number and
2:43:02therefore M must be an even number
2:43:05because the square of an odd number is
2:43:07odd now if m is even m^2 must divide by
2:43:124 for if m equals the product of 2 and P
2:43:17then m^2 equals the product of 4 and
2:43:21p^2 thus we shall have the product of 4
2:43:24and p^2 is equal to the product of 2 and
2:43:28n^2 where p is half of M hence the
2:43:33product of 2 and p^2 is equal to
2:43:37n^2 and therefore n / P will also be
2:43:41theare < TK of 2 but then we can repeat
2:43:45the argument if n is equal to the
2:43:47product of 2 and q p / Q will also be
2:43:51the square root of two and so on through
2:43:54an unending series of numbers that are
2:43:57each half of its
2:43:59predecessor but this is
2:44:01impossible if we divide a number by two
2:44:04and then have the half and so on we must
2:44:07reach an odd number after a finite
2:44:10number of steps or we may put the
2:44:12argument even more simply by assuming
2:44:15that the m / n we start with is in its
2:44:19lowest terms in that case M and N cannot
2:44:23both be even yet we have seen that if
2:44:27m^2 / n^2 = 2 they must be thus there
2:44:33cannot be any fraction m / n whose
2:44:37square is
2:44:392 thus no fraction will Express exactly
2:44:42the length of the diagonal of a square
2:44:45whose side is 1 in long this seems like
2:44:48a challenge thrown out by nature to
2:44:51arithmetic however the arithmetician May
2:44:54boast as Pythagoras did about the power
2:44:58of numbers nature seems able to baffle
2:45:01him by exhibiting lengths which no
2:45:04numbers can estimate in terms of the
2:45:06unit but the problem did not remain in
2:45:09this geometrical
2:45:11form as soon as algebra was invented the
2:45:14same problem arose as regards the
2:45:16solution of equations though here it
2:45:19took on a wider form since it also
2:45:21involved complex
2:45:23numbers it is clear that fractions can
2:45:26be found which approach nearer and
2:45:28nearer to having their Square equal to
2:45:30two we can form an ascending series of
2:45:33fractions all of which have their
2:45:34squares less than two but differing from
2:45:38two in their later members by less than
2:45:41any assigned
2:45:42amount that is to say suppose I assign
2:45:46some small amount in advance say 1
2:45:49billionth it will be found that all the
2:45:52terms of our series after a certain one
2:45:54say the 10th have squares that differ
2:45:57from Two by less than this
2:46:00amount and if I had assigned a still
2:46:03smaller amount it might have been
2:46:05necessary to go further along the series
2:46:09but we should have reached sooner or
2:46:11later a term in the series say the 20th
2:46:14at after which all terms would have had
2:46:17squares differing from Two by less than
2:46:20this still smaller
2:46:22amount if we set to work to extract the
2:46:25square root of two by the usual
2:46:27arithmetical rule we shall obtain an
2:46:30unending decimal which taken to so and
2:46:33so many places exactly fulfills the
2:46:36above
2:46:37conditions we can equally well form a
2:46:40descending series of fractions whose
2:46:42squares are all greater than two but
2:46:45Greater by continually smaller amounts
2:46:47as we come to later terms of the series
2:46:50and differing Sooner or Later by less
2:46:53than any assigned amount in this way we
2:46:56seem to be drawing a coordin around the
2:46:58square root of two and it may seem
2:47:00difficult to believe that it can
2:47:02permanently Escape us nevertheless it is
2:47:06not by this method that we shall
2:47:08actually reach the square root of
2:47:12two if we divide all ratios into two
2:47:15classes according as their squares are
2:47:18less than two or not we find that among
2:47:20those whose squares are not less than
2:47:23two all have their squares greater than
2:47:26two there is no maximum to the ratios
2:47:29whose square is less than two and no
2:47:32minimum to those whose square is greater
2:47:35than
2:47:36two there is no lower limit short of
2:47:38zero to the difference between the
2:47:41numbers whose square is a little less
2:47:43than two and the the numbers who square
2:47:46is a little greater than two we can in
2:47:50short divide all ratios into two classes
2:47:54such that all the terms in one class are
2:47:56less than all in the other and there is
2:47:59no maximum to the one class and there's
2:48:01no minimum to the
2:48:03other between these two classes where
2:48:06the square root of two ought to be there
2:48:08is
2:48:09nothing thus our Cordon though we have
2:48:12drawn it as tight as possible has been
2:48:15drawn in the wrong place and has not
2:48:18caught the square root of
2:48:21two the above method of dividing all the
2:48:23terms of a series into two classes of
2:48:26which the one holy precedes the other
2:48:29was brought into prominence by
2:48:31dedicant footnote
2:48:35onein second edition brunwick 1892 end
2:48:39of footnote one and is therefore called
2:48:42a dedicant cut with respect to what
2:48:45happens at the point of section there
2:48:48are four
2:48:49possibilities one there may be a maximum
2:48:52to the lower section and a minimum to
2:48:54the upper section two there may be a
2:48:57maximum to the one and no minimum to the
2:49:00other three there may be no maximum to
2:49:03the one but a minimum to the other four
2:49:06there may be neither a maximum to the
2:49:08one nor a minimum to the
2:49:10other of these four cases the first is
2:49:14Illustrated by any Series in which there
2:49:16are consecutive terms in the series of
2:49:19integers for instance a lower section
2:49:21must end with some number n and the
2:49:24upper section must then begin with n + 1
2:49:29the second case will be Illustrated in
2:49:31the series of ratios if we take as our
2:49:34lower section all ratios up to and
2:49:36including one and in our upper section
2:49:39all ratios greater than one the third
2:49:42case is Illustrated if we take for our
2:49:45lower section all ratios less than one
2:49:48and for our upper section all ratios
2:49:51from one upward including one
2:49:54itself the fourth case as we have seen
2:49:57is Illustrated if we put in our lower
2:49:59section all ratios whose square is less
2:50:03than two and in our upper section all
2:50:06ratios whose square is greater than
2:50:09two we may neglect the first of our four
2:50:12cases since it only ARIS in series where
2:50:16there are a consecutive
2:50:18terms in the second of our four cases we
2:50:20may say that the maximum of the lower
2:50:22section is the lower limit of the upper
2:50:26section or of any set of terms chosen
2:50:29out of the upper section in such a way
2:50:31that no term of the upper section is
2:50:34before all of them in the third of our
2:50:37four cases we say that the minimum of
2:50:39the upper section is the upper limit of
2:50:42the lower
2:50:43section or of any set of terms chosen
2:50:46out of the lower section in such a way
2:50:49that no term of the lower section is
2:50:51after all of them in the fourth case we
2:50:55may say that there is a gap neither the
2:50:58upper section nor the lower has a limit
2:51:01or a last term in this case we may also
2:51:06say that we have an irrational section
2:51:09since sections of the series of ratios
2:51:12have gaps when they correspond to
2:51:16irrationals what delayed the theory of
2:51:18irrationals was a mistaken belief that
2:51:20there must be limits of series of ratios
2:51:23the notion of limit is of the utmost
2:51:26importance and before proceeding further
2:51:29it will be well to Define
2:51:31it a term X is said to be an upper limit
2:51:35of a class Alpha with respect to a
2:51:37relation P if one alpha has no maximum m
2:51:41p two every member of Alpha which
2:51:45belongs to the field of P precedes x
2:51:47three every member of the field of P
2:51:50which precedes X precedes some member of
2:51:53alpha by proceeds we mean has the
2:51:56relation
2:51:57P2 this presupposes the following
2:52:00definition of a
2:52:01maximum a term X is said to be a maximum
2:52:05of a class Alpha with respect to a
2:52:08relation P if x is a member of Alpha and
2:52:12of the field of P and does not have the
2:52:15relation P to any other member of
2:52:20alpha these definitions do not demand
2:52:23that the terms to which they are applied
2:52:25should be quantitative for example given
2:52:29a series of moments of time arranged by
2:52:31earlier and later their maximum if any
2:52:35will be the last of the moments but if
2:52:37they arranged by later and earlier their
2:52:40maximum if any will be the first of the
2:52:43moments
2:52:45the minimum of a class with respect to P
2:52:48is its maximum with respect to the
2:52:50converse of p and the lower limit with
2:52:54respect to P is the upper limit with
2:52:57respect to the converse of
2:53:00P the Notions of limit and maximum do
2:53:04not essentially demand that the relation
2:53:06in respect to which they are defined
2:53:08should be serial but they have few
2:53:11important applications except to cases
2:53:15when the relation is serial or quasy
2:53:18serial a notion which is often important
2:53:20is the notion upper limit or maximum to
2:53:23which we may give the name upper
2:53:26boundary thus the upper boundary of a
2:53:28set of terms chosen out of a series is
2:53:31their last member if they have one but
2:53:35if not it is the first term after all of
2:53:37them if there is such a term if there is
2:53:41neither a maximum nor a limit there is
2:53:44no upper boundary the lower limit is the
2:53:47lower limit or
2:53:50minimum reverting to the four kinds of
2:53:52dedin section we see that in the case of
2:53:55the first three kinds each section has a
2:53:58boundary upper or lower as the case may
2:54:00be while in The Fourth Kind neither has
2:54:03a
2:54:04boundary it is also clear that whenever
2:54:06the lower section has an upper boundary
2:54:09the upper section has a lower boundary
2:54:12in the second and third cases the two
2:54:14two boundaries are identical in the
2:54:16first they are consecutive terms of the
2:54:18series a series is called DED aindian
2:54:22when every section has a boundary upper
2:54:24or lower as the case may
2:54:27be we have seen that the series of
2:54:30ratios in order of magnitude is not
2:54:34dedan from the habit of being influenced
2:54:37by spatial imagination people have
2:54:39supposed that series must have limits in
2:54:42cases where it seems odd if they do not
2:54:45thus perceiving that there was no
2:54:46rational limit to the ratios whose
2:54:49square is less than two they allowed
2:54:51themselves to postulate an irrational
2:54:54limit which was to fill the dedin Gap
2:54:57dedin in the above mentioned work set up
2:55:00the axom that the Gap must always be
2:55:02filled that is that every section must
2:55:04have a boundary it is for this reason
2:55:07that Series where his Axiom is verified
2:55:09are called
2:55:11Dean but there are an infinite number of
2:55:13series for which it is not
2:55:15verified the method of postulating what
2:55:19we want has many advantages they are the
2:55:22same as the advantages of theft over
2:55:25honest
2:55:25toil let us leave them to others and
2:55:28proceed with our honest
2:55:31toil it is clear that an irrational
2:55:33dedicant cut in some way represents an
2:55:37irrational in order to make use of this
2:55:39which to begin with is no more than a
2:55:42vague feeling we must find some way of
2:55:45eliciting from it a precise definition
2:55:48and in order to do this we must disabuse
2:55:51our minds of the notion that an
2:55:53irrational must be the limit of a set of
2:55:56ratios just as ratios whose denominator
2:56:00is one are not identical with integers
2:56:03so those rational numbers which can be
2:56:05greater or less than irrationals or can
2:56:08have irrationals as their limits must
2:56:10not be identified with ratios we have to
2:56:13define a new kind of numbers called real
2:56:16numbers of which some will be rational
2:56:18and some irrational those that are
2:56:21rational correspond to ratios in the
2:56:24same kind of way in which the ratio n /
2:56:271 corresponds to the integer n but they
2:56:30are not the same as
2:56:32ratios in order to decide what they are
2:56:34to be let us observe that an irrational
2:56:37is represented by an irrational cut and
2:56:40a cut is represented by its lower
2:56:43section
2:56:44let us confine ourselves to Cuts in
2:56:46which the lower section has no maximum
2:56:49in this case we will call the lower
2:56:51section a
2:56:52segment then those segments that
2:56:54correspond to ratios are those that
2:56:57consist of all ratios less than the
2:56:59ratio they correspond to which is their
2:57:02boundary while those that represent
2:57:05irrationals are those that have no
2:57:07boundary segments both those that have
2:57:10boundaries and those that do not are
2:57:12such that of any two pertaining to one
2:57:14series one must be part of the other
2:57:18hence they can all be arranged in a
2:57:20series by the relation of whole and
2:57:23part A Series in which there are
2:57:25dedicant gaps that is in which there are
2:57:28segments that have no boundary will give
2:57:30rise to more segments than it has terms
2:57:33since each term will Define a segment
2:57:36having that term for a
2:57:38boundary and then the segments without
2:57:40boundaries will be
2:57:42extra we we now in a position to define
2:57:45a real number and an irrational number a
2:57:49real number is a segment of the series
2:57:52of ratios in order of
2:57:55magnitude an irrational number is a
2:57:58segment of the series of ratios which
2:58:00has no
2:58:02boundary a rational real number is a
2:58:05segment of the series of ratios which
2:58:08has a
2:58:10boundary thus a rational real number
2:58:13consists of all ratios less than a
2:58:16certain ratio and it is the rational
2:58:19real number corresponding to that
2:58:22ratio the real number one for instance
2:58:25is the class of proper
2:58:27fractions in the cases in which we
2:58:30naturally supposed that an irrational
2:58:32must be the limit of a set of ratios the
2:58:35truth is that it is the limit of the
2:58:38corresponding set of rational real
2:58:40numbers in the series of segments
2:58:43ordered by whole and part for example
2:58:47the square root of two is the upper
2:58:49limit of all those segments of the
2:58:51series of ratios that correspond to
2:58:54ratios whose square is less than two
2:58:57more simply still the square root of two
2:59:00is the segment consisting of all those
2:59:03ratios whose square is less than
2:59:06two it is easy to prove that the series
2:59:09of segments of any series is
2:59:12Dean for given any set of segments their
2:59:16boundary will be their logical sum that
2:59:19is the class of all those terms that
2:59:22belong to at least one segment of the
2:59:24set footnote one for a folder treatment
2:59:28of the subject of segments and D Indian
2:59:30relations see principia Mathematica
2:59:34Volume 2 Star 210 to
2:59:37214 for folder treatment of real numbers
2:59:40see the same work volume three stars 310
2:59:44and following and principles of
2:59:46mathematics chapters 33 and 34 end of
2:59:51footnote
2:59:52one the above definition of real numbers
2:59:55is an example of construction as against
2:59:59postulation of which we had another
3:00:01example in the definition of cardinal
3:00:04numbers the great advantage of this
3:00:07method is that it requires no new
3:00:10assumptions but enables us to proceed
3:00:13deductively
3:00:14from the original apparatus of
3:00:17logic there is no difficulty in defining
3:00:20addition and multiplication for real
3:00:22numbers as above defined given two real
3:00:26numbers me and new each being a class of
3:00:29ratios take any member of mu and any
3:00:33member of new and add them together
3:00:36according to the rule for the addition
3:00:39of
3:00:40ratios form the class of all such sums
3:00:43obtainable
3:00:44by varying the selected members of mu
3:00:46and new this gives a new class of ratios
3:00:50and it is easy to prove that this new
3:00:52class is a segment of the series of
3:00:55ratios we Define it as the sum of mu and
3:00:59new we may State the definition more
3:01:02shortly as
3:01:03follows the arithmetical sum of two real
3:01:07numbers is the class of the arithmetical
3:01:10sums of a member of one and a member of
3:01:13the other
3:01:14chosen in all possible
3:01:18ways we can Define the arithmetical
3:01:20product of two real numbers in exactly
3:01:22the same way by multiplying a member of
3:01:25the one by a member of the other in all
3:01:29possible ways the class of ratios thus
3:01:32generated is defined as the product of
3:01:35the two real numbers in all such
3:01:37definitions the series of ratios is to
3:01:40be defined as excluding Z and and
3:01:44infinity there is no difficulty in
3:01:47extending our definitions to positive
3:01:49and negative real numbers and their
3:01:52addition and
3:01:54multiplication complex numbers though
3:01:56capable of a geometrical interpretation
3:01:59are not demanded by geometry in the same
3:02:02imperative way in which irrationals are
3:02:06demanded a complex number means a number
3:02:09involving the square root of a negative
3:02:12number whether inte fractional or
3:02:15real since the square of a negative
3:02:18number is positive a number whose square
3:02:20is to be negative has got to be a new
3:02:23sort of number using the letter i for
3:02:27the square root of -1 any number
3:02:30involving the square root of a negative
3:02:32number can be expressed in the form the
3:02:36sum of X and the product of Y and I
3:02:39where X and Y are real the part the
3:02:43product of Y and I is called the
3:02:46imaginary part of this number X being
3:02:49the real part the reason for the phrase
3:02:52real numbers is that they're contrasted
3:02:54with such as are
3:02:56imaginary complex numbers have been for
3:02:59a long time habitually used by
3:03:02mathematicians in spite of the absence
3:03:04of any precise
3:03:06definition it has been simply assumed
3:03:08that they would obey the usual
3:03:10arithmetical rules and on this
3:03:12assumption their employment has been
3:03:14found
3:03:15profitable they are required less for
3:03:18geometry than for algebra and
3:03:21Analysis we desire for example to be
3:03:24able to say that every quadratic
3:03:26equation has two roots and every cubic
3:03:29equation has three and so on but if we
3:03:33are confined to real numbers such an
3:03:35equation as x^2 + 1 equals 0 has no
3:03:40roots and such an equation as X cubed -
3:03:441 = 0 has only one every generalization
3:03:48of number has first presented itself as
3:03:51needed for some simple problem negative
3:03:54numbers are needed in order that
3:03:56subtraction might always be possible
3:03:59since a minus B would be meaningless if
3:04:02a were less than b fractions were needed
3:04:05in order that division might always be
3:04:07possible and complex numbers are needed
3:04:10in order that extraction of roots and
3:04:12solution of of equations may be always
3:04:16possible but extensions of number are
3:04:18not created by the mere need for them
3:04:21they are created by definition and it is
3:04:24to the definition of complex numbers
3:04:27that we must now turn our
3:04:30attention a complex number may be
3:04:33regarded and defined as simply an
3:04:35ordered couple of real numbers here as
3:04:38elsewhere many definitions are possible
3:04:41all that is necessary is that the the
3:04:43definitions adopted shall lead to
3:04:45certain properties in the case of
3:04:48complex numbers if they are defined as
3:04:50ordered couples of real numbers we
3:04:52secure at once some of the properties
3:04:55required namely that two real numbers
3:04:58are required to determine a complex
3:05:00number and that among these we can
3:05:02distinguish a first and a second and
3:05:05that two complex numbers are only
3:05:07identical when the first real number
3:05:09involved in the one is equal to the
3:05:11first involved in the other and the
3:05:13second to the second what is needed
3:05:16further can be secured by defining the
3:05:18rules of addition and
3:05:20multiplication we are to have the sum of
3:05:24the sum of X and the product of Y and I
3:05:27with the sum of X Prime and the product
3:05:31of Y Prime and I is equal to the sum of
3:05:36the sum of X and X Prime with the
3:05:39product of the sum of Y and Y Prime with
3:05:44I the product of the sum of X and the
3:05:49product of Y and I with the sum of X
3:05:53Prime and the product of Y Prime and I
3:05:57is equal to the sum of the subtraction
3:06:01of the product of X and X Prime from the
3:06:06product of Y and Y Prime with the
3:06:09product of the sum of the product of X
3:06:12and Y Prime
3:06:14and the product of X Prime and Y with
3:06:18I thus we shall Define that given two
3:06:21ordered couples of real numbers x y and
3:06:25x Prime y Prime their sum is to be the
3:06:28couple x + x Prime y + y Prime and their
3:06:32product is to be the couple x * X Prime
3:06:36- y * y Prime x * y Prime + x Prime y by
3:06:42these definitions we shall secure that
3:06:45our ordered couples shall have the
3:06:47properties we desire for example take
3:06:50the product of the two couples 0 Y and 0
3:06:54y Prime this will by the above rule be
3:06:58the couple the negative product of Y and
3:07:01Y Prime and zero thus the square of the
3:07:04couple 01 will be the couple 1
3:07:090 now those couples in which the second
3:07:11term is zero are those which according
3:07:14to the usual nomenclature have their
3:07:17imaginary part
3:07:18zero in the notation X plus the product
3:07:22of Y and I they are X plus the product
3:07:26of zero and I which it is natural to
3:07:29write simply X just as it is natural but
3:07:32erroneous to identify ratios whose
3:07:34denominator is Unity with integers so it
3:07:38is natural but erroneous to identify
3:07:41complex numbers who whose imaginary part
3:07:44is zero with real numbers although this
3:07:47is an error in theory it is a
3:07:49convenience in
3:07:51practice X plus the product of0 and I
3:07:54may be replaced simply by X and 0 + y
3:07:58and the product of I by Y and the
3:08:01product of I provided we remember that
3:08:04the X is not really a real number but a
3:08:07special case of a complex number and
3:08:10when Y is one the of Y and I may of
3:08:14course be replaced by
3:08:16I thus the couple 01 is represented by I
3:08:20and the couple -1 0 is represented by -1
3:08:25now our rules of multiplication make the
3:08:28square of the ordered pair 01 equal to
3:08:31the square of the ordered pair -1 0 that
3:08:34is the square of I is1 this is what we
3:08:38desired to secure thus our definitions
3:08:41serve all necessary
3:08:44purposes it is easy to give a
3:08:46geometrical interpretation of complex
3:08:48numbers in the geometry of the plane
3:08:51this subject was agreeably expounded by
3:08:54w k Clifford in his common sense of the
3:08:57exact Sciences a book of great Merit but
3:09:01written before the importance of purely
3:09:03logical definitions had been
3:09:06realized complex numbers of a higher
3:09:08order though much less useful and
3:09:10important than those we have been
3:09:12defining
3:09:13have certain uses that are not without
3:09:15importance in Geometry as may be seen
3:09:18for example in Dr Whitehead's universal
3:09:21algebra the definition of complex
3:09:23numbers of order n is obtained by an
3:09:27obvious extension of the definition we
3:09:29have given we Define a complex number of
3:09:33order n as a on many relation whose
3:09:37domain consists of certain real numbers
3:09:39and whose Converse domain consists of
3:09:42the integers from 1 to n footnote one
3:09:46confer the principles of mathematics
3:09:48section 360 page 379 end of footnote
3:09:531 this is what would ordinarily be
3:09:56indicated by the notation the order
3:09:59andle X1 X2 X3 and so on to xn where the
3:10:05suffixes denote correlation with the
3:10:07integers used as suffixes and the
3:10:10correlation is one many not necessarily
3:10:1311 one because x subr and x Subs may be
3:10:18equal when R and S are not
3:10:21equal the above definition with a
3:10:23suitable rule of multiplication will
3:10:26serve all the purposes for which complex
3:10:28numbers of higher orders are
3:10:31needed we have now completed our review
3:10:34of those extensions of number which do
3:10:36not involve Infinity the application of
3:10:39number to infinite collections must be
3:10:42our next next topic end of chapter
3:10:567 chapter eight of introduction to
3:11:00mathematical Philosophy by Bertrand
3:11:02Russell this LibriVox recording is in
3:11:05the public
3:11:06domain infinite cardinal
3:11:09numbers the definition of cardinal
3:11:11numbers which we gave in Chapter 2 was
3:11:14applied in chapter 3 to finite numbers
3:11:17that is to the ordinary natural numbers
3:11:20to these we gave the name inductive
3:11:22numbers because we found that they are
3:11:25to be defined as numbers which obey
3:11:27mathematical induction starting from
3:11:30zero but we have not yet considered
3:11:33collections which do not have an
3:11:35inductive number of terms nor have we
3:11:37inquired whether such collections can be
3:11:40said to have a number at all
3:11:43this is an ancient problem which has
3:11:45been solved in our own day chiefly by
3:11:48gor caner in the present chapter we
3:11:52shall attempt to explain the theory of
3:11:54trans finite or infinite cardinal
3:11:56numbers as it results from a combination
3:11:59of his discoveries with those of fragga
3:12:02on The Logical theory of
3:12:04numbers it cannot be said to be certain
3:12:08that there are in fact any infinite
3:12:10collections in the world the assump that
3:12:13there are is what we call the Axiom of
3:12:16infinity although various ways suggest
3:12:19themselves by which we might hope to
3:12:21prove this Axiom there is reason to fear
3:12:24that they are all facius and that there
3:12:27is no conclusive logical reason for
3:12:30believing it to be
3:12:32true at the same time there is certainly
3:12:35no logical reason against infinite
3:12:38Collections and we are therefore
3:12:40justified in logic in investigating the
3:12:43hypothesis that there are such
3:12:46collections the Practical form of this
3:12:48hypothesis for our present purposes is
3:12:51the assumption that if n is any
3:12:54inductive number n is not equal to n +
3:12:591 various subtleties arise in
3:13:01identifying this form of our assumption
3:13:04with the form that asserts the existence
3:13:06of infinite collections but we will
3:13:08leave these out of account until in a
3:13:11later chapter we come to consider the
3:13:14axium of Infinity on its own account for
3:13:17the present we shall merely assume that
3:13:20if n is an inductive number n is not
3:13:23equal to n + 1 this is involved in
3:13:26piano's assumption that no two inductive
3:13:29numbers have the same successor for if
3:13:33Nal n +1 then n minus one and N have the
3:13:37same successor namely n thus we are
3:13:42assuming nothing nothing that was not
3:13:43involved in Pano's primitive
3:13:46propositions let us now consider the
3:13:48collection of the inductive numbers
3:13:51themselves this is a perfectly well-
3:13:53defined class in the first place a
3:13:56cardinal number is a set of classes
3:13:59which are all similar to each other and
3:14:01are not similar to anything except each
3:14:04other we then Define as the inductive
3:14:07numbers those among Cardinals which
3:14:10belong to the posterity of zero
3:14:13with respect to the relation of n to n +
3:14:16one that is those which possess every
3:14:19property possessed by zero and by the
3:14:22successors of possessors meaning by the
3:14:26successor of n the number n plus one
3:14:30thus the class of inductive numbers is
3:14:32perfectly definite by our general
3:14:35definition of cardinal numbers the
3:14:37number of terms in the class of
3:14:39inductive numbers is to be defined as
3:14:42all those classes that are similar to
3:14:45the class of inductive numbers that is
3:14:49this set of classes is the number of the
3:14:52inductive numbers according to our
3:14:56definitions now it is easy to see that
3:14:59this number is not one of the inductive
3:15:02numbers if n is any inductive number the
3:15:05number of numbers from 0 to n both
3:15:08included is n + 1 therefore
3:15:13the total number of inductive numbers is
3:15:15greater than n no matter which of the
3:15:18inductive numbers n may be if we arrange
3:15:22the inductive numbers in a series in
3:15:24order of magnitude this Series has no
3:15:27last term but if n is an inductive
3:15:30number every series whose field has n
3:15:33terms has a last term as it is easy to
3:15:37prove such differences might be
3:15:39multiplied at lib thus the number of
3:15:43inductive numbers is a new number
3:15:46different from all of them not
3:15:48possessing all inductive
3:15:50properties it may happen that zero has a
3:15:53certain property and that if n has it so
3:15:56has n+ one and yet that this new number
3:16:00does not have it the difficulties that
3:16:03so long delay the theory of infinite
3:16:05numbers were largely due to the fact
3:16:08that some at least of the inductive
3:16:11properties were wrongly judged to be
3:16:13such as must belong to all numbers
3:16:17indeed it was thought that they could
3:16:19not be denied without
3:16:21contradiction the first step in
3:16:23understanding infinite numbers consists
3:16:25in realizing the mistaken of this
3:16:30view the most noteworthy and astonishing
3:16:33difference between an inductive number
3:16:36and this new number is that this new
3:16:38number is unchanged by adding one or
3:16:41subtracting one or doubling or having or
3:16:45any of a number of other operations
3:16:48which we think of as necessarily making
3:16:50a number larger or
3:16:52smaller the fact of being not altered by
3:16:55the addition of one is used by Cantor
3:16:58for the definition of what he calls
3:17:01transfinite cardinal numbers but for
3:17:04various reasons some of which will
3:17:06appear as we proceed it is better to
3:17:09Define an infinite Cardinal number as
3:17:11one which does not possess all inductive
3:17:14properties that is simply as one which
3:17:17is not an inductive
3:17:19number nevertheless the property of
3:17:22being unchanged by the addition of one
3:17:25is a very important one and we must
3:17:27dwell on it for a
3:17:29time to say that a class has a number
3:17:33which is not altered by the addition of
3:17:34one is the same thing as to say that if
3:17:38we take a term X which does not belong
3:17:40to the class we can find a one one
3:17:44relation whose domain is the class and
3:17:47whose Converse domain is obtained by
3:17:49adding x to the
3:17:51class for in that case the class is
3:17:54similar to the sum of itself and the
3:17:57term X that is to a class having one
3:18:00extra term so that it has the same
3:18:03number as a class with one extra term so
3:18:07that if n is this number n is equal to n
3:18:11+ 1
3:18:13in this case we shall also have Nal n
3:18:15minus one that is there will be one one
3:18:19relations whose domains consist of the
3:18:22whole class and whose Converse domains
3:18:25consist of just one term short of the
3:18:28whole
3:18:29class it can be shown that the cases in
3:18:32which this happens are the same as the
3:18:34Apparently more General cases in which
3:18:37some part short of the whole can be put
3:18:40into one one relation
3:18:42with the whole when this can be done the
3:18:45correlator by which it is done may be
3:18:48said to reflect the whole class into a
3:18:50part of itself for this reason such
3:18:53classes will be called reflexive thus a
3:18:57reflexive class is one which is similar
3:19:00to a proper part of itself a proper part
3:19:04is a part short of the
3:19:06whole a reflexive Cardinal number is the
3:19:10Cardinal number of a reflexive
3:19:12class we have now to consider this
3:19:15property of reflexiveness
3:19:18one of the most striking instances of a
3:19:21reflection is Roy's illustration of the
3:19:24map he imagines it decided to make a map
3:19:27of England upon a part of the surface of
3:19:31England a map if it is accurate has a
3:19:34perfect one one correspondents with its
3:19:37original thus our map which is part is
3:19:41in one one relation with the hole and
3:19:44must contain the same number of points
3:19:46as the hole which must therefore be a
3:19:48reflexive
3:19:50number Roy is interested in the fact
3:19:52that the map if it is correct must
3:19:55contain a map of the map which must in
3:19:58turn contain a map of the map of the map
3:20:01and so on add
3:20:03infinum this point is interesting but
3:20:06need not occupy us at this moment in
3:20:08fact we shall do well to pass from
3:20:10picturesque illustrations to such as are
3:20:13more completely definite and for this
3:20:15purpose we cannot do better than to
3:20:18consider the number series
3:20:21itself the relation of n to n + one
3:20:25confined to inductive numbers is 1 one
3:20:29has the whole of the inductive numbers
3:20:30for its domain and all except zero for
3:20:33its Converse domain thus the whole class
3:20:36of inductive numbers is similar to what
3:20:39the same class becomes when we omit
3:20:42zero consequently it is a reflexive
3:20:45class according to the definition and
3:20:48the number of its terms is a reflexive
3:20:51number again the relation of n to 2N
3:20:55confined to inductive numbers is one one
3:20:59has the whole of the inductive numbers
3:21:01for its domain and the even inductive
3:21:03numbers alone for its Converse domain
3:21:07hence the total number of inductive
3:21:09numbers is the same as the number of
3:21:11even inductive
3:21:13numbers this property was used by lies
3:21:16and many others as a proof that infinite
3:21:19numbers are impossible it was thought
3:21:22self-contradictory that the part should
3:21:24be equal to the whole but this is one of
3:21:27those phrases that depends for their
3:21:29plausibility upon an unperceived
3:21:32vagueness the word equal has many
3:21:35meanings but if it is taken to mean what
3:21:37we have called similar there is no
3:21:40contradiction since an infinite
3:21:42collection can perfectly well have Parts
3:21:44similar to
3:21:46itself those who regard this as
3:21:48impossible have unconsciously as a rule
3:21:51attributed to numbers in general
3:21:54properties which can only be proved by
3:21:56mathematical induction and which only
3:21:59their familiarity makes us regard
3:22:02mistakenly as true beyond the region of
3:22:04the
3:22:06finite whenever we can reflect a class
3:22:09into a part of itself the same Rel
3:22:12will necessarily reflect that part into
3:22:15a smaller part and so on ADD
3:22:18infinum for example we can reflect as
3:22:22we've just seen all the inductive
3:22:24numbers into the even numbers we can by
3:22:28the same relation that of n to n reflect
3:22:32the even numbers into the multiples of
3:22:34four these into the multiples of eight
3:22:37and so
3:22:39on this is an abstract analog to Roy's
3:22:42problem of the map the even numbers are
3:22:45a map of all the inductive numbers the
3:22:48multiples of four are a map of the map
3:22:52the multiples of eight are a map of the
3:22:54map of the map and so
3:22:57on if we had applied the same process to
3:23:00the relation of n to n +1 our map would
3:23:04have consisted of all the inductive
3:23:06numbers except zero the map of the map
3:23:11would have consisted Ed of all from two
3:23:13onward the map of the map of the map of
3:23:16all from three onward and so
3:23:19on the chief use of such illustrations
3:23:23is in order to become familiar with the
3:23:25idea of reflexive classes so that
3:23:28apparently paradoxical arithmetical
3:23:31propositions can be readily translated
3:23:33into the language of Reflections and
3:23:36classes in which the air of paradox is
3:23:39much
3:23:40less
3:23:42it will be useful to give a definition
3:23:44of the number which is that of the
3:23:46inductive Cardinals for this purpose we
3:23:49will first Define the kind of series
3:23:51exemplified by the inductive Cardinals
3:23:54in order of
3:23:55magnitude this kind of series which is
3:23:57called a progression has already been
3:24:00considered in chapter 1 it is a series
3:24:03which can be generated by a relation of
3:24:05consecutiveness
3:24:06every member of the series is to have a
3:24:09successor but there is to be just one
3:24:11one which has no predecessor and every
3:24:13member of the series is to be in the
3:24:16posterity of this term with respect to
3:24:18the relation immediate
3:24:20predecessor these characteristics may be
3:24:23summed up in the following definition
3:24:26footnote one confer principia
3:24:28Mathematica Volume 2 Star
3:24:32123 end of footnote
3:24:351 a progression is a one one relation
3:24:39such that there is just one term
3:24:41belonging to the domain but not to the
3:24:44converse domain and the domain is
3:24:46identical with the posterity of this one
3:24:50term it is easy to see that a
3:24:53progression so defined satisfies piano's
3:24:56five
3:24:57axioms the term belonging to the domain
3:24:59but not to the converse domain will be
3:25:01what he calls zero the term to which a
3:25:04term has the one one relation will be
3:25:07the successor of the term and the domain
3:25:10of the one one relation
3:25:12will be what he calls
3:25:14number taking his five aums in turn we
3:25:17have the following
3:25:19translations one zero is a number
3:25:22becomes the member of the domain which
3:25:25is not a member of the converse domain
3:25:27is a member of the
3:25:29domain this is equivalent to the
3:25:32existence of such a member which is
3:25:34given in our
3:25:35definition we will call this member the
3:25:38first
3:25:39term two this successor of any number is
3:25:43a number becomes the term to which a
3:25:46given member of the domain has the
3:25:48relation in question is again a member
3:25:51of the domain this is proved as follows
3:25:55by the definition every member of the
3:25:57domain is a member of the posterity of
3:26:00the first term hence the successor of a
3:26:03member of the domain must be a member of
3:26:06the posterity of the first term because
3:26:09the posterity of a term always contains
3:26:12its own successors by the general
3:26:14definition of posterity and therefore a
3:26:18member of the domain because by the
3:26:20definition of the domain the posterity
3:26:23of the first term is the same as the
3:26:27domain three no two numbers have the
3:26:30same
3:26:31successor this is only to say that the
3:26:33relation is one many which it is by
3:26:36definition being one
3:26:38one four zero is not the successor of
3:26:42any number becomes the first term is not
3:26:44a member of the converse domain which is
3:26:47again an immediate result of the
3:26:50definition five this is mathematical
3:26:53induction and becomes every member of
3:26:56the domain belongs to the posterity of
3:26:59the first term which was part of our
3:27:02definition thus progressions as we have
3:27:05Define them have the five formal
3:27:07properties from which piano deduces
3:27:09arithmetic it is easy to show that two
3:27:12progressions are similar in the sense
3:27:14defined for similarity of relations in
3:27:17chapter 6 we can of course derive a
3:27:20relation which is serial from the one
3:27:23one relation by which we Define a
3:27:25progression the method used is that
3:27:28explained in chapter 4 and the relation
3:27:31is that of a term to a member of its
3:27:33proper posterity with respect to the
3:27:35original one one
3:27:38relation two transitive asymmetrical
3:27:40relations which generate progressions
3:27:43are similar for the same reasons for
3:27:46which the corresponding one one
3:27:47relations are similar the class of all
3:27:51such transitive generators of
3:27:53progressions is a serial number in the
3:27:57sense of chapter six it is in fact the
3:28:00smallest of infinite serial numbers the
3:28:03number to which Cantor has given the
3:28:05name Omega by which he has made it
3:28:08famous but we are concerned for the
3:28:10moment with cardinal numbers since two
3:28:13progressions are similar relations it
3:28:16follows that their domains or their
3:28:18fields which are the same as their
3:28:20domains are similar
3:28:22classes the domains of progressions form
3:28:24a cardinal number since every class
3:28:27which is similar to The Domain of a
3:28:29progression is easily shown to be itself
3:28:32the domain of a
3:28:33progression this Cardinal number is the
3:28:36smallest of the infinite cardinal
3:28:37numbers it is the one to which Cantor
3:28:40has a appropriated the Hebrew Alf with
3:28:43suffix zero to distinguish it from
3:28:46larger infinite Cardinals which have
3:28:48other
3:28:49suffixes thus the name of the smallest
3:28:52of infinite Cardinals is Al
3:28:55Subzero to say that a class has Olive
3:28:58Subzero terms is the same thing as to
3:29:01say that it is a member of olive Subzero
3:29:04and this is the same thing as to say
3:29:06that the members of the class can be
3:29:08arranged in a
3:29:09progression it is OB VI that any
3:29:11progression remains a progression if we
3:29:14omit a finite number of terms from it or
3:29:17every other term or all except every
3:29:1910th or every hundredth
3:29:21term these methods of thinning out a
3:29:24progression do not make it cease to be a
3:29:27progression and therefore do not
3:29:29diminish the number of its terms which
3:29:31remains all of
3:29:33subzero in fact any selection from a
3:29:36progression is a progression if it has
3:29:38no last term however sparsely it may be
3:29:43distributed take say inductive numbers
3:29:45of the form n the N or n to the end to
3:29:49the N such numbers grow very rare in the
3:29:53higher parts of the number series and
3:29:55yet there are just as many of them as
3:29:59there are inductive numbers all together
3:30:02namely Al if
3:30:04null conversely we can add terms to the
3:30:08inductive numbers without increasing
3:30:10their number
3:30:12take for example ratios one might be
3:30:15inclined to think that there must be
3:30:18many more ratios than integers since
3:30:21ratios whose denominator is one
3:30:24correspond to the integers and seem to
3:30:27be only an infinite decimal portion of
3:30:30ratios but in actual fact the number of
3:30:33ratios or fractions is exactly the same
3:30:37as the number of inductive numbers
3:30:39namely Alf sub
3:30:41Z this is easily seen by arranging
3:30:44ratios in a series on the following plan
3:30:48if the sum of numerator and denominator
3:30:51in one is less than in the other put the
3:30:54one before the other if the sum is equal
3:30:57in the two but first the one with the
3:31:00smaller
3:31:01numerator this gives us the series 1 12
3:31:052 1/3 3 1/4 2/3 3 halves 4 1 and so
3:31:15on this series is a progression and all
3:31:19ratios occur in it sooner or later hence
3:31:22we can arrange all ratios in a
3:31:25progression and their number is
3:31:27therefore Olive
3:31:30Subzero it is not the case however that
3:31:33all infinite collections have Olive
3:31:35Subzero terms the number of real numbers
3:31:39for example is greater than Olive
3:31:41Subzero it is in fact 2 to the olive
3:31:45Subzero and it is not hard to prove that
3:31:492 to the N is greater than n even when n
3:31:53is
3:31:53infinite the easiest way of proving this
3:31:56is to prove first that if a class has n
3:31:59members it contains two to the n
3:32:03subclasses in other words that there are
3:32:05two to the end ways of selecting some of
3:32:08its members including the extreme cases
3:32:11where we select all or none and secondly
3:32:15that the number of subclasses contained
3:32:17in the class is always greater than the
3:32:20number of members of the class of these
3:32:23two propositions the first is familiar
3:32:26in the case of finite numbers and is not
3:32:29hard to extend to infinite numbers the
3:32:32proof of the second is so simple and so
3:32:36instructive that we shall give
3:32:38it in the first place it is clear that
3:32:42the number of subclasses of a given
3:32:44class say Alpha is at least as great as
3:32:47the number of members since each member
3:32:51constitutes a
3:32:52subass and we thus have a correlation of
3:32:55all the members with some of the sub
3:32:58classes hence it follows that if the
3:33:01number of subclasses is not equal to the
3:33:04number of members it must be greater now
3:33:07it is easy to prove that the number is
3:33:09not equal by showing that given any one
3:33:12one relation whose domain is the members
3:33:16and whose Converse domain is contained
3:33:18among the set of subclasses there must
3:33:21be at least one subclass not belonging
3:33:24to the converse domain the proof is as
3:33:27follows footnote one this proof is taken
3:33:31from Cantor with some simplifications C
3:33:39J 1
3:33:421892 page 77 end of footnote
3:33:461 when a 1 one correlation R is
3:33:49established between all the members of
3:33:51Alpha and some of the sub classes it may
3:33:55happen that a given member X is
3:33:57correlated with a subass of which it is
3:34:00a member or again it may happen that X
3:34:03is correlated with a subass of which it
3:34:06is not a
3:34:07member let us form the whole class beta
3:34:10say of those members X which are
3:34:14correlated with subclasses of which they
3:34:16are not members this is a subass of
3:34:19Alpha and it is not correlated with any
3:34:22member of
3:34:23alpha for taking first the members of
3:34:26beta each of them is by the definition
3:34:29of beta correlated with some subass of
3:34:32which it is not a member and is
3:34:34therefore not correlated with beta
3:34:37taking next to the terms which are not
3:34:39members of beta each of them by the
3:34:41definition of beta is correlated with
3:34:44some subclass of which it is a member
3:34:47and therefore again is not correlated
3:34:49with
3:34:50beta thus no member of alpha is
3:34:52correlated with beta since R was any one
3:34:56one correlation of all members with some
3:35:00subclasses it follows that there is no
3:35:02correlation of all members with all
3:35:06subclasses it does not matter to the
3:35:08proof If beta has no members all that
3:35:11happens in that case is that the subass
3:35:14which is shown to be omitted is the null
3:35:17class hence in any case the number of
3:35:21subclasses is not equal to the number of
3:35:24members and therefore by what was said
3:35:27earlier it is greater combining this
3:35:30with the proposition that if n is the
3:35:33number of members 2 to the N is the
3:35:36number of
3:35:37subclasses we have the theorem that 2 to
3:35:40the n is always greater than n even when
3:35:44n is
3:35:46infinite it follows from this
3:35:48proposition that there is no maximum to
3:35:50the infinite cardinal numbers however
3:35:53great an infinite number n may be 2 to
3:35:56the N will be still
3:35:59greater the arithmetic of infinite
3:36:01numbers is somewhat surprising until one
3:36:03becomes a custom to it we have for
3:36:07example Alpha Sub 0 + 1 isal to Alpha
3:36:11Sub 0 Alp sub 0+ n is equal to Alpha Sub
3:36:150 where n is any inductive number Alpha
3:36:19sub 0^ SAR is equal to Alpha Sub
3:36:230 this follows from The Case of the
3:36:25ratios for since a ratio is determined
3:36:28by a pair of inductive numbers it is
3:36:31easy to see that the number of ratios is
3:36:34the square of the number of inductive
3:36:37numbers that is it is Al subz
3:36:412 but we saw that it is also
3:36:45lf0 LF Sub 0 to the N is equal to l of
3:36:49Sub 0 where n is any inductive number
3:36:53this follows from L of sub 0^ s is equal
3:36:55to l of Sub 0 by induction for if L of
3:36:59Sub 0 to the N is equal to l of Sub 0
3:37:02then L of Sub 0 to the n + 1 is equal to
3:37:06Al sub 0^ 2 which is equal to Al subz
3:37:11but 2 to the ALF Sub 0 is greater than
3:37:16Alf
3:37:17subz in fact as we shall see later 2 to
3:37:21the AL subz is a very important number
3:37:24namely the number of terms in a series
3:37:27which has continuity in the sense in
3:37:30which this word is used by
3:37:32Cantor assuming space and time to be
3:37:34continuous in this sense as we commonly
3:37:37do in analytical geometry and kinematics
3:37:41this will be the number of points in
3:37:42space or of instance in time it will
3:37:46also be the number of points in any
3:37:47finite portion of space whether line
3:37:50area or
3:37:52volume after Al of subzero 2 to the AL
3:37:56subz is the most important and
3:37:58interesting of infinite cardinal
3:38:01numbers although addition and
3:38:03multiplication are always possible with
3:38:06infinite Cardinals subtraction and
3:38:08division no longer give de definite
3:38:10results and cannot therefore be employed
3:38:14as they are employed in elementary
3:38:17arithmetic take subtraction to begin
3:38:19with so long as the number subtracted is
3:38:22finite all goes well if the other number
3:38:25is reflexive it remains
3:38:28unchanged thus Al Sub 0 minus n is equal
3:38:33to Al Sub 0 if n is finite so far
3:38:37subtraction gives a perfectly definite
3:38:39result but it is otherwise when we
3:38:42subtract Alf subz from itself we may
3:38:45then get any result from 0 up to ALF
3:38:49subn this is easily seen by examples
3:38:53from the inductive numbers take away the
3:38:55following collections of Al of subn
3:38:58terms one all the inductive numbers
3:39:01remainder zero two all the inductive
3:39:04numbers from n onwards remainder the
3:39:07numbers from 0 to n minus one numbering
3:39:10n terms in
3:39:12all three all the odd numbers remainder
3:39:16all the even numbers numbering Al subn
3:39:19terms all these are different ways of
3:39:22subtracting Alf subn from Alf subn and
3:39:26all give different
3:39:28results as regards division very similar
3:39:31results follow from the fact that Alf
3:39:33subn is unchanged when multiplied by two
3:39:36or three or any finite number n or by
3:39:41Alf subn it follows that Al subn divided
3:39:44by Al subn may have any value from one
3:39:48up to ALF
3:39:50subn from the ambiguity of subtraction
3:39:53and division it results that negative
3:39:55numbers and ratios cannot be extended to
3:39:58infinite numbers addition multiplication
3:40:01and
3:40:02exponentiation proceed quite
3:40:04satisfactorily but the inverse
3:40:06operations subtraction Division and
3:40:09extraction of roots are ambiguous and
3:40:12the Notions that depend upon them fail
3:40:14when infinite numbers are
3:40:16concerned the characteristic by which we
3:40:18defined finitude was mathematical
3:40:20induction that is we defined a number as
3:40:24finite when it obeys mathematical
3:40:26induction starting from zero and a class
3:40:29as finite when its number is
3:40:31finite this definition yields the sort
3:40:33of result that a definition ought to
3:40:35yield namely that the finite numbers are
3:40:39those that occur in the ordinary number
3:40:41series 0 1 2 3 and so on but in the
3:40:46present chapter the infinite numbers we
3:40:48have discussed have not merely been
3:40:50non-inductive they have also been
3:40:53reflexive caner used reflexiveness as
3:40:56the definition of the infinite and
3:40:58believes that it is equivalent to non-
3:41:01inductiv that is to say he believes that
3:41:04every class and every Cardinal is either
3:41:07inductive or
3:41:08reflexive this may be true and may very
3:41:11possibly be capable of proof but the
3:41:14proofs hitherto offered by Cantor and
3:41:16others including the present author in
3:41:19former days are fallacious for reasons
3:41:22which will be explained when we come to
3:41:24consider the multiplicative AUM at
3:41:27present it is not known whether there
3:41:30are classes and Cardinals which are
3:41:32neither reflexive nor
3:41:35inductive if n were such a cardinal we
3:41:38should not have Nal n +1
3:41:40but n would not be one of the natural
3:41:42numbers and would be lacking in some of
3:41:45the inductive properties all known
3:41:48infinite classes and Cardinals are
3:41:50reflexive but for the present it is well
3:41:53to preserve an open mind as to whether
3:41:55there are instances hither to unknown of
3:41:58classes and Cardinals which are neither
3:42:00reflexive nor
3:42:03inductive meanwhile we adopt the
3:42:05following
3:42:06definitions a finite class or Cardinal
3:42:09is is one which is
3:42:11inductive an infinite class or Cardinal
3:42:14is one which is not
3:42:16inductive all reflexive classes and
3:42:19Cardinals are infinite but it is not
3:42:22known at present whether all infinite
3:42:24classes and Cardinals are
3:42:26reflexive we shall return to this
3:42:28subject in chapter 12 end of chapter
3:42:388
3:42:45chapter nine of introduction to
3:42:47mathematical Philosophy by Bertrand
3:42:50Russell this LibriVox recording is in
3:42:53the public
3:42:55domain infinite series and
3:42:58ordinals an infinite series may be
3:43:01defined as a series of which the field
3:43:04is an infinite class we have already had
3:43:06occasion to consider one kind of
3:43:09infinite series namely
3:43:11progressions in this chapter we shall
3:43:13consider the subject more
3:43:15generally the most noteworthy
3:43:17characteristic of an infinite series is
3:43:20that its serial number can be altered
3:43:22merely by rearranging its
3:43:25terms in this respect there's a certain
3:43:27oppositeness between Cardinal and serial
3:43:30numbers it is possible to keep the
3:43:32Cardinal number of a reflexive class
3:43:35unchanged in spite of adding terms to it
3:43:39on the other hand hand it is possible to
3:43:41change the serial number of a series
3:43:44Without adding or taking away any terms
3:43:47by mere
3:43:49rearrangement at the same time in the
3:43:51case of any infinite series it is also
3:43:54possible as with Cardinals to add terms
3:43:57without altering the serial number
3:44:00everything depends upon the way in which
3:44:02they are
3:44:04added in order to make matters clear it
3:44:07will be best to begin with examples
3:44:11let us consider various different kinds
3:44:12of series which can be made out of the
3:44:16inductive numbers arranged on various
3:44:18plans we start with the series 1 2 3 4
3:44:23and so on to n and so on which as we
3:44:27have seen represents the smallest of
3:44:30infinite serial numbers the sort that
3:44:32Cantor calls Omega let us proceed to
3:44:35thin out this series by repeatedly
3:44:37performing the operation of removing
3:44:39moving to the end the first even number
3:44:42that occurs we thus obtain in succession
3:44:45the various series 1 3 4 5 and so on to
3:44:50n and so on to two 1 3 5 6 and so on to
3:44:55n + 1 and so on to 2
3:44:594 1
3:45:01357 and so on to n+ 2 and so on to 2 4 6
3:45:08and so on
3:45:10if we imagine this process carried on as
3:45:12long as possible we finally reach the
3:45:15series 1 35 7 and so on to 2 N + 1 and
3:45:21so on to 2 4 6 8 and so on to 2N and so
3:45:27on in which we have first all the odd
3:45:30numbers and then all the even
3:45:33numbers the serial numbers of these
3:45:35various series are Omega + 1 Omega + 2
3:45:40Omega + 3 and so on to 2
3:45:44Omega each of these numbers is greater
3:45:48than any of its predecessors in the
3:45:50following sense one serial number is
3:45:53said to be greater than another if any
3:45:55series having the first number contains
3:45:58a part having the second number but no
3:46:01series having the second number contains
3:46:03a part having the first
3:46:05number if we compare the two series 1 2
3:46:093 4 and so on to n and so on 1 3 4 5 and
3:46:15so on to n + 1 and so on to two we see
3:46:20that the first is similar to the part of
3:46:22the second which omits the last term
3:46:25namely the number two but the second is
3:46:28not similar to any part of the first
3:46:31this is obvious but is easily
3:46:33demonstrated thus the second series has
3:46:36a greater serial number than the first
3:46:38according to the definition
3:46:40that is Omega + 1 is greater than
3:46:43Omega but if we add a term at the
3:46:46beginning of a progression instead of
3:46:48the end we still have a progression thus
3:46:511 + Omega is equal to Omega thus 1 +
3:46:56Omega is not equal to Omega + 1 this is
3:47:00a characteristic of relation arithmetic
3:47:02generally if mu and new are two relation
3:47:05numbers the general rule is that mu+ new
3:47:09is not equal to New Plus mu the case of
3:47:12finite ordinals in which there is
3:47:14equality is quite
3:47:17exceptional the series we finally
3:47:19reached just now consisted of first all
3:47:22the odd numbers and then all the even
3:47:24numbers in it serial number is 2
3:47:27Omega this number is greater than Omega
3:47:30or Omega plus n where n is
3:47:33finite it is to be observed that in
3:47:36accordance with the general definition
3:47:38of Order each of of these Arrangements
3:47:40of integers is to be regarded as
3:47:42resulting from some definite relation
3:47:45for example the one which merely removes
3:47:48two to the end will be defined by the
3:47:50following
3:47:51relation X and Y are finite integers and
3:47:55either Y is two and X is not two or
3:47:59neither is two and X is less than
3:48:02y the one which puts first all the odd
3:48:04numbers and then all the even ones will
3:48:07be defined by X and Y y are finite
3:48:10integers and either X is odd and Y is
3:48:13even or X is less than y and both are
3:48:17odd or both are even we shall not
3:48:20trouble as a rule to give these formula
3:48:23in future but the fact that they could
3:48:26be given is
3:48:28essential the number which we have
3:48:30called 2 Omega the number of a series
3:48:33consisting of two progressions is
3:48:36sometimes called the product of Omega
3:48:38and two
3:48:40multiplication like addition depends
3:48:42upon the order of the factors a
3:48:45progression of couples gives a series
3:48:47such as x sub 1 y sub 1 x sub 2 y sub 2
3:48:52x sub3 y sub3 and so on to x subn y subn
3:48:58and so on which is itself a
3:49:01progression but a couple of progressions
3:49:03gives a series which is twice as long as
3:49:06a progression it is therefore necessary
3:49:08to distinguish between 2 Omega and the
3:49:12product of Omega and two usage is
3:49:15variable we shall use two Omega for a
3:49:19couple of progressions and the product
3:49:21of Omega and two for a progression of
3:49:24couples and this decision of course
3:49:27governs our general interpretation of
3:49:29the product of Alpha and beta when Alpha
3:49:31and beta are relation numbers the
3:49:34product of Alpha and beta will have to
3:49:36stand for a suitably constructed sum of
3:49:39Alpha relations each having beta
3:49:43terms we can proceed indefinitely with
3:49:46the process of thinning out the
3:49:47inductive numbers for example we can
3:49:50place first the odd numbers then their
3:49:52doubles then the doubles of these and so
3:49:55on WE thus obtain the series 1 3 5 7 and
3:50:00so on to 2 6 10 14 and so on to 4 12 20
3:50:0628 and so on to 8 24 4 40 56 and so on
3:50:12of which the number is Omega squared
3:50:15since it is a progression of
3:50:18progressions any one of the progressions
3:50:20in this new series can of course be
3:50:21thinned out as we thinned out our
3:50:23original progression we can proceed to
3:50:26Omega cubed Omega to the 4th power and
3:50:29so on to Omega to the Omega power and so
3:50:33on however far we have gone we can
3:50:36always go
3:50:37further the series of all the ordinals
3:50:40that can be obtained in this way that is
3:50:43all that can be obtained by thinning out
3:50:44a progression is itself longer than any
3:50:47series that can be obtained by
3:50:49rearranging the terms of a progression
3:50:52this is not difficult to prove the
3:50:54Cardinal number of the class of such
3:50:56ordinals can be shown to be greater than
3:50:58alive subz it is the number which cancer
3:51:01calls alive sub one the ordinal number
3:51:05of the series of all ordinals that can
3:51:07be made out of an Al sub 0 taken in
3:51:10order of magnitude is called Omega sub
3:51:13one thus a series whose ordinal number
3:51:16is Omega sub one as a field whose
3:51:19Cardinal number is Alf sub
3:51:22one we can proceed from Omega sub one
3:51:25and Alpha sub one to Omega sub 2 and
3:51:27Alpha sub 2 by a process exactly
3:51:30analogous to that by which we Advanced
3:51:32from Omega and Alf subz to Omega sub one
3:51:36and Alf sub one and there is nothing to
3:51:39to prevent us from advancing
3:51:40indefinitely in this way to new
3:51:42Cardinals and new
3:51:44ordinals it is not known whether 2 to
3:51:47the ALF subn is equal to any of the
3:51:50Cardinals in the series of
3:51:52alfs it is not even known whether it is
3:51:54comparable with them in magnitude for a
3:51:57we know it may be neither equal to nor
3:51:59greater nor less than any one of the
3:52:03alfs this question is connected with the
3:52:05multiplicative Axiom of which we shall
3:52:08treat later
3:52:10all the series we have been considering
3:52:12so far in this chapter have been what is
3:52:14called well-ordered a well ordered
3:52:17series is one which has a beginning and
3:52:20has consecutive terms and has a term
3:52:23next after any selection of its terms
3:52:26provided there are any terms after the
3:52:28selection this excludes on the one hand
3:52:31compact Series in which there are terms
3:52:34between any two and on the other hand
3:52:37series which have no beginning or in
3:52:39which there are subordinate Parts having
3:52:42no beginning the series of negative
3:52:44integers in order of magnitude having no
3:52:47beginning but ending
3:52:49with1 is not well ordered but taken in
3:52:52the reverse order beginning with
3:52:54negative one it is well-ordered being in
3:52:57fact a
3:52:59progression the definition is a
3:53:02well-ordered series is one in which
3:53:04every subclass except of course the null
3:53:07class has a first term
3:53:10an ordinal number means the relation
3:53:13number of a well-ordered series it is
3:53:15thus a species of serial
3:53:18number among well-ordered series a
3:53:21generalized form of mathematical
3:53:22induction applies a property may be said
3:53:25to be trans finitely hereditary if when
3:53:29it belongs to a certain selection of the
3:53:31terms in a series it belongs to their
3:53:33immediate successor provided they have
3:53:36one in a well-ordered series A trans
3:53:39finitely hereditary property belonging
3:53:41to the first term of the series belongs
3:53:44to the whole series this makes it
3:53:46possible to prove many propositions
3:53:49concerning well-ordered series which are
3:53:51not true of all
3:53:54series it is easy to arrange the
3:53:56inductive numbers in series which are
3:53:58not well ordered and even to arrange
3:54:00them in compact series for example we
3:54:03can adopt the following plan consider
3:54:06the decimals from 1/10th inclus to One
3:54:10exclusive arranged in order of magnitude
3:54:14these form a compact series between any
3:54:16two there are always an infinite number
3:54:18of others now emit the dot at the
3:54:21beginning of each and we have a compact
3:54:24series consisting of all finite integers
3:54:27except such as divide by 10 if we wish
3:54:31to include those that divide by 10 there
3:54:33is no difficulty instead of starting
3:54:36with 1/10th we will include all decimals
3:54:39less than one but when we remove the dot
3:54:41we will transfer to the right any zeros
3:54:44that occur at the beginning of our
3:54:46decimal omitting these and returning to
3:54:49the ones that have no zeros at the
3:54:51beginning we can state the rule for the
3:54:53arrangement of our integers as follows
3:54:56of two integers that do not begin with
3:54:58the same digit the one that begins with
3:55:00the smaller digit comes first of two
3:55:03that do begin with the same digit but
3:55:04differ at the second digit the one with
3:55:07the smaller digit comes first
3:55:09but first of all the one with no second
3:55:11digit and so on generally if two
3:55:15integers agree as regards the first n
3:55:17digits but not as regards the n+ one
3:55:21that one comes first which has either no
3:55:23n plus one digit or a smaller one than
3:55:26the other this rule of arrangement as
3:55:29the reader can easily convince himself
3:55:31gives rise to a compact series
3:55:33containing all the integers not
3:55:35divisible by 10 and as we saw there is
3:55:39no difficulty about including those that
3:55:41are divisible by 10 it follows from this
3:55:45example that it is possible to construct
3:55:47compact series having Alf null terms in
3:55:51fact we've already seen that there are
3:55:53Alf null ratios and ratios in order of
3:55:55magnitude form a compact series thus we
3:55:59have here another example we shall
3:56:01resume this topic in the next
3:56:04chapter of the usual formal laws of
3:56:07addition multiplication and
3:56:09exponentiation all are obeyed by
3:56:11transfinite Cardinals but only some are
3:56:14obeyed by transfinite
3:56:15ordinals and those that are obeyed by
3:56:17them are obeyed by all relation numbers
3:56:21by the usual formal laws we mean the
3:56:23following one the communative law Alpha
3:56:26plus beta is equal to Beta plus Alpha
3:56:29and the product of Alpha and beta is
3:56:32equal to the product of beta and Alpha
3:56:35to the associative law the sum of the SU
3:56:38sum of Alpha and beta with gamma is
3:56:41equal to the sum of alpha with the sum
3:56:44of beta and gamma and the product of the
3:56:48product of Alpha and beta with gamma is
3:56:52equal to the product of alpha with the
3:56:55product of beta and gamma three the
3:56:58distributive law the product of alpha
3:57:01with the sum of beta and gamma is equal
3:57:04to the sum of the product of Alpha and
3:57:07beta with the product of Alpha and
3:57:11Gamma when the commutative law does not
3:57:13hold the above form of the distributive
3:57:15law must be distinguished from the
3:57:19product of the sum of beta and gamma
3:57:21with Alpha is equal to the sum of the
3:57:26product of beta and Alpha with the
3:57:28product of gamma and
3:57:30Alpha as we shall see immediately one
3:57:33form may be true and the other false
3:57:36four the laws of exponentiation
3:57:39the product of alpha to the beta power
3:57:42with Alpha to the gamma power is equal
3:57:45to Alpha raised to the sum of beta and
3:57:49gamma power the product of alpha to the
3:57:52gamma power with beta to the gamma power
3:57:55is equal to the product of Alpha and
3:57:58beta raised to the gamma power and Alpha
3:58:03raised to the beta power raised to the
3:58:05gamma power is equal to Alpha raised to
3:58:09the product of beta and gamma
3:58:12power all these laws hold for Cardinals
3:58:15whether finite or infinite and for
3:58:17finite ordinals but when we come to
3:58:19infinite ordinals or indeed to relation
3:58:21numbers in general some hold and some do
3:58:23not the communative law does not hold
3:58:26the associative law does hold the
3:58:29distributive law adopting the convention
3:58:31we have adopted above as regards the
3:58:33order of the factors in a product holds
3:58:36in the form the sum of beta and gamma
3:58:39taken as the product with Alpha is equal
3:58:42to the sum of the product of beta and
3:58:45Alpha with the product of gamma and
3:58:48Alpha but not in the form the product of
3:58:51alpha with the sum of beta and gamma is
3:58:54equal to the sum of the product of Alpha
3:58:58and beta and of gamma and Alpha the
3:59:02exponential laws Alpha raised to the
3:59:05beta power taken as the product with
3:59:08Alpha to the gamma power is equal to
3:59:11Alpha raised to the beta + gamma power
3:59:14and Alpha raised to the beta power
3:59:17raised to the gamma power is equal to
3:59:20Alpha raised to the product of beta and
3:59:24gamma still hold but not the law Alpha
3:59:28raised to the gamma power taken as the
3:59:30product with beta raised to the gamma
3:59:32power is equal to the product of Alpha
3:59:35and beta raised to the gamma power
3:59:39which is obviously connected with the
3:59:41communative law for
3:59:43multiplication the definitions of
3:59:45multiplication and
3:59:47exponentiation that are assumed in the
3:59:49above propositions are somewhat
3:59:52complicated the reader who wishes to
3:59:54know what they are and how the above
3:59:55laws are proved must consult the second
3:59:58volume of principia Mathematica star
4:00:00numbers 172 to
4:00:04176 ordinal transfinite arithmetic was
4:00:07developed by Cantor at an earlier stage
4:00:09than Cardinal transfinite arithmetic
4:00:11because it has various technical
4:00:13mathematical uses which led him to it
4:00:17but from the point of view of the
4:00:18philosophy of mathematics it is less
4:00:21important and less fundamental than the
4:00:23theory of transfinite
4:00:25Cardinals Cardinals are essentially
4:00:27simpler than ordinals and it is a
4:00:29curious historical accident that they
4:00:31first appeared as an abstraction from
4:00:33the latter and only gradually came to be
4:00:35studied on their own account this does
4:00:38not apply to fraga's work in which
4:00:40Cardinals finite and transfinite were
4:00:42treated in complete independence of
4:00:44ordinals but it was Canter's work that
4:00:47made the world aware of the subject
4:00:49while fragas remained almost unknown
4:00:52probably in the main on account of the
4:00:54difficulty of his symbolism and
4:00:56mathematicians like other people have
4:00:58more difficulty in understanding and
4:01:00using Notions which are comparatively
4:01:02simple in The Logical sense than in
4:01:05manipulating more complex Notions which
4:01:07are more kin to their ordinary practice
4:01:10for these reasons it was only gradually
4:01:13that the true importance of cardinals in
4:01:15mathematical philosophy was recognized
4:01:18the importance of ordinals though by no
4:01:20means small is distinctly less than that
4:01:23of Cardinals and is very largely merged
4:01:26in that of the more General conception
4:01:28of relation numbers end of chapter
4:01:37n chapter 10 of introduction to
4:01:40mathematical Philosophy by Bertrand
4:01:42Russell this LibriVox recording is in
4:01:45the public
4:01:47domain limits and
4:01:49continuity the conception of a limit is
4:01:52one of which the importance in
4:01:54mathematics has been found continually
4:01:56greater than have been thought the whole
4:01:59of the differential and integral
4:02:00calculus indeed practically everything
4:02:03in higher mathematics depends upon
4:02:06limits formerly it was suppos that
4:02:09infinite tmals were involved in the
4:02:11foundations of these subjects but vros
4:02:14showed that this is an
4:02:16error wherever infinitesimals were
4:02:18thought to occur what really occurs is a
4:02:21set of finite quantities having zero for
4:02:24their lower
4:02:26limit it used to be thought that limit
4:02:28was an essentially quantitative notion
4:02:31namely the notion of a quantity to which
4:02:33others approached nearer and nearer so
4:02:36that among those others there would be
4:02:39some differing by less than any assigned
4:02:42quantity but in fact the notion of limit
4:02:45is a purely ordinal notion not involving
4:02:48quantity at all except by accident when
4:02:51the series concerned happens to be
4:02:54quantitative a given point on a line may
4:02:56be the limit of a set of points on the
4:02:58line without it's being necessary to
4:03:00bring in coordinates or measurement or
4:03:02anything quantitative the Cardinal
4:03:05number Alf subz is the limit in the
4:03:08order of magnitude of the cardinal
4:03:10numbers 1 2 3 and so on to n and so on
4:03:14although the numerical difference
4:03:16between iive Subzero and a finite
4:03:18Cardinal is constant and infinite from a
4:03:21quantitative point of view finite
4:03:23numbers get no nearer to Olive Subzero
4:03:25as they grow larger what makes Olive
4:03:28Subzero the limit of the finite numbers
4:03:31is the fact that in the series it comes
4:03:34immediately after them which is an
4:03:36ordinal fact not a quantitative ative
4:03:39fact there are various forms of the
4:03:41notion of limit of increasing complexity
4:03:45the simplest and most fundamental form
4:03:47from which the rest are derived has been
4:03:49already defined but we will here repeat
4:03:52the definitions which lead to it in a
4:03:54general form in which they do not demand
4:03:56that the relation concerned shall be
4:03:58serial the definitions are as follows
4:04:02the Minima of a class Alpha with respect
4:04:05to a relation P are those members of alp
4:04:08and the field of P if any to which no
4:04:11member of alpha has the relation
4:04:15P the Maxima with respect to P are the
4:04:18Minima with respect to the converse of
4:04:21P the sequence of a class Alpha with
4:04:24respect to a relation P are the Minima
4:04:27of the successors of Alpha and the
4:04:29successors of alpha are those members of
4:04:32the field of P to which every member of
4:04:35the common part of Alpha and the field
4:04:37of P P has the relation
4:04:40P the Precedence with respect to P are
4:04:44the sequence with respect to the
4:04:46converse of
4:04:48P the upper limits of alpha with respect
4:04:51to P are the sequence provided Alpha has
4:04:55no maximum but if Alpha has a maximum it
4:04:58has no upper
4:05:00limits the lower limits with respect to
4:05:03P are the upper limits with respect to
4:05:06the converse of P
4:05:09whenever P has connexity a class can
4:05:12have at most one maximum one minimum one
4:05:15sequent and so on thus in the cases we
4:05:18are concerned with in practice we can
4:05:20speak of the limit if any when p is a
4:05:25Serial relation we can greatly simplify
4:05:28the above definition of a limit we can
4:05:31in that case Define first the boundary
4:05:34of a class Alpha that is its limits or
4:05:37Maxim Maxum and then proceed to
4:05:40distinguish the case where the boundary
4:05:42is the limit from the case where it is a
4:05:45maximum for this purpose it is best to
4:05:48use the notion of a
4:05:50segment we will speak of the segment of
4:05:53P defined by a class Alpha as all those
4:05:57terms that have the relation P to some
4:06:00one or more of the members of
4:06:03alpha this will be a segment in the
4:06:06sense defined in chapter s indeed every
4:06:10segment in the sense they're defined is
4:06:12the segment defined by some class
4:06:15Alpha if p is serial the segment defined
4:06:19by Alpha consists of all the terms that
4:06:22precede some term or other of
4:06:25alpha if Alpha has a maximum the segment
4:06:28will be all the predecessors of the
4:06:31maximum but if Alpha has no maximum
4:06:35every member of alpha precedes some
4:06:37other member of Alpha and the whole of
4:06:39alpha is therefore included in the
4:06:42segment defined by
4:06:44Alpha take for example the class
4:06:47consisting of the fractions 1/2 3/4s 78
4:06:521516 and so on that is of all fractions
4:06:57of the form 1us one of 2 to the N for
4:07:02different finite values of
4:07:04n this series of fractions has no
4:07:07maximum
4:07:08and it is clear that the segment of
4:07:11which it defines in the whole series of
4:07:14fractions in order of magnitude is the
4:07:16class of all proper fractions or again
4:07:21consider the prime numbers considered as
4:07:24a selection from the Cardinals finite
4:07:26and infinite in order of
4:07:29magnitude in this case the segment
4:07:32defined consists of all finite
4:07:35integers assuming that P is serial the
4:07:38boundary of a class Alpha will be the
4:07:41term X if it exists whose predecessors
4:07:44are the segment defined by
4:07:47Alpha a maximum of alpha is a boundary
4:07:50which is a member of alpha an upper
4:07:53limit of alpha is a boundary which is
4:07:56not a member of
4:07:58alpha if a class has no boundary it has
4:08:01neither maximum nor limit this is the
4:08:05case of an irrational dedicant cut or of
4:08:08what is called a
4:08:10gap thus the upper limit of a set of
4:08:13terms Alpha with respect to a series p
4:08:17is that term X if it exists which comes
4:08:20after all the alphas but as such that
4:08:24every earlier term comes before some of
4:08:27the
4:08:28alphas we may Define all the upper
4:08:31limiting points of a set of terms beta
4:08:34as all those that are the upper limits
4:08:37of sets of terms chosen out of
4:08:41beta we shall of course have to
4:08:43distinguish upper limiting points from
4:08:46lower limiting
4:08:47points if we consider for example the
4:08:50series of ordinal numbers 1 2 and three
4:08:54and so on up to Omega Omega + 1 and so
4:08:58on up to the product of two and Omega up
4:09:01to the product of 2 Omega + 1 and so on
4:09:06up to the product of 3 and Omega and so
4:09:09on up to Omega squar and so on up to
4:09:13Omega cubed and so on the upper limiting
4:09:16points of the field of this series are
4:09:19those that have no immediate
4:09:21predecessors that is one Omega the
4:09:24product of two and Omega the product of
4:09:26three and Omega and so on up to Omega
4:09:30squar Omega 2 plus Omega and so on up to
4:09:34the product of 2 and Omega SAR and so on
4:09:38up to Omega cubed and so on the upper
4:09:41limiting points of the field of this new
4:09:43series will be 1 Omega squar the product
4:09:47of two and Omega squar and so on up to
4:09:49Omega cubed the sum of Omega cubed and
4:09:52Omega squar and so on on the other hand
4:09:56the series of ordinals and indeed every
4:09:59well-ordered series has no lower
4:10:01limiting points because there are no
4:10:04terms except the last that have no
4:10:07immediate
4:10:08successors but if we consider such a
4:10:10series as the series of ratios every
4:10:14member of this series is both an upper
4:10:16and a lower limiting point for suitably
4:10:19chosen sets if we consider the series of
4:10:22real numbers and select out of it the
4:10:25rational real numbers this set the
4:10:28rationals will have all the real numbers
4:10:31as upper and lower limiting points the
4:10:34limiting points of a set are called its
4:10:36first derivative ative and the limiting
4:10:39points of the first derivative are
4:10:40called the second derivative and so
4:10:44on with regard to limits we may
4:10:47distinguish various grades of what may
4:10:49be called continuity in a series the
4:10:52word continuity had been used for a long
4:10:55time but had remained without any
4:10:58precise definition until the time of
4:11:00dedin and Cantor each of these two men
4:11:04gave a precise significance to the term
4:11:07but can's definition is narrower than
4:11:09dakins a series which has cantorian
4:11:12continuity must have dedan continuity
4:11:15but the converse does not hold the first
4:11:18definition that would naturally occur to
4:11:20a man seeking a precise meaning for the
4:11:23continuity of series would be to Define
4:11:26it as consisting in what we have called
4:11:28compactness that is in the fact that
4:11:31between any two terms of the series
4:11:33there are others but this would be an
4:11:35inadequate definition because of the
4:11:38existence of gaps in series such as the
4:11:41series of
4:11:42ratios we saw in chapter 7 that there
4:11:45are innumerable ways in which the series
4:11:47of ratios can be divided into two parts
4:11:51of which one wholly precedes the other
4:11:53and of which the first has no last term
4:11:56while the second has no first term such
4:11:59a state of affairs seems contrary to the
4:12:01vague feeling we have as to what should
4:12:04characterize continuity and what is more
4:12:08it shows that the series of ratios is
4:12:10not the sort of series that is needed
4:12:13for many mathematical
4:12:15purposes take geometry for example we
4:12:18wish to be able to say that when two
4:12:21straight lines cross each other they
4:12:23have a point in common but if the series
4:12:26of points on a line were similar to the
4:12:29series of ratios the two lines might
4:12:32cross in a gap and have no point in
4:12:35common this is a crude example but many
4:12:39others might be given to show that
4:12:41compactness is inadequate as a
4:12:44mathematical definition of
4:12:46continuity it was the needs of geometry
4:12:50as much as anything that led to the
4:12:52definition of deian
4:12:55continuity it will be remembered that we
4:12:57defined a series as dedan when every
4:13:01subclass of the field has a boundary it
4:13:04is sufficient to assume that there is
4:13:06always an upper boundary or that there
4:13:08is always a lower boundary if one of
4:13:11these is assumed the other can be
4:13:13deduced that is to say a series is D
4:13:17indan when there are no gaps the absence
4:13:20of gaps may arise either through terms
4:13:23having successors or through the
4:13:25existence of limits in the absence of
4:13:28Maxima thus a finite series or a
4:13:31well-ordered series is D indan and so is
4:13:34the series of real
4:13:36numbers the former sort of datan series
4:13:39is excluded by assuming that our series
4:13:42is compact in that case our series must
4:13:45have a property which may for many
4:13:47purposes be fittingly called
4:13:50continuity thus we are led to the
4:13:53definition a Series has dedan continuity
4:13:57when it is dedan and
4:14:00compact but this definition is still too
4:14:02wide for many
4:14:04purposes suppose for example that we
4:14:07desire to be able to assign such
4:14:09properties to geometrical space as shall
4:14:12make it certain that every Point can be
4:14:14specified by means of coordinates which
4:14:16are real numbers this is not ensured by
4:14:20datan continuity
4:14:22alone we want to be sure that every
4:14:24point which cannot be specified by
4:14:26rational coordinates can be specified as
4:14:30the limit of a progression of points
4:14:33whose coordinates are rational and this
4:14:35is a further property which our
4:14:37definition does not enable us to
4:14:41deduce we are thus led to a closer
4:14:43investigation of series with respect to
4:14:46limits this investigation was made by
4:14:48Cantor and formed the basis of his
4:14:51definition of continuity although in its
4:14:54simplest form this definition somewhat
4:14:56conceals the considerations which have
4:14:58given rise to it we shall therefore
4:15:02first travel through some of cantor's
4:15:04conceptions in this subject before
4:15:06giving his definition of
4:15:08continuity Cantor defines a series as
4:15:11perfect when all its points are limiting
4:15:14points and all its limiting points
4:15:16belong to it but this definition does
4:15:19not express quite accurately what he
4:15:21means there is no correction required so
4:15:24far as concerns the property that all
4:15:27its points are to be limiting points
4:15:30this is a property belonging to compact
4:15:32series and to no others if all points
4:15:35are to be upper limiting or all lower
4:15:37limiting
4:15:39points but if it is only assumed that
4:15:41they are limiting points one way without
4:15:44specifying which there will be other
4:15:46series that will have the property in
4:15:49question for example the series of
4:15:51decimals in which a decimal ending in a
4:15:54recurring nine is distinguished from the
4:15:56corresponding terminating decimal and
4:15:58placed immediately before it such a
4:16:01series is very nearly compact but has
4:16:05exceptional terms which are consecutive
4:16:07and of which the first has no immediate
4:16:09predecessor while the second has no
4:16:11immediate
4:16:13successor apart from such series the
4:16:15series in which every point is a
4:16:17limiting Point are compact
4:16:19series and this holds without
4:16:21qualification if it is specified that
4:16:24every point is to be an upper limiting
4:16:26point or that every point is to be a
4:16:29lower limiting
4:16:31Point although Cantor does not
4:16:33explicitly consider the matter we must
4:16:36distinguish different different kinds of
4:16:37limiting points according to the nature
4:16:40of the smallest subseries by which they
4:16:42can be
4:16:43defined canor assumes that they are to
4:16:46be defined by progressions or by
4:16:48regressions which are the Converses of
4:16:51progressions when every member of our
4:16:53series is the limit of a progression or
4:16:56regression caner calls our series
4:16:59condensed in itself
4:17:01in we come now to the second property by
4:17:04which Perfection was to be defined
4:17:07namely the property which caner calls
4:17:09that of being closed
4:17:13aen this as we saw was first defined as
4:17:17consisting in the fact that all the
4:17:19limiting points of a series belong to it
4:17:22but this only has any effective
4:17:24significance if our series is given as
4:17:27contained in some other larger series as
4:17:31is the case for example with a selection
4:17:33of real numbers and limiting points are
4:17:36taken in relation to the larger
4:17:38series otherwise if a series is
4:17:41considered simply on its own account it
4:17:44cannot fail to contain its limiting
4:17:46points what Cantor means is not exactly
4:17:49what he says indeed on other occasions
4:17:52he says something rather different which
4:17:54is what he
4:17:56means what he really means is that every
4:17:59subordinate series which is of the sort
4:18:01that might be expected to have a limit
4:18:04does have a limit within the given
4:18:06series
4:18:07that is every subordinate series which
4:18:09has no maximum has a limit that is every
4:18:13subordinate Series has a
4:18:15boundary bantor does not State this for
4:18:18every subordinate series but only for
4:18:20progressions and
4:18:22regressions it is not clear how far he
4:18:24recognizes that this is a
4:18:27limitation thus finally we find that the
4:18:30definition we want is the
4:18:32following a series is said to be closed
4:18:35of G
4:18:37when every progression or regression
4:18:40contained in the series has a limit in
4:18:42the series we then have the further
4:18:46definition a series is perfect when it
4:18:49is condensed in itself and closed that
4:18:52is when every term is the limit of a
4:18:55progression or regression and every
4:18:57progression or regression contained in
4:18:59the series has a limit in the
4:19:03series in Seeking a definition of
4:19:05continuity what Cantor has in mind is
4:19:08the search for a definition which shall
4:19:10apply to the series of real numbers and
4:19:13to any series similar to that but to
4:19:16know
4:19:17others for this purpose we have to add a
4:19:20further
4:19:21property among the real numbers some are
4:19:24rational some are irrational although
4:19:27the number of irrationals is greater
4:19:29than the number of rationals yet there
4:19:31are rationals between any two real
4:19:34numbers however little the two May
4:19:36differ
4:19:37the number of rationals as we saw is Alf
4:19:41Subzero this gives a further property
4:19:44which suffices to characterize
4:19:45continuity completely namely the
4:19:48property of containing a class of allf
4:19:51null members in such a way that some of
4:19:53this class occur between any two terms
4:19:56of our series however near
4:19:58together this property added to
4:20:01Perfection suffices to define a class of
4:20:04series which are all similar and are in
4:20:07fact a serial number this class Cantor
4:20:10defines as that of continuous
4:20:13series we may slightly simplify his
4:20:16definition to begin with we say a median
4:20:20class of a series is a subclass of the
4:20:23field such that members of it are to be
4:20:26found between any two terms of the
4:20:28series thus the rationals are a median
4:20:31class in the series of real numbers it
4:20:34is obvious that there cannot be median
4:20:37classes except in compact
4:20:40series we then find that cantor's
4:20:43definition is equivalent to the
4:20:45following a series is continuous when
4:20:48one it is Dak Indian two it contains a
4:20:52median class having Alf Subzero
4:20:55terms to avoid confusion we shall speak
4:20:58of this kind as cantoran
4:21:01continuity it will be seen that it
4:21:03implies D Indian continuity but the
4:21:06Converse is not the case all series
4:21:09having cantorian continuity are similar
4:21:13but not all series having dakan
4:21:17continuity the Notions of limit and
4:21:20continuity which we have been defining
4:21:22must not be confounded with the Notions
4:21:24of the limit of a function for
4:21:27approaches to a given argument or the
4:21:29continuity of a function in the
4:21:32neighborhood of a given argument these
4:21:34are different Notions very very
4:21:36important but derivative from the above
4:21:39and more
4:21:40complicated the continuity of motion if
4:21:42motion is continuous is an instance of
4:21:45the continuity of a function on the
4:21:48other hand the continuity of space and
4:21:50time if they are continuous is an
4:21:53instance of the continuity of series or
4:21:56to speak more cautiously of a kind of
4:21:59continuity which can by sufficient
4:22:02mathematical manipulation be reduced to
4:22:05the continuity of
4:22:07Series in view of the fundamental
4:22:09importance of motion in Applied
4:22:11Mathematics as well as for other reasons
4:22:14it will be well to deal briefly with the
4:22:17Notions of limits and continuity as
4:22:19applied to functions but this subject
4:22:22will be best reserved for a separate
4:22:25chapter the definitions of continuity
4:22:28which we have been considering namely
4:22:30those of datakind and Cantor do not
4:22:33correspond very closely to the vague
4:22:35idea which is is associated with the
4:22:37word in the mind of the man in the
4:22:39street or the
4:22:41philosopher they conceive continuity
4:22:44rather as absence of separateness the
4:22:47sort of general obliteration of
4:22:49distinctions which characterizes a thick
4:22:51fog a fog gives an impression of
4:22:54vastness without definite multiplicity
4:22:57or
4:22:58division it is this sort of thing that a
4:23:00metaphysician means by continuity
4:23:03declaring it very truly to be
4:23:05characteristic of his mental life and of
4:23:08that of children and
4:23:10animals the general idea vaguely
4:23:13indicated by the word continuity when so
4:23:16employed or by the word flux is one
4:23:19which is certainly quite different from
4:23:21that which we have been
4:23:23defining take for example the series of
4:23:26real
4:23:26numbers each is what it is quite
4:23:29definitely and
4:23:31uncompromisingly it does not pass over
4:23:34by imperceptible degrees into an another
4:23:37it is a hard separate unit and its
4:23:40distance from every other unit is finite
4:23:44though it can be made less than any
4:23:46given amount assigned in
4:23:49advance the question of the relation
4:23:51between the kind of continuity existing
4:23:54among the real numbers and the kind
4:23:56exhibited for example by what we see at
4:23:59a given time is a difficult and
4:24:02intricate one it is not to be maintained
4:24:04that the two kinds are simp identical
4:24:07but it may I think be very well
4:24:09maintained that the mathematical
4:24:11conception which we have been
4:24:12considering in this chapter gives the
4:24:15abstract logical scheme to which it must
4:24:17be possible to bring empirical material
4:24:20by suitable manipulation if that
4:24:23material is to be called continuous in
4:24:26any precisely definable
4:24:28sense it would be quite impossible to
4:24:30justify this thesis within the limits of
4:24:32the present volume the reader who is
4:24:35interested May read an attempt to
4:24:37justify it as regards time in particular
4:24:40by the present author in the monist for
4:24:421914 to
4:24:441915 as well as in parts of our
4:24:46knowledge of the external world with
4:24:49these indications we must leave this
4:24:51problem interesting as it is in order to
4:24:55return to topics more closely connected
4:24:57with
4:24:58mathematics end of chapter
4:25:0510
4:25:08chapter 11 of introduction to
4:25:10mathematical Philosophy by berand
4:25:13Russell this LibriVox recording is in
4:25:16the public
4:25:18domain limits and continuity of
4:25:21functions in this chapter we shall be
4:25:24concerned with the definition of the
4:25:26limit of a function if any as the
4:25:28argument approaches a given value and
4:25:31also with the definition of what is
4:25:33meant by a continuous function
4:25:37both of these ideas are somewhat
4:25:39Technical and would hardly demand
4:25:41treatment in a mere introduction to
4:25:44mathematical philosophy but for the fact
4:25:47that especially through the so-called
4:25:48infinitesimal calculus wrong views upon
4:25:51our present topics have become so firmly
4:25:54embedded in the minds of professional
4:25:57philosophers that a prolonged and
4:25:59considerable effort is required for
4:26:02their
4:26:03uprooting it has been thought ever since
4:26:05the time time of lies that the
4:26:07differential and integral calculus
4:26:10required infinitesimal
4:26:12quantities mathematicians especially
4:26:15vros proved that this is an error but
4:26:19errors Incorporated for example in what
4:26:21Hegel has to say about mathematics die
4:26:24hard and philosophers have tended to
4:26:26ignore the work of such men as
4:26:30firos limiting continuity of functions
4:26:33in works on ordinary mathematics are
4:26:35defined in terms involving number this
4:26:38is not essential as Dr Whitehead has
4:26:40shown footnote one C principia
4:26:43Mathematica Volume 2 Star numbers 230 to
4:26:48234 end of footnote 1 we will however
4:26:52begin with the definitions in the
4:26:54textbooks and proceed afterwards to show
4:26:57how these definitions can be generalized
4:27:00so as to apply to Series in general and
4:27:03not only to such as our numerical or
4:27:06numerically
4:27:08measurable let us consider any ordinary
4:27:11mathematical function f
4:27:13ofx where X and F ofx are both real
4:27:16numbers and F ofx is one valued that is
4:27:20when X is given there is only one value
4:27:23that F ofx can have we call X the
4:27:27argument and F ofx the value of the
4:27:30argument
4:27:32X when a function is what we call
4:27:34continuous the the rough idea for which
4:27:37we are seeking a precise definition is
4:27:40that small differences in X shall
4:27:42correspond to small differences in F ofx
4:27:45and if we make the differences in X
4:27:48small enough we can make the differences
4:27:50in F ofx fall below any assigned
4:27:53amount we do not want if a function is
4:27:56to be continuous that there shall be
4:27:58sudden jumps so that for some value of x
4:28:02any change however small will make a
4:28:05change in F of X which exceeds some
4:28:08assigned finite
4:28:09amount the ordinary simple functions of
4:28:12mathematics have this property it
4:28:14belongs for example to x^2 x cub and so
4:28:19on to log of x sin of X and so
4:28:24on but it is not at all difficult to
4:28:26Define discontinuous functions take as a
4:28:29non mathematical example the place of
4:28:32birth of the youngest person living at
4:28:34time T the this is a function of T its
4:28:37value is constant from the time of one
4:28:39person's birth to the time of the next
4:28:42birth and then the value changes
4:28:45suddenly from one birthplace to the
4:28:47other an analogous mathematical example
4:28:50would be the integer next below X where
4:28:53X is a real
4:28:55number this function remains constant
4:28:58from one integer to the next and then
4:29:00gives a sudden
4:29:01jump the actual fact is that though
4:29:05continuous functions are more familiar
4:29:07they are the Exceptions there are
4:29:09infinitely more discontinuous functions
4:29:12than continuous
4:29:14ones many functions are discontinuous
4:29:16for one or several values of the
4:29:19variable but continuous for all other
4:29:22values take as an example s of 1
4:29:26/x the function sin Theta passes through
4:29:30all values from -1 to 1 every time that
4:29:34Theta passes from
4:29:36piun / 2 to piun / 2 or from piun / 2 to
4:29:413 piun / 2 or generally from 2 nus one
4:29:47taken as the product with pi over 2 2 n
4:29:51+ one taken as the product with pi over
4:29:54two where n is any
4:29:57integer now if we consider 1 /x when X
4:30:01is very small we see that as X
4:30:03diminishes 1/x grows faster and faster
4:30:07so that it passes more and more quickly
4:30:09through the cycle of values from one
4:30:11multiple of Pi / 2 to another as X
4:30:14becomes smaller and smaller consequently
4:30:18s of 1 /x passes more and more quickly
4:30:22from1 to 1 and back again as X grows
4:30:26smaller in fact if we take any interval
4:30:29containing zero say the interval from
4:30:32negative Epsilon to positive Epsilon
4:30:34where Epsilon is some very small number
4:30:37s of 1x will go through an infinite
4:30:40number of oscillations in this interval
4:30:43and we cannot diminish the oscillations
4:30:45by making the interval
4:30:47smaller thus round about the argument
4:30:51zero the function is
4:30:53discontinuous it is easy to manufacture
4:30:56functions which are discontinuous in
4:30:58several places or in Alf Subzero places
4:31:01or
4:31:02everywhere examples will be found in any
4:31:05book on the theory of functions of a
4:31:07real
4:31:08variable proceeding now to seek a
4:31:11precise definition of what is meant by
4:31:13saying that a function is continuous for
4:31:15a given argument when argument and value
4:31:17are both real numbers let us first
4:31:20Define a neighborhood of a number X as
4:31:23all the numbers from xus Epsilon to X+
4:31:27Epsilon where Epsilon is some number
4:31:30which in important cases will be very
4:31:34small it is it is clear that continuity
4:31:37at a given point has to do with what
4:31:39happens in any neighborhood of that
4:31:42point however
4:31:43small what we desire is this if a is the
4:31:47argument for which we wish our function
4:31:49to be continuous let us first Define a
4:31:51neighborhood Alpha say containing the
4:31:54value F of a which the function has for
4:31:57the argument a we desire that if we take
4:32:01a sufficiently small neighborhood
4:32:03containing a all values arguments
4:32:06throughout this neighborhood shall be
4:32:08contained in the neighborhood of alpha
4:32:11no matter how small we may have made
4:32:14Alpha that is to say if we decree that
4:32:17our function is not to differ from F of
4:32:19a by more than some very tiny amount we
4:32:23can always find a stretch of real
4:32:25numbers having a in the middle of it
4:32:28such that throughout this stretch F ofx
4:32:31will not differ from F of alpha by more
4:32:34than the prescribed tiny
4:32:37amount and this is to remain true
4:32:39whatever tiny amount we may select hence
4:32:42we are led to the following
4:32:44definition the function f ofx is said to
4:32:47be continuous for the argument a if for
4:32:51every positive number Sigma different
4:32:54from zero but as small as we please
4:32:57there exists a positive number Epsilon
4:33:00different from zero such that for all
4:33:02values of Delta which are numerically
4:33:05less footnote 1 a number said to be
4:33:08numerically less than Epsilon when it
4:33:10lies between negative Epsilon and
4:33:12positive Epsilon end of footnote one
4:33:16then Epsilon the difference F of the sum
4:33:19of a and
4:33:21Delta minus F of a is numerically less
4:33:25than
4:33:26Sigma in this definition Sigma first
4:33:29defines a neighborhood of f of a namely
4:33:32the neighborhood from the difference of
4:33:35of f of a from Sigma to the sum of f of
4:33:38a and sigma the definition proceeds to
4:33:42say that we can by means of Epsilon
4:33:45Define a neighborhood namely that from
4:33:47the difference of a from Epsilon to the
4:33:50sum of a and Epsilon such that for all
4:33:54arguments within this neighborhood the
4:33:56value of the function lies within the
4:33:58neighborhood from the difference of f of
4:34:00a from Sigma to the sum of f of a and
4:34:04sigma
4:34:06if this can be done however Sigma may be
4:34:08chosen the function is continuous for
4:34:11the argument
4:34:13a so far we have not defined the limit
4:34:16of a function for a given
4:34:18argument if we had done so we could have
4:34:21defined the continuity of a function
4:34:23differently a function is continuous at
4:34:25a point where its value is the same as
4:34:28the limit of its value for approaches
4:34:31either from above or From
4:34:33Below but it is only the the
4:34:35exceptionally tame function that has a
4:34:39definite limit as the argument
4:34:41approaches a given
4:34:42point the general rule is that a
4:34:45function oscillates and that given any
4:34:47neighborhood of a given argument however
4:34:50small a whole stretch of values will
4:34:52occur for arguments within this
4:34:55neighborhood as this is the general rule
4:34:58let us consider it
4:35:00first let us consider what may happen as
4:35:03the argument approaches some value a
4:35:06from below that is to say we wish to
4:35:09consider what happens for arguments
4:35:11contained in the interval from the
4:35:14difference of a from Epsilon to a where
4:35:18Epsilon is some number which in
4:35:20important cases will be very
4:35:23small the values of the function for
4:35:25arguments from the difference of a from
4:35:29Epsilon to a a excluded will be a set of
4:35:33real numbers which will Define a a
4:35:35certain section of the set of real
4:35:37numbers namely the section consisting of
4:35:41those numbers that are not greater than
4:35:43all the values for arguments from the
4:35:46difference of a from Epsilon to a given
4:35:50any number in this section there are
4:35:53values at least as great as this number
4:35:56for arguments between the difference of
4:35:58a from Epsilon and a that is for
4:36:01arguments that fall very little short of
4:36:04a if if Epsilon is very
4:36:07small let us take all possible epsilons
4:36:10and all possible corresponding sections
4:36:14the common part of all these sections we
4:36:16will call the ultimate section as the
4:36:19argument approaches a to say that a
4:36:22number Z belongs to the ultimate section
4:36:26is to say that however small we may make
4:36:29Epsilon there are arguments between the
4:36:32difference of a from Epsilon and a
4:36:35for which the value of the function is
4:36:38not less than
4:36:40Z we may apply exactly the same process
4:36:43two upper sections that is two sections
4:36:47that go from some point up to the top
4:36:49instead of the bottom up to some
4:36:52point here we take those numbers that
4:36:55are not less than all those values for
4:36:58arguments from the difference of a from
4:37:00Epsilon to a this defines an upper
4:37:04section which will vary as Epsilon
4:37:07varies taking the common part of all
4:37:10such sections for all possible epsilons
4:37:13we obtain the ultimate upper
4:37:15section to say that a number Z belongs
4:37:18to the ultimate upper section is to say
4:37:21that however small we make Epsilon there
4:37:25are arguments between the difference of
4:37:27a from Epsilon and a for which the value
4:37:30of the function is not greater than
4:37:34Z if a term belongs both to the ultimate
4:37:37section and to the ultimate upper
4:37:40section we shall say that it belongs to
4:37:42the ultimate
4:37:43oscillation we may illustrate the matter
4:37:46by considering once more the function
4:37:49sin of 1 /x as X approaches the value
4:37:53zero we shall assume in order to fit in
4:37:57with the above definitions that this
4:37:59value is approached From
4:38:01Below let us begin with the ultimate
4:38:04section
4:38:05between negative Epsilon and0 whatever
4:38:09Epsilon may be the function will assume
4:38:12the value one for certain arguments but
4:38:14will never assume any greater value
4:38:17hence the ultimate section consists of
4:38:20all real numbers positive and negative
4:38:22up to and including one that is it
4:38:26consists of all negative numbers
4:38:28together with zero together with the
4:38:31positive numbers up to and including
4:38:33one similarly the ultimate upper section
4:38:38consists of all positive numbers
4:38:40together with zero together with the
4:38:42negative numbers down to and including
4:38:46-1 thus the ultimate oscillation
4:38:49consists of all real numbers from -1 to
4:38:521 both
4:38:55included we may say generally that the
4:38:58ultimate oscillation of a function as
4:39:01the argument approaches a From Below
4:39:03consists of all those numbers X which
4:39:05are such that however near we come to a
4:39:09we shall still find values as great as X
4:39:12and values as small as
4:39:16X the ultimate oscillation may contain
4:39:19no terms or one term or many terms in
4:39:22the first two cases the function has a
4:39:24definite limit four approaches from
4:39:26below if the ultimate oscillation has
4:39:29one term this is fairly
4:39:31obvious it is equally true if it has
4:39:34none
4:39:35for it is not difficult to prove that if
4:39:38the ultimate oscillation is null the
4:39:40boundary of the ultimate section is the
4:39:43same as that of the ultimate upper
4:39:45section and may be defined as the limit
4:39:48of the function for approaches From
4:39:51Below but if the ultimate oscillation
4:39:54has many terms there is no definite
4:39:56limit to the function four approaches
4:39:59From
4:39:59Below in this case we can take the lower
4:40:03and upper boundaries of the ultimate
4:40:05oscillation that is the lower boundary
4:40:07of the ultimate upper section and the
4:40:10upper boundary of the ultimate section
4:40:12as the lower and upper limits of its
4:40:14ultimate values for approaches From
4:40:17Below similarly we obtain lower and
4:40:20upper limits of the ultimate values for
4:40:23approaches from
4:40:24above thus we have in the general case
4:40:28four limits to a function four
4:40:30approaches to a given
4:40:32argument the limit for a given AR
4:40:34argument a only exists when all these
4:40:37four are equal and is then their common
4:40:41value if it is also the value for the
4:40:44argument a the function is continuous
4:40:47for this
4:40:48argument this may be taken as defining
4:40:51continuity it is equivalent to our
4:40:53former
4:40:56definition we can Define the limit of a
4:40:58function for a given argument if it
4:41:01exists without passing through the
4:41:03ultimate oscillation and the four limits
4:41:05of the general case the definition
4:41:08proceeds in that case just as the
4:41:10earlier definition of continuity
4:41:13preceded let us Define the limit for
4:41:15approaches from below if there is to be
4:41:19a definite limit for approaches to a
4:41:21from below it is necessary and
4:41:23sufficient that given any small number
4:41:26Sigma two values for arguments
4:41:28sufficiently near to a but both less
4:41:31than a will differ by less than Sigma
4:41:36that is if Epsilon is sufficiently small
4:41:39and our arguments both lie between the
4:41:41difference of a from Epsilon and a a
4:41:45excluded then the difference between the
4:41:47values for these arguments will be less
4:41:50than
4:41:51Sigma this is to hold for any Sigma
4:41:54however small in that case the function
4:41:57has a limit four approaches From
4:42:00Below similarly we Define the case when
4:42:04there is a limit for approaches from
4:42:08above these two limits even when both
4:42:10exist need not be identical and if they
4:42:14are identical they still need not be
4:42:16identical with the value for the
4:42:19argument
4:42:20a it is only in this last case that we
4:42:24call the function continuous for the
4:42:27argument
4:42:28a a function is called continuous
4:42:31without
4:42:32qualification when it is continuous for
4:42:35every
4:42:36argument another slightly different
4:42:38method of reaching the definition of
4:42:40continuity is the
4:42:42following let us say that a function
4:42:44ultimately converges into a class Alpha
4:42:47if there is some real number such that
4:42:50for this argument and all arguments
4:42:52greater than this the value of the
4:42:54function is a member of the class
4:42:57Alpha similarly we shall say that a
4:43:00function converges into Alpha as the
4:43:02argument approaches X From Below
4:43:05if there is some argument y less than x
4:43:09such that throughout the interval from y
4:43:11included to X excluded the function has
4:43:15values which are members of
4:43:18alpha we may now say that a function is
4:43:20continuous for the argument a for which
4:43:23it has the value F of a if it satisfies
4:43:27four conditions namely one given any
4:43:31real number less than F of a the
4:43:34function converges into the successors
4:43:36of this number as the argument
4:43:39approaches a From
4:43:41Below two given any real number greater
4:43:45than F of a the function converges into
4:43:48the predecessors of this number as the
4:43:51argument approaches a From
4:43:53Below three and four similar conditions
4:43:57for approaches to a from
4:44:00above the advantages of this form of
4:44:03definition is that it analyzes the
4:44:05conditions of continuity into four
4:44:08derived from considering arguments and
4:44:10values respectively greater or less than
4:44:13the argument and value for which
4:44:15continuity is to be defined we may now
4:44:19generalize our definitions so as to
4:44:22apply to series which are not numerical
4:44:24or known to be numerically measurable
4:44:28the case of motion is a convenient one
4:44:30to bear in mind there is a story by HG
4:44:34Wells which will illustrate from the
4:44:37case of motion the difference between
4:44:39the limit of a function for a given
4:44:41argument and its value for the same
4:44:45argument the hero of the story who
4:44:48possessed without his knowledge the
4:44:50power of realizing his wishes was being
4:44:53attacked by a policeman but on
4:44:55ejaculating go to he found that the
4:44:59policeman
4:45:00disappeared if F of T was the
4:45:03policeman's position at time T and T
4:45:06subz the moment of the ejaculation the
4:45:09limit of the policeman's positions as T
4:45:12approached to T subz from below would be
4:45:16in contact with the hero whereas the
4:45:19value for the argument T subz was
4:45:23undefined but such occurrences are
4:45:25supposed to be rare in the real world
4:45:28and it is assumed though without
4:45:30adequate evidence that all motions are
4:45:33continuous that is that given any body
4:45:37if F of T is its position at time t f of
4:45:40T is a continuous function of T it is
4:45:44the meaning of continuity involved in
4:45:46such statements which we now wish to
4:45:49Define as simply as
4:45:52possible the definitions given for the
4:45:54case of functions where the argument and
4:45:56value are real numbers can readily be
4:45:59adapted for more General
4:46:02use let p and Q be two relations which
4:46:06it is well to imagine serial though it
4:46:08is not necessary to our definitions that
4:46:10they should be
4:46:12so let R be a One Mini relation whose
4:46:15domain is contained in the field of P
4:46:18while its Converse domain is contained
4:46:20in the field of Q then R is in a
4:46:23generalized sense a function whose
4:46:26arguments belong to the field of Q while
4:46:29its values belong to the field of
4:46:31P suppose for example that we are
4:46:34dealing with a particle moving on a
4:46:37line let Q be the time series P the
4:46:41series of points on our line from left
4:46:43to right are the relation of the
4:46:45position of our particle on the line at
4:46:47time a to the time a so that the r of a
4:46:52is its position at time
4:46:54a this illustration may be borne in mind
4:46:57throughout our
4:47:00definitions we shall say that the
4:47:02function R is continuous for the
4:47:04argument a if given any interval Alpha
4:47:08on the P series containing the value of
4:47:10the function for the argument a there is
4:47:13an interval on the Q Series containing a
4:47:17not as an end point and such that
4:47:20throughout this interval the function
4:47:22has values which are members of alpha we
4:47:26mean by an interval all the terms
4:47:28between any two that is if X and Y are
4:47:32two members of the field of P
4:47:34and X has the relation P to Y we shall
4:47:38mean by the P interval X to Y all terms
4:47:42Z such that X has the relation P to Z
4:47:46and Z has the relation P to Y together
4:47:50when so stated with X or Y
4:47:54themselves we can easily Define the
4:47:56ultimate section and the ultimate
4:47:59oscillation to define the ultimate
4:48:01section for approaches to the argument a
4:48:03from From Below take any argument Y
4:48:06which precedes a that is has the
4:48:09relation Q to a take the values of the
4:48:13function for all arguments up to and
4:48:15including Y and form the section of P
4:48:19defined by these values that is those
4:48:22members of the P series which are
4:48:24earlier than or identical with some of
4:48:26these
4:48:28values form all such sections for all
4:48:31y's that precede a and take their common
4:48:34part this will be the ultimate section
4:48:38the ultimate upper section and the
4:48:40ultimate oscillation are then defined
4:48:42exactly as in the previous
4:48:46case the adaptation of the definition of
4:48:49convergence and the resulting
4:48:50alternative definition of continuity
4:48:53offers no difficulty of any
4:48:56kind we say that a function R is
4:48:59ultimately Q convergent into Alpha if
4:49:02there is a member y y of the converse
4:49:05domain of R and the field of Q such that
4:49:09the value of the function for the
4:49:11argument Y and for any argument to which
4:49:13y has the relation Q is a member of
4:49:17alpha we say that r q converges into
4:49:21Alpha as the argument approaches a given
4:49:23argument a if there is a term y having
4:49:27the relation Q to a and belonging to the
4:49:31converse domain of R and such that
4:49:34the value of the function for any
4:49:36argument in the Q interval from y
4:49:39inclusive to a exclusive belongs to
4:49:44Alpha of the four conditions that a
4:49:46function must fulfill in order to be
4:49:48continuous for the argument a the first
4:49:52is putting B for the value for the
4:49:54argument a given any term having the
4:49:57relation P to b r q converges into the
4:50:01successors of B with respect to p
4:50:04as the argument approaches a from
4:50:07below the second condition is obtained
4:50:09by replacing P by its Converse the third
4:50:13and fourth are obtained from the first
4:50:14and second by replacing Q by its
4:50:18Converse there is thus nothing in the
4:50:21Notions of the limit of a function or
4:50:23the continuity of a function that
4:50:25essentially involves number both can be
4:50:29defined generally and many propositions
4:50:32about them can be proved for any two
4:50:34series one being the argument series and
4:50:37the other the value
4:50:39series it will be seen that the
4:50:42definitions do not involve infinite
4:50:44decimals they involve infinite classes
4:50:47of intervals growing smaller without any
4:50:50limit short of zero but they do not
4:50:53involve any intervals that are not
4:50:56finite this is analogous to the fact
4:50:59that if a line an inch long be haved
4:51:02then haved again and so on indefinitely
4:51:05we never reach infinite decimals in this
4:51:08way after n bisections the length of our
4:51:12bit is 1/ 2 to the N power of an inch
4:51:17and this is finite whatever finite
4:51:19number n may be the process of
4:51:22successive bisection does not lead to
4:51:24divisions whose ordinal number is
4:51:27infinite since it is essentially a one
4:51:30by one process thus infinite decimals
4:51:33are not to be reached in this way
4:51:36confusions on such topics have had much
4:51:38to do with the difficulties which have
4:51:41been found in the discussion of infinity
4:51:44and
4:51:45continuity end of chapter
4:51:5211 chapter 12 of introduction to
4:51:56mathematical Philosophy by Bertrand
4:51:58Russell this LibriVox recording is in
4:52:01the public
4:52:02domain
4:52:04selections and the multiplicative
4:52:06Axiom in this chapter we have to
4:52:09consider an axiom which can be
4:52:11enunciated but not proved in terms of
4:52:13logic and which is convenient though not
4:52:16indispensable in certain portions of
4:52:19mathematics it is convenient in the
4:52:22sense that many interesting propositions
4:52:24which it seems natural to suppose true
4:52:27cannot be proved without its help but it
4:52:30is not indispensable because even
4:52:32without those propositions
4:52:34the subjects in which they occur still
4:52:36exist though in a somewhat mutilated
4:52:40form before enunciating the
4:52:42multiplicative Axiom we must first
4:52:45explain the theory of selections and the
4:52:47definition of multiplication when the
4:52:50number of factors may be
4:52:53infinite in defining the arithmetical
4:52:55operations the only correct procedure is
4:52:58to construct an actual class or relation
4:53:01in the case of relation numbers having
4:53:03the required number of terms this
4:53:06sometimes Demands a certain amount of
4:53:08Ingenuity but it is essential in order
4:53:11to prove the existence of the number
4:53:13defined take as the simplest example the
4:53:16case of
4:53:17addition suppose we are given a cardinal
4:53:20number mu and a class Alpha which has mu
4:53:24terms how shall we Define mu plus
4:53:28mu for this purpose we must have two
4:53:30classes having me terms and they must
4:53:33not
4:53:34overlap we can construct such classes
4:53:37from alpha in various ways of which the
4:53:39following is perhaps the simplest form
4:53:43first all the ordered couples whose
4:53:45first term is a class consisting of a
4:53:48single member of Alpha and whose second
4:53:51term is the null
4:53:52class then secondly form all the ordered
4:53:55couples whose first term is the null
4:53:58class and whose second term is a class
4:54:00consisting of a single member of alpha
4:54:04these two classes of couples have no
4:54:06member in common and The Logical sum of
4:54:09the two classes will have mu plus mu
4:54:12terms exactly analogously we can Define
4:54:15mu plus new given that mu is the number
4:54:19of some class Alpha and new is the
4:54:21number of some class
4:54:24beta such definitions as a rule are
4:54:27merely a question of a suitable
4:54:29technical device but in the case of
4:54:31multiplication where the number of fact
4:54:33factors may be infinite important
4:54:35problems arise out of the
4:54:37definition multiplication when the
4:54:40number of factors is finite offers no
4:54:42difficulty given two classes Alpha and
4:54:45beta of which the first has mu terms and
4:54:48the second new terms we can Define mu *
4:54:51new as the number of ordered couples
4:54:54that can be formed by choosing the first
4:54:56member out of Alpha and the second out
4:54:58of beta it will be seen that this
4:55:01definition does not require that Alpha
4:55:03and beta should not overlap it even
4:55:06remains adequate when Alpha and beta are
4:55:09identical for example let Alpha be the
4:55:12class whose members are x sub1 x sub 2 x
4:55:15sub3 then the class which is used to
4:55:18define the product mu * mu is the class
4:55:21of couples the ordered pair x sub1 x
4:55:25sub1 the ordered pair x sub1 x sub 2 the
4:55:29ordered pair x sub1 x sub3 the ordered
4:55:33pair x sub 2 x sub1 the ordered pair x
4:55:36sub 2 x sub2 the ordered pair x sub2 x
4:55:40sub3 the ordered pair x sub3 x sub 1 the
4:55:45ordered pair x sub3 x sub 2 the ordered
4:55:48pair x sub3 x sub
4:55:513 this definition remains applicable
4:55:54when mu or new or both are infinite and
4:55:57it can be extended step by step to three
4:56:00or four or any finite number of factors
4:56:04no difficulty arises as regards this
4:56:06definition except that it cannot be
4:56:09extended to an infinite number of
4:56:12factors the problem of multiplication
4:56:14when the number of factors may be
4:56:16infinite arises in this way suppose we
4:56:20have a class Kappa consisting of classes
4:56:24suppose the number of terms in each of
4:56:26these classes is given how shall we
4:56:29Define the product of all these
4:56:32numbers if we can frame our definition
4:56:35generally it will be applicable whether
4:56:37Kappa is finite or infinite it is to be
4:56:41observed that the problem is to be able
4:56:43to deal with the case when Kappa is
4:56:45infinite not with the case when its
4:56:48members
4:56:49are if Kappa is not infinite the method
4:56:52defined above is just as applicable when
4:56:54its members are infinite as when they
4:56:56are finite it is the case when Kappa is
4:56:59infinite even though its members may be
4:57:01finite that we have to find a way of
4:57:04dealing
4:57:05with the following method of defining
4:57:08multiplication generally is due to Dr
4:57:11Whitehead it is explained and treated at
4:57:14length in principia Mathematica volume 1
4:57:17star number 80 and following and volume
4:57:20two St number
4:57:23114 let us suppose to begin with that
4:57:26Kappa is a class of classes no two of
4:57:29which overlap say the constituencies in
4:57:32a country where there is no plural
4:57:34voting each constituency being
4:57:35considered as a class of Voters let us
4:57:38now set to work to choose one term out
4:57:41of each class to be its representative
4:57:43as constituencies do when they elect
4:57:46members of parliament assuming that by
4:57:48law each constituency has to elect a man
4:57:51who is a voter in that
4:57:53constituency we thus arrive at a class
4:57:56of Representatives who make up our
4:57:58Parliament one being selected out of
4:58:01each
4:58:02constituency how many different possible
4:58:04ways of choosing a parliament are there
4:58:08each constituency can select any one of
4:58:10its voters and therefore if there are me
4:58:13voters in a constituency it can make me
4:58:16choices the choices of the different
4:58:18constituencies are independent thus it
4:58:21is obvious that when the total number of
4:58:23constituencies is finite the number of
4:58:26possible parliaments is obtained by
4:58:28multiplying together the number of
4:58:30Voters in the various
4:58:32constituencies when we do not know
4:58:34whether the number of constituencies is
4:58:36finite or infinite we may take the
4:58:38number of possible parliaments as
4:58:40defining the product of the numbers of
4:58:42the separate
4:58:44constituencies this is the method by
4:58:46which infinite products are defined we
4:58:49must now drop our illustration and
4:58:51proceed to exact
4:58:53statements let Kappa be a class of
4:58:55classes and let us assume to begin with
4:58:58that no two members of Kappa overlap
4:59:00that is if Alpha and beta are two
4:59:02different different members of Kappa
4:59:04then no member of the one is a member of
4:59:07the other we shall call a Class A
4:59:09selection from Kappa when it consists of
4:59:12just one term from each member of Kappa
4:59:15that is Mu is a selection from Kappa if
4:59:19every member of me belongs to some
4:59:21member of Kappa and if Alpha be any
4:59:24member of Kappa mu and Alpha have
4:59:27exactly one term in common the class of
4:59:30all selections from Kappa we shall call
4:59:33the multiplicative class of Kappa the
4:59:36number of terms in the multiplicative
4:59:38class of Kappa that is the number of
4:59:41possible selections from Kappa is
4:59:43defined as the product of the numbers of
4:59:45the members of
4:59:47Kappa this definition is equally
4:59:49applicable whether Kappa is finite or
4:59:53infinite before we can be wholly
4:59:55satisfied with these definitions we must
4:59:57remove the Restriction that no two
4:59:59members of Kappa are to
5:00:01overlap for this purpose instead of
5:00:04defining first a class called a
5:00:06selection we will Define first a
5:00:09relation which we will call a
5:00:11selector a relation R will be called a
5:00:14selector from Kappa if from every member
5:00:18of Kappa it picks out one term as the
5:00:21representative of that member that is if
5:00:25given any member Alpha of Kappa there is
5:00:28just one term X which is a member of
5:00:31Alpha and has the relation R to a and
5:00:35this is to be all that R does the formal
5:00:38definition is a selector from a class of
5:00:42classes Kappa is a one many relation
5:00:46having Kappa for its Converse domain and
5:00:48such that if x has the relation to Alpha
5:00:52then X is a member of
5:00:54alpha if R is a selector from Kappa and
5:00:59Alpha is a member of Kappa and X is the
5:01:02term which has the relation R to Alpha
5:01:05we call X the representative of alpha in
5:01:08respect of the relation
5:01:10r a selection from Kappa will now be
5:01:14defined as the domain of a selector and
5:01:16the multiplicative class as before will
5:01:19be the class of
5:01:21selections but when the members of Kappa
5:01:24overlap there may be more selectors than
5:01:26selections since a term X which belongs
5:01:29to two classes Alpha and beta may be
5:01:32selected once to represent Alpha and
5:01:35once to represent beta giving rise to
5:01:38different selectors in the two cases but
5:01:40to the same
5:01:41selection for purposes of defining
5:01:44multiplication it is the selectors we
5:01:46require rather than the
5:01:49selections thus we Define the product of
5:01:52the numbers of the members of a class of
5:01:54classes Kappa is the number of selectors
5:01:58from
5:01:59Kappa we can Define exponentiation by an
5:02:02adapt a of the above plan we might of
5:02:06course Define mu raised to the new power
5:02:09as the number of selectors from new
5:02:11classes Each of which has me terms but
5:02:14there are objections to this definition
5:02:17derived from the fact that the
5:02:18multiplicative Axiom of which we shall
5:02:21speak shortly is unnecessarily involved
5:02:24if it is
5:02:25adopted we adopt instead the following
5:02:29construction let Alpha be a class having
5:02:32me terms and beta a class having new
5:02:35terms let y be a member of beta and form
5:02:39the class of all ordered couples that
5:02:41have y for their second term and a
5:02:44member of alpha for their first term
5:02:47there will be me such couples for a
5:02:49given y since any member of alpha may be
5:02:53chosen for the first term and Alpha has
5:02:56mu
5:02:56members if we now form all the classes
5:02:59of this sort that result from varying y
5:03:03we obtain together new classes since y
5:03:06may be any member of beta and beta has
5:03:10new
5:03:11members these new classes are each of
5:03:14them a class of couples namely all the
5:03:17couples that can be formed of a variable
5:03:19member of Alpha and a fixed member of
5:03:22beta we Define mu raised to the new
5:03:25power as the number of selectors from
5:03:28the class consisting of these new
5:03:30classes or we may equally well Define mu
5:03:33raised to the new power as the number of
5:03:36selections for since our classes of
5:03:39couples are mutually exclusive the
5:03:41number of selectors is the same as the
5:03:43number of
5:03:45selections a selection from our class of
5:03:47classes will be a set of ordered couples
5:03:51of which there will be exactly one
5:03:53having any given member of beta for its
5:03:55second term and the first term may be
5:03:58any member of alpha thus mu raised to
5:04:01the new power is defined by the
5:04:03selectors from a certain set of new
5:04:06classes each having new terms but the
5:04:09set is one having a certain structure
5:04:11and a more manageable composition than
5:04:13is the case in
5:04:15general the relevance of this to the
5:04:17multiplicative aium will appear
5:04:20shortly what applies to exponentiation
5:04:23applies also to the product of two
5:04:26cardinals we might Define the product of
5:04:29mu and new as the sum of the numbers of
5:04:32of new classes each having new terms but
5:04:36we prefer to Define it as the number of
5:04:38ordered couples to be formed consisting
5:04:40of a member of alpha followed by a
5:04:43member of beta where Alpha has mu terms
5:04:46and beta has new terms this definition
5:04:50also is designed to evade the necessity
5:04:53of assuming the multiplicative
5:04:56Axiom with our definitions we can prove
5:04:59the usual formal laws of multiplication
5:05:02and
5:05:03exponentiation but there is one thing we
5:05:06cannot prove we cannot prove that a
5:05:09product is only zero when one of its
5:05:11factors is
5:05:13zero we can prove this when the number
5:05:15of factors is finite but not when it is
5:05:19infinite in other words we cannot prove
5:05:22that given a class of classes none of
5:05:24which is null there must be selectors
5:05:26from them or that given a class of
5:05:29mutually exclusive classes there must
5:05:32must be at least one class consisting of
5:05:35one term out of each of the given
5:05:37classes these things cannot be proved
5:05:40and although at First Sight they seem
5:05:42obviously true yet reflection brings
5:05:44gradually increasing doubt until at last
5:05:47we become content to register the
5:05:50Assumption and its consequences as we
5:05:53register the axium of parallels without
5:05:55assuming that we can know whether it is
5:05:58true or false the Assumption Loosely
5:06:01worded is that selectors and selections
5:06:04exist when we should expect them there
5:06:07are many equivalent ways of stating it
5:06:09precisely we may begin with the
5:06:12following given any class of mutually
5:06:15exclusive classes of which none is null
5:06:18there is at least one class which has
5:06:21exactly one term in common with each of
5:06:23the given
5:06:25classes this proposition we will call
5:06:28the multiplicative Axiom footnote one C
5:06:31principia Mathematica volume 1 star
5:06:34number 88 also volume 3 star numbers 257
5:06:39to
5:06:40258 end of footnote one we will give
5:06:44first various equivalent forms of the
5:06:46proposition and then consider certain
5:06:48ways in which its truth or falsehood is
5:06:50of interest to mathematics the
5:06:53multiplicative Axiom is equivalent to
5:06:55the proposition that a product is only
5:06:58zero when at least one of its factors is
5:07:00zero that is that if any number of
5:07:04cardinal numbers be multiplied together
5:07:06the result cannot be zero unless one of
5:07:09the numbers concerned is
5:07:11zero the multiplicative axum is
5:07:13equivalent to the proposition that if R
5:07:16be any relation and Kappa any class
5:07:19contained in the converse domain of R
5:07:21then there is at least one one many
5:07:23relation implying R and having Kappa for
5:07:27its Converse
5:07:28domain the multiplicative axium is
5:07:31equivalent to the assumption
5:07:33that if Alpha be any class and Kappa all
5:07:35the subclasses of alpha with the
5:07:37exception of the null class then there
5:07:40is at least one selector from
5:07:42Kappa this is the form in which the
5:07:45Axiom was first brought to the notice of
5:07:47the Learned World by zero in
5:07:54hiset footnote one maemes analan volume
5:07:5959 Pages 514 to
5:08:02116 in this form we shall speak of it as
5:08:06Zero's Axiom end of footnote
5:08:091 zero regards the axiim as an
5:08:12unquestionable truth it must be
5:08:15confessed that until he made it explicit
5:08:18mathematicians had used it without a
5:08:20qualm but it would seem that they had
5:08:22done so
5:08:23unconsciously and the credit due to
5:08:25zerel for having made it explicit is
5:08:28entirely independent of the question
5:08:30whether it is true or false
5:08:34the multiplicative axium has been shown
5:08:36by zero in the above mentioned proof to
5:08:38be equivalent to the proposition that
5:08:41every class can be well ordered that is
5:08:44can be arranged in a series in which
5:08:46every subclass has a first term except
5:08:49of course the null class the full proof
5:08:52of this proposition is difficult but it
5:08:55is not difficult to see the general
5:08:57principle upon which it proceeds it uses
5:09:01the form which we call Zera aium that is
5:09:04it assumes that given any class Alpha
5:09:07there is at least one one many relation
5:09:10R whose Converse domain consists of all
5:09:13existing subclasses of Alpha and which
5:09:16is such that if x has the relation R
5:09:19toai then X is a member of
5:09:22Cai such a relation picks out a
5:09:25representative from each
5:09:27subass of course it will often happen
5:09:29that two subclasses have the same
5:09:31representative
5:09:33what zero does in effect is to count off
5:09:36the members of alpha one by one by means
5:09:38of R and transfinite induction we put
5:09:42first the representative of alpha call
5:09:44it X sub1 then take the representative
5:09:48of the class consisting of all of alpha
5:09:51except X sub1 call it xub 2 it must be
5:09:56different from X sub1 because every
5:09:59representative is a member of its class
5:10:02and X sub1 is shut out from this class
5:10:05proceed similarly to take away x sub 2
5:10:09and let X sub3 be the representative of
5:10:12what is
5:10:13left in this way we first obtain a
5:10:16progression x sub 1 x sub 2 and so on to
5:10:19X subn and so on assuming that Alpha is
5:10:22not
5:10:23finite we then take away the whole
5:10:25progression let x sub Omega be the
5:10:28representative of what is left of alpha
5:10:32in this way we can go on until nothing
5:10:34is left the successive Representatives
5:10:38will form a well-ordered series
5:10:40containing all the members of alpha the
5:10:42above is of course only a hint of the
5:10:45general lines of the proof this
5:10:47proposition is called Cello's
5:10:51theorem the multiplicative axium is also
5:10:54equivalent to the assumption that of any
5:10:56two cardinals which are not equal one
5:10:58must be the greater if the axium is
5:11:01false there will be Cardinals mu and new
5:11:04such that mu is neither less than equal
5:11:07to nor greater than
5:11:09new we have seen that Alf sub one and
5:11:13two raised to the alive Subzero power
5:11:16possibly form an instance of such a
5:11:18pair many other forms of the axium might
5:11:21be given but the above are the most
5:11:23important of the forms known at present
5:11:26as to the truth or falsehood of the
5:11:28axium in any of its forms nothing is
5:11:31known at
5:11:33present the propositions that depend
5:11:35upon the Axiom without being known to be
5:11:38equivalent to it are numerous and
5:11:40important take first the connection of
5:11:43addition and
5:11:44multiplication we naturally think that
5:11:46the sum of new mutually exclusive
5:11:49classes each having mu terms must have
5:11:52the product of mu and new terms when new
5:11:56is finite this can be proved but when
5:11:58new is infinite it cannot be proved
5:12:00without the multiplic axium except where
5:12:04owing to some special Circumstance the
5:12:06existence of certain selectors can be
5:12:09proved the way the multiplicative axium
5:12:12enters in is as follows suppose we have
5:12:15two sets of new mutually exclusive
5:12:18classes each having new terms and we
5:12:21wish to prove that the sum of one set
5:12:24has as many terms as the sum of the
5:12:26other in order to prove this we must
5:12:29establish a one one relation
5:12:32now since there are in each case new
5:12:34classes there is some one one relation
5:12:37between the two sets of classes but what
5:12:40we want is a one- one relation between
5:12:43their
5:12:44terms let us consider some one one
5:12:46relation s between classes then if Kappa
5:12:51and Lambda are the two sets of classes
5:12:53and Alpha is some member of Kappa there
5:12:56will be a member beta of Lambda which
5:12:59will be the correlate of alpha with
5:13:01respect to S now Alpha and beta each
5:13:05have new terms and are therefore similar
5:13:08there are accordingly one one
5:13:10correlations of Alpha and beta the
5:13:13trouble is that there are so many in
5:13:16order to obtain a 1 one correlation of
5:13:18the sum of Kappa with the sum of Lambda
5:13:21we have to pick out one selection from a
5:13:24set of classes of correlators one class
5:13:27of the set being all the one one
5:13:29correlators of alpha with beta if Kappa
5:13:32and Lambda are infinite we cannot in
5:13:35general know that such a selection
5:13:37exists unless we can know that the
5:13:39multiplicative Axiom is true hence we
5:13:43cannot establish the usual kind of
5:13:45connection between addition and
5:13:49multiplication this fact has various
5:13:51curious consequences to begin with we
5:13:54know that LF subz raised to the second
5:13:57power is equal to the product of Alf
5:13:59subz and Alf Sub 0 which is equal to ALF
5:14:03subz it is commonly inferred from this
5:14:06that the sum of Alf Subzero classes each
5:14:09having Alf Subzero members must itself
5:14:12have Alf Subzero members but this
5:14:15inference is fallacious since we do not
5:14:18know that the number of terms in such a
5:14:20sum is iive subz * alive subz nor
5:14:24consequently that it is Olive
5:14:27Subzero this has a bearing upon the
5:14:29theory of transfinite ordinals it is
5:14:32easy to prove that an ordinal which has
5:14:34Olive Subzero predecessors must be one
5:14:37of what canor calls the second class
5:14:40that is such that a series having this
5:14:43ordinal number will have alive Subzero
5:14:46terms in its
5:14:47field it is also easy to see that if we
5:14:51take any progression of ordinals of the
5:14:53second class the predecessors of their
5:14:56limit form at most the sum of Al of
5:14:59subzero classes each having Al of
5:15:02subzero terms it is inferred then
5:15:05maliciously unless the multiplicative
5:15:07axum is true that the predecessors of
5:15:10the limit are all if Subzero in number
5:15:13and therefore that the limit is a number
5:15:15of the second class that is to say it is
5:15:19supposed to be proved that any
5:15:21progression of ordinals of the second
5:15:23class has a limit which is again an
5:15:26ordinal of the second class this
5:15:29proposition with the Cory that Omega sub
5:15:32one the smallest ordinal of the third
5:15:34class is not the limit of any
5:15:37progression is involved in most of the
5:15:39recognized theory of ordinals of the
5:15:41second
5:15:42class in view of the way in which the
5:15:45multiplicative axium is involved the
5:15:48proposition and its corollary cannot be
5:15:50regarded as
5:15:51proved they may be true or they may not
5:15:55all that can be said at present is that
5:15:58we do not know thus the greater part of
5:16:01the the are of ordinals of the second
5:16:03class must be regarded as
5:16:06unproved another illustration may help
5:16:09to make the point clearer we know that
5:16:12the product of two and Alf Sub 0 equals
5:16:15Alf Sub 0 hence we might suppose that
5:16:18the sum of all of subzero pairs must
5:16:21have all Subzero terms but this though
5:16:24we can prove that it is sometimes the
5:16:26case cannot be proved to happen always
5:16:29unless we assume the multiplicative
5:16:32axium this is illustrated by the
5:16:34millionaire who bought a pair of socks
5:16:37whenever he bought a pair of boots and
5:16:39never at any other time and who had such
5:16:42a passion for buying both that at last
5:16:45he had Olive Subzero pairs of boots and
5:16:47also zero pairs of socks the problem is
5:16:51how many boots had he and how many socks
5:16:55one would naturally suppose that he had
5:16:57twice as many boots and twice as many
5:16:59socks as he had pairs of each
5:17:02and that therefore he had Al of subzero
5:17:04of each since that number is not
5:17:06increased by
5:17:08doubling but this is an instance of the
5:17:10difficulty already noted of connecting
5:17:13the sum of new classes each having me
5:17:16terms with the product of me and
5:17:19new sometimes this can be done sometimes
5:17:22it cannot in our case it can be done
5:17:24with the boots but not with the socks
5:17:27except by some very artificial device
5:17:30the reason for the difference is this
5:17:32among boots we can distinguish right and
5:17:35left and therefore we can make a
5:17:36selection of one out of each pair namely
5:17:40we can choose all the right boots or all
5:17:42the left boots but with socks no such
5:17:45principle of selection suggests itself
5:17:48and we cannot be sure unless we assume
5:17:50the multiplicative Axiom that there is
5:17:53any class consisting of one sock out of
5:17:56each pair hence the
5:17:59problem we may put the m in another way
5:18:03to prove that a class has all Subzero
5:18:05terms it is necessary and sufficient to
5:18:08find some way of arranging its terms in
5:18:10a progression there is no difficulty in
5:18:13doing this with the boots the pairs are
5:18:16given as forming an Al of subzero and
5:18:19therefore as the field of a
5:18:20progression within each pair take the
5:18:23left boot first and the right second
5:18:26keeping the order of the pair unchanged
5:18:29in this way we obtain a progression of
5:18:31all the boots but with the socks we
5:18:34shall have to choose arbitrarily with
5:18:37each pair which to put first and an
5:18:39infinite number of arbitrary choices is
5:18:42an
5:18:44impossibility unless we can find a rule
5:18:46for selecting that is a relation which
5:18:49is a selector we do not know that a
5:18:52selection is even theoretically
5:18:54possible of course in the case of
5:18:57objects in space like socks we always
5:19:00can find some princip of selection for
5:19:03example take the centers of mass of the
5:19:05socks there will be points p in space
5:19:09such that with any pair the centers of
5:19:11mass of the two socks are not both at
5:19:13exactly the same distance from P thus we
5:19:16can choose from each pair that sock
5:19:19which has its Center of mass nearer to
5:19:22P but there is no theoretical reason why
5:19:25a method of selection such as this
5:19:27should always be possible and the case
5:19:29of socks with a little Goodwill on the
5:19:32part of the reader may serve to show how
5:19:34a selection might be
5:19:37impossible it is to be observed that if
5:19:40it were impossible to select one out of
5:19:42each pair of socks it would follow that
5:19:45the socks could not be arranged in a
5:19:47progression and therefore that there
5:19:49were not all of subzero of
5:19:51them this case illustrates that if mu is
5:19:55an infinite number one set of me pairs
5:19:58may not contain the same number of terms
5:20:00as in another set of new pairs for given
5:20:04allive Subzero pairs of boots there are
5:20:06certainly allive Subzero pairs of boots
5:20:09but we cannot be sure of this in the
5:20:11case of the socks unless we assume the
5:20:14multiplicative Axiom or fall back upon
5:20:17some fortuitous geometrical method of
5:20:19selection such as the
5:20:21above another important problem
5:20:24involving the multiplicative axum is the
5:20:26relation of reflexiveness to non-
5:20:29inductiv it will be remembered that in
5:20:32chapter 8 we pointed out that a
5:20:35reflexive number must be
5:20:37non-inductive but that the converse so
5:20:40far as is known at present can only be
5:20:43proved if we assume the multiplicative
5:20:45Axiom the way in which this comes about
5:20:47is as follows it is easy to prove that a
5:20:51reflexive class is one which contains
5:20:53subclasses having Al of subzero terms
5:20:57the class May of course itself have Al
5:20:59of subzero terms
5:21:02thus we have to prove if we can that
5:21:04given any non-inductive class it is
5:21:08possible to choose a progression out of
5:21:10its
5:21:11terms now there is no difficulty in
5:21:13showing that a non-inductive class must
5:21:16contain more terms than any inductive
5:21:19class or what comes to the same thing
5:21:22that if Alpha is a non-inductive class
5:21:25and new is any inductive number there
5:21:28are subclasses of alpha that have new
5:21:30terms terms thus we can form sets of
5:21:33finite subclasses of alpha first one
5:21:36class having no terms then classes
5:21:39having one term as many as there are
5:21:41members of alpha then classes having two
5:21:44terms and so on WE thus get a
5:21:47progression of sets of subclasses each
5:21:50set consisting of all those that have a
5:21:53given finite number of
5:21:56terms so far we have not used the
5:21:58multiplicative axom but we have only
5:22:01proved that the number of collections of
5:22:03subclasses of Alpha is a reflexive
5:22:06number that is that if mu is the number
5:22:09of members of alpha so that 2 raised to
5:22:12the MU power is the number of subclasses
5:22:14of Alpha and two raised to 2 ra to the
5:22:18MU power is the number of collections of
5:22:22subclasses then provided mu is not
5:22:24inductive two rays to two rays to the MU
5:22:28power must be
5:22:29reflexive but this is a long way from
5:22:32what we set out to
5:22:34prove in order to advance Beyond this
5:22:37point we must employ the multiplicative
5:22:40axom from each set of subclasses let us
5:22:44choose out one omitting the subass
5:22:46consisting of the null class alone that
5:22:49is to say we select one subass
5:22:51containing one term Alpha sub one say
5:22:55one containing two terms Alpha sub 2 say
5:22:58one containing three Alpha sub3 say and
5:23:01so on we can do this if the
5:23:04multiplicative Axiom is assumed
5:23:06otherwise we do not know whether we can
5:23:08always do it or not we have now a
5:23:11progression Alpha sub 1 Alpha sub 2
5:23:14Alpha sub 3 and so on of subclasses of
5:23:18Alpha instead of a progression of
5:23:20collections of
5:23:22subclasses thus We Are One Step nearer
5:23:24to our goal we now know that assuming
5:23:27the multiplicative Axiom if mu is a
5:23:30non-inductive number number 2 raised to
5:23:32the MU power must be a reflexive number
5:23:35the next step is to notice that although
5:23:38we cannot be sure that new members of
5:23:40alpha come in at any one specified stage
5:23:43in the progression Alpha sub one alpha
5:23:46sub 2 Alpha sub3 and so on we can be
5:23:49sure that new members keep on coming in
5:23:51from time to time let us illustrate the
5:23:55class Alpha sub one which consists of
5:23:58one term is a new beginning let the one
5:24:01term be X sub1 the class Alpha sub 2
5:24:05consisting of two terms may or may not
5:24:07contain x sub 1 if it does it introduces
5:24:11one new term if it does not it must
5:24:14introduce two new terms say x sub 2 x
5:24:18sub3 in this case it is possible that
5:24:21Alpha sub3 consists of x sub1 x sub 2 x
5:24:25sub3 and so introduces no new terms but
5:24:29in that case Alpha Al sub 4 must
5:24:31introduce a new term the First new
5:24:35classes of alpha sub one alpha sub 2
5:24:37Alpha sub3 and so on up to Alpha sub new
5:24:41contain at the very most 1 plus 2 plus 3
5:24:45plus so on up to new
5:24:48terms that is the product of new with
5:24:51new + one divided by two terms thus it
5:24:56would be possible if there were no
5:24:58repetitions in the First new classes to
5:25:01go on with only repetitions from the new
5:25:03plus one class to the product of new and
5:25:07new plus one over twoth class but by
5:25:10that time the old terms would no longer
5:25:13be sufficiently numerous to form a next
5:25:15class with the right number of members
5:25:18that is the product of new and new + 1
5:25:22ided by two taken as the sum with one
5:25:26therefore new terms must come in at this
5:25:29point if not sooner
5:25:32it follows that if we omit from our
5:25:34progression Alpha sub 1 alpha sub2 alpha
5:25:37sub3 and so on all those classes that
5:25:40are composed entirely of members that
5:25:42have occurred in previous classes we
5:25:45shall still have a progression let our
5:25:48new progression be called beta sub 1
5:25:50beta sub 2 Beta
5:25:52sub3 and so on we shall have Alpha sub 1
5:25:56is equal to Beta sub 1 and Alpha sub 2
5:25:58is equal to Beta sub 2 because Alpha sub
5:26:02one and Alpha sub 2 must introduce new
5:26:05terms we may or may not have Alpha sub3
5:26:08is equal to Beta sub3 but speaking
5:26:12generally beta sub mu will be Alpha sub
5:26:15new where new is some number greater
5:26:18than mu that is the betas are sum of the
5:26:22alphas now these betas are such that any
5:26:25one of them say beta sub mu contains
5:26:28members which have not occurred in any
5:26:31of the previous betas let gamma sub mu
5:26:34be the part of beta subm which consists
5:26:37of new members thus we get a new
5:26:40progression gamma sub 1 gamma sub 2
5:26:44gamma sub3 and so on again gamma sub 1
5:26:49will be identical with beta sub one and
5:26:51with Alpha sub one if Alpha sub 2 does
5:26:54not contain the one member of alpha sub
5:26:56one we shall have gamma sub 2 is equal
5:27:00to Beta sub 2 which is equal to Alpha
5:27:03sub 2 but if Alpha sub 2 does contain
5:27:06this one member gamma 2 will consist of
5:27:09the other member of alpha sub
5:27:122 this new progression of gamas consists
5:27:15of mutually exclusive classes hence a
5:27:18selection from them will be a
5:27:21progression that is if x sub1 is the
5:27:24member of Y sub One X sub2 is a member
5:27:28of Y sub 2 x sub3 is a member of Y sub3
5:27:33and so on then x sub1 x sub2 x sub3 and
5:27:38so on is a progression and is a subass
5:27:42of
5:27:43alpha assuming the multiplicative axium
5:27:46such a selection can be made thus by
5:27:49twice using this Axiom we can prove that
5:27:53if the Axiom is true every non-inductive
5:27:56Cardinal must be
5:27:58reflexive this could ALS Al be deduced
5:28:01from zella's theorem that if the Axiom
5:28:04is true every class can be well ordered
5:28:08for a well-ordered series must have
5:28:10either a finite or a reflexive number of
5:28:13terms in its
5:28:15field there is one advantage in the
5:28:18above direct argument as against
5:28:20deduction from Zero's theorem that the
5:28:24above argument does not demand the
5:28:26universal truth of the multiplicative
5:28:28Axiom but only its truth truth as
5:28:30applied to a set of olive Subzero
5:28:34classes it may happen that the axium
5:28:36holds for alive Subzero classes though
5:28:40not for a larger number of
5:28:42classes for this reason it is better
5:28:45when it is possible to content ourselves
5:28:48with the more restricted
5:28:50assumption the Assumption made in the
5:28:52above direct argument is that a product
5:28:55of olive Subzero factors is never zero
5:28:59unless one of the factor is
5:29:01zero we may State the Assumption in the
5:29:04form all of subz is a multipliable
5:29:08number where a number new is defined as
5:29:12multipliable when a product of new
5:29:15factors is never zero unless one of the
5:29:18factors is
5:29:20zero we can prove that a finite number
5:29:23is always
5:29:24multipliable but we cannot prove that
5:29:26any infinite number is so the multip
5:29:30multiplicative axom is equivalent to the
5:29:32assumption that all cardinal numbers are
5:29:36multipliable but in order to identify
5:29:38the reflexive with the non-inductive or
5:29:41to deal with the problem of the boots
5:29:43and socks or to show that any
5:29:45progression of numbers of the second
5:29:47class is of the second class we only
5:29:50need the very much smaller assumption
5:29:54that Alf Subzero is
5:29:57multipliable it is not improbable that
5:30:00there is much to be discovered in regard
5:30:02to the topics discussed in the present
5:30:05chapter cases may be found where
5:30:07propositions which seem to involve the
5:30:09multiplicative Axiom can be proved
5:30:12without it it is conceivable that the
5:30:15multiplicative axium in its general form
5:30:18may be shown to be false from this point
5:30:21of view Zero's theorem offers the best
5:30:24hope the Continuum or some still more
5:30:27dense series might be proved to be
5:30:30incapable of having its terms well
5:30:32ordered which would prove the
5:30:34multiplicative axium false in virtue of
5:30:37Zero's
5:30:38theorem but so far no method of
5:30:41obtaining such results has been
5:30:43discovered and the subject remains
5:30:45wrapped in
5:30:47obscurity end of chapter
5:30:5912
5:31:01chapter 13 of introduction to
5:31:03mathematical Philosophy by Bertrand
5:31:06Russell this LibriVox recording is in
5:31:08the public
5:31:10domain the axium of infinity and logical
5:31:15types the axm of infinity is an
5:31:18assumption which may be enunciated as
5:31:21follows if n be any inductive Cardinal
5:31:24number there is at least one class of
5:31:27individuals having n terms
5:31:31if this is true it follows of course
5:31:34that there are many classes of
5:31:36individuals having n terms and that the
5:31:39total number of individuals in the world
5:31:42is not an inductive
5:31:44number for by the aium there is at least
5:31:47one class having n + one terms from
5:31:51which it follows that there are many
5:31:53classes of n terms and that n is not the
5:31:56number of individuals in the world
5:32:00since n is any inductive number it
5:32:03follows that the number of individuals
5:32:05in the world must if R axium be true
5:32:09exceed any inductive number in view of
5:32:12what we found in the preceding chapter
5:32:15about the possibility of cardinals which
5:32:17are neither inductive nor reflexive we
5:32:20cannot infer from our Axiom that there
5:32:23are at least Al if Subzero individuals
5:32:26unless we assume the multiplicative
5:32:29Axiom
5:32:30but we do not know that there are at
5:32:32least Olive Subzero classes of classes
5:32:35since the inductive Cardinals are
5:32:38classes of classes and form a
5:32:40progression if our Axiom is
5:32:42true the way in which the need for this
5:32:45Axiom arises may be explained as
5:32:48follows one of piano's assumptions is
5:32:52that no two inductive Cardinals have the
5:32:53same successor that is that we shall not
5:32:57have M +1 is equal to n + 1 1 unless m
5:33:02equal n if M and N are inductive
5:33:05Cardinals in chapter 8 we had occasion
5:33:08to use what is virtually the same as the
5:33:11above Assumption of Panos namely that if
5:33:14n is an inductive Cardinal n is not
5:33:17equal to n +
5:33:191 it might be thought that this could be
5:33:21proved we can prove that if Alpha is an
5:33:25inductive class and N is the number of
5:33:28members of alpha then then n is not
5:33:30equal to n +
5:33:321 this proposition is easily proved by
5:33:35induction and might be thought to imply
5:33:39the other but in fact it does not since
5:33:42there might be no such class as
5:33:45Alpha what it does imply is this if n is
5:33:49an inductive Cardinal such that there is
5:33:51at least one class having n members then
5:33:55n is not equal to n +
5:33:571 the axum of infinity assures us
5:34:01whether truly or falsely that there are
5:34:04classes having n members and thus
5:34:06enables us to assert that n is not equal
5:34:10to n +
5:34:111 but without this axom we should be
5:34:14left with the possibility that n and n +
5:34:18one might both be the null
5:34:22class let us illustrate this possibility
5:34:25by an
5:34:26example Suppose there were exactly nine
5:34:29in individuals in the world as to what
5:34:32is meant by the word individual I must
5:34:35ask the reader to be
5:34:37patient then the inductive Cardinals
5:34:39from 0 up to 9 would be such as we
5:34:42expect but 10 defined as 9 + 1 would be
5:34:46the null
5:34:48class it will be remembered that n + one
5:34:51may be defined as follows n+1 is the
5:34:55collection of all those classes which
5:34:58have a term X such that when X is taken
5:35:01away there remains a class of n
5:35:04terms now applying this definition we
5:35:07see that in the case supposed 9 + 1 is a
5:35:12class consisting of no classes that is
5:35:14it is the null class the same will be
5:35:17true of 9 + 2 or generally of 9 + n
5:35:21unless n is zero thus 10 and all
5:35:25subsequent inductive Cardinals will all
5:35:28be identical since they will all be the
5:35:30null class in such a case the inductive
5:35:33Cardinals will not form a progression
5:35:36nor will it be true that no two have the
5:35:39same successor for nine and 10 will both
5:35:43be succeeded by the null class 10 being
5:35:47itself the null
5:35:48class it is in order to prevent such
5:35:51arithmetical catastrophes that we
5:35:54require the Axiom of
5:35:57infinity as a matter of fact so so long
5:36:00as we are content with the arithmetic of
5:36:02finite integers we do not introduce
5:36:05either infinite integers or infinite
5:36:08classes or series of finite integers or
5:36:11ratios it is possible to obtain all
5:36:13desired results without the axim of
5:36:16infinity that is to say we can deal with
5:36:19the addition multiplication and
5:36:21exponentiation of finite integers and of
5:36:24ratios but we cannot deal with infinite
5:36:27integers or with irrationals
5:36:30thus the theory of the transfinite and
5:36:32the theory of real numbers fails us how
5:36:36these various results come about must
5:36:39now be
5:36:41explained assuming that the number of
5:36:43individuals in the world is n the number
5:36:46of classes of individuals will be two
5:36:48raised to the N
5:36:50power this is in virtue of the general
5:36:53proposition mentioned in chapter 8 that
5:36:56the number of classes contained in a
5:36:58class which has n members is 2 raised to
5:37:02the N
5:37:03power now 2 raised to the N power is
5:37:06always greater than n hence the number
5:37:09of classes in the world is greater than
5:37:11the number of
5:37:13individuals if now we suppose the number
5:37:16of individuals to be nine as we did just
5:37:19now the number of classes will be 2
5:37:21raised to the 9 power that is
5:37:25512 thus if we take our numbers as being
5:37:28applied to to the counting of classes
5:37:31instead of to the counting of
5:37:33individuals our arithmetic will be
5:37:35normal until we reach
5:37:3852 the first number to be null will be
5:37:42513 and if we advance to classes of
5:37:44classes we shall do still better the
5:37:48number of them will be 2 raised to the
5:37:51512
5:37:52power a number which is so large as to
5:37:55stagger imagination since it has about
5:37:58150 three
5:38:00digits and if we advance to classes of
5:38:03classes of classes we shall obtain a
5:38:06number represented by two raised to a
5:38:09power which has about 153 digits the
5:38:13number of digits in this number will be
5:38:15about 3 times 10 raised to the 152nd
5:38:20power in a time of paper shortage it is
5:38:22undesirable to write out this number and
5:38:25if we want larger ones we can obtain
5:38:28them by traveling further along the
5:38:30logical
5:38:32hierarchy in this way any assigned
5:38:34inductive Cardinal can be made to find
5:38:37its place among numbers which are not
5:38:39null merely by traveling along the
5:38:43hierarchy for a sufficient distance
5:38:46footnote one on this subject see
5:38:48principia Mathematica Volume 2 Star
5:38:51number 120 and following on the
5:38:55corresponding problems as regards ratio
5:38:58see that same work volume 3 star numbers
5:39:02303 and following end of footnote
5:39:07one as regards ratios we have a very
5:39:10similar State of Affairs if a ratio mu
5:39:13over new is to have the expected
5:39:16properties there must be enough objects
5:39:19of whatever sort is being counted to
5:39:21ensure that the null class does not
5:39:24suddenly obtrude
5:39:25itself but this can be ensured for any
5:39:28given ratio mu over new without the
5:39:31Axiom of infinity merely by traveling up
5:39:34the hierarchy a sufficient
5:39:36distance if we cannot succeed by
5:39:39counting individuals we can try counting
5:39:41classes of
5:39:43individuals if we still do not succeed
5:39:45we can try classes of classes and so on
5:39:49ultimately however few individuals there
5:39:52may be in the world we shall reach a
5:39:54stage where there are many more than me
5:39:57objects whatever inductive number Mew
5:40:00may
5:40:01be even if there were no individuals at
5:40:04all this would still be true for there
5:40:06would then be one class namely the null
5:40:10class two classes of classes namely the
5:40:13null class of classes and the class
5:40:15whose only member is the null class of
5:40:18individuals four classes of classes of
5:40:21classes 16 at the next stage
5:40:2665,536 at the next stage and so
5:40:29on thus no such assumption as the axium
5:40:33of infinity is required in order to
5:40:35reach any given ratio or any given
5:40:38inductive
5:40:40Cardinal it is when we wish to deal with
5:40:43the whole class or series of inductive
5:40:45Cardinals or of ratios that the Axiom is
5:40:49required we need the whole class of
5:40:51inductive Cardinals in order to
5:40:53establish the existence of olive Subzero
5:40:56in the whole series in order to
5:40:58establish the consistence of
5:41:00progressions for these results it is
5:41:03necessary that we should be able to make
5:41:05a single class or Series in which no
5:41:08inductive Cardinal is null we need the
5:41:12whole series of ratios in order of
5:41:14magnitude in order to Define real
5:41:16numbers as
5:41:17segments this definition will not give
5:41:20the desired result unless the series of
5:41:22ratios is compact which it cannot be if
5:41:25the total number of ratios at the stage
5:41:28concerned is
5:41:30finite it would be natural to suppose as
5:41:33I supposed myself in former days that by
5:41:36means of constructions such as we have
5:41:39been considering the Axiom of infinity
5:41:41could be proved it may be said let us
5:41:45assume that the number of individuals is
5:41:48n where n may be zero without spoiling
5:41:52our argument then if we form the
5:41:54complete set of individuals classes
5:41:57classes of classes and so on all taken
5:42:01together the number of terms in our
5:42:04whole set will be the sum of N and 2
5:42:08raised to the N power and 2 raised to 2
5:42:11raised to the N power and so on at
5:42:14infinum which is Alf
5:42:18Subzero thus taking all kinds of objects
5:42:21together and not confining ourselves to
5:42:24objects of any one type we shall
5:42:27certainly obtain an infinite class
5:42:29and shall therefore not need the Axiom
5:42:32of infinity so it might be
5:42:35said now before going into this argument
5:42:39the first thing to observe is that there
5:42:41is an air of Hocus Pocus about it
5:42:44something reminds one of The Conjurer
5:42:46who brings things out of a hat the man
5:42:49who has lent his hat is quite sure there
5:42:51wasn't a live rabbit in it before but he
5:42:54is at a loss to say how the rabbit got
5:42:57there so the reader if he has a robust
5:43:00sense of reality will feel convinced
5:43:03that it is impossible to manufacture an
5:43:05infinite collection out of a finite
5:43:08collection of individuals though he may
5:43:11be unable to say where the flaw is in
5:43:14the above
5:43:15construction it would be a mistake to
5:43:18lay too much stress on such feelings of
5:43:20Hocus Pocus like other emotions they may
5:43:23easily lead us
5:43:25astray but they afford a prima fasy
5:43:27ground for scre scrutinizing very
5:43:30closely any argument which arouses
5:43:33them and when the above argument is
5:43:35scrutinized it will in my opinion be
5:43:38found to be
5:43:40facius though the fallacy is a subtle
5:43:43one and by no means easy to avoid
5:43:46consistently the fallacy involved is the
5:43:49fallacy which may be called confusion of
5:43:52types to explain the subject of types
5:43:55fully would require a whole volume more
5:43:59ever it is the purpose of this book to
5:44:01avoid those parts of the subjects which
5:44:03are still obscure and controversial
5:44:05isolating for the convenience of
5:44:07beginners those parts which can be
5:44:09accepted as embodying mathematically
5:44:12ascertained truths now the theory of
5:44:15types emphatically does not belong to
5:44:17the finished and certain part of our
5:44:20subject much of this theory is still
5:44:22inate confused and
5:44:25obscure but the need of some doctrine of
5:44:28types is less doubtful than the precise
5:44:30form the doctrine should take and in
5:44:33connection with the axium of infinity it
5:44:36is particularly easy to see the
5:44:39necessity of some such Doctrine this
5:44:42necessity results for example from the
5:44:45contradiction of the greatest Cardinal
5:44:48we saw in chapter 8 that the number of
5:44:50classes contained in a given class is
5:44:53always greater than the number of
5:44:55members of the class and we inferred
5:44:58that there is no greatest Cardinal
5:45:01number but if we could as we suggested a
5:45:04moment ago add together into one class
5:45:07the individuals classes of individuals
5:45:10classes of classes of individuals and so
5:45:13on we should obtain a class of which its
5:45:17own subclasses would be
5:45:20members the class consisting of all
5:45:23objects that can be counted of whatever
5:45:25sort must if there be such a class have
5:45:28the Cardinal number which is the
5:45:30greatest
5:45:32possible since all its subclasses will
5:45:34be members of it there cannot be more of
5:45:37them than there are members hence we
5:45:40arrive at a
5:45:42contradiction when I first came upon
5:45:45this contradiction in the year
5:45:471901 I attempted to discover some flaw
5:45:50in cantor's proof that there is no
5:45:52greatest Cardinal which we gave in
5:45:54chapter 8 applying this proof to the
5:45:57supposed class of all imaginable objects
5:46:01I was led to a new and simpler
5:46:03contradiction namely the
5:46:06following the comprehensive class we are
5:46:08considering which is to embrace
5:46:10everything must Embrace itself as one of
5:46:13its members in other words if there is
5:46:17such a thing as everything then
5:46:19everything is something and is a member
5:46:22of the class
5:46:24everything but normally a class is not a
5:46:27member of itself man kind for example is
5:46:30not a man form now the assemblage of all
5:46:34classes which are not members of
5:46:37themselves this is a class is it a
5:46:40member of itself or not if it is it is
5:46:44one of those classes that are not
5:46:46members of themselves that is it is not
5:46:49a member of itself if it is not it is
5:46:53not one of those classes that are not
5:46:55members of themselves that is it is is a
5:46:59member of
5:47:00itself thus of the two hypotheses that
5:47:03it is and that it is not a member of
5:47:05itself each implies its
5:47:09contradictory this is a
5:47:11contradiction there is no difficulty in
5:47:14manufacturing similar contradictions at
5:47:17one's Liberty the solution of such
5:47:19contradictions by the theory of types is
5:47:22set forth fully in principia Mathematica
5:47:25footnote 1 volume 1 introduction chapter
5:47:28chapter 2 Star 12 and star 20 Volume 2
5:47:33prefatory statement end of footnote 1
5:47:37and also more briefly in articles by the
5:47:40present author in the American Journal
5:47:42of mathematics footnote one mathematical
5:47:45logic as based on the theory of types
5:47:48volume 30 1908 Pages 222 to
5:47:54262 end of footnote 1 and in the review
5:47:59the metaphysique at the morale footnote
5:48:013 l paradoxes theic 1906 Pages
5:48:07627 to
5:48:09650 end of footnote 2 for the present an
5:48:13outline of the solution must
5:48:16suffice the fallacy consists in the
5:48:19formation of what we may call impure
5:48:21classes that is classes which are not
5:48:24pure as to
5:48:26type as we shall see in a later chapter
5:48:29classes are logical fictions and a
5:48:32statement which appears to be about a
5:48:34class will only be significant if it is
5:48:37capable of translation into a form in
5:48:40which no mention is made of the
5:48:43class this places a limitation upon the
5:48:46ways in which what are nominally though
5:48:48not really names for classes can occur
5:48:52significantly a sentence or set of
5:48:55symbols in which such pseudonames occur
5:48:58in wrong ways is not false but strictly
5:49:01devoid of meaning the supposition that a
5:49:05class is or that it is not a member of
5:49:08itself is meaningless in just this way
5:49:12and more generally to suppose that one
5:49:15class of individuals is a member or is
5:49:18not a member of another class of
5:49:20individuals will be to suppose nonsense
5:49:24and to construct symbolically any class
5:49:27whose members are are not all of the
5:49:29same grade in logical hierarchy is to
5:49:33use symbols in a way which makes them no
5:49:36longer symbolize
5:49:39anything thus if there are n individuals
5:49:42in the world and two rays to the N power
5:49:45classes of individuals we cannot form a
5:49:48new class consisting of both individuals
5:49:51and classes and having the sum of N and
5:49:562 raised to the N power members
5:49:59in this way the attempt to escape from
5:50:01the need for the Axiom of infinity
5:50:03breaks down I do not pretend to have
5:50:06explained the doctrine of types or done
5:50:08more than indicate in rough outline why
5:50:12there is need of such a Doctrine I have
5:50:15aimed only at saying just so much as was
5:50:19required in order to show that we cannot
5:50:22prove the existence of infinite numbers
5:50:25and classes by such conjurers methods as
5:50:28we have been
5:50:30examining there remain however certain
5:50:33other possible methods which must be
5:50:37considered various arguments professing
5:50:40to prove the existence of infinite
5:50:42classes are given in the principles of
5:50:45mathematics section
5:50:47339 page
5:50:50357 in so far as these arguments assume
5:50:54that if n is an inductive Cardinal n is
5:50:57not equal to n +1 they have been already
5:51:01dealt with there is an argument
5:51:04suggested by a passage in Plato's
5:51:06Parmenides to the effect that if there
5:51:09is such a number as one then one has
5:51:13being but one is not identical with
5:51:16being and therefore one and being are
5:51:19two and therefore there is such a number
5:51:22as two and two together with one and
5:51:26being gives a class of three terms and
5:51:29so
5:51:30on this argument is fallacious partly
5:51:33because being is not a term having any
5:51:35definite meaning and still more because
5:51:39if a definite meaning were invented for
5:51:41it it would be found that numbers do not
5:51:43have being they are in fact what are
5:51:46called logical fictions as we shall see
5:51:49when we come to consider the definition
5:51:51of
5:51:53classes the argument that the number of
5:51:56numbers from 0 to n both inclusive is n
5:52:00+1 depends upon the assumption that up
5:52:03to and including n no number is equal to
5:52:07its successor which as we have seen will
5:52:11not be always true if the Axiom of
5:52:14infinity is false it must be understood
5:52:18that the equation n is equal to the sum
5:52:21of N and one which might be true for a
5:52:24finite n if n exceeded the total number
5:52:28of individuals in the world is quite
5:52:31different from the same equation as
5:52:34applied to a reflexive
5:52:36number as applied to a reflexive number
5:52:40it means that given a class of n terms
5:52:43this class is similar to that obtained
5:52:46by adding another term but as applied to
5:52:49a number which is too great for the
5:52:51actual World it merely means that there
5:52:54is no class of individuals and no class
5:52:57of n plus one
5:52:59individuals it does not mean that if we
5:53:02Mount the hierarchy of types
5:53:04sufficiently far to secure the existence
5:53:07of a class of n terms we shall then find
5:53:10this class similar to one of n+ one
5:53:13terms for if n is inductive this will
5:53:17not be the case quite independently of
5:53:20the truth or falsehood of the axium of
5:53:24infinity there is an argument employed
5:53:27by both Bano footnote one
5:53:32Bano 13 end of footnote one and DED
5:53:37footnote 2 DED
5:53:41VIN number 66 end of footnote 2 to prove
5:53:46the existence of reflexive classes the
5:53:50argument in brief is this an object is
5:53:53not identical with the idea of the
5:53:55object but there is at least in the
5:53:58realm of being an idea of any object the
5:54:03relation of an object to the idea of it
5:54:06is one one and ideas are only some among
5:54:11objects hence the relation idea of
5:54:14constitutes a reflection of the whole
5:54:16class of objects into a part of itself
5:54:20namely into that part which consists of
5:54:23ideas accordingly the class of objects
5:54:26and the class of ideas are both both
5:54:29infinite this argument is interesting
5:54:32not only on its own account but because
5:54:35the mistakes in it or what I judge to be
5:54:37mistakes are of a Kind which it is
5:54:40instructive to note the main error
5:54:43consists in assuming that there is an
5:54:45idea of every object it is of course
5:54:48exceedingly difficult to decide what is
5:54:51meant by un idea but let us assume that
5:54:54we
5:54:55know we are then to suppose that
5:54:58starting say with Socrates there is the
5:55:01idea of Socrates and so on ADD
5:55:05infinitum now it is plain that this is
5:55:07not the case in the sense that all these
5:55:10ideas have actual empirical existence in
5:55:13people's minds beyond the third or
5:55:15fourth stage they become
5:55:17mythical if the argument is to be upheld
5:55:21the ideas intended must be platonic
5:55:24ideas laid up in heaven for certainly
5:55:27they are not on Earth but then it at
5:55:30once becomes doubtful whether there are
5:55:33such
5:55:33ideas if we are to know that there are
5:55:37it must be on the basis of some logical
5:55:39Theory proving that it is necessary to a
5:55:43thing that there should be an idea of it
5:55:46we certainly cannot obtain this result
5:55:49empirically or apply it as dedin does to
5:55:53M gonin Vault the world of my thoughts
5:55:58if we were concerned to examine fully
5:56:00the relation of idea and object we
5:56:04should have to enter upon a number of
5:56:06psychological and logical inquiries
5:56:09which are not relevant to our main
5:56:11purpose but a few further points should
5:56:14be noted if idea is to be understood
5:56:17logically it may be identical with the
5:56:20object or it may stand for a description
5:56:24in the sense to be explained in a
5:56:26subsequent chapter
5:56:28in the former case the argument fails
5:56:31because it was essential to the proof of
5:56:33reflexiveness that object and idea
5:56:36should be distinct in the second case
5:56:39the argument also fails because the
5:56:42relation of object and description is
5:56:44not one one there are innumerable
5:56:47correct descriptions of Any Given
5:56:50object Socrates for example may be
5:56:54described as the master of Plato or as
5:56:57the philosopher who drank the hemlock or
5:56:59as the husband of
5:57:01zanthi if to take up the remaining
5:57:04hypothesis idea is to be interpreted
5:57:08psychologically it must be maintained
5:57:10that there is not any one definite
5:57:12psychological entity which could be
5:57:15called the idea of the object there are
5:57:18innumerable beliefs and attitudes Each
5:57:21of which could be called an idea of the
5:57:24object in the sense in which we might
5:57:26say my idea idea of Socrates is quite
5:57:29different from yours but there is not
5:57:32any Central entity except Socrates
5:57:34himself to bind together various ideas
5:57:38of
5:57:39Socrates and thus there is not any such
5:57:42one- one relation of idea and object as
5:57:45the argument
5:57:47supposes nor of course as we have
5:57:50already noted is it true psychologically
5:57:53that there are ideas in however extended
5:57:56a sense in more than a tiny proportion
5:57:59of the things in the world for all these
5:58:02reasons the above argument in favor of
5:58:05the logical existence of reflexive
5:58:07classes must be
5:58:11rejected it might be thought that
5:58:13whatever may be said of logical
5:58:15arguments the empirical arguments
5:58:18derivable from space and time the
5:58:20diversity of colors and so on are quite
5:58:24sufficient to prove the actual existence
5:58:26of an infinite number of
5:58:28particulars I do not believe this we
5:58:31have no reason except Prejudice for
5:58:33believing in the infinite extent of
5:58:36space and time at any rate in the sense
5:58:38in which space and time are physical
5:58:41facts not mathematical fictions we
5:58:44naturally regard space and time as
5:58:46continuous or at least as compact but
5:58:50this again is mainly
5:58:52Prejudice the theory of Quant in physics
5:58:55whether true or false illustrates the
5:58:58fact that physics can never afford proof
5:59:01of continuity though it might quite
5:59:03possibly afford
5:59:05disproof the senses are not sufficiently
5:59:08exact to distinguish between continuous
5:59:10motion and Rapid discreet succession as
5:59:14anyone may discover in a
5:59:16cinema a world in which all motion
5:59:18consisted of a series of small finite
5:59:21jerks would be empirically
5:59:23indistinguishable from one in which
5:59:25motion was continuous
5:59:28it would take up too much space to
5:59:30defend these thesis adequately for the
5:59:33present I am merely suggesting them for
5:59:35the reader
5:59:36consideration if they are valid it
5:59:38follows that there is no empirical
5:59:40reason for believing the number of
5:59:42particulars in the world to be infinite
5:59:45and that there never can be also that
5:59:48there is at present no empirical reason
5:59:51to believe the number to be finite
5:59:54though it is theoretically conceivable
5:59:56that someday there might be evidence
5:59:58pointing though not conclusively in that
6:00:03direction from the fact that the
6:00:05infinite is not self-contradictory but
6:00:07is also not demonstrable logically we
6:00:10must conclude that nothing can be known
6:00:12a prior as to whether the number of
6:00:15things in the world is finite or
6:00:19infinite the conclusion is therefore to
6:00:22adopt a li nitan phraseology that some
6:00:25of the possible worlds are finite some
6:00:28infinite and we have no means of knowing
6:00:31to which of these two kinds our actual
6:00:33world
6:00:34belongs the axim of infinity will be
6:00:37true in some possible worlds and false
6:00:40in others whether it is true or false in
6:00:43this world we cannot
6:00:46tell throughout this chapter the
6:00:48synonyms individual and particular have
6:00:51been used without
6:00:52explanation it would be impossible to
6:00:55explain them adequately without a longer
6:00:57disquisition on the theory of types than
6:01:00would be appropriate to the present work
6:01:03but a few words before we leave this
6:01:04topic may do something to diminish the
6:01:07obscurity which would otherwise envelop
6:01:09the meaning of these
6:01:11words in an ordinary statement we can
6:01:14distinguish a verb expressing an
6:01:16attribute or relation from the
6:01:19substantives which express the subject
6:01:21of the attribute or the terms of the
6:01:24relation Caesar lived ascribes an
6:01:27attribute to Caesar Brutus killed Caesar
6:01:30expresses a relation between Brutus and
6:01:33Caesar using the word subject in a
6:01:36generalized sense we may call both
6:01:39Brutus and Caesar subjects of this
6:01:42proposition the fact that Brutus is
6:01:45grammatically the subject and Caesar
6:01:47object is logically irrelevant since the
6:01:50same occurrence may be expressed in the
6:01:52words Caesar was killed by Brutus where
6:01:55Caesar is the grammatical subject
6:01:58for example we may say killing is a
6:02:00relation which holds between Brutus and
6:02:03Caesar but in such cases the grammar is
6:02:06misleading and in a straightforward
6:02:08statement following the rules that
6:02:10should guide philosophical grammar
6:02:12Brutus and Caesar will appear as
6:02:14subjects and killing as the
6:02:17verb we are thus led to the conception
6:02:19of terms which when they occur in
6:02:22propositions can only occur as subjects
6:02:25and never in any other way
6:02:28this is part of the old Scholastic
6:02:30definition of substance but persistence
6:02:33through time which belonged to that
6:02:35notion forms no part of the notion with
6:02:38which we are concerned we shall Define
6:02:41proper names as those terms which can
6:02:44occur only as subjects in propositions
6:02:48using subject in the extended sense just
6:02:51explained we shall further Define
6:02:53individuals or particulars as the
6:02:56objects that can be named by proper
6:02:58names it would be better to Define them
6:03:00directly rather than by means of the
6:03:03kind of symbols by which they are
6:03:05symbolized but in order to do that we
6:03:08should have to plunge deeper into
6:03:10metaphysics than is desirable here it is
6:03:14of course possible that there is an
6:03:16endless regress that whatever appears as
6:03:19a particular is really on closer
6:03:21scrutiny a class or some kind of complex
6:03:25if this be the case the ACT Max of
6:03:27infinity must of course be true but if
6:03:31it be not the case it must be
6:03:33theoretically possible for analysis to
6:03:35reach ultimate subjects and it is these
6:03:38that give the meaning of particulars or
6:03:42individuals it is to the number of these
6:03:44that the axim of infinity is assumed to
6:03:48apply if it is true of them it is true
6:03:51of classes of them and classes of
6:03:53classes of them and so on similarly if
6:03:57it is false of them it is false
6:04:00throughout this
6:04:01hierarchy hence it is natural to
6:04:04enunciate the Axiom concerning them
6:04:06rather than concerning any other stage
6:04:08in the
6:04:09hierarchy but whether the Axiom is true
6:04:12or false there seems no known method of
6:04:16discovering end of chapter
6:04:2613
6:04:30chapter 14 of introduction to
6:04:33mathematical Philosophy by berand
6:04:36Russell this LibriVox recording is in
6:04:38the public
6:04:40domain incompatibility and the theory of
6:04:45deduction we have now explored somewhat
6:04:48hastily it is true that part of the
6:04:50philosophy of mathematics which does not
6:04:53demand a Critical examination of the
6:04:56idea of class
6:04:58in the preceding chapter however we
6:05:01found ourselves confronted by problems
6:05:03which make such an examination
6:05:06imperative before we can undertake it we
6:05:09must consider certain other parts of the
6:05:11philosophy of mathematics which we have
6:05:14hitherto ignored in a synthetic
6:05:17treatment the parts which we shall now
6:05:19be concerned with come first they are
6:05:22more fundamental than anything that we
6:05:24have discussed hither to three three
6:05:27topics will concern us before we reach
6:05:29the theory of classes namely one the
6:05:32theory of deduction two propositional
6:05:35functions three
6:05:37descriptions of these the third is not
6:05:40logically presupposed in the theory of
6:05:42classes but it is a simpler example of
6:05:45the kind of theory that is needed in
6:05:48dealing with
6:05:49classes it is the first topic the theory
6:05:52of deduction that will concern Us in the
6:05:54present
6:05:56chapter
6:05:57mathematics is a deductive science
6:06:00starting from certain premises it
6:06:02arrives by a strict process of deduction
6:06:06at the various theorems which constitute
6:06:08it it is true that in the past
6:06:11mathematical deductions were often
6:06:13greatly lacking in
6:06:15rigor it is true also that perfect rigor
6:06:19is a scarcely attainable ideal
6:06:22nevertheless in so far as rigor is
6:06:24lacking in a mathematical proof the
6:06:27proof is
6:06:28defective it is no defense to urge that
6:06:31Common Sense shows the result to be
6:06:34correct for if we were to rely upon that
6:06:37it would be better to dispense with
6:06:39argument altogether rather than bring
6:06:42fallacy to the rescue of common sense no
6:06:45appeal to Common Sense or Intuition or
6:06:48anything except strict deductive logic
6:06:51ought to be needed in mathematics after
6:06:53the premises have been laid down Kant
6:06:57having observed that the geometers of
6:06:59his day could not prove their theorems
6:07:01by uned argument but required an appeal
6:07:04to the figure invented a theory of
6:07:07mathematical reasoning According to
6:07:09which the inference is never strictly
6:07:11logical but always requires the support
6:07:14of what is called
6:07:16intuition the whole trend of modern
6:07:19mathematics with its increased pursuit
6:07:21of rigor has been against this
6:07:25Theory the things in the the mathematics
6:07:27of cone which cannot be proved cannot be
6:07:30known for example the Axiom of
6:07:33parallels what can be known in
6:07:35mathematics and by mathematical methods
6:07:38is what can be deduced from Pure
6:07:41logic what else is to belong to human
6:07:44knowledge must be ascertained otherwise
6:07:46empirically through the senses or
6:07:48through experience in some form but not
6:07:51a
6:07:52priori the positive grounds for this
6:07:54thesis are to be found in principial
6:07:57Mathematica pass him a controversial
6:07:59defense of it is given in the principles
6:08:02of mathematics we cannot here do more
6:08:05than refer the reader to those Works
6:08:07since the subject is too vast for Hasty
6:08:10treatment meanwhile we shall assume that
6:08:12all mathematics is deductive and proceed
6:08:15to inquire as to what is involved in
6:08:20deduction in deduction we have one or
6:08:22more propositions called premises from
6:08:25which we infer a prop proposition called
6:08:27the
6:08:28conclusion for our purposes it will be
6:08:31convenient when there are originally
6:08:34several premises to amalgamate them into
6:08:36a single proposition so as to be able to
6:08:40speak of the premise as well as of the
6:08:44conclusion thus we may regard deduction
6:08:47as a process by which we pass from
6:08:50knowledge of a certain proposition the
6:08:52premise to knowledge of a certain other
6:08:55proposition the conclusion
6:08:57IUS but we shall not regard such a
6:09:00process as logical deduction unless it
6:09:03is correct that is unless there is such
6:09:07a relation between premise and
6:09:09conclusion that we have a right to
6:09:11believe the conclusion if we know the
6:09:14premise to be
6:09:15true it is this relation that is chiefly
6:09:18of interest in The Logical theory of
6:09:22deduction in order to be able validly to
6:09:25infer the truth of of a proposition we
6:09:28must know that some other proposition is
6:09:30true and that there is between the two a
6:09:33relation of the sort called implication
6:09:36that is that as we say the premise
6:09:39implies the
6:09:41conclusion we shall Define this relation
6:09:44shortly or we may know that a certain
6:09:47other proposition is false and that
6:09:50there is a relation between the two of
6:09:52the sort called disjunction expressed by
6:09:56P or Q footnote one we shall use the
6:09:59letters P Q R S T to denote variable
6:10:05propositions end of footnote one so that
6:10:08the knowledge that the one is false
6:10:10allows us to infer that the other is
6:10:13true again what we wish to infer may be
6:10:16the falsehood of some proposition not
6:10:18its truth this may be inferred from the
6:10:21truth of another proposition provided we
6:10:24know that the two are incompatible that
6:10:27is that if one is true the other is
6:10:30false it may also be inferred from the
6:10:33falsehood of another proposition in just
6:10:36the same circumstances in which the
6:10:38truth of the other might have been
6:10:40inferred from the truth of the one that
6:10:43is from the falsehood of P we may infer
6:10:46the falsehood of Q when Q implies
6:10:50P all these four are cases of inference
6:10:54when our minds are fixed upon inference
6:10:56it seems natural to take implication as
6:10:59the Primitive fundamental relation since
6:11:02this is the relation which must hold
6:11:04between p and Q if we are to be able to
6:11:07infer the truth of Q from the truth of
6:11:12P but for technical reasons this is not
6:11:15the best primitive idea to
6:11:17choose before proceeding to primitive
6:11:20ideas and definitions let us consider
6:11:23further the various functions of
6:11:25propositions suggested by the above
6:11:28mentioned relations of
6:11:31propositions the simplest of such
6:11:33functions is the negative not P this is
6:11:37that function of P which is true when p
6:11:39is false and false when p is
6:11:42true it is convenient to speak of the
6:11:45truth of a proposition or its falsehood
6:11:48as its truth value footnote 2 this term
6:11:51is due to frga end of footnote
6:11:542 that is truth is the truth value of a
6:11:58true proposition and falsehood of a
6:12:01false one thus not P has the opposite
6:12:05truth value to P we may take next
6:12:09disjunction P or Q This is a function
6:12:13whose truth value is truth when p is
6:12:16true and also when Q is true but is
6:12:19falsehood when both p and Q are
6:12:22false next we may take conjunction p and
6:12:26and Q This has Truth for its truth value
6:12:29when both p and Q are both true
6:12:33otherwise it has falsehood for its truth
6:12:37value take next
6:12:39incompatibility that is p and Q are not
6:12:43both
6:12:44true this is the negation of conjunction
6:12:48it is also the disjunction of the
6:12:50negations of p and Q that is it is not P
6:12:54or not Q
6:12:57its truth value is truth when p is false
6:13:00and likewise when Q is false its truth
6:13:03value is falsehood when p and Q are both
6:13:08true last take implication that is p
6:13:12implies Q or if P then
6:13:16Q this is to be understood in the widest
6:13:19sense that will allow us to infer the
6:13:21truth of Q if we know the truth of P
6:13:25thus we interpret it as has meaning
6:13:27unless p is false Q is true or either p
6:13:31is false or Q is true the fact that
6:13:35implies is capable of other meanings
6:13:38does not concern us this is the meaning
6:13:41which is convenient for
6:13:43us that is to say p implies Q is to mean
6:13:47not P or Q it's truth value is to be
6:13:51truth if p is false likewise if Q is
6:13:54true and is to be falsehood if p is true
6:13:58and Q is
6:14:00false we thus have five functions
6:14:03negation disjunction conjunction
6:14:06incompatibility and
6:14:07implication we might have added others
6:14:10for example joint falsehood not p and
6:14:13not q but the above five will
6:14:17suffice negation differs from the other
6:14:20four in being a function of one
6:14:22proposition whereas the others are
6:14:24functions of two but all five are agreed
6:14:27in this that their truth value depends
6:14:30only upon that of the propositions which
6:14:33are their
6:14:34arguments given the truth or falsehood
6:14:37of P or of p and Q as the case may be we
6:14:41are given the truth or falsehood of the
6:14:43negation disjunction conjunction
6:14:46incompatibility or
6:14:48implication a function of propositions
6:14:50which has this property is called a
6:14:53truth
6:14:55function the whole meaning of a truth
6:14:57function is exhausted by the statement
6:14:59of the circumstances under which it is
6:15:02true or false not P for example is
6:15:06simply that function of P which is true
6:15:08when p is false and false when p is true
6:15:12there is no further meaning to be
6:15:13assigned to it the same applies to P or
6:15:17q and the rest it follows that two truth
6:15:21functions which have the same truth
6:15:22value for all values of the argument are
6:15:25indistinct
6:15:26uable for example p and Q is the
6:15:30negation of not P or not q and vice
6:15:35versa thus either of these may be
6:15:37defined as the negation of the other
6:15:40there is no further meaning in a truth
6:15:42function over and above the conditions
6:15:45under which it is true or
6:15:48false it is clear that the above five
6:15:51truth functions are not all independent
6:15:54we can Define some of them in terms of
6:15:57others there is no great difficulty in
6:15:59reducing the number to two the two
6:16:02chosen in principia Mathematica are
6:16:04negation and
6:16:06disjunction implication is then defined
6:16:08as not P or Q incompatibility as not P
6:16:14or not Q conjunction as the negation of
6:16:18incompatibility but it has been shown by
6:16:20Sheffer footnote one transactions of the
6:16:23American mathematical Society Volume 14
6:16:27pages 481 to
6:16:30488 end of footnote 1 that we can be
6:16:34content with one primitive idea for all
6:16:36five and by theod footnote 2 proceedings
6:16:41of the Cambridge philosophical Society
6:16:43volume 19 number one January 1917 end of
6:16:48footnote 2 that this enables us to
6:16:51reduce the Primitive propositions
6:16:53required in the theory of deduction to
6:16:56two nonformal principles and one formal
6:16:59one for this purpose we may take as our
6:17:03one indefinable either incompatibility
6:17:06or joint falsehood we will choose the
6:17:09former our primitive idea now is a
6:17:12certain truth function called
6:17:15incompatibility which we will denote by
6:17:19PQ negation can be at once defined as
6:17:23the incompatibility of a proposition
6:17:25with itself
6:17:26that is not p is defined as
6:17:31p/p disjunction is the incompatibility
6:17:34of not p and not q that is it is the
6:17:39incompatibility of p with P slash the
6:17:42incompatibility of Q with
6:17:45Q implication is the incompatibility of
6:17:48p and not q that is p/q is incompatible
6:17:53with Q conjunction the negation of
6:17:57incompatibility that is it is the
6:18:01incompatibility of p and Q slash the
6:18:03incompatibility of p and
6:18:06Q thus all our four other functions are
6:18:10defined in terms of
6:18:13incompatibility it is obvious that there
6:18:15is no limit to the manufacturer of Truth
6:18:17functions either by introducing more
6:18:19arguments or by repeating
6:18:22arguments what we are concerned with is
6:18:24the connection of this this subject with
6:18:28inference if we know that P is true and
6:18:31that P implies Q we can proceed to
6:18:34assert
6:18:35Q there is always unavoidably something
6:18:39psychological about inference inference
6:18:42is a method by which we arrive at new
6:18:44knowledge and what is not psychological
6:18:46about it is the relation which allows us
6:18:49to infer correctly but the actual
6:18:52passage from the assertion of P to the
6:18:54assertion of Q is is a psychological
6:18:57process and we must not seek to
6:18:59represent it in purely logical
6:19:02terms in mathematical practice when we
6:19:05infer we have always some expression
6:19:08containing variable propositions say p
6:19:10and Q which is known in virtue of its
6:19:13form to be true for all values of p and
6:19:17Q we have also some other expression
6:19:20part of the former which is also known
6:19:23to be true for all values of p and q and
6:19:27in virtue of the principles of inference
6:19:29we are able to drop this part of our
6:19:31original expression and assert what is
6:19:35left this somewhat abstract account may
6:19:38be made clearer by a few
6:19:41examples let us assume that we know the
6:19:44five formal principles of deduction
6:19:46enumerated in principia
6:19:48Mathematica M niod has reduced these to
6:19:51one but as it is a complicated
6:19:53proposition we will begin with the five
6:19:56these five propositions are as follows
6:20:00one p or P implies P that is if either p
6:20:04is true or p is true then p is
6:20:08true two Q implies P or q that is the
6:20:14disjunction P or Q is true when one of
6:20:17its Alternatives is
6:20:19true three P or Q implies Q or P this
6:20:25would not not be required if we had a
6:20:27theoretically more perfect notation
6:20:29since in the conception of disjunction
6:20:31there is no order involved so that P or
6:20:34q and Q or P should be
6:20:37identical but since our symbols in any
6:20:40convenient form inevitably introduce an
6:20:43order we need suitable assumptions for
6:20:45showing that the order is
6:20:48irrelevant four if either p is true or Q
6:20:52or R is true then either Q is true or P
6:20:57or R is true the twist in this
6:21:00proposition serves to increase its
6:21:02deductive
6:21:04power five if Q implies R then P or Q
6:21:08implies P or
6:21:10R these are the formal principles of
6:21:13deduction employed in principia
6:21:16Mathematica a formal principle of
6:21:18deduction has a double use and it is in
6:21:21order to make this clear that we have
6:21:23cited the above five propositions
6:21:26it has a use as the premise of an
6:21:28inference and a use as establishing the
6:21:31fact that the premise implies the
6:21:34conclusion in the schema of an inference
6:21:36we have a proposition p and a
6:21:39proposition P implies Q from which we
6:21:42infer
6:21:43Q now when we are concerned with the
6:21:46principles of deduction our apparatus of
6:21:49primitive propositions has to yield both
6:21:52the p and the P implies Q of our
6:21:55inference
6:21:56that is to say our rules of deduction
6:21:59are to be used not only as rules which
6:22:02is their use for establishing P implies
6:22:04q but also as substantive premises that
6:22:08is as the P of our
6:22:11schema suppose for example we wish to
6:22:14prove that if P implies Q then if Q
6:22:17implies R it follows that P implies
6:22:20R we have here a relation of three
6:22:23propositions which state implications
6:22:26put P sub 1 is identical to P implies q
6:22:30p sub 2 is identical to Q implies R and
6:22:34P sub3 is identical to P implies
6:22:38R then we have to prove that P sub 1
6:22:41implies that P sub 2 implies P
6:22:44sub3 now take the fifth of our above
6:22:47principles substitute not P for p and
6:22:51remember that not P or Q is by
6:22:54definition the same
6:22:56as P implies Q thus our fifth principle
6:23:00reads if Q implies R then P implies Q
6:23:04implies P implies r that is p sub 2
6:23:08implies that P sub 1 implies P
6:23:12sub3 call this proposition
6:23:15a but the fourth of our principles when
6:23:18we substitute not P not Q for p and Q
6:23:22and remember the definition of
6:23:24implication becomes
6:23:26if P implies that Q implies R then Q
6:23:29implies that P implies R writing P sub 2
6:23:33in place of p p sub 1 in place of Q and
6:23:37P sub3 in place of R this becomes if P
6:23:41sub 2 implies that P sub 1 implies P
6:23:45sub3 then P sub 1 implies that P sub 2
6:23:48implies P sub3 call this
6:23:52B now we proved by means of our fifth
6:23:55principle that P sub 2 implies that P
6:23:58sub 1 implies P sub3 which was what we
6:24:01called
6:24:02a thus we have here an instance of the
6:24:05schema of inference since a represents
6:24:08the P of our scheme and B represents the
6:24:11P implies Q hence we arrive at Q namely
6:24:16P sub 1 implies that P sub 2 implies P
6:24:19sub3 which was the proposition to be
6:24:23proved in this proof the adap ation of
6:24:25our fifth principle which yields a
6:24:28occurs as a substantive premise while
6:24:31the adaptation of our fourth principle
6:24:33which yields B is used to give the form
6:24:36of the inference the formal and material
6:24:40Employments of premises in the theory of
6:24:42deduction are closely intertwined and it
6:24:45is not very important to keep them
6:24:47separated provided we realize that they
6:24:50are in theory
6:24:53distinct the earliest method of AR ring
6:24:55at new results from a premise is one
6:24:58which is Illustrated in the above
6:25:00deduction but which itself can hardly be
6:25:03called
6:25:04deduction the Primitive propositions
6:25:07whatever they may be are to be regarded
6:25:10as asserted for all possible values of
6:25:12the variable propositions P QR which
6:25:16occur in them we may therefore
6:25:18substitute for say p any expression
6:25:22whose value is always a proposition for
6:25:25example not p s implies T and so on by
6:25:31means of such substitutions we really
6:25:33obtain sets of special cases of our
6:25:36original proposition but from a
6:25:38practical point of view we obtain what
6:25:40are virtually new
6:25:43propositions the legitimacy of
6:25:45substitutions of this kind has to be
6:25:47ensured by means of a nonformal
6:25:50principle of
6:25:51inference footnote one no such principle
6:25:55is enunciated in principia Mathematica
6:25:58or in M niod's article mentioned above
6:26:01but this would seem to be an Omission
6:26:04end of footnote
6:26:06one we may now State the one formal
6:26:09principle of inference to which n Cod
6:26:12has reduced the five given above for
6:26:15this purpose we will first show how
6:26:18certain truth functions can be defined
6:26:20in terms of
6:26:22incompatibility we saw already that
6:26:25p is incompatible with q/q means P
6:26:29implies Q we now observe that P is
6:26:33incompatible with
6:26:35q/r means P implies both q and R for
6:26:40this expression means p is incompatible
6:26:42with the incompatibility of Q and R that
6:26:46is p implies that q and R are not
6:26:50incompatible that is p implies that q
6:26:53and R are both true
6:26:56for as we saw the conjunction of Q and R
6:26:59is the negation of their
6:27:02incompatibility observe next that the
6:27:04incompatibility of t with
6:27:07t/t means T implies itself this is a
6:27:12particular case of p is incompatible
6:27:15with
6:27:17q/q let us WR P bar for the negation of
6:27:21P thus the bar of P p/ s will mean the
6:27:27negation of p/ s that is it will mean
6:27:31the conjunction of P and
6:27:33S it follows that the incompatibility of
6:27:38s/q with the bar of
6:27:41PS expresses the
6:27:43incompatibility of
6:27:46s/q with the conjunction of P and S in
6:27:50other words it states that if P and S
6:27:53are both true sash Q is false that is s
6:27:58and Q are both true in still simpler
6:28:02words it states that P and S jointly
6:28:05imply s and Q
6:28:08jointly now put P equals p is
6:28:12incompatible with
6:28:14QR Pi is equal to T is incompatible with
6:28:20t/t Q is equal to the incompatibility of
6:28:25s/q with the bar of
6:28:29p/s then mot's sole formal principle of
6:28:33deduction is the incompatibility of p
6:28:37with
6:28:38PIQ in other words P implies both pi and
6:28:44Q he employs in addition one non-formal
6:28:47principle belonging to the theory of
6:28:49types which need not concern us and one
6:28:53corresponding to the principle that
6:28:55given p and given that p implies Q we
6:28:58can assert Q This principle is if p is
6:29:03incompatible with
6:29:05RQ is true and P is true then Q is
6:29:10true from this apparatus the whole
6:29:13theory of deduction follows except in so
6:29:16far as we are concerned with deduction
6:29:18from or to the existence or the
6:29:21universal truth of propositional
6:29:24functions which we shall consider in the
6:29:26next
6:29:28chapter there is if I am not mistaken a
6:29:31certain confusion in the minds of some
6:29:33authors as to the relation between
6:29:36propositions in virtue of which un
6:29:38inference is
6:29:39valid in order that it may be valid to
6:29:42infer Q from P it is only necessary that
6:29:46P should be true and that the
6:29:48proposition not P or Q should be true
6:29:52whenever this is the case it is clear
6:29:55that Q must be true but inference will
6:29:58only in fact take place when the
6:30:00proposition not P or Q is known
6:30:04otherwise than through knowledge of not
6:30:07P or knowledge of Q whenever p is false
6:30:11not P or Q is true but is useless for
6:30:15inference which requires that P should
6:30:18be
6:30:18true whenever Q is already known to be
6:30:21true not P or Q is of course also also
6:30:25known to be true but is again useless
6:30:28for inference since Q is already known
6:30:31and therefore does not need to be
6:30:33inferred in fact inference only arises
6:30:36when not P or Q can be known without our
6:30:40knowing already which of the two
6:30:42Alternatives it is that makes the
6:30:45disjunction
6:30:46true now the circumstances under which
6:30:49this occurs are those in which certain
6:30:51relations of form exist between p and Q
6:30:55for example we know that if R implies
6:30:58the negation of s then s implies the
6:31:01negation of R between R implies not S
6:31:05and S implies not R there is a formal
6:31:08relation which enables us to know that
6:31:10the first implies the second without
6:31:13having first to know that the first is
6:31:15false or to know that the second is
6:31:19true it is under such circumstances that
6:31:22the relation of implication is
6:31:24practically use useful for drawing
6:31:27inferences but this formal relation is
6:31:29only required in order that we may be
6:31:32able to know that either the premise is
6:31:34false or the conclusion is true it is
6:31:38the truth of not P or q that is required
6:31:41for the validity of the inference what
6:31:44is required further is only required for
6:31:46the Practical feasibility of the
6:31:48inference Professor CI Lewis footnote
6:31:51one see mind volume 21
6:31:551912 Pages 522 to
6:31:59531 and volume 23
6:32:021914 Pages 240 to
6:32:05247 end of footnote 1 has especially
6:32:09studied the narrower formal relation
6:32:12which we may call formal deducibility he
6:32:15urges that the wider relation that
6:32:17expressed by not P or Q should not be
6:32:20called
6:32:21implication that is however a matter of
6:32:24words words provided our use of words is
6:32:26consistent it matters little how we
6:32:29Define them the essential point of
6:32:32difference between the theory which I
6:32:34advocate and the theory advocated by
6:32:36Professor Lewis is this he maintains
6:32:39that when one proposition Q is formally
6:32:42deducible from another P the relation
6:32:45which we perceive between them is one
6:32:48which he calls strict implication which
6:32:51is not the relation expressed by not P
6:32:54or q but a narrower relation holding
6:32:57only when there are certain Formal
6:32:59Connections between p and Q I maintain
6:33:03that whether or not there be such a
6:33:05relation as he speaks of it is in any
6:33:07case one that mathematics does not need
6:33:10and therefore one that on General
6:33:12grounds of economy ought not to be
6:33:15admitted into our apparatus of
6:33:17fundamental
6:33:18Notions that whenever the relation of
6:33:20formal deducibility holds between two
6:33:23propositions it is the case that we can
6:33:25see that either the first is false or
6:33:28the second is true and that nothing
6:33:30Beyond this fact is necessary to be
6:33:32admitted into our premises and that
6:33:36finally the reasons of detail which
6:33:38Professor Lewis produces against the
6:33:40view which I advocate can all be met in
6:33:43detail and depend for their plausibility
6:33:46Upon A covert and unconscious Assumption
6:33:49of the point of view which I reject I
6:33:52conclude therefore that the there is no
6:33:55need to admit as a fundamental notion
6:33:57any form of implication not expressable
6:34:00as a truth
6:34:01function end of chapter
6:34:1014 chapter 15 of introduction to
6:34:13mathematical Philosophy by Bertrand
6:34:16Russell this LibriVox recording is in
6:34:18the public
6:34:20domain propositional
6:34:23functions when in the preceding chapter
6:34:25we were discussing propositions we did
6:34:28not attempt to give a definition of the
6:34:30word proposition but although the word
6:34:32cannot be formally defined it is
6:34:35necessary to say something as to its
6:34:37meaning in order to avoid the very
6:34:39common confusion with propositional
6:34:41functions which are to be the topic of
6:34:43the present
6:34:44chapter we mean by a proposition
6:34:47primarily a form of Words which
6:34:49expresses what is either true or false
6:34:52we say primarily because I do not wish
6:34:55to exclude other than verbal symbols or
6:34:58even mere thoughts if they have a
6:35:00symbolic
6:35:01character but I think the word
6:35:03proposition should be limited to what
6:35:05may in some sense be called symbols and
6:35:08further to such symbols as give
6:35:10expression to truth and
6:35:13falsehood thus two and two are four and
6:35:16two and two are five will be
6:35:18propositions and so will Socrates is a
6:35:21man and Socrates is not a man
6:35:24the statement whatever numbers A and B
6:35:28may be the square of the sum of A and B
6:35:32is equal to the sum of a 2 and 2ab and
6:35:37b^2 is a proposition but the bare
6:35:41formula the square of the sum of A and B
6:35:45is equal to the sum of a 2 and 2 a b and
6:35:49b^2 alone is not since it asserts
6:35:52nothing definite in unless we are
6:35:54further told or LED to suppose that A
6:35:58and B are to have all possible values or
6:36:02are to have such and such values the
6:36:05former of these is tacitly assumed as a
6:36:07rule in the enunciation of mathematical
6:36:10formula which thus become
6:36:12propositions but if no such assumption
6:36:15were made they would be propositional
6:36:17functions a propositional function in
6:36:20fact is an expression containing one or
6:36:23more un determined constituents such
6:36:26that when values are assigned to these
6:36:29constituents the expression becomes a
6:36:32proposition in other words it is a
6:36:35function whose values are
6:36:37propositions but this latter definition
6:36:39must be used with caution a descriptive
6:36:42function for example the hardest
6:36:44proposition in A's mathematical treaties
6:36:47will not be a propositional function
6:36:50although its values are
6:36:52propositions but in such a case the
6:36:54propositions are only described in a
6:36:57propositional function the values must
6:36:59actually enunciate
6:37:02propositions examples of propositional
6:37:04functions are easy to give X as a human
6:37:07is a propositional function so long as X
6:37:11remains undetermined it is neither true
6:37:13nor false but when a value is assigned
6:37:16to X it becomes a true or false
6:37:19proposition any mathematical equation is
6:37:22a propositional function
6:37:25so long as the variables have no
6:37:27definite value the equation is merely an
6:37:30expression awaiting determination in
6:37:32order to become a true or false
6:37:36proposition if it is an equation
6:37:38containing one variable it becomes true
6:37:41when the variable is made equal to a
6:37:44root of the equation otherwise it
6:37:46becomes
6:37:47false but if it is an identity it will
6:37:51be true when the variable is any number
6:37:54the equation to a curve in a line or to
6:37:56a surface in space is a propositional
6:37:59function true for values of the
6:38:01coordinates belonging to points on the
6:38:03curve or Surface false for other
6:38:07values expressions of traditional logic
6:38:09such as all a is B are propositional
6:38:13functions A and B have to be determined
6:38:16as definite classes before such
6:38:19Expressions become true or
6:38:21false the notion of cases or instances
6:38:24depends upon propositional functions
6:38:27consider for example the kind of process
6:38:29suggested by what is called
6:38:32generalization and let us take some very
6:38:34primitive example say lightning is
6:38:37followed by Thunder we have a number of
6:38:40instances of this that is a number of
6:38:43propositions such as this is a flash of
6:38:46lightning and is followed by
6:38:48Thunder what are these occurrences
6:38:51instances of they are instances of the
6:38:54propositional function if x is a flash
6:38:57of lightning X is followed by
6:39:00Thunder the process of generalization
6:39:03with whose validity we are fortunately
6:39:05not concerned consists in passing from a
6:39:08number of such instances to the
6:39:10universal truth of the propositional
6:39:13function if x is a flash of lightning X
6:39:16is followed by
6:39:18Thunder it will be found that in an
6:39:20analogous way propositional functions
6:39:23are always always involved whenever we
6:39:25talk of instances or cases or
6:39:29examples we do not need to ask or
6:39:32attempt to answer the question what is a
6:39:35propositional function a propositional
6:39:38function standing all alone may be taken
6:39:41to be a mere schema a mere shell an
6:39:44empty receptacle for meaning not
6:39:47something already
6:39:49significant we are concerned with
6:39:51propositional functions broadly speaking
6:39:53in two ways
6:39:54first as involved in the Notions true in
6:39:57all cases and true in some cases
6:40:00secondly as involved in the theory of
6:40:02classes and
6:40:03relations the second of these topics we
6:40:06will postpone to a later chapter the
6:40:08first must occupy us
6:40:11now when we say that something is always
6:40:14true or true in all cases it is clear
6:40:17that the something involved cannot be a
6:40:20proposition a proposition is just true
6:40:23or false and there is an end of the
6:40:25matter there are no instances or cases
6:40:28of Socrates is a man or Napoleon died at
6:40:32St
6:40:33Helena these are propositions and it
6:40:35would be meaningless to speak of their
6:40:37being true in all cases this phrase is
6:40:40only applicable to propositional
6:40:44functions take for example the sort of
6:40:46thing that is often said when causation
6:40:48is being discussed we are not concerned
6:40:51with the truth or falsehood of what is
6:40:52said but only with its logical
6:40:55analysis we are told that a is in every
6:40:59instance followed by B now if there are
6:41:02instances of a a must be some general
6:41:05concept of which it is significant to
6:41:07say x sub one is a x sub 2 is a x sub3
6:41:12is a and so on where X sub1 x sub 2 x
6:41:17sub3 are particulars which are not
6:41:20identical with one
6:41:21another this applies for for example to
6:41:24our previous case of
6:41:26lightning we say that lightning a is
6:41:29followed by Thunder B but the separate
6:41:33flashes are particulars not identical
6:41:36but sharing the common property of being
6:41:38lightning the only way of expressing a
6:41:41common property generally is to say that
6:41:44a common property of a number of objects
6:41:47is a propositional function which
6:41:49becomes true when any one of those
6:41:51objects is taken as the value of of the
6:41:55variable in this case all the objects
6:41:57are instances of the truth of the
6:42:00propositional function for a
6:42:03propositional function though it cannot
6:42:05itself be true or false is true in
6:42:08certain instances and false in certain
6:42:11others unless it is always true or
6:42:14always
6:42:16false when to return to our example we
6:42:19say that a is in every instance followed
6:42:22by B we mean that whatever X may be if x
6:42:26is an a it is followed by a b that is we
6:42:31are asserting that a certain
6:42:33propositional function is always
6:42:37true sentences involving such words as
6:42:41all every a the some require
6:42:45propositional functions for their
6:42:48interpretation the way in which
6:42:50propositional functions occur can be
6:42:53explained by means of two of the above
6:42:55words namely all and
6:43:00some there are in the last analysis only
6:43:04two things that can be done with a
6:43:05propositional function one is to assert
6:43:08that it is true in all cases the other
6:43:11is to assert that it is true in at least
6:43:14one case or in some cases as we shall
6:43:17say assuming that there is to be no
6:43:20necessary implication of a plurality of
6:43:22cases
6:43:25all the other uses of propositional
6:43:27functions can be reduced to these
6:43:29two when we say that a propositional
6:43:32function is true in all cases or always
6:43:36as we shall also say without any
6:43:38temporal suggestion we mean that all its
6:43:41values are
6:43:43true if 5x is the function and a is the
6:43:47right sort of object to be an argument
6:43:49to F ofx then 5 of a is to be true
6:43:54however a may have been chosen for
6:43:57example if a is human a is Mortal is
6:44:00true whether a is human or not in fact
6:44:04every proposition of this form is true
6:44:08thus the propositional function if x is
6:44:10human X is Mortal is always true or true
6:44:15in all cases or again the statement
6:44:18there are no unicorns is the same as the
6:44:22statement the propositional function X
6:44:24is not a unicorn is true in all
6:44:28cases the assertions in the preceding
6:44:30chapter about propositions for example P
6:44:33or Q implies Q or P are really
6:44:37assertions that certain propositional
6:44:39functions are true in all cases we do
6:44:43not assert the above principle for
6:44:45example as being true only of this or
6:44:48that particular P or q but as being true
6:44:51of any P or Q Q concerning which it can
6:44:55be made
6:44:57significantly the condition that a
6:44:59function is to be significant for a
6:45:01given argument is the same as the
6:45:03condition that it shall have a value for
6:45:05that argument either true or
6:45:08false the study of the conditions of
6:45:10significance belongs to the doctrine of
6:45:12types which we shall not pursue beyond
6:45:15the sketch given in the preceding
6:45:18chapter not only the principles of
6:45:20deduction but all the Primitive
6:45:22propositions of logic consist of
6:45:25assertions that certain propositional
6:45:27functions are always
6:45:29true if this were not the case they
6:45:32would have to mention particular things
6:45:34or concepts Socrates or redness or east
6:45:37or west or what not and clearly it is
6:45:41not the province of logic to make
6:45:43assertions which are true concerning one
6:45:45thing or concept but not concerning
6:45:48another it is part of the definition of
6:45:51logic but not the whole of its
6:45:52definition that all its propositions are
6:45:55completely General that is they all
6:45:59consist of the assertion that some
6:46:01propositional function containing no
6:46:03constant terms is always
6:46:06true we shall return in our final
6:46:08chapter to the discussion of
6:46:10propositional functions containing no
6:46:12constant
6:46:13terms for the present we will proceed to
6:46:16the other thing that is to be done with
6:46:18a propositional function namely the
6:46:21assertion that it is sometimes true that
6:46:23is true in at least one
6:46:27instance when we say there are men that
6:46:30means that the propositional function X
6:46:32is a man is sometimes true when we say
6:46:36some men are Greeks that means that the
6:46:38propositional function X is a man and a
6:46:41Greek is sometimes true when we say
6:46:45cannibals still exist in Africa that
6:46:48means that the propositional function X
6:46:50is a cannibal now in Africa is sometimes
6:46:54true that is is true for some values of
6:46:58x to say there are at least n
6:47:01individuals in the world is to say that
6:47:04the propositional function Alpha is a
6:47:07class of individuals and a member of the
6:47:09Cardinal number n is sometimes true or
6:47:13as we may say is true for certain values
6:47:16of
6:47:18alpha this form of expression is more
6:47:20convenient when it is necessary to
6:47:22indicate which is the variable
6:47:24constituent which we are taking as the
6:47:26argument to our propositional function
6:47:29for example the above propositional
6:47:32function which we may shorten to Alpha
6:47:35is a class of n individuals contains two
6:47:38variables Alpha and
6:47:40N the axum of infinity in the language
6:47:43of propositional functions is the
6:47:46propositional function if n is an
6:47:48inductive number it is true for some
6:47:51values of alpha that Alpha is a class of
6:47:54n individuals is true for all possible
6:47:58values of n here there is a subordinate
6:48:01function Alpha is a class of n
6:48:04individuals which is said to be in
6:48:06respect of alpha sometimes true and the
6:48:10assertion that this happens if n is an
6:48:12inductive number is said to be in
6:48:15respect of n always
6:48:18true the statement that a function f ofx
6:48:22is always true is the negation of the
6:48:25statement that not f of x is sometimes
6:48:27true and the statement that f of x is
6:48:30sometimes true is the negation of the
6:48:32statement that not f of x is always
6:48:36true thus the statement all men are
6:48:39Immortals is the negation of the
6:48:41statement that the function X is an
6:48:43immortal man is sometimes true and the
6:48:47statement there are unicorns is the
6:48:49negation of the statement that the
6:48:51function X is not a unicorn is always
6:48:54true footnote one the method of
6:48:58deduction is given in principia
6:49:00Mathematica volume one star N9 end of
6:49:03footnote
6:49:04one we say that f of x is never true or
6:49:08always false if not V of X is always
6:49:12true we can if we choose take one of the
6:49:15pair always sometimes as a primitive
6:49:19idea and Define the other by means of
6:49:21the one and
6:49:24negation thus if we choose sometimes as
6:49:27our primitive idea we can Define f of x
6:49:31is always true is to mean it is false
6:49:34that not Fe of X is sometimes
6:49:37true footnote two for linguistic reasons
6:49:41to avoid suggesting either the plural or
6:49:44the singular it is often convenient to
6:49:46say f of x is not always false rather
6:49:51than f of x sometimes or F ofx is
6:49:54sometimes true end of footnote
6:49:582 but for reasons connected with the
6:50:00theory of types it seems more correct to
6:50:03take both always and sometimes as
6:50:06primitive ideas and Define by their
6:50:08means the negations of propositions in
6:50:11which they occur that is to say assuming
6:50:15that we have already defined or adopted
6:50:18as a primitive idea the negation of
6:50:20propositions of the type to which X
6:50:23belongs we Define the negation of f ofx
6:50:27always is not F ofx
6:50:31sometimes and the negation of f ofx
6:50:34sometimes is not F ofx
6:50:38always in like manner we can redefine
6:50:41disjunction and the other truth
6:50:43functions as appli to propositions
6:50:45containing apparent variables in terms
6:50:48of the definitions and primitive ideas
6:50:50for propositions containing no apparent
6:50:54variables propositions containing no
6:50:56apparent variables are called Elementary
6:51:00propositions from these we can mount up
6:51:02step by step using such methods as have
6:51:06just been indicated to the theory of
6:51:08Truth functions as applied to
6:51:10propositions containing 1 2 3 and so on
6:51:14variables or any number up to n where n
6:51:18is any assigned finite
6:51:21number the forms which are taken as
6:51:24simplest in traditional formal logic are
6:51:27really far from being so and all involve
6:51:30the assertion of all values or some
6:51:33values of a compound propositional
6:51:36function take to begin with all s is p
6:51:41we will take it that s is defined by a
6:51:44propositional function f of x and P by a
6:51:47propositional function s of X for
6:51:50example if s is men V of X will be X is
6:51:55human if p is Mortals s of X will be
6:52:00there is a time at which X
6:52:02dies then all S's p means P of X implies
6:52:08s of X is always true it is to be
6:52:11observed that all s is p does not apply
6:52:15only to those terms that actually are
6:52:17s's it says something equally about
6:52:20terms which are not S's suppose we come
6:52:24across an X of which we do not know
6:52:26whether it is an s or not still our
6:52:29statement all s is p tells us something
6:52:33about X namely that if x is an S then X
6:52:37is a
6:52:39p and this is every bit as true when X
6:52:42is not an S as when X is an S if it were
6:52:47not equally true in both cases the
6:52:49reductio ad absurdum would not be a
6:52:52valid method
6:52:53for the essence of this method consists
6:52:56in using implications in cases where as
6:52:59it afterwards turns out the hypothesis
6:53:01is
6:53:02false we may put the matter another way
6:53:06in order to understand all s is p it is
6:53:09not necessary to be able to enumerate
6:53:12what terms are s's provided we know what
6:53:15is meant by being an S and what by being
6:53:19a p we can understand completely what is
6:53:22actually affirmed by all s is
6:53:25p however little we may know of actual
6:53:28instances of
6:53:30either this shows that it is not merely
6:53:33the actual terms that are s's that are
6:53:35relevant in the statement all s is p but
6:53:39all the terms concerning which the
6:53:41supposition that they are s's is
6:53:45significant that is all the terms that
6:53:48are s's together with all the terms that
6:53:50are not s's that that is the whole of
6:53:54the appropriate logical
6:53:56type what applies to statements about
6:53:59all applies also to statements about
6:54:01some there are men for example means
6:54:05that X is human is true for some values
6:54:08of X here all values of X that is all
6:54:12values for which X is human is
6:54:14significant whether true or false are
6:54:17relevant and not only those that in fact
6:54:20are human this becomes obvious if we
6:54:23consider how we could prove such a
6:54:25statement to be
6:54:26false every assertion about all or some
6:54:30thus involves not only the arguments
6:54:32that make a certain function true but
6:54:35all that make it significant that is all
6:54:38for which it has a value at all whether
6:54:41true or
6:54:43false we may now proceed with our
6:54:46interpretation of the traditional forms
6:54:48of the old-fashioned formal logic we
6:54:51assume that s is those terms X for which
6:54:54V of X is true and P is those for which
6:54:58s of X is
6:54:59true as we shall see in a later chapter
6:55:02all classes are derived in this way from
6:55:05propositional
6:55:06functions then all s is p means f of x
6:55:11implies s of X is always true some s is
6:55:15p means f of x and S of X is sometimes
6:55:20true no s is p means means 5x implies
6:55:25not s of X is always
6:55:27true some s is not p means 5x and not SX
6:55:34is sometimes
6:55:36true it will be observed that the
6:55:39propositional functions which are here
6:55:41asserted for all or some values are not
6:55:44f of x and S of X themselves but truth
6:55:47functions of F ofx and S of X for the
6:55:51same argument X
6:55:53the easiest way to conceive of the sort
6:55:56of thing that is intended is to start
6:55:58not from 5x and SX in general but from 5
6:56:02a and side a where a is some
6:56:05constant suppose we are considering all
6:56:08men are mortal we will begin with if
6:56:11Socrates is human Socrates is mortal and
6:56:15then we will regard Socrates as replaced
6:56:18by a variable X wherever Socrates
6:56:21occurs the the object to be secured is
6:56:24that although X remains a variable
6:56:27without any definite value yet it is to
6:56:30have the same value in 5x as in s of X
6:56:35when we are asserting that 5x implies s
6:56:37of X is always
6:56:39true this requires that we shall start
6:56:42with a function whose values are such as
6:56:455 a implies s of a rather than with two
6:56:49separate functions 5 ofx and S of x
6:56:53for if we start with two separate
6:56:55functions we can never secure that the X
6:56:58while remaining undetermined shall have
6:57:00the same value in
6:57:03both for brevity we say 5x always
6:57:06implies s x when we mean that 5x implies
6:57:11s x is always
6:57:13true propositions of the form 5x always
6:57:17implies s of X are called formal
6:57:21implications this name is given equally
6:57:24if there are several
6:57:27variables the above definitions show how
6:57:30far removed from the simplest forms are
6:57:32such propositions as all s is p with
6:57:36which traditional logic
6:57:38begins it is typical of the lack of
6:57:41analysis involved that traditional logic
6:57:44treats all S's p as a proposition of the
6:57:47same form as X is P for example it
6:57:51treats all men are mortal as of the same
6:57:54form as Socrates is
6:57:57Mortal as we have just seen the first is
6:58:00of the form f of x always implies s of X
6:58:04while the second is of the form s of
6:58:07X the emphatic separation of these two
6:58:10forms which was affected by piano and
6:58:13Fraga was a very vital advance in
6:58:16symbolic
6:58:19logic it will be seen that all s is is p
6:58:23and no s is p do not really differ in
6:58:26form except by the substitution of not s
6:58:29of X for S of X and that the same
6:58:32applies to some s is p and some s is not
6:58:36P it should be observed that the
6:58:39traditional rules of conversion are
6:58:41faulty if we adopt the view which is the
6:58:44only technically tolerable one that such
6:58:47propositions as all s is p do not
6:58:50involve the existence of s's that is do
6:58:54not require that there should be terms
6:58:56which are s's the above definitions Le
6:59:00to the result that if 5x is always false
6:59:04that is if there are no s's then all s
6:59:07is p and no s's P will both be true
6:59:11whatever P may
6:59:12be for according to the definition in
6:59:15the last chapter 5x implies SX means not
6:59:215x or s x which is always true if not 5x
6:59:26is always
6:59:28true at the first moment this result
6:59:31might lead the reader to desire
6:59:32different definitions but a little
6:59:35practical experience soon shows that any
6:59:37different definitions would be
6:59:39inconvenient and would conceal the
6:59:41important
6:59:43ideas the proposition 5x always implies
6:59:46s of X and 5x is sometimes true is
6:59:51essentially comp composite and it would
6:59:53be very awkward to give this as the
6:59:55definition of all s is P for them we
6:59:59should have no language left for 5x
7:00:01always implies s of X which is needed
7:00:05100 times for once that the other is
7:00:08needed but with our definitions all s is
7:00:11p does not imply some s is p since the
7:00:15first allows the non-existence of s and
7:00:18the second does
7:00:19not thus conversion per accident ends
7:00:23becomes invalid and some moods of the
7:00:25syllogism are fallacious for example
7:00:29darti all m is s all m is p therefore
7:00:33some s is p which fails if there is no
7:00:38M the notion of existence has several
7:00:42forms one of which will occupy Us in the
7:00:45next chapter but the fundamental form is
7:00:48that which is derived immediately from
7:00:50the notion of sometimes
7:00:52true we say that an argument a satisfies
7:00:56a function V of x if V of a is
7:01:00true this is the same sense in which the
7:01:02roots of an equation are said to satisfy
7:01:05the
7:01:06equation now if f of x is sometimes true
7:01:10we may say there are X's for which it is
7:01:12true or we may say arguments satisfying
7:01:16F ofx
7:01:18exist this is the fundamental meaning of
7:01:20the word existence
7:01:23other meanings are either derived from
7:01:25this or embody mere confusion of thought
7:01:28we may correctly say men exist meaning
7:01:32that X is a man is sometimes true but if
7:01:36we make a pseudo syllogism men exist
7:01:39Socrates is a man therefore Socrates
7:01:42exists we are talking nonsense since
7:01:46Socrates is not like men merely an
7:01:48undetermined argument to a given
7:01:50propositional function
7:01:53the fallacy is closely analogous to that
7:01:56of the argument men are numerous
7:01:58Socrates is a man therefore Socrates is
7:02:02numerous in this case it is obvious that
7:02:05the conclusion is nonsensical but in the
7:02:07case of existence it is not obvious for
7:02:11reasons which will appear more fully in
7:02:13the next
7:02:14chapter for the present let us merely
7:02:17note the fact that though it is correct
7:02:19to say men exist it is incorrect or
7:02:22rather meaningless to ascribe existence
7:02:24to a given particular ex who happens to
7:02:27be a man generally terms satisfying f of
7:02:31x exist means f of x is sometimes true
7:02:36but a exists where a is a term
7:02:38satisfying Fe of X is a mere noise or
7:02:41shape devoid of
7:02:44significance it will be found that by
7:02:46bearing in mind this simple fallacy we
7:02:49can solve many ancient philos opical
7:02:52puzzles concerning the meaning of
7:02:55existence another set of Notions as to
7:02:58which philosophy has allowed itself to
7:03:01fall into hopeless confusions through
7:03:03not sufficiently separating propositions
7:03:06and propositional functions are the
7:03:08Notions of modality necessary possible
7:03:12and
7:03:13impossible sometimes contingent or
7:03:15assertoric is used instead of
7:03:18possible the traditional view was that
7:03:21among true propositions some were
7:03:23necessary While others were merely
7:03:25contingent or oser toic while among
7:03:28false propositions some were impossible
7:03:31namely those whose contradictories were
7:03:33necessary While others merely happened
7:03:36not to be
7:03:38true in fact however there was never any
7:03:41clear account of what was added to Truth
7:03:44by the conception of
7:03:46necessity in the case of propositional
7:03:48functions the three-fold division is
7:03:51obvious
7:03:53if f ofx is an undetermined value of a
7:03:56certain propositional function it will
7:03:58be necessary if the function is always
7:04:01true possible if it is sometimes true
7:04:05and impossible if it is never
7:04:07true this sort of situation arises in
7:04:10regard to probability for example
7:04:14suppose a ball X is drawn from a bag
7:04:16which contains a number of balls if all
7:04:19the balls are white X is white is
7:04:22necessary if some are white it is
7:04:25possible if none it is
7:04:29impossible here all that is known about
7:04:31X is that it satisfies a certain
7:04:34propositional function namely X was a
7:04:37ball in the
7:04:38bag this is a situation which is General
7:04:42in probability problems and not uncommon
7:04:44in Practical life for example when a
7:04:48person calls of whom we know nothing
7:04:50except that he brings a letter of
7:04:52introduction from our friend so and so
7:04:55in all such cases as in regard to
7:04:57modality in general the propositional
7:05:00function is relevant for Clear thinking
7:05:04in many very diverse directions the
7:05:06habit of keeping propositional functions
7:05:09sharply separated from propositions is
7:05:12of the utmost importance and the failure
7:05:14to do so in the past has been a disgrace
7:05:18to
7:05:19philosophy end of chapter
7:05:3315 chapter 16 of introduction to
7:05:36mathematical Philosophy by berand
7:05:39Russell this LibriVox recording is in
7:05:41the public
7:05:43domain
7:05:45descriptions we dealt in the preceding
7:05:47chapter with the words all and some in
7:05:51this chapter we shall consider the word
7:05:53' in the singular and in the next
7:05:56chapter we shall consider the word ' in
7:05:59the plural it may be thought excessive
7:06:02to devote two chapters to one word but
7:06:05to the philosophical mathematician it is
7:06:08a word of very great
7:06:11importance like Browning's Garian with
7:06:13the enclitic day I would give the
7:06:16doctrine of this word if I were dead
7:06:18from the waist down and not merely in a
7:06:20prison
7:06:23we have already had occasion to mention
7:06:25descriptive functions that is such
7:06:28Expressions as the father of X or the
7:06:31sign of X these are to be defined by
7:06:34first defining
7:06:37descriptions a description may be of two
7:06:39sorts definite and indefinite or
7:06:43ambiguous an indefinite description is a
7:06:46phrase of the form a so and so and a
7:06:49definite description is a phrase of the
7:06:51form the so and so in the
7:06:54singular let us begin with the
7:06:57former who did you meet I met a man that
7:07:01is a very indefinite
7:07:03description we are therefore not
7:07:05departing from usage in our terminology
7:07:09our question is what do I really assert
7:07:11when I assert I met a
7:07:14man let us assume for the moment that my
7:07:17assertion is true and that in fact I met
7:07:20Jones
7:07:22it is clear that what I assert is not I
7:07:25met Jones I may say I met a man but it
7:07:28was not Jones in that case though I lie
7:07:32I do not contradict myself as I should
7:07:35do if when I say I met a man I really
7:07:38mean that I met
7:07:40Jones it is clear also that the person
7:07:43to whom I am speaking can understand
7:07:46what I say even if he is a foreigner and
7:07:49has never heard of Jones
7:07:53but we may go further not only Jones but
7:07:56no actual man enters into my statement
7:07:59this becomes obvious when the statement
7:08:01is false since then there is no more
7:08:04reason why Jones should be supposed to
7:08:06enter into the proposition than why
7:08:08anyone else
7:08:10should indeed the statement would remain
7:08:12significant though it could not possibly
7:08:15be true even if there were no man at
7:08:18all I met a unicorn or I met a sea
7:08:21serpent
7:08:22is a perfectly significant assertion if
7:08:24we know what it would be to be a unicorn
7:08:27or a sea serpent that is what is the
7:08:30definition of these fabulous
7:08:33monsters thus it is only what we may
7:08:35call the concept that enters into the
7:08:38proposition in the case of unicorn for
7:08:41example there is only the concept there
7:08:44is not also somewhere among the shades
7:08:47something unreal which may be called a
7:08:50unicorn
7:08:52therefore since it is significant though
7:08:54false to say I met a unicorn it is clear
7:08:57that this proposition rightly analyzed
7:09:00does not contain a constituent a
7:09:03unicorn though it does contain the
7:09:05concept
7:09:07unicorn the question of unreality which
7:09:10confronts us at this stage is a very
7:09:12important one misel by grammar the great
7:09:16majority of those logicians who have
7:09:18dealt with this question have dealt with
7:09:20it on mistake taking lines they have
7:09:23regarded grammatical form as a Sher
7:09:25guide in analysis than in fact it
7:09:28is and they have not known what
7:09:30differences in grammatical form are
7:09:33important I met Jones and I met a man
7:09:36would count traditionally as
7:09:38propositions of the same form but in
7:09:40actual fact they are of quite different
7:09:43forms the first name is an actual person
7:09:46Jones while the second involves a
7:09:48propositional function and becomes when
7:09:50made EXP it the function I met x and x
7:09:54is human is sometimes true it will be
7:09:57remembered that we adopted the
7:09:58convention of using sometimes as not
7:10:01implying more than once this proposition
7:10:05is obviously not of the form I met X
7:10:08which accounts for the existence of the
7:10:10proposition I met a unicorn in spite of
7:10:13the fact that there is no such thing as
7:10:15a
7:10:16unicorn for want of the apparatus of
7:10:19propositional functions many logicians
7:10:22have been driven to the conclusion that
7:10:24there are unreal objects it is argued
7:10:27for example by Minong footnote
7:10:34one 1904 end of footnote one that we can
7:10:39speak about the Golden Mountain the
7:10:41round square and so on we can make true
7:10:45propositions of which these are the
7:10:46subjects hence they must have some kind
7:10:49of logical being since otherwise the
7:10:52propositions in which they occur would
7:10:54be
7:10:54meaningless in such theories it seems to
7:10:57me there is a failure of that feeling
7:11:00for reality which ought to be preserved
7:11:03even in the most abstract
7:11:06studies logic I should maintain must no
7:11:10more admit a unicorn than zoology can
7:11:13for logic is concerned with the real
7:11:15world just as truly as zoology is though
7:11:19with its more abstract and General
7:11:22features to say that unicorns have an
7:11:24existence in heraldry or in literature
7:11:27or in imagination is a most pitiful and
7:11:31poultry
7:11:32evasion what exists in heraldry is not
7:11:35an animal made of Flesh and Blood moving
7:11:38and breathing of its own
7:11:39initiative what exists is a picture or a
7:11:42description in words similarly to
7:11:46maintain that Hamlet for example exists
7:11:48in his own world namely in the world of
7:11:51Shakespeare's imagination just as truly
7:11:54as say Napoleon existed in the Ordinary
7:11:57World is to say something deliberately
7:12:00confusing or else confused to a degree
7:12:03which is scarcely
7:12:05credible there is only one world the
7:12:08real world Shakespeare's imagination is
7:12:11part of it and the thoughts that he had
7:12:13in writing Hamlet are real so are the
7:12:16thoughts that we have in reading the
7:12:18play but it is of the very essence of
7:12:21fiction that only the thoughts feelings
7:12:24and so on in Shakespeare and in his
7:12:26readers are real and that there is not
7:12:29in addition to them an objective
7:12:32Hamlet when you have taken account of
7:12:34all the feelings roused by Napoleon in
7:12:37writers and readers of History you have
7:12:39not touched the actual man but in the
7:12:42case of Hamlet you have come to an end
7:12:44of him if no one thought about Hamlet
7:12:47there would be nothing left of him if no
7:12:49one had thought about Napoleon in he
7:12:51would have soon seen to it that someone
7:12:54did the sense of reality is vital in
7:12:57logic and whoever juggles with it by
7:13:00pretending that Hamlet has another kind
7:13:02of reality is doing a disservice to
7:13:06thought a robust sense of reality is
7:13:10very necessary in framing a correct
7:13:12analysis of propositions about unicorns
7:13:15golden mountains round squares and other
7:13:18such pseudo objects
7:13:22in obedience to the feeling of reality
7:13:25we shall insist that in the analysis of
7:13:28propositions nothing unreal is to be
7:13:31admitted but after all if there is
7:13:34nothing unreal how it may be asked could
7:13:37we admit anything
7:13:38unreal the reply is that in dealing with
7:13:41propositions we are dealing in the first
7:13:44instance with symbols and if we
7:13:46attribute significance to groups of
7:13:48symbols which have no significance we
7:13:50shall fall into the error of admitting
7:13:53unrealities in the only sense in which
7:13:55this is possible namely as objects
7:13:59described in the proposition I met a
7:14:01unicorn the whole four words together
7:14:04make a significant proposition and the
7:14:07word unicorn by itself is significant in
7:14:10just the same sense as the word
7:14:13man but the two words a unicorn do not
7:14:16form a subordinate group having a
7:14:18meaning of its own thus if we falsely
7:14:21attribute meaning to these two words we
7:14:24find ourselves saddled with a unicorn
7:14:27and with the problem how there can be
7:14:29such a thing in a world where there are
7:14:32no
7:14:33unicorns a unicorn is an indefinite
7:14:36description which describes nothing it
7:14:39is not an indefinite description which
7:14:41describes something
7:14:44unreal such a proposition as X is unreal
7:14:47only has meaning when X is a description
7:14:51definite or
7:14:52indefinite in that case the proposition
7:14:54will be true if x is a description which
7:14:57describes
7:14:58nothing but whether the description X
7:15:01describes something or describes nothing
7:15:04it is in any case not a constituent of
7:15:07the proposition in which it occurs like
7:15:10a unicorn just now it is not a
7:15:13subordinate group having a meaning on
7:15:15its own all this results from the fact
7:15:19that when X is a description X is unreal
7:15:22or X does not exist is not nonsense but
7:15:26is always significant and sometimes
7:15:29true we may now proceed to Define
7:15:32generally the meaning of propositions
7:15:35which contain ambiguous
7:15:37descriptions suppose we wish to make
7:15:40some statement about a so and so where
7:15:42so and so are those objects that have a
7:15:45certain property fi that is those
7:15:48objects X for which the proposition
7:15:51function 5x is true for example if we
7:15:54take a man as our instance of a so and
7:15:57so 5x will be X is
7:16:00human let us now wish to assert the
7:16:03property s of a so and so that is we
7:16:07wish to assert that a so and so has that
7:16:10property which X has when s of X is true
7:16:14for example in the case of I met a man s
7:16:18of X will be I met X
7:16:22now the proposition that a so and so has
7:16:24the property s is not a proposition of
7:16:27the form
7:16:29SX if it were a so and so would have to
7:16:32be identical with X for a suitable X and
7:16:36although in a sense this may be true in
7:16:38some cases it is certainly not true in
7:16:41such a case as a
7:16:43unicorn it is just this fact that the
7:16:46statement that a so and so has the
7:16:48property s is not of the form s of X
7:16:52which makes it possible for a so and so
7:16:54to be in a certain clearly definable
7:16:57sense
7:16:58unreal the definition is as
7:17:01follows the statement that an object
7:17:04having the property fi has the property
7:17:06s means the joint assertion of F ofx and
7:17:11S of X is not always
7:17:14false so far as logic goes this is the
7:17:17same proposition as might be expressed
7:17:20by some fi are SI but rhetorically there
7:17:24is a difference because in the one case
7:17:26there is a suggestion of Singularity and
7:17:29in the other case of
7:17:31plurality this however is not the
7:17:33important point the important point is
7:17:36that when rightly analyzed propositions
7:17:40verbally about a so and so are found to
7:17:43contain no constituent represented by
7:17:45this phrase and that is why such
7:17:48propositions can be significant even
7:17:50when there is no such thing as a so and
7:17:54so the definition of existence as
7:17:57applied to ambiguous descriptions
7:18:00results from what was said at the end of
7:18:02the preceding chapter we say that men
7:18:05exist or a man exists if the
7:18:07propositional function X is human is
7:18:10sometimes true and generally a so and so
7:18:14exists if x is so and so is sometimes
7:18:18true we may put this in other languag
7:18:20anguage the proposition Socrates is a
7:18:23man is no doubt equivalent to Socrates
7:18:26is human but it is not the very same
7:18:30proposition the is of Socrates is human
7:18:33expresses the relation of subject and
7:18:35predicate the is of Socrates is a man
7:18:38expresses
7:18:40identity it is a disgrace to the human
7:18:43race that it has chosen to employ the
7:18:46same word is for these two entirely
7:18:49different ideas a disgrace which a
7:18:51symbolical logical language of course
7:18:55remedies the identity in Socrates is a
7:18:58man is identity between an object named
7:19:01accepting Socrates as a name subject to
7:19:03qualifications explained later and an
7:19:06object Ambiguously
7:19:08described an object Ambiguously
7:19:10described will exist when at least one
7:19:13such proposition is true that is when
7:19:16there is at least one true proposition
7:19:19of the form X is a so and so where X is
7:19:22a
7:19:23name it is characteristic of ambiguous
7:19:27as opposed to definite
7:19:29descriptions that there may be any
7:19:31number of true propositions of the above
7:19:33form Socrates is a man Plato is a man
7:19:37and so
7:19:38on thus a man exists follows from
7:19:41Socrates or Plato or anyone else with
7:19:44definite descriptions on the other hand
7:19:47the corresponding form of proposition
7:19:49namely X is the so and so where X is a
7:19:53name can only be true for one value of x
7:19:56at most this brings us to the subject of
7:20:00definite descriptions which are to be
7:20:02defined in a way analogous to that
7:20:05employed for ambiguous descriptions but
7:20:08rather more
7:20:10complicated we come now to the main
7:20:13subject of the present chapter namely
7:20:15the definition of the wordthe in the
7:20:19singular one very important point about
7:20:21the definition of a so and so applies
7:20:24equally to the so and so the definition
7:20:28to be sought is a definition of
7:20:30propositions in which this phrase occurs
7:20:33not a definition of the phrase itself in
7:20:36isolation in the case of ASO and so this
7:20:40is fairly obvious no one could suppose
7:20:43that a man was a definite object which
7:20:46could be defined by
7:20:48itself Socrates is a man Plato is a man
7:20:51Aristotle is a man but we cannot infer
7:20:54that a man means the same as Socrates
7:20:57means and also the same as Plato means
7:21:00and also the same as Aristotle means
7:21:03since these three names have different
7:21:05meanings nevertheless when we have
7:21:08enumerated all the men in the world
7:21:10there is nothing left of which we can
7:21:12say this is a man and not only so but it
7:21:16is the a man the quintessential entity
7:21:19that is just just an indefinite Man
7:21:22without being anybody in
7:21:24particular it is of course quite clear
7:21:27that whatever there is in the world is
7:21:29definite if it is a man it is one
7:21:31definite man and not any other thus
7:21:34there cannot be such an entity as a man
7:21:37to be found in the world as opposed to
7:21:40specific men and accordingly it is
7:21:43natural that we do not Define a man
7:21:46itself but only the propositions in
7:21:48which it occurs
7:21:51in the case of the so and so this is
7:21:53equally true though at First Sight less
7:21:57obvious we may demonstrate that this
7:21:59must be the case by a consideration of
7:22:02the difference between a name and a
7:22:04definite
7:22:05description take the proposition Scott
7:22:07is the author of Waverly we have here a
7:22:11name Scott and a description the author
7:22:13of Waverly which are asserted to apply
7:22:16to the same person the distinction
7:22:19between a man name and all other symbols
7:22:22may be explained as
7:22:24follows a name is a simple symbol whose
7:22:28meaning is something that can only occur
7:22:30as subject that is something of the kind
7:22:34that in Chapter 13 we defined as an
7:22:37individual or a particular and a simple
7:22:41symbol is one which has no parts that
7:22:44are
7:22:45symbols thus Scott is a simple symbol
7:22:49because though it has Parts namely
7:22:51separate letters these parts are not
7:22:54symbols on the other hand the author of
7:22:56Waverly is not a simple symbol because
7:23:00the separate words that compose the
7:23:02phrase are Parts which are
7:23:04symbols if as may be the case whatever
7:23:08seems to be an individual is really
7:23:10capable of further analysis we shall
7:23:13have to content ourselves with what may
7:23:15be called relative individuals which
7:23:18will be terms that throughout the cont
7:23:20context in question are never analyzed
7:23:22and never occur otherwise than as
7:23:25subjects and in that case we shall have
7:23:28correspondingly to content ourselves
7:23:30with relative names from the standpoint
7:23:33of our present problem namely the
7:23:35definition of descriptions this problem
7:23:38whether there are absolute names or only
7:23:40relative names may be ignored since it
7:23:43concerns different stages in the
7:23:45hierarchy of types whereas we have to
7:23:48compare such couples as Scott and the
7:23:51author
7:23:52Waverly which both apply to the same
7:23:54object and do not raise the problem of
7:23:57types we may therefore for the moment
7:24:00treat names as capable of being absolute
7:24:04nothing that we shall have to say will
7:24:06depend upon this assumption but the
7:24:08wording may be a little shortened by
7:24:11it we have then two things to compare
7:24:15one a name which is a simple symbol
7:24:19directly designated ating an individual
7:24:21which is its meaning and having this
7:24:24meaning in its own right independently
7:24:27of the meanings of all other
7:24:29words two a description which consists
7:24:33of several words whose meanings are
7:24:35already fixed and from which results
7:24:38whatever is to be taken as the meaning
7:24:41of the
7:24:43description a proposition containing a
7:24:45description is not identical with what
7:24:48that proposition becomes when a name is
7:24:51substituted even if the name names the
7:24:54same object as the description
7:24:57describes Scott is the author of Waverly
7:25:00is obviously a different proposition
7:25:02from Scott is Scott the first is a fact
7:25:05in literary history the second a trivial
7:25:09truism and if we put anyone other than
7:25:12Scott in place of the author of Waverly
7:25:15our proposition would become false and
7:25:17would therefore certainly no longer be
7:25:20the same
7:25:22proposition but it may be said our
7:25:24proposition is essentially of the same
7:25:26form as say Scott is Sir Walter in which
7:25:30two names are said to apply to the same
7:25:33person the reply is that if Scott is Sir
7:25:37Walter really means the person named
7:25:39Scott is the person named Sir Walter
7:25:42then the names are being used as
7:25:44descriptions that is the individual
7:25:47instead of being named is being
7:25:49described described as the person having
7:25:51that
7:25:52name this is a way in which names are
7:25:55frequently used in practice and there
7:25:57will as a rule be nothing in the
7:25:59phraseology to show whether they are
7:26:02being used in this way or as
7:26:05names when a name is used directly
7:26:08merely to indicate what we are speaking
7:26:11about it is no part of the fact asserted
7:26:14or of the falsehood if our assertion
7:26:16happens to be false it is merely part
7:26:19part of the symbolism by which we
7:26:21express our thought what we want to
7:26:24express is something which might for
7:26:27example be translated into a foreign
7:26:29language it is something for which the
7:26:32actual words are a vehicle but of which
7:26:34they are no
7:26:36part on the other hand when we make a
7:26:39proposition about the person called
7:26:41Scott the actual name Scott enters into
7:26:44what we are asserting and not merely
7:26:46into the language used in making the
7:26:49assertion
7:26:50our proposition will now be a different
7:26:52one if we substitute the person called
7:26:55Sir Walter but so long as we are using
7:26:58names as names whether we say Scott or
7:27:01whether we say Sir Walter is as
7:27:03irrelevant to what we are asserting as
7:27:06whether we speak English or
7:27:08French thus so long as names are used as
7:27:12names Scott is Sir Walter is the same
7:27:15trivial proposition as Scott is Scott
7:27:19this completes the proof that Scott is
7:27:21the author of Waverly is not the same
7:27:23proposition as results from substituting
7:27:26a name for the author of Waverly no
7:27:29matter what name may be
7:27:33substituted when we use a variable and
7:27:36speak of a propositional function f of x
7:27:39say the process of applying General
7:27:41statements about X to particular cases
7:27:45will consist in substituting a name for
7:27:48the letter x assuming that fee is a
7:27:51function which has individuals for its
7:27:54arguments suppose for example that fee
7:27:57of X is always true Let It Be say the
7:28:01law of identity X is identical with
7:28:04X then we may substitute for X any name
7:28:07we choose and we shall obtain a true
7:28:10proposition assuming for the moment that
7:28:13Socrates Plato and Aristotle are names a
7:28:16very rash assumption we can infer from
7:28:19the law of identity that Socrates is
7:28:22Socrates Plato is Plato and Aristotle is
7:28:26Aristotle but we shall commit a fallacy
7:28:29if we attempt to infer without further
7:28:32premises that the author of Waverly is
7:28:35the author of
7:28:36Waverly this results from what we have
7:28:39just proved that if we substitute a name
7:28:42for the author of Waverly in a
7:28:44proposition the proposition we obtain is
7:28:47a different one that is to say
7:28:50applying the result to our present case
7:28:52if x is a name X is identical with X is
7:28:56not the same proposition as the author
7:28:59of Waverly is the author of Waverly no
7:29:02matter what name X may
7:29:05be thus from the fact that all
7:29:07propositions of the form X is identical
7:29:10with X are true we cannot infer without
7:29:14more Ado that the author of Waverly is
7:29:17the author of Waverly in fact
7:29:20propositions of the form the so and so
7:29:23is the so and so are not always true it
7:29:27is necessary that the so and so should
7:29:29exist a term which will be explained
7:29:32shortly it is false that the present
7:29:34King of France is the present King of
7:29:36France or that the round square is the
7:29:39round Square when we substitute a
7:29:42description for a name propositional
7:29:44functions which are always true may
7:29:47become false if the descript describes
7:29:51nothing there is no mystery in this as
7:29:54soon as we realize what was proved in
7:29:56the preceding paragraph that when we
7:29:58substitute a description the result is
7:30:01not a value of the propositional
7:30:03function in
7:30:05question we are now in a position to
7:30:08Define propositions in which a definite
7:30:10description occurs the only thing that
7:30:13distinguishes the so and so from a so
7:30:16and so is the implication of uniqueness
7:30:20we cannot speak of the inhabitant of
7:30:22London because inhabiting London is an
7:30:25attribute which is not unique we cannot
7:30:28speak about the present King of France
7:30:30because there is none but we can speak
7:30:33about the present King of England the
7:30:35propositions about the so and so always
7:30:38imply the corresponding propositions
7:30:40about a so and so with the addendum that
7:30:43there is not more than one so and
7:30:47so such a proposition as Scott is the
7:30:50author of Waverly could not be true if
7:30:53Waverly had never been written or if
7:30:55several people had written it and no
7:30:57more could any other proposition
7:30:59resulting from a propositional function
7:31:01of X by the substitution of the author
7:31:05of Waverly for X we may say that the
7:31:09author of Waverly means the value of x
7:31:12for which X wrote Waverly is true thus
7:31:15the proposition the author of Waverly
7:31:17was Scotch for example
7:31:19involves One X wrote Waverly is not
7:31:23always false two if X and Y wrote
7:31:28Waverly X and Y are identical is always
7:31:32true three if x wrote wly X was Scotch
7:31:37is always
7:31:39true these three propositions translated
7:31:42into ordinary language State one at
7:31:45least one person wrote Waverly two at
7:31:49most one person wrote
7:31:51Waverly three whoever wrote Waverly was
7:31:56Scotch all these three are implied by
7:31:59the author of Waverly was
7:32:02Scotch conversely the three together but
7:32:05no two of them imply that the author of
7:32:08Waverly was
7:32:10Scotch hence the three together may be
7:32:12taken as defining what is meant by the
7:32:15proposition the author of Waverly was
7:32:18scotch
7:32:20we may somewhat simplify these three
7:32:23propositions the first and second
7:32:25together are equivalent to there is a
7:32:29term C such that X wrote Waverly is true
7:32:33when X is C and is false when X is not C
7:32:38in other words there is a term C such
7:32:42that X wrote Waverly is always
7:32:45equivalent to X is C two proposition are
7:32:49equivalent when both are true or both
7:32:52are
7:32:54false we have here to begin with two
7:32:57functions of x x wrote Waverly and X is
7:33:01C and we form a function of C by
7:33:04considering the equivalence of these two
7:33:06functions of X for all values of X we
7:33:10then proceed to assert that the
7:33:12resulting function of C is sometimes
7:33:15true that is that it is true for at at
7:33:19least one value of C it obviously cannot
7:33:22be true for more than one value of
7:33:25C these two conditions together are
7:33:28defined as giving the meaning of the
7:33:30author of Waverly
7:33:33exists we may now Define the term
7:33:35satisfying the function V of X
7:33:38exists this is the general form of which
7:33:41the above is a particular case the
7:33:44author of Waverly is the term satisfying
7:33:47the function X wrote Waverly and the so
7:33:50and so will always involve reference to
7:33:53some propositional function namely that
7:33:56which defines the property that makes a
7:33:58thing so and so our definition is as
7:34:02follows the term satisfying the function
7:34:05f ofx exists means there is a term C
7:34:09such that f ofx is always equivalent to
7:34:13X is
7:34:15C in order to define the author of
7:34:18Waverly was Scott
7:34:19we have still to take account of the
7:34:21third of our three propositions namely
7:34:24whoever wrote Waverly was
7:34:26Scotch this will be satisfied by merely
7:34:29adding that the C in question is to be
7:34:33Scotch thus the author of Waverly was
7:34:35Scotch is there is a term C such that
7:34:40One X wrote Waverly is always equivalent
7:34:43to X is C to C is
7:34:48scotch and generally the term satisfying
7:34:52f of x satisfies s of X is defined as
7:34:56meaning there is a term C such that one
7:35:00f of x is always equivalent to X is c 2
7:35:05s c is
7:35:08true this is the definition of
7:35:10propositions in which descriptions
7:35:14occur it is possible to have much
7:35:17knowledge concerning a term described
7:35:20that is to know many propositions
7:35:21concerning the so and so without
7:35:24actually knowing what the so and so is
7:35:27that is without knowing any proposition
7:35:29of the form X is the so and so where X
7:35:33is a
7:35:34name in a detective story propositions
7:35:38about the man who did the deed are
7:35:40accumulated in the hope that ultimately
7:35:44they will suffice to demonstrate that it
7:35:46was a who did the deed we may even go so
7:35:49far as to say that in all such knowledge
7:35:52as can be expressed in words with the
7:35:55exception of this and that and a few
7:35:57other words of which the meaning varies
7:35:59on different occasions no names in the
7:36:02strict sense occur but what seem like
7:36:05names are really
7:36:07descriptions we may inquire
7:36:09significantly whether Homer existed
7:36:12which we could not do if Homer were a
7:36:15name the proposition though so and so
7:36:18exists is significant whether true or
7:36:21false but if a is the so and so where a
7:36:24is a name the words a exists are
7:36:29meaningless it is only a descriptions
7:36:31definite or indefinite that existence
7:36:34can be significantly asserted for if a
7:36:37is a name it must name something what
7:36:41does not name anything is not a name and
7:36:45therefore if intended to be a name is a
7:36:48symbol a void of meaning whereas a
7:36:51description like the present King of
7:36:53France does not become incapable of
7:36:56occurring significantly merely on the
7:36:59ground that it describes nothing the
7:37:02reason being that it is a complex symbol
7:37:06of which the meaning is derived from
7:37:08that of its constituent
7:37:10symbols and so when we ask whether Homer
7:37:14existed we are using the word Homer as
7:37:17an abbreviated description
7:37:19we may replace it by say the author of
7:37:22The Iliad and the Odyssey the same
7:37:25considerations apply to almost all uses
7:37:29of what look like proper
7:37:31names when descriptions occur in
7:37:34propositions it is necessary to
7:37:36distinguish what may be called primary
7:37:39and secondary
7:37:41occurrences the abstract distinction is
7:37:44as follows a description has a primary
7:37:47occurrence which when the proposition in
7:37:50which it occurs results from
7:37:52substituting the description for X in
7:37:55some propositional function Fe of x a
7:37:59description has a secondary occurrence
7:38:02when the result of substituting the
7:38:04description for X in Fe of X gives only
7:38:08part of the proposition concerned an
7:38:11instance will make this
7:38:12clearer consider the present King of
7:38:15France is
7:38:16bald here the present King of France has
7:38:19a primary occurrence and the proposition
7:38:21is
7:38:22false every proposition in which a
7:38:25description which describes nothing as a
7:38:28primary occurrence is
7:38:30false but now consider the present King
7:38:33of France is not bald this is
7:38:37ambiguous if we are first to take X's
7:38:40bald then substitute the present King of
7:38:43France for x and then deny the result
7:38:46the occurrence of the present King of
7:38:48France
7:38:49is secondary and our proposition is
7:38:52true but if we are to take X is not bald
7:38:56and substitute the present King of
7:38:58France for X then the present King of
7:39:00France has a primary occurrence and the
7:39:03proposition is
7:39:04false confusion of primary and secondary
7:39:08occurrences is a ready source of
7:39:10fallacies where descriptions are
7:39:14concerned descriptions occur in
7:39:16mathematics chiefly in the form of
7:39:18descriptive functions that is the term
7:39:21having the relation R to y or the r of Y
7:39:26as we may say on the analogy of the
7:39:29father of Y and similar
7:39:32phrases to say the father of Y is Rich
7:39:35for example is to say that the following
7:39:37propositional function of C C is Rich
7:39:41and X begat Y is always equivalent to
7:39:44X's C is sometimes true that is is true
7:39:48for at least one value of
7:39:50C it obviously cannot be true for more
7:39:53than one value the theory of
7:39:56descriptions briefly outlined in the
7:39:58present chapter is of the utmost
7:40:01importance both in logic and in theory
7:40:03of knowledge but for purposes of
7:40:06mathematics the more philosophical parts
7:40:09of the theory are not essential and have
7:40:12therefore been omitted in the above
7:40:14account which has confined itself to the
7:40:16barest mathematical requisite
7:40:19isit end of chapter
7:40:3216 chapter 17 of introduction to
7:40:36mathematical Philosophy by bertr and
7:40:38Russell this LibriVox recording is in
7:40:41the public
7:40:43domain
7:40:45classes in the present chapter we shall
7:40:48be concerned with the in the plural the
7:40:52inhabitants of London the sons of rich
7:40:54men and so on in other words we shall be
7:40:57concerned with classes we saw in Chapter
7:41:012 that a cardinal number is to be
7:41:03defined as a class of classes and in
7:41:06chapter 3 that the number one is to be
7:41:09defined as the class of all unit classes
7:41:13that is of all that have just one member
7:41:16as we should say but for the vicious
7:41:20circle of course when the number one is
7:41:22defined as the class of all unit classes
7:41:25unit classes must be defined so as not
7:41:28to assume that we know what is meant by
7:41:31one in fact they are defined in a way
7:41:34closely analogous to that used for
7:41:36descriptions namely a class Alpha is
7:41:40said to be a unit class if the
7:41:42propositional function X is an alpha is
7:41:46always equivalent to X is C regarded as
7:41:50a function of C is not always false that
7:41:54is in more ordinary language if there is
7:41:57a term C such that X will be a member of
7:42:00alpha when X is C but not
7:42:05otherwise this gives us a general
7:42:07definition of a unit class if we already
7:42:09know what a class is in
7:42:12general either two we have in dealing
7:42:14with arithmetic treated class as a
7:42:16primitive idea but for reasons set forth
7:42:19in Chapter 13 if for no others we cannot
7:42:23accept class as a primitive idea we must
7:42:27seek a definition on the same lines as
7:42:30the definition of descriptions that is a
7:42:33definition which will assign a meaning
7:42:35to propositions in whose verbal or
7:42:38symbolic expression words or symbols
7:42:41apparently representing classes occur
7:42:44but which will assign a meaning that
7:42:46altogether eliminates all mention of
7:42:49classes from a right analysis of such
7:42:53propositions we shall then be able to
7:42:55say that the symbols for classes are
7:42:57mere conveniences not representing
7:43:00objects called classes and that classes
7:43:03are in fact like descriptions logical
7:43:06fictions or as we say incomplete
7:43:12symbols the theory of classes is less
7:43:14complete than the theory of descriptions
7:43:17and there are reasons reasons which we
7:43:19shall give in outline for regarding the
7:43:21definition of classes that will be
7:43:24suggested as not finally
7:43:27satisfactory some further subtlety
7:43:29appears to be required but the reasons
7:43:32for regarding the definition which will
7:43:34be offered as being approximately
7:43:36correct and on the right lines are
7:43:39overwhelming the first thing is to
7:43:41realize why classes cannot be regarded
7:43:44as part of the ultimate Furniture of the
7:43:46world it is difficult to explain
7:43:49precisely what one means by this
7:43:51statement but one consequence which it
7:43:53implies may be used to elucidate its
7:43:56meaning if we had a complete symbolic
7:43:59language with a definition for
7:44:01everything definable and an undefined
7:44:03symbol for everything indefinable the
7:44:06undefined symbols in this language would
7:44:08represent symbolically what I mean by
7:44:11the ultimate fature of the world I am
7:44:14maintaining that no symbols either for
7:44:16class in general or or for particular
7:44:19classes would be included in this
7:44:21apparatus of undefined
7:44:23symbols on the other hand all the
7:44:26particular things there are in the world
7:44:28would have to have names which would be
7:44:30included among undefined
7:44:33symbols we might try to avoid this
7:44:35conclusion by the use of descriptions
7:44:37take say the last thing Caesar saw
7:44:40before he died this is a description of
7:44:43some particular we might use it as in
7:44:47one perfectly legitimate sense a
7:44:49definition of that
7:44:51particular but if a is a name of the
7:44:54same particular a proposition in which a
7:44:57occurs is not as we saw in the preceding
7:45:00chapter identical with what this
7:45:02proposition becomes when for a we
7:45:06substitute the last thing Caesar saw
7:45:08before he
7:45:09died if our language does not contain
7:45:12the name a or some other name for the
7:45:15same particular we shall have no means
7:45:17of expressing the proposition which we
7:45:20expressed by means of a as opposed to
7:45:23the one that we expressed by means of
7:45:26the
7:45:27description thus descriptions would not
7:45:29enable a perfect language to dispense
7:45:31with names for all
7:45:33particulars in this respect we are
7:45:35maintaining classes differ from
7:45:37particulars and need not be represented
7:45:40by undefined symbols our first business
7:45:43is to give the reasons for this
7:45:46opinion we have already already seen
7:45:48that classes cannot be regarded as a
7:45:50species of individuals on account of the
7:45:53contradiction about classes which are
7:45:55not members of themselves explained in
7:45:57Chapter 13 and because we can prove that
7:46:01the number of classes is greater than
7:46:03the number of
7:46:05individuals we cannot take classes in
7:46:07the pure extensional way as simply heaps
7:46:10or
7:46:11conglomerations if we were to do that we
7:46:13should find it impossible to understand
7:46:16how there can be such a class as as the
7:46:18null class which has no members at all
7:46:20and cannot be regarded as a heap we
7:46:24should also find it very hard to
7:46:26understand how it comes about that a
7:46:28class which has only one member is not
7:46:30identical with that one member I do not
7:46:33mean to assert or to deny that there are
7:46:36such entities as heaps as a mathematical
7:46:39logician I am not called upon to have an
7:46:41opinion on this point all that I am
7:46:44maintaining is that if there are such
7:46:46things as heaps we cannot identify them
7:46:49with the classes composed of their
7:46:53constituents we shall come much nearer
7:46:55to a satisfactory theory if we try to
7:46:57identify classes with propositional
7:47:00functions every class as we explained in
7:47:02Chapter 2 is defined by some
7:47:04propositional function which is true of
7:47:07the members of the class and false of
7:47:09other things but if a class can be
7:47:12defined by one propositional function it
7:47:15can equally well be defined by any other
7:47:17which is true whenever the first is true
7:47:20and false whenever the first is
7:47:22false for this reason the class cannot
7:47:25be identified with any one such
7:47:27propositional function rather than with
7:47:30any other and given a propositional
7:47:32function there are always many others
7:47:35which are true when it is true and false
7:47:38when it is
7:47:39false we can say that two propositional
7:47:42functions are formally equivalent when
7:47:44this happens two propositions are
7:47:47equivalent when both are true or both
7:47:49false two propositional functions 5x s
7:47:53of X are formally equivalent when FX is
7:47:57always equivalent to S of X it is the
7:48:00fact that there are other functions
7:48:02formally equivalent to a given function
7:48:05that makes it impossible to identify a
7:48:07class with a function for we wish
7:48:10classes to be such that no two distinct
7:48:12classes have exactly the same members
7:48:15and therefore two formally equivalent
7:48:17functions will have to determine the
7:48:19same
7:48:21class when we have decided that classes
7:48:23cannot be things of the same sort as
7:48:26their members that they cannot be just
7:48:28heaps or Aggregates and also that they
7:48:31cannot be identified with propositional
7:48:33functions it becomes very difficult to
7:48:35see what they can be if they are to be
7:48:39more than symbolic
7:48:40fictions and if we can find any way of
7:48:43dealing with them as symbolic fictions
7:48:45we increase the logical security of our
7:48:48position since we avoid the need of
7:48:51assuming that there are classes without
7:48:53being compelled to make the opposite
7:48:55assumption that there are no classes we
7:48:58merely abstain from both
7:49:01assumptions this is an example of aam
7:49:03Razor namely entities are not to be
7:49:06multiplied without
7:49:08necessity but when we refuse to assert
7:49:11that there are classes we must not be
7:49:14supposed to be asserting dogmatically
7:49:16that there are none
7:49:18we are merely agnostic as regards them
7:49:21like place we can
7:49:25say
7:49:27hypoth let us set forth the conditions
7:49:29that a symbol must fulfill if it is to
7:49:32serve as a class I think the following
7:49:35conditions will be found necessary and
7:49:38sufficient one every propositional
7:49:41function must determine a class
7:49:43consisting of those Arguments for which
7:49:45the function is true given any
7:49:48proposition true or false say about
7:49:51Socrates we can imagine Socrates
7:49:54replaced by Plato or Aristotle or a
7:49:57gorilla or the man in the moon or any
7:50:00other individual in the world in general
7:50:03some of these substitutions will give a
7:50:06true proposition and some a false one
7:50:09the class determined will consist of all
7:50:11those substitutions that give a true one
7:50:15of course we have still to decide what
7:50:17we mean by all those which and so on all
7:50:21that we are observing at present is that
7:50:23a class is rendered determinant by a
7:50:26propositional function and that every
7:50:28propositional function determines an
7:50:31appropriate
7:50:32class two formally equivalent
7:50:35propositional functions must determine
7:50:37the same class and two which are not
7:50:39formally equivalent must determine
7:50:41different classes that is a class is
7:50:44determined by its membership and no two
7:50:47different class class can have the same
7:50:49membership if a class is determined by a
7:50:51function f of x we say that a is a
7:50:55member of the class if V OFA is
7:50:58true three we must find some way of
7:51:02defining not only classes but classes of
7:51:05classes we saw in Chapter 2 that
7:51:08cardinal numbers are to be defined as
7:51:10classes of
7:51:11classes the ordinary phrase of
7:51:13elementary mathematics the combinations
7:51:16of N Things M at a time represents a
7:51:19class of classes namely the class of all
7:51:23classes of M terms that can be selected
7:51:25out of a given class of n
7:51:28terms without some symbolic method of
7:51:30dealing with classes of classes
7:51:32mathematical logic would break down four
7:51:37it must under all circumstances be
7:51:39meaningless not false to suppose a Class
7:51:43A Member of itself or not a member of
7:51:45itself this results from the
7:51:47contradiction which we discussed in
7:51:49Chapter
7:51:5113 five lastly and this is the condition
7:51:55which is most difficult of fulfillment
7:51:58it must be possible to make propositions
7:52:01about all the classes that are composed
7:52:03of individuals or about all the classes
7:52:06that are composed of objects of any one
7:52:09logical type if this were not the case
7:52:13many uses of classes would go astray for
7:52:16example Ma mathematical
7:52:18induction in defining the posterity of a
7:52:21given term we need to be able to say
7:52:23that a member of the posterity belongs
7:52:26to all hereditary classes to which the
7:52:29given term
7:52:30belongs and this requires the sort of
7:52:32totality that is in question the reason
7:52:36there is a difficulty about this
7:52:38condition is that it can be proved to be
7:52:40impossible to speak of all the
7:52:43propositional functions that can have
7:52:45arguments of a given type
7:52:48we will to begin with ignore this last
7:52:50condition and the problems which it
7:52:52raises the first two conditions may be
7:52:55taken together they state that there is
7:52:58to be one class no more and no less for
7:53:01each group of formally equivalent
7:53:03propositional functions for example the
7:53:07class of men is to be the same as that
7:53:09of featherless bipeds or rational
7:53:11animals or yahoos or whatever other
7:53:14characteristic may be preferred for
7:53:16defining a human
7:53:18being now when we say that two formally
7:53:21equivalent propositional functions may
7:53:22not be identical although they Define
7:53:25the same class we may prove the truth of
7:53:28the assertion by pointing out that a
7:53:30statement may be true of the one
7:53:31function and false of the other for
7:53:34example I believe that all men are
7:53:36mortal may be true while I believe that
7:53:39all rational animals are mortal may be
7:53:42false since I may believe falsely that
7:53:45the Phoenix is an immortal r
7:53:48animal thus we are led to consider
7:53:50statements about functions or more
7:53:53correctly functions of
7:53:56functions some of the things that may be
7:53:59said about a function may be regarded as
7:54:01said about the class defined by the
7:54:03function whereas others cannot the
7:54:06statement all men are mortal involves
7:54:08the functions X is human and X is Mortal
7:54:12or if we choose we can say that it
7:54:14involves the classes men and
7:54:17Mortals we can interpret the statement
7:54:20in either way because its truth value is
7:54:23unchanged if we substitute for X is
7:54:25human or for X is Mortal any formally
7:54:29equivalent
7:54:30function but as we have just seen the
7:54:33statement I believe that all men are
7:54:35mortal cannot be regarded as being about
7:54:37the class determined by either function
7:54:40because its truth value may be changed
7:54:43by the substitution of a formally
7:54:45equivalent function which leaves the
7:54:47class
7:54:49unchanged we will call a statement
7:54:51involving a function V ofx an
7:54:54extensional function of the function f
7:54:56of x if it is like all men are mortal
7:55:00that is if its truth value is unchanged
7:55:03by the substitution of any formally
7:55:05equivalent
7:55:07function and when a function of a
7:55:09function is not extensional we will call
7:55:12it
7:55:13intentional so that I believe that all
7:55:16men are mortal is is an intentional
7:55:18function of X is human or X is
7:55:22Mortal thus extensional functions of a
7:55:25function X May for practical purposes be
7:55:29regarded as functions of the class
7:55:31determined by X while intentional
7:55:34functions cannot be so
7:55:37regarded it is to be observed that all
7:55:39these specific functions of functions
7:55:42that we have occasion to introduce in
7:55:44mathematical logic are
7:55:46extensional thus for example the two
7:55:49fundamental functions of functions are f
7:55:51of x is always true and f of x is
7:55:55sometimes true each of these has its
7:55:58truth value unchanged if any formally
7:56:01equivalent function is substituted for f
7:56:03of x in the language of classes if Alpha
7:56:07is the class determined by f of x f of x
7:56:10is always true is equivalent to
7:56:13everything is a member of Alpha and f of
7:56:16x is times true is equivalent to Alpha
7:56:19has members or better Alpha has at least
7:56:23one
7:56:24member take again the condition dealt
7:56:27with in the preceding chapter for the
7:56:29existence of the term satisfying f of x
7:56:33the condition is that there is a term C
7:56:36such that f of x is always equivalent to
7:56:40X is
7:56:41C this is obviously extensional it is
7:56:45equivalent to the assertion that the
7:56:47class defined by the function f of x is
7:56:49a unit class that is a class having one
7:56:53member in other words a class which is a
7:56:57member of
7:56:58one given a function of a function which
7:57:02may or may not be extensional we can
7:57:05always derive from it a connected and
7:57:08certainly extensional function of the
7:57:10same function by the following
7:57:13plan let our original function of a
7:57:15function be one which attributes to f of
7:57:18x the property F then consider the
7:57:21assertion there is a function having the
7:57:23property F and formally equivalent to f
7:57:27of
7:57:28x this is an extensional function of f
7:57:31of x it is true when our original
7:57:33statement is true and it is formally
7:57:36equivalent to the original function of F
7:57:38ofx if this original function is
7:57:41extention but when the original function
7:57:44is intentional the new one is more often
7:57:47true than the old
7:57:49one for example consider again I believe
7:57:52that all men are mortal regarded as a
7:57:55function of X is
7:57:58human the derived extensional function
7:58:00is there is a function formally
7:58:03equivalent to X as human and such that I
7:58:06believe that whatever satisfies it is
7:58:09Mortal this remains true when we
7:58:11substitute X is a rational animal for X
7:58:15is human even if I believe falsely that
7:58:17the Phoenix is rational and
7:58:21Immortal we give the name of derived
7:58:23extensional function to the function
7:58:25constructed as above namely to the
7:58:28function there is a function having the
7:58:30property F and formally equivalent to f
7:58:33of x where the original function was the
7:58:36function f of x has the property
7:58:40F we may regard the derived extensional
7:58:43function as having for its argument the
7:58:46class determin by the function f ofx and
7:58:49as asserting F of this
7:58:51class this may be taken as the
7:58:54definition of a proposition about a
7:58:56class that is we may Define to assert
7:59:00that the class determined by the
7:59:02function f ofx has the property f is to
7:59:05assert that F ofx satisfies the
7:59:08extensional function derived from
7:59:11F this gives a meaning to any statement
7:59:14about a class which can be made
7:59:16significantly about a function and it
7:59:18will be found that technically it yields
7:59:21the results which are required in order
7:59:24to make a theory symbolically
7:59:26satisfactory footnote one see principia
7:59:29Mathematica volume 1 Pages 75 to 84 and
7:59:34star 20 end of footnote
7:59:371 what we have said just now as regards
7:59:41the definition of classes is sufficient
7:59:43to satisfy our first four conditions the
7:59:47way in which it secures the third and
7:59:48fourth namely the possibility of classes
7:59:51of classes and the impossibility of a
7:59:54class being or not being a member of
7:59:56itself is somewhat technical it is
7:59:59explained in principia Mathematica but
8:00:01may be taken for granted here it results
8:00:04that but for our fifth condition we
8:00:06might regard our task as completed but
8:00:09this condition at once the most
8:00:11important and the most difficult is not
8:00:14fulfilled in virtue of anything we have
8:00:17said as
8:00:18yet the difficulty is connected with the
8:00:21theory of types and must be briefly
8:00:24discussed footnote 2 the reader who
8:00:27desires a fuller discussion should
8:00:29consult principia Mathematica
8:00:31introduction chapter 2 also star 12 end
8:00:36of footnote
8:00:372 we saw in Chapter 13 that there is a
8:00:41hierarchy of logical types and that it
8:00:44is a fallacy to allow an object
8:00:46belonging to one of these to be
8:00:48substituted for an object belonging to
8:00:50another now it is not difficult to show
8:00:53that the various functions which can
8:00:55take a given object a as argument are
8:00:59not all of one type let us call them all
8:01:02a
8:01:02functions we may take first those among
8:01:05them which do not involve reference to
8:01:07any collection of
8:01:09functions these we will call predicative
8:01:12a functions if we now proceed to
8:01:14functions involving reference to to the
8:01:16totality of predicative a functions we
8:01:19shall incur a fallacy if we regard these
8:01:23as of the same type as the predicative a
8:01:26functions take such an everyday
8:01:29statement as a is a typical Frenchman
8:01:32how shall we Define a typical Frenchman
8:01:35we may Define him as one possessing all
8:01:38qualities that are possessed by most
8:01:40Frenchmen but unless we confine all
8:01:43qualities to such as do not involve pay
8:01:46reference to any totality of qualities
8:01:49we shall have to observe that most
8:01:51Frenchmen are not typical in the above
8:01:54sense and therefore the definition shows
8:01:57that to be not typical is essential to a
8:01:59typical
8:02:01Frenchman this is not a logical
8:02:03contradiction since there is no reason
8:02:05why there should be any typical
8:02:06Frenchmen but it illustrates the need
8:02:09for separating off qualities that
8:02:11involve reference to a totality of
8:02:13qualities from those that do not
8:02:17whenever buy statements about all or
8:02:19some of the values that a variable can
8:02:21significantly take we generate a new
8:02:24object this new object must not be among
8:02:27the values which our previous variable
8:02:29could take since if it were the totality
8:02:32of values over which the variable could
8:02:34range would only be definable in terms
8:02:37of itself and we should be involved in a
8:02:40vicious
8:02:41circle for example if I say Napoleon had
8:02:44all the qualities that make a great
8:02:46General en I must Define qualities in
8:02:49such a way that it will not include what
8:02:52I am now saying that is having all the
8:02:55qualities that make a great General must
8:02:57not be itself a quality in the sense
8:03:00supposed this is fairly obvious and is
8:03:03the principle which leads to the theory
8:03:05of types by which Vicious Circle
8:03:07paradoxes are avoided as applied to a
8:03:11functions we may suppose that qualities
8:03:13is to mean predicative functions then
8:03:17when I say Napoleon had all the
8:03:18qualities and so on I mean Napoleon
8:03:22satisfied all the predicative functions
8:03:24and so
8:03:25on this statement attributes a property
8:03:28to Napoleon but not a predicative
8:03:31property thus we escape the Vicious
8:03:34Circle but wherever all functions which
8:03:37occurs the functions in question must be
8:03:40limited to one type if a vicious circle
8:03:43is to be avoided and as Napoleon and the
8:03:46typical Frenchmen have shown the type is
8:03:48not rendered determinant by that of the
8:03:51argument it would require a much Fuller
8:03:54discussion to set forth this point fully
8:03:57but what has been said May suffice to
8:03:59make it clear that the functions which
8:04:01can take a given argument are of an
8:04:04infinite series of types we could by
8:04:07various technical devices construct a
8:04:10variable which would run through the
8:04:12first n of these types where n is finite
8:04:15but we cannot construct a variable which
8:04:17will run through them all and if we
8:04:20could that mere fact would at once
8:04:22generate a new type of function with the
8:04:24same arguments and would set the whole
8:04:27process going
8:04:29again we call predicative a functions
8:04:32the first type of a functions a
8:04:34functions involving reference to the
8:04:36totality of the first type we call the
8:04:39second type and so on no variable a
8:04:43function can run through all these
8:04:45different types it must stop short at
8:04:47some definite
8:04:50one these considerations are relevant to
8:04:53our definition of the derived
8:04:55extensional function we there spoke of a
8:04:58function formally equivalent to F ofx it
8:05:01is necessary to decide upon the type of
8:05:03our function any decision will do but
8:05:06some decision is
8:05:09unavoidable let us call the supposed
8:05:11formally equivalent function s then s
8:05:15appears as a variable
8:05:17and must be of some determinant
8:05:19type all that we know necessarily about
8:05:22the type of fee is that it takes
8:05:24arguments of a given type that it is say
8:05:28an a function but this as we have just
8:05:31seen does not determine its type if we
8:05:34are to be able as our fifth requisite
8:05:37demands to deal with all classes whose
8:05:40members are of the same type as a we
8:05:43must be able to Define all such classes
8:05:46by by means of functions of some one
8:05:48type that is to say there must be some
8:05:51type of a function say the nth such that
8:05:55any a function is formally equivalent to
8:05:57some a function of the nth
8:06:00type if this is the case then any
8:06:03extensional function which holds of all
8:06:06a functions of the nth type will hold of
8:06:09any a function
8:06:10whatever it is chiefly as a technical
8:06:13means of embodying an assumption leading
8:06:16to to this result that classes are
8:06:19useful the assumption is called the
8:06:21axium of reducibility and may be stated
8:06:24as
8:06:26follows there is a type TA of a
8:06:29functions such that given any a function
8:06:33it is formally equivalent to some
8:06:35function of the type in
8:06:38question if this axium is assumed we use
8:06:42functions of this type in defining our
8:06:44Associated extensional function
8:06:46statements about all a classes that is
8:06:49all classes defined by a functions can
8:06:52be reduced to statements about all a
8:06:54functions of the type
8:06:56to so long as only extensional functions
8:07:00of functions are involved this gives us
8:07:02in practice results which would
8:07:04otherwise have required The Impossible
8:07:06notion of all a functions one particular
8:07:10region where this is vital is
8:07:12mathematical
8:07:14induction the ax of reducibility
8:07:17involves all that is really essential in
8:07:19the theory of classes it is therefore
8:07:22worthwhile to ask whether there is any
8:07:25reason to suppose it
8:07:28true this Axiom like the multiplicative
8:07:31Axiom and the Axiom of infinity is
8:07:34necessary for certain results but not
8:07:36for the bare existence of deductive
8:07:39reasoning the theory of deduction as
8:07:42explained in chapter 14 and the laws for
8:07:45propositions involve in all and some are
8:07:48of the very texture of mathematical
8:07:51reasoning without them or something like
8:07:53them we should not merely not obtain the
8:07:56same results but we should not obtain
8:07:59any results at
8:08:01all we cannot use them as hypotheses and
8:08:04deduce hypothetical consequences for
8:08:07they are rules of deduction as well as
8:08:10premises they must be absolutely true or
8:08:13else what we deduce according to them
8:08:16does not even follow from the
8:08:18premises on the other hand the axium of
8:08:21reducibility like our two previous
8:08:23mathematical axioms could perfectly well
8:08:26be stated as a hypothesis whenever it is
8:08:29used instead of being assumed to be
8:08:32actually true we can deduce its
8:08:35consequences hypothetically we can also
8:08:38deduce the consequences of supposing it
8:08:41false it is therefore only convenient
8:08:44not necessary and in view of the
8:08:46complication of the theory of types and
8:08:50of the uncertainty of all except its
8:08:52most general principles it is impossible
8:08:55as yet to say whether there may not be
8:08:58some way of dispensing with the Axiom of
8:09:01reducibility
8:09:03altogether however assuming the
8:09:05correctness of the theory outlined above
8:09:07what can we say as to the truth or
8:09:10falsehood of the
8:09:12Axiom the Axiom we may observe is a
8:09:16generalized form of light's identity of
8:09:19indiscernibles lighten is assumed as a
8:09:22logical principle that two different
8:09:24subjects must differ as to predicates
8:09:28now predicates are only some among what
8:09:30we called predicative functions which
8:09:33will include also relations to given
8:09:35terms in various properties not to be
8:09:37reckoned as
8:09:39predicates thus Liz's assumption is a
8:09:42much stricter and narrower one than ours
8:09:45not of course according to his logic
8:09:48which regarded all propositions as
8:09:50reducible to the subject predicate
8:09:53form but there is no good reason for
8:09:56believing his form so far as I can
8:09:59see there might quite well as a matter
8:10:02of abstract logical possibility be two
8:10:05things which had exactly the same
8:10:07predicates in the narrow sense in which
8:10:09we have been using the word
8:10:12predicate how does our Axiom look when
8:10:14we pass beyond predicate in this narrow
8:10:17sense in the actual World there seems no
8:10:20way of doubting its empirical truth as
8:10:22regards particulars owing to spacio
8:10:25temporal
8:10:26differentiation no two particulars have
8:10:29exactly the same spatial and temporal
8:10:31relations to all other
8:10:34particulars but this is as it were an
8:10:37accident a fact about the world in which
8:10:39we happen to find ourselves pure logic
8:10:42and pure mathematics which is the same
8:10:45thing aims at being true in linan
8:10:48phraseology in all possible worlds not
8:10:52only this Higgly pigly job blot of a
8:10:54world in which chance has imprisoned us
8:10:58there is a certain lordliness which the
8:11:00logician should preserve he must not
8:11:02condescend to derive arguments from the
8:11:05things he sees about
8:11:07him viewed from this strictly logical
8:11:10point of view I do not see any reason to
8:11:13believe that the axiim of reducibility
8:11:15is logically necessary which is what
8:11:18would be meant by saying that it is true
8:11:20in all possible Worlds the admission of
8:11:24this Axiom into a system of logic is
8:11:27therefore a defect even if the Axiom is
8:11:30empirically
8:11:32true it is for this reason that the
8:11:34theory of classes cannot be regarded as
8:11:37being as complete as the theory of
8:11:40descriptions there is need of further
8:11:42work on the theory of types in the hope
8:11:44of arriving at a doctrine of classes
8:11:46which does not require such a dubious
8:11:50assumption but it is reasonable to
8:11:52regard the theory outlined in the
8:11:54present chapter as right in its main
8:11:56lines that is in its reduction of
8:11:59propositions nominally about classes to
8:12:02propositions about their defining
8:12:04functions the avoidance of classes as
8:12:07entities by this method must it would
8:12:09seem be sound in principle however the
8:12:12detail may still require
8:12:14adjustment it is because this seems
8:12:17indubitable that we have included the
8:12:19theory of classes in spite of our desire
8:12:21to exclude as far as possible whatever
8:12:24seemed open to Serious
8:12:27doubt the theory of classes as above
8:12:30outlined reduces itself to one Axiom and
8:12:33one
8:12:34definition for the sake of definiteness
8:12:37we will here repeat them the axium is
8:12:40there is a type towel such that if Fe is
8:12:43a function which can take a given an
8:12:45object a as argument then there is a
8:12:48function C of the type to which is
8:12:52formally equivalent to F the definition
8:12:55is if V is a function which can take a
8:12:58given object a as argument and to the
8:13:02type mentioned in the above Axiom then
8:13:05to say that the class determined by fee
8:13:07has the property f is to say that there
8:13:10is a function of type too formally
8:13:12equivalent to fee and having the
8:13:15property
8:13:16F end of chapter
8:13:2917 chapter 18 of introduction to
8:13:33mathematical Philosophy by Bertrand
8:13:35Russell this LibriVox recording is in
8:13:38the public
8:13:40domain mathematics and
8:13:43logic mathematics and logic historically
8:13:46speaking have been entirely distinct
8:13:49studies mathematics has been connected
8:13:51with science logic with Greek but both
8:13:55have developed in modern times logic has
8:13:58become more mathematical and Mathematics
8:14:00has become more
8:14:02logical the consequence is that it has
8:14:05now become wholly impossible to draw a
8:14:07line between the two in fact the two are
8:14:10one they differ as boy and man logic is
8:14:14the Youth of math mathematics and
8:14:16Mathematics is the manhood of
8:14:19logic this view is presented by
8:14:22logicians who having spent their time in
8:14:25the study of classical texts are
8:14:27incapable of following a piece of
8:14:29symbolic reasoning and by mathematicians
8:14:33who have learned a technique without
8:14:35troubling to inquire into its meaning or
8:14:39justification both types are now
8:14:41fortunately growing rarer so much of
8:14:44modern mathem iCal work is obviously on
8:14:47the borderline of logic so much of
8:14:49modern logic is symbolic and formal that
8:14:52the very close relationship of logic and
8:14:55Mathematics has become obvious to every
8:14:57instructed
8:14:59student the proof of their identity is
8:15:02of course a matter of
8:15:04detail starting with premises which
8:15:06would be universally admitted to belong
8:15:08to logic and arriving by deduction at
8:15:11results which as obviously belong to
8:15:14mathematics we find that there is no
8:15:16point at which a sharp line can be drawn
8:15:19with logic to the left and Mathematics
8:15:21to the
8:15:22right if there are still those who do
8:15:25not admit the identity of logic in
8:15:27mathematics we may challenge them to
8:15:29indicate at what point in the successive
8:15:32definitions and deductions of principia
8:15:35Mathematica they consider that logic
8:15:37ends and Mathematics begins it will then
8:15:41be obvious that any answer must be quite
8:15:44arbitrary
8:15:46in the earlier chapters of this book
8:15:49starting from the natural numbers we
8:15:51have first defined Cardinal number and
8:15:53shown how to generalize the conception
8:15:55of number and have then analyzed the
8:15:58conceptions involved in the definition
8:16:00until we found ourselves dealing with
8:16:02the fundamentals of logic in a synthetic
8:16:05deductive treatment these fundamentals
8:16:08come first and the natural numbers are
8:16:11only reached after a long
8:16:13journey such treatment though form more
8:16:16correct than that which we have adopted
8:16:19is more difficult for the reader because
8:16:21the ultimate logical Concepts and
8:16:23propositions with which it starts are
8:16:26remote and unfamiliar as compared with
8:16:29the natural
8:16:30numbers also they represent the present
8:16:33Frontier of knowledge Beyond which is
8:16:36the still unknown and the Dominion of
8:16:38knowledge over them is not as yet very
8:16:43secure it used to be said that
8:16:46mathematics is the science of quantity
8:16:49quantity is a vague word but for the
8:16:52sake of argument we may replace it by
8:16:54the word
8:16:55number the statement that mathematics is
8:16:58the science of number would be untrue in
8:17:00two different
8:17:01ways on the one hand there are
8:17:04recognized branches of mathematics which
8:17:06have nothing to do with number all
8:17:09geometry that does not use coordinates
8:17:11or measurement for example projective
8:17:13and descriptive geometry down to the
8:17:15point at which coordinates are
8:17:16introduced does not have to do with
8:17:19number or even with quantity in the
8:17:21sense of greater and less on the other
8:17:24hand through the definition of cardinals
8:17:27through the theory of induction and
8:17:29ancestral relations through the general
8:17:31theory of series and through the
8:17:33definitions of the arithmetical
8:17:35operations it has become possible to
8:17:38generalize much that used to be proved
8:17:41only in connection with
8:17:43numbers the result is that that what was
8:17:45formerly the single study of arithmetic
8:17:48has now become divided into numbers of
8:17:50separate studies no one of which is
8:17:53specially concerned with
8:17:55numbers the most Elementary properties
8:17:57of numbers are concerned with one one
8:17:59relations and similarity between
8:18:02classes addition is concerned with the
8:18:04construction of mutually exclusive
8:18:06classes respectively similar to a set of
8:18:08classes which are not known to be
8:18:10mutually
8:18:12exclusive multiplication is merged in
8:18:14the theory of selections that is of a
8:18:17certain kind of one many relations
8:18:20finitude Is merged in the general study
8:18:22of ancestral relations which yields the
8:18:25whole theory of mathematical
8:18:27induction the ordinal properties of the
8:18:30various kinds of number series and the
8:18:32elements of the theory of continuity of
8:18:35functions and the limits of functions
8:18:38can be generalized so as no longer to
8:18:41involve any essential reference to
8:18:43numbers it is a principle in all formal
8:18:46reasoning to generalize to the utmost
8:18:49since we thereby secure that a given
8:18:51process of deduction shall have more
8:18:54widely applicable
8:18:56results we are therefore in thus
8:18:58generalizing the reasoning of arithmetic
8:19:01merely following a precept which is
8:19:03universally admitted in
8:19:06mathematics and in thus generalizing we
8:19:08have an effect created a set of new
8:19:11deductive systems in which traditional
8:19:13arithmetic is at once dissolved and
8:19:17enlarged but whether any one of these
8:19:19new deductive systems for example the
8:19:22theory of selections is to be said to
8:19:24belong to logic or to arithmetic is
8:19:27entirely arbitrary and incapable of
8:19:30being decided
8:19:33rationally we are thus brought face to
8:19:36face with the question what is this
8:19:38subject which may be called
8:19:41indifferently either mathematics or
8:19:43logic is there any way in which we can
8:19:46Define
8:19:47it certain characteristics of the
8:19:50subject are clear to begin with we do
8:19:52not in this subject deal with particular
8:19:55things or particular properties we deal
8:19:58formally with what can be said about any
8:20:02or any
8:20:03property we are prepared to say that one
8:20:06and one are two but not that Socrates
8:20:09and Plato are two because in our
8:20:12capacity of logicians or pure
8:20:13mathematicians we have never heard of
8:20:16Socrates and
8:20:17Plato a world in which there were no
8:20:20such individuals would still be a world
8:20:23in which one and one are two it is not
8:20:26open to us as pure mathematicians or
8:20:29logicians to mention anything at all
8:20:32because if we do we introduce something
8:20:34irrelevant and not
8:20:36formal we may make this clear by
8:20:38applying it to the case of the
8:20:40syllogism traditional logic says all men
8:20:43are mortal so is a man therefore
8:20:46Socrates is
8:20:48Mortal now it is clear that what we mean
8:20:51to assert to begin with is only that the
8:20:54premises imply the conclusion not that
8:20:57the premises and the conclusion are
8:20:59actually true even the most traditional
8:21:02logic points out that the actual truth
8:21:04of the premises is irrelevant to
8:21:07logic thus the first change to be made
8:21:10in the above traditional syllogism is to
8:21:12State it in the form if all men are
8:21:15mortal and Socrates is a man then
8:21:18Socrates is
8:21:19Mortal we may now observe that it is
8:21:22intended to convey that this argument is
8:21:24valid in virtue of its form not in
8:21:27virtue of the particular terms occurring
8:21:29in
8:21:30it if we had omitted Socrates is a man
8:21:33from our premises we should have had a
8:21:36nonformal argument only admissible
8:21:39because Socrates is in fact a man in
8:21:42that case we could not have General ized
8:21:45the
8:21:45argument but when as above the argument
8:21:48is formal nothing depends upon the terms
8:21:51that occur in
8:21:53it thus we may substitute Alpha for men
8:21:56beta for Mortals X for Socrates where
8:22:00Alpha and beta are any classes whatever
8:22:03and X is any
8:22:05individual we then arrive at the
8:22:07statement No Matter What possible values
8:22:09X and Alpha and beta may have if all
8:22:12Alphas are betas and X is an alpha then
8:22:16X is a beta in other words the
8:22:19propositional function if all Alphas are
8:22:22beta and X is an alpha then X is a beta
8:22:26is always
8:22:27true here at last we have a proposition
8:22:30of logic the one which is only suggested
8:22:33by the traditional statement about
8:22:35Socrates and men and
8:22:38Mortals it is clear that if formal
8:22:41reasoning is what we are aiming at we
8:22:43shall always arrive ultimately at
8:22:45statements like the above in which no
8:22:48actual things or properties are
8:22:51mentioned this will happen through the
8:22:53mere desire not to waste our time
8:22:56proving in a particular case what can be
8:22:58proved generally it would be ridiculous
8:23:01to go through a long argument about
8:23:03Socrates and then go through precisely
8:23:06the same argument again about
8:23:08Plato if our argument is one say which
8:23:11holds of all men we shall prove it
8:23:14concerning X x with the hypothesis if x
8:23:16is a
8:23:17man with this hypothesis the argument
8:23:20will retain its hypothetical validity
8:23:22even when X is not a
8:23:25man but now we shall find that our
8:23:27argument would still be valid If instead
8:23:30of supposing x to be a man we were
8:23:32supposed him to be a monkey or a goose
8:23:35or a prime
8:23:36minister we shall therefore not waste
8:23:38our time taking as our premise X as a
8:23:41man but shall take X as an alpha where
8:23:45Alpha is any class of individuals or f
8:23:48of x where Fe is any propositional
8:23:51function of some assigned
8:23:53type thus the absence of all mention of
8:23:56particular things or properties in logic
8:23:58or pure mathematics is a necessary
8:24:01result of the fact that this study is as
8:24:04we say purely
8:24:06formal at this point we find ourselves
8:24:09faced with a problem which is easier to
8:24:12State than to solve the problem is is
8:24:15what are the constituents of a logical
8:24:18proposition I do not know the answer but
8:24:20I propose to explain how the problem
8:24:25arises take say the proposition Socrates
8:24:28was before
8:24:30Aristotle here it seems obvious that we
8:24:32have a relation between two terms and
8:24:35that the constituents of the proposition
8:24:37as well as of the corresponding fact are
8:24:40simply the two terms and the relation
8:24:43that is so rates Aristotle and
8:24:46before I ignore the fact that Socrates
8:24:49and Aristotle are not simple also the
8:24:52fact that what appear to be their names
8:24:54are really truncated descriptions
8:24:56neither of these facts is relevant to
8:24:58the present
8:25:00issue we may represent the general form
8:25:02of such propositions by XR Y which may
8:25:06be read X has the relation R to
8:25:10Y this general form may occur in logical
8:25:13propositions but no particular instance
8:25:16of it can occur are we to infer that the
8:25:19general form itself is a constituent of
8:25:21such logical
8:25:23propositions given a proposition such as
8:25:26Socrates is before Aristotle we have
8:25:29certain constituents and also a certain
8:25:32form but the form is not itself a new
8:25:35constituent if it were we should need a
8:25:38new form to embrace both it and the
8:25:40other
8:25:42constituents we can in fact turn all the
8:25:44constituents of a proposition into
8:25:46variables while keeping the form
8:25:50unchanged this is what we do when we use
8:25:52such a schema as XR Y which stands for
8:25:56any one of a certain class of
8:25:58propositions namely those asserting
8:26:00relations between two
8:26:03terms we can proceed to General
8:26:05assertions such as XR Y is sometimes
8:26:09true that is there are cases where dual
8:26:12relations hold
8:26:15this assertion will belong to logic or
8:26:17mathematics in the sense in which we are
8:26:20using the word but in this assertion we
8:26:22do not mention any particular things or
8:26:25particular
8:26:26relations no particular things or
8:26:29relations can ever enter into a
8:26:32proposition of pure logic we are left
8:26:34with pure forms as the only possible
8:26:37constituents of logical
8:26:39propositions I do not wish to assert
8:26:42positively that pure forms for for
8:26:44example the form XR y do actually enter
8:26:49into propositions of the kind we are
8:26:51considering the question of the analysis
8:26:53of such propositions is a difficult one
8:26:56with conflicting considerations on the
8:26:58one side and on the other we cannot
8:27:01Embark upon this question now but we may
8:27:04accept as a first
8:27:06approximation The View that forms are
8:27:09what enter into logical propositions as
8:27:11their
8:27:12constituents and we may explain though
8:27:15not formally defined what we mean by the
8:27:17form of a proposition as
8:27:20follows the form of a proposition is
8:27:24that in it that remains unchanged when
8:27:27every constituent of the proposition is
8:27:29replaced by
8:27:31another thus Socrates is earlier than
8:27:34Aristotle has the same form as Napoleon
8:27:37is greater than Wellington though every
8:27:40constituent of the two propositions is
8:27:43different we may thus lay down as a
8:27:46necessary though not sufficient
8:27:48characteristic of logical or
8:27:50mathematical propositions that they are
8:27:52to be such as can be obtained from a
8:27:55proposition containing no variables that
8:27:58is no such words as all Su a the and so
8:28:03on by turning every constituent into a
8:28:06variable and asserting that the result
8:28:08is always true or sometimes true or that
8:28:11it is always true in respect of some of
8:28:13the variable Ables that the result is
8:28:16sometimes true in respect of the others
8:28:18or any variant of these
8:28:20forms and another way of stating the
8:28:23same thing is to say that logic or
8:28:26mathematics is concerned only with forms
8:28:29and is concerned with them only in the
8:28:32way of stating that they are always or
8:28:34sometimes true with all the permutations
8:28:37of always and sometimes that may
8:28:41occur there are in every language some
8:28:44words whose sole function is to indicate
8:28:46form these words broadly speaking are
8:28:50commonest in languages having fewest
8:28:53inflections take Socrates is human here
8:28:57is is not a constituent of the
8:28:59proposition but merely indicates the
8:29:01subject predicate form similarly in
8:29:04Socrates is earlier than
8:29:07Aristotle is and then merely indicate
8:29:10form the proposition is the same as
8:29:13Socrates precedes Aristotle in which
8:29:16these words have disappeared and the
8:29:18form is otherwise
8:29:20indicated form as a rule can be
8:29:23indicated otherwise than by specific
8:29:25words the order of the words can do most
8:29:28of what is wanted but this principle
8:29:31must not be pressed for example it is
8:29:34difficult to see how we could
8:29:36conveniently Express molecular forms of
8:29:39propositions that is what we call truth
8:29:41functions without any word at all
8:29:44we saw in chapter 14 that one word or
8:29:47symbol is enough for this purpose namely
8:29:50a word or symbol expressing
8:29:54incompatibility but without even one we
8:29:56should find ourselves in
8:29:58difficulties this however is not the
8:30:01point that is important for our present
8:30:03purpose what is important for us is to
8:30:06observe that form may be the one concern
8:30:09of a general proposition even when no
8:30:12word or symbol in that proposition
8:30:14designates the form if we wish to speak
8:30:17about the form itself we must have a
8:30:20word for it but if as in mathematics we
8:30:23wish to speak about all propositions
8:30:25that have the form a word for the form
8:30:27will usually be found not
8:30:29indispensable probably in theory it is
8:30:32never
8:30:35indispensable assuming as I think we may
8:30:38that the forms of propositions can be
8:30:40represented by the forms of the
8:30:42propositions in which they are expressed
8:30:45without any special word for forms we
8:30:47should arrive at a language in which
8:30:49everything formal belonged to syntax and
8:30:52not to
8:30:53vocabulary in such a language we could
8:30:56express all the propositions of
8:30:58mathematics even if we did not know one
8:31:01single word of the language the language
8:31:04of mathematical logic if it were
8:31:06perfected would be such a language we
8:31:09should have symbols for variables such
8:31:12as X and R and Y arranged in various
8:31:15ways and the way of arrangement would
8:31:18indicate that something was being said
8:31:21to be true of all values or some values
8:31:24of the
8:31:25variables we should not need to know any
8:31:28words because they would only be needed
8:31:30for giving values to the variables which
8:31:33is the business of the applied
8:31:35mathematician not of the pure
8:31:37mathematician or
8:31:38logician it is one of the marks of a
8:31:41proposition of logic that given a
8:31:44suitable language such a proposition can
8:31:46be asserted in such a language by a
8:31:49person who knows the syntax without
8:31:51knowing a single word of the
8:31:55vocabulary but after all there are words
8:31:58that Express form such as is and then
8:32:02and in every symbolism hitherto invented
8:32:04for mathematical logic there are symbols
8:32:07having constant formal meanings we may
8:32:10take as an example the symbol for
8:32:12incompatibility
8:32:14which is employed in building up truth
8:32:16functions such words or symbols may
8:32:19occur in Logic the question is how are
8:32:23we to Define
8:32:25them such words or symbols Express what
8:32:28are called logical constants logical
8:32:31constants may be defined exactly as we
8:32:33defined forms in fact they are in
8:32:35essence the same thing a fundamental
8:32:38logical constant will be that which is
8:32:41in common among a number of propositions
8:32:44any one of which can result from any
8:32:46other by substitution of terms one for
8:32:50another for example Napoleon is greater
8:32:53than Wellington results from Socrates is
8:32:56earlier than Aristotle by the
8:32:58substitution of Napoleon for Socrates
8:33:01Wellington for Aristotle and greater for
8:33:05earlier some propositions can be
8:33:07obtained in this way from the Prototype
8:33:10Socrates is earlier than Aristotle and
8:33:12some cannot
8:33:14those that can are those that are of the
8:33:17form XR y that is Express dual
8:33:22relations we cannot obtain from the
8:33:24above prototype by term for term
8:33:26substitution such propositions as
8:33:29Socrates is human or the Athenians gave
8:33:31the hemlock to Socrates because the
8:33:34first is of the subject predicate form
8:33:36and the second expresses a three-term
8:33:39relation if we are to have any words in
8:33:41our Pure logical language they must be
8:33:44such as Express logical constants and
8:33:47logical constants will always either be
8:33:50or be derived from what is in common
8:33:53among a group of propositions derivable
8:33:56from each other in the above manner by
8:33:58term for term
8:34:00substitution and this which is in common
8:34:03is what we call
8:34:06form in this sense all the constants
8:34:09that occur in pure mathematics are
8:34:10logical constants the number one for
8:34:13example example is derivative from
8:34:15propositions of the form there is a term
8:34:18C such that f of x is true when and only
8:34:22when X is
8:34:23C this is a function of fee and various
8:34:27different propositions result from
8:34:29giving different values to fee we may
8:34:32with a little omission of intermediate
8:34:34steps not relevant to our present
8:34:36purpose take the above function of fee
8:34:39as what is meant by the class determined
8:34:42by Fe is a unit class or the class
8:34:45determined by fee is a member of one one
8:34:49being a class of
8:34:51classes in this way propositions in
8:34:54which one occurs acquire a meaning which
8:34:56is derived from a certain constant
8:34:59logical
8:35:00form and the same will be found to be
8:35:02the case with all mathematical
8:35:05constants all our logical constants or
8:35:08symbolic abbreviations whose full use in
8:35:11a proper context is defined by means of
8:35:13logical
8:35:15constants but although all logical or
8:35:18mathematical propositions can be
8:35:20expressed wholly in terms of logical
8:35:23constants together with variables it is
8:35:26not the case that conversely all
8:35:29propositions that can be expressed in
8:35:31this way are
8:35:32logical we have found so far a necessary
8:35:36but not a sufficient Criterion of
8:35:38mathematical propositions we have
8:35:41sufficiently defined the character of
8:35:43the primitive ideas in terms of which
8:35:46all the ideas of mathematics can be
8:35:48defined but not of the Primitive
8:35:51propositions from which all the
8:35:53propositions of mathematics can be
8:35:56deduced this is a more difficult matter
8:35:59as to which it is not yet known what the
8:36:01full answer
8:36:03is we may take the axium of infinity as
8:36:07an example of a proposition which though
8:36:09it can be enunciated in logical terms
8:36:12cannot be asserted by logic to be
8:36:14true all the propositions of logic have
8:36:17a characteristic which used to be
8:36:19expressed by saying that they were
8:36:21analytic or that their contradictories
8:36:24were self-contradictory
8:36:25this mode of statement however is not
8:36:29satisfactory the law of contradiction is
8:36:31merely one among logical propositions it
8:36:34has no special preeminence and the proof
8:36:37that the contradictory of some
8:36:38proposition is self-contradictory is
8:36:41likely to require other principles of
8:36:44deduction besides the law of
8:36:47contradiction nevertheless the
8:36:50characteristic of logical propositions
8:36:52that we are in search of is the one
8:36:55which was felt and intended to be
8:36:57defined by those who said that it
8:36:59consisted in deducibility from the law
8:37:01of
8:37:03contradiction this characteristic which
8:37:05for the moment we may call toy obviously
8:37:09does not belong to the assertion that
8:37:11the number of individuals in the
8:37:13universe is n whatever number n may
8:37:16be but for the diversity of types it
8:37:20would be possible to prove logically
8:37:22that there are classes of n terms where
8:37:25n is any finite integer or even that
8:37:27there are classes of olive subn terms
8:37:31but owing to types such proofs as we saw
8:37:33in Chapter 13 are
8:37:36fallacious we are left to empirical
8:37:38observation to determine whether there
8:37:41are as many as n individuals in the
8:37:44world among possible worlds in the
8:37:47libnan sense there will be worlds having
8:37:501 2 3 and so on
8:37:53individuals there does not even seem any
8:37:55logical necessity why there should be
8:37:58even one
8:37:59individual footnote one the Primitive
8:38:02propositions in principia Mathematica
8:38:05are such as to allow the inference that
8:38:07at least one individual exists but I now
8:38:10view this as a defect in logical purity
8:38:14end of footnote 1 why in fact there
8:38:17should be any world at
8:38:18all the ontological proof of the
8:38:21existence of God if it were valid would
8:38:24establish The Logical necessity of at
8:38:26least one
8:38:27individual but it is generally
8:38:29recognized as invalid and in fact rests
8:38:33upon a mistaken view of existence that
8:38:36is it fails to realize that existence
8:38:39can only be asserted of something
8:38:41described not of something named so that
8:38:44it is meaningless to argue from this is
8:38:46the so and so and the so and so exists
8:38:50to this exists if we reject the
8:38:53ontological argument we seem driven to
8:38:56conclude that the existence of a world
8:38:58is an
8:38:59accident that is it is not logically
8:39:03necessary if that be so no principle of
8:39:06logic can assert existence except under
8:39:08a hypothesis that is none can be of the
8:39:12form the propositional function so and
8:39:14so is sometimes
8:39:16true propositions of this form when they
8:39:19occur in logic will have to occur as
8:39:22hypotheses or Consequences of hypotheses
8:39:26not as complete asserted
8:39:28propositions the complete asserted
8:39:31propositions of logic will all be such
8:39:34as affirm that some propositional
8:39:36function is always
8:39:38true for example it is always true that
8:39:42if P implies Q q and Q implies R then P
8:39:46implies r or that if all Alphas are
8:39:50betas and X is an alpha then X is a
8:39:54beta such propositions may occur in
8:39:56logic and their truth is independent of
8:39:59the existence of the
8:40:01universe we may lay it down that if
8:40:03there were no Universe all General
8:40:06propositions would be true for the
8:40:08contradictory of a general proposition
8:40:11as we saw in chapter 15 is a proposition
8:40:14asserting existence and would therefore
8:40:18always be false if no Universe
8:40:21existed logical propositions are such as
8:40:24can be known a priori without study of
8:40:27the actual world we only know from a
8:40:30study of empirical facts that Socrates
8:40:32is a man but we know the correctness of
8:40:35the syllogism in its abstract form that
8:40:39is when it is stated in terms of
8:40:41variables without needing any appeal to
8:40:45experience this is a characteristic not
8:40:48of logical propositions in themselves
8:40:51but of the way in which we know them it
8:40:53has however a bearing upon the question
8:40:56what their nature may be since there are
8:40:59some kinds of propositions which it
8:41:01would be very difficult to suppose we
8:41:03could know without
8:41:06experience it is clear that the
8:41:08definition of logic or mathematics must
8:41:11be sought by trying to give a new
8:41:12definition
8:41:13of the old notion of analytic
8:41:15propositions although we can no longer
8:41:17be satisfied to Define logical
8:41:20propositions as those that follow from
8:41:22the law of contradiction we can and must
8:41:25still admit that they are a wholly
8:41:28different class of propositions from
8:41:30those that we come to know
8:41:32empirically they all had the
8:41:33characteristic which a moment ago we
8:41:36agreed to call
8:41:38tautology this combined with the fact
8:41:41that they can be expressed wholly in
8:41:42terms of variables and logical constants
8:41:45a logical constant being something which
8:41:47remains constant in a proposition even
8:41:50when all its constituents are changed
8:41:52will give the definition of logic or
8:41:55pure
8:41:56mathematics for the moment I do not know
8:41:58how to define toy footnote one the
8:42:02importance of topology for a definition
8:42:04of mathematics was pointed out to me by
8:42:07my former pupil ludvig Vicken Stein who
8:42:10was working on the problem I do not know
8:42:13whether he assault it or even whether he
8:42:15is alive or dead end of footnote
8:42:18one it would be easy to offer a
8:42:20definition which might seem satisfactory
8:42:22for a while but I know of none that I
8:42:25feel to be
8:42:26satisfactory in spite of feeling
8:42:28thoroughly familiar with the
8:42:29characteristic of which a definition is
8:42:32wanted at this point therefore for the
8:42:34moment we reach the frontier of
8:42:37knowledge on our backward Journey Into
8:42:39The Logical foundations of
8:42:41mathematics
8:42:44we come now to an end of our somewhat
8:42:46summary introduction to mathematical
8:42:49philosophy it is impossible to convey
8:42:51adequately the ideas that are concerned
8:42:53in this subject so long as we abstain
8:42:56from the use of logical symbols since
8:42:59ordinary language has no words that
8:43:01naturally Express exactly what we wish
8:43:04to express it is necessary so long as we
8:43:07adhere to ordinary language to strain
8:43:10words into unusual meanings and and the
8:43:13reader is sure after a time if not at
8:43:15first to lapse into attaching the usual
8:43:18meanings to words thus arriving at wrong
8:43:21Notions as to what is intended to be
8:43:24said moreover ordinary grammar and
8:43:27syntax is extraordinarily
8:43:30misleading this is the case for example
8:43:33as regards numbers 10 men is
8:43:36grammatically the same form as white men
8:43:39so that 10 might be thought to be an
8:43:41adjective qualifying men
8:43:43it is the case again wherever
8:43:45propositional functions are involved and
8:43:48in particular as regards existence and
8:43:51descriptions because language is
8:43:53misleading as well as because it is
8:43:55diffuse and inexact when applied to
8:43:58Logic for which it was never intended
8:44:01logical symbolism is absolutely
8:44:03necessary to any exact or thorough
8:44:06treatment of our
8:44:07subject those readers therefore who wish
8:44:10to acquire a Mastery of the principles
8:44:12of mathematics will it is to be hoped
8:44:15not shrink from the labor of mastering
8:44:17the symbols a labor which is in fact
8:44:20much less than might be thought as the
8:44:23above Hasty survey must have made
8:44:25evident there are innumerable unsolved
8:44:27problems in the subject and much work
8:44:29needs to be done if any student is led
8:44:33into a serious study of mathematical
8:44:35logic by this little book it will serve
8:44:39the chief purpose for which it has been
8:44:41written
8:44:43end of chapter 18 and end of
8:44:46introduction to mathematical Philosophy
8:44:48by berand
8:45:03Russell

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