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You Truly Don't Understand What Just Happened to Mathematics — Transcript

by Prof Gio | Giordano Scarciotti · 2,125 words · 372 segments · language en · Watch on YouTube

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0:00Imagine that tomorrow morning you wake
0:01up to the news that an American flag has
0:04appeared on Mars. There is a photograph
0:06and newspapers around the world are
0:07announcing it as our biggest
0:09achievement.
0:10Except there is one small tiny problem.
0:13No one built a rocket. No one invented
0:16new propulsion systems, developed new
0:18materials, or figured out how to keep
0:20astronauts alive on the journey. The
0:22flag just magically appeared. Now, would
0:25that be an achievement? Well,
0:28technically yes, there is a flag on
0:29Mars, but would humanity be any closer
0:32to becoming a spacefaring civilization?
0:35I'm not sure about that. And this is
0:37exactly the mistake I think we're making
0:40with mathematics. Most people that think
0:42that mathematical research is about
0:44solving problems. The harder the
0:46problem, the greater the achievement
0:48when somebody finally solves it. But
0:51that's a fundamental misunderstanding of
0:53how mathematics advances. Over the next
0:5510 minutes, I want to challenge the way
0:58you think about mathematics and explain
1:00why this misunderstanding could have two
1:03serious consequences for the future of
1:05science. Artificial intelligence is
1:07solving hundreds of problems that have
1:09resisted the world's greatest
1:10mathematicians for decades. OpenAI
1:13recently announced a solution to one of
1:15the famous Millennium Prize problems,
1:17followed by hundreds of other
1:18mathematical results. And mathematicians
1:20are upset.
1:22Why so? Isn't solving problems exactly
1:25what mathematicians are supposed to do?
1:27If you read the comments online, the
1:29explanation is apparently very simple.
1:31Mathematicians are terrified because AI
1:34is coming for their jobs. And frankly, I
1:36find this explanation deeply concerning
1:39because it is missing something
1:41enormously important.
1:43Let me tell you a story. In 1736, a
1:46mathematician called Leonhard Euler
1:48investigated what may be one of the most
1:50pointless problems in the history of
1:52mathematics. In the city of Königsberg,
1:55there was a river with two islands and
1:57seven bridges connecting the different
1:59parts of the city. And people wondered
2:01whether it was possible to take a walk
2:03across all seven bridges without
2:05crossing any bridge more than once.
2:08That's it. That was the problem. And
2:10Euler, one of the greatest
2:12mathematicians who ever lived, decided
2:14to investigate it. He eventually proved
2:16that that the answer was no, but you
2:19cannot do it.
2:20Well, that's fantastic. One of the
2:22greatest minds in human history had
2:24established that you couldn't take a
2:26particular walk around a particular
2:28city. An achievement that was, for all
2:30practical purposes, completely useless.
2:33But that's not the point. To solve this
2:35ridiculous problem, Euler realized that
2:38he could ignore almost everything about
2:40the city. He didn't need to know how
2:42long the bridges were, where the roads
2:44went, and even the shape of the river.
2:46All that mattered was how the different
2:48pieces of land were connected. So, he
2:51developed a mathematical way of
2:53representing those connections,
2:54stripping away all the irrelevant
2:56details.
2:58And that idea helped establish an
3:00entirely new branch of mathematics,
3:01graph theory.
3:03Today, graph theory is fundamental to
3:05computer networks, internet routing,
3:07logistics, transportation systems, and
3:10countless algorithms that we rely on
3:12every single day. All of this can trace
3:15part of its mathematical history back to
3:17a man wondering whether he could cross
3:19seven bridges without repeating one.
3:22Now, imagine Euler had access to an
3:24incredibly powerful AI. He types in the
3:27question, "Can I cross all seven bridges
3:29without repetitions?" And the AI
3:31immediately responds, "No, it's not
3:34possible." Well, that's brilliant.
3:36Problem solved. Except, would we have
3:38discovered graph theory?
3:40Maybe, or maybe not. Because the
3:42important thing Euler produced was not
3:44the answer. It was the mathematics he
3:46developed while searching for it. And
3:48this is something about mathematical
3:49research that I think is rarely
3:51explained to the public.
3:53Mathematics is really weird. A huge part
3:56of mathematical research is not about
3:58finding answers. It's about developing
4:01new mathematics in the process of
4:03looking for them.
4:04Mathematicians spend years investigating
4:07questions with no obvious practical
4:09application. And sometimes, in the
4:11struggle to answer them, they discover
4:13ideas that turn out to be far more
4:15important than the original problem.
4:17Which brings me back to the flag on
4:19Mars. Think about the Apollo program.
4:22What was the objective? Well, definitely
4:24put a human on the moon. But, in working
4:27towards that objective, humanity made
4:29enormous scientific and engineering
4:31progress. Computers, guidance systems,
4:34new materials, propulsion, manufacturing
4:36techniques, medical discoveries, all
4:38sorts of technologies that have uses
4:41well beyond putting someone on the moon.
4:44Now, imagine we could achieve the same
4:46objective by simply teleporting Neil
4:48Armstrong on the lunar surface.
4:50We would have reached moon, but missed
4:52out much of the technological
4:54development that made the mission so
4:56valuable. And this is the danger of
4:58treating mathematical problems like a
5:00gigantic checklist. One problem solved,
5:03another achievement unlocked. Look at
5:05how much science we're doing. Except,
5:07are we?
5:08Recently, OpenAI announced the a
5:10solution to the Navier-Stokes Millennium
5:12Prize Problem. Now, the Navier-Stokes
5:14equations are incredibly useful. They
5:16describe how fluids move, and we use
5:19them for everything from designing
5:21airplanes to predicting the weather.
5:23But, what about the Navier-Stokes
5:25Millennium Prize Problem? Practically,
5:27that problem is useless. Utility was not
5:31the reason this problem was selected as
5:33a Millennium Prize Problem. The reasons
5:36were its ability to stimulate deep
5:38mathematical research and its potential
5:40to advance mathematical understanding.
5:43The point was never simply to get the
5:45answer. In fact, the problem asks a very
5:47abstract mathematical question. Can
5:50these equations develop singularities,
5:52which are points where the mathematical
5:54description breaks down? OpenAI
5:56constructed a very particular example
5:59where this happens by applying a
6:00carefully constructed external force to
6:02a fluid initially at rest.
6:05That's an impressive demonstration of
6:07what AI can do, but that does not
6:09automatically make it an important
6:11advance in mathematics. In fact, what we
6:14have actually gained from it? Can we now
6:16design better airplanes, predict the
6:18weather more accurately, understand how
6:20real fluids behave in ways we couldn't
6:22before?
6:24No.
6:24What we have is an incredibly
6:26complicated mathematical construction
6:28that proves that something about a very
6:29particular, rather artificial scenario.
6:32And don't misunderstand me, there is
6:34nothing wrong with investigating
6:36abstract mathematical questions. That's
6:38precisely what mathematicians have been
6:40doing for centuries. But remember Euler
6:43and his seven bridges. The answer to
6:45that problem was completely useless.
6:48What made Euler's work extraordinary was
6:50graph theory, the new mathematics that
6:52came out of it. So, the question is,
6:55where is the graph theory of OpenAI's
6:57Navier-Stokes solution? Perhaps there
6:59are some generally useful mathematical
7:00ideas deep inside the proof, which
7:03others will eventually discover and
7:05develop. But right now, we're
7:07celebrating the fact that an AI has
7:09apparently solved a famous problem
7:11without having established the whether
7:13the solution has given us anything
7:15valuable. The point is that mathematics
7:17is not like other research fields. There
7:19is a fundamental difference between
7:21solving an abstract mathematical problem
7:23and solving a practical scientific
7:26challenge. If an AI discovers a cure for
7:28cancer or makes a nuclear fusion
7:31commercially viable, that's an
7:32extraordinary achievement with immediate
7:35implications for society. But most pure
7:38mathematical problems don't work like
7:39that. The history of mathematics is full
7:42of seemingly pointless questions that
7:44ended up changing the world. Not because
7:46somebody finally answered them, but
7:48because of the mathematics developed
7:50along the way. Sometimes becoming useful
7:53centuries after the mathematicians who
7:55developed it were dead. And while we are
7:58celebrating all these problems being
8:00solved, there is an even more
8:02fundamental and important problem.
8:04Imagine you are a 17 years old. You are
8:07exceptionally good at mathematics. You
8:09love solving problems. You love the
8:11challenge. And you are thinking about
8:13becoming a mathematician. You know, it's
8:16going to take years of studying, a
8:18university degree, a PhD, maybe several
8:21years as a post-doctoral researcher. And
8:23after all that, if you're talented, hard
8:26working, and a little lucky, perhaps you
8:29will eventually make some important
8:31mathematical discoveries of your own.
8:33But then, you open your phone, and every
8:35week there is another headline. AI
8:37solves a problem mathematicians couldn't
8:39crack for decades. AI discovers hundreds
8:42of new theorems. Another mathematical
8:43problem solved in hours that may have
8:46taken humans years. And you're sitting
8:48there, age 17, wondering what the world
8:51is going to look like by the time you
8:53finish your PhD. Perhaps another 10
8:56years from now.
8:57Would you still want to become a
8:58mathematician?
9:00I'm not sure about that.
9:02And before somebody says, "Well, people
9:04will study mathematics before they enjoy
9:06it." Yes, of course, but let's be
9:08serious for a moment. Research is not an
9:10hobby that people do for a couple of
9:13hours on Sunday afternoon. It takes
9:15years of enormous personal commitment,
9:17often with considerably uncertainty
9:19about your career. And one of the
9:21biggest motivations for becoming a
9:23researcher is the possibility of
9:25contributing something genuinely new to
9:27human knowledge. Now, imagine growing up
9:30in a world where you are constantly
9:31being told that whatever problem you're
9:33working on, an AI will probably solve it
9:36before you even start understanding it.
9:39What does that do to your motivation? I
9:41supervise PhD students and I can tell
9:43you that doing research is hard. You can
9:45spend months, sometimes years working on
9:48something and make no visible progress.
9:51Sometimes you begin to wonder whether
9:53you are capable of solving the problem
9:55at all. What keeps you going is the
9:57possibility that eventually you will
9:58understand something nobody understood
10:00before. Now, imagine that in the middle
10:03of all the struggle you discover that
10:05some AI has already solved your problem
10:07and then the next one and then the next.
10:10How many times does that need to happen
10:12before you begin to wonder whether you
10:14are dedicating your life to a race you
10:16have already lost?
10:18And the consequence wouldn't be
10:19immediate. Mathematicians don't just
10:22appear out of nowhere. Mathematics
10:24depends on an entire ecosystem of
10:26students, researchers, and professors
10:29built up over generations. Now, imagine
10:32that over the next 20 years a
10:33significant number of talented students
10:36decide not to enter that ecosystem. Not
10:38because they dislike mathematics, but
10:40because they have been convinced that
10:42there is no future in contributing to
10:44it.
10:45Initially, nothing dramatic happens. The
10:47existing professors are still there.
10:49Universities continue publishing a
10:52research. AI keeps producing more
10:54mathematical results than ever before.
10:57Everything looks fantastic.
10:59Except that slowly the community of
11:01people capable of independently
11:03understanding advanced mathematics
11:06begins to shrink and eventually we could
11:08find ourselves in a rather awkward
11:10situation. We have access to more
11:12mathematical knowledge than any
11:14generation in human history, but fewer
11:16and fewer humans are capable of
11:18understanding it. And at this point you
11:20may say, "So what? If AI can do
11:23mathematics better than humans, why do
11:25we need mathematicians at all?"
11:28Well, imagine an AI designs a completely
11:30new airplane. It produces an incredibly
11:32sophisticated mathematical analysis
11:34demonstrating that the design is safe.
11:37Except that the mathematics is so
11:38complicated that nobody really
11:40understands it. So, we ask another AI to
11:42check the calculation and well,
11:44everything looks good. Everything looks
11:46correct. But what if the original model
11:48was wrong? What if an important physical
11:50effect that was missing from the
11:52equations? The calculation could be
11:54flawless, but the aircraft could still
11:57be unsafe. And we need people who
11:59understand mathematics deeply enough to
12:02recognize those problems. People who can
12:04question assumptions, challenge
12:07established methods, and develop
12:09entirely new ways of thinking when the
12:11existing ones aren't good enough. That
12:14expertise takes years to develop. If we
12:16stop developing it, we may eventually
12:18become completely dependent on systems
12:21we can no longer challenge. And I find
12:23that prospect generally frightening.
12:26This is actually why I decided to make
12:28this video. This morning I opened Reddit
12:31and I saw yet another comment dismissing
12:33mathematicians as a bunch of people
12:35terrified of losing their jobs. And
12:38frankly, it made me angry because this
12:40is so much bigger than that. Reducing
12:42these concerns to mathematicians
12:44protecting their jobs is incredibly
12:46short-sighted. Of course, some
12:48mathematicians will be worried about
12:50their careers. Despite what one may
12:52think, they are human beings and they
12:55have every right to worry about their
12:57livelihoods. But the much bigger issue
12:59is what happens to mathematics itself.
13:01Mathematics has given us an
13:03extraordinary ability to understand and
13:05transform the world. And that didn't
13:07happen because somebody handed us a long
13:10list of correct answers. It happened
13:12because generations of researchers
13:14developed new mathematical ideas
13:16building on the work of those who came
13:18before them. And because new generation
13:20of students kept joining that process.
13:23And those are the two things we cannot
13:25afford to lose. Thank you for watching.
13:30>> [music]
13:42[music]

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