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0:00Your brain is nothing but an extraordinary pattern recognition machine. It is

0:05because it had to be. To survive, our ancestors had to process environmental data

0:11in real time and quickly figure out whether the rustling in the bushes was just

0:17the wind or a hungry lion. The ancestors who couldn't do this, well, those clumsy

0:23ones died out before passing their genes to the next generation. So, the genes we

0:29inherited belong to those who could spot these dangers earliest and most accurately.

0:35And our brain, thanks to these genes filtered over generations, is amazing, but

0:41since it was hyper-optimized for the conditions of that era, in the hyper-fast-paced

0:47modern world, it evolved into a pattern detection machine that we realize is full of

0:53vulnerabilities. And math, the nightmare of so many of us in school today, is really

0:59just the most abstract, refined, and uniquely human form that this ancient evoluti

1:05onary ability has reached. In math, too, we start with very basic truths and build

1:11incredibly complex systems on top of them, and then we detect the patterns

1:17and rules that emerge within those systems. And

1:20because nature starts with very simple systems like subatomic particles

1:25or basic laws of nature and is built up step by step on top of them, it

1:30essentially follows the same logic as math. In fact, as we've talked about before,

1:36for some, math isn't just some toy language. On the contrary,

1:40it's the ultimate truth underlying all sciences and nature.

1:45But totally regardless of whether math is an invention or a discovery,

1:50nature exhibits very similar patterns to many branches of math. Therefore,

1:56math becomes an incredibly powerful tool for understanding and explaining

2:01nature. Now, since there are no other animal species that can compete with us in doi

2:07ng math, we've assumed until now that human intelligence would be the sole intellige

2:13nce on Earth capable of practicing math, discovering new math, and solving mathemati

2:19cal problems. But we were wrong. Because even though it takes inspiration from human

2:25intelligence, AI, which has a completely different architecture

2:30and working logic than ours,

2:32is starting to outperform human mathematicians in an ever-growing number of fields.

2:38So much so that AI isn't just settling for memorizing previously solved problems

2:44and solving their variants anymore. It has reached a point where it can

2:48completely autonomously solve and prove solutions to professor-level problems

2:54that even the world's brightest minds haven't been able to solve for decades.

2:59I want to point out: just a few years ago,

3:02these AIs were incapable of completing even simple sentences.

3:06Remember that. So the pace of development in this field is truly abnormal,

3:12and the problems AI has already solved could be enough to

3:17completely reinvent the future of math. Let's take a look.

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4:56Now, about the AIs that are taking math right out of our hands. To

5:01understand just how serious things are,

5:04we can look at the planar unit distance problem posed in 1946 by Paul Erdős,

5:10one of the math world's most eccentric and legendary names. The problem is actually

5:17incredibly simple. Take a huge, blank piece of paper. Start placing random dots

5:23on it. Now count the number of pairs of these dots that are exactly 1 unit apart,

5:29say 1 cm. The question is this:

5:31as the number of dots you place continues to increase to infinity, how fast would

5:37you say the number of dot pairs exactly 1 unit apart can increase at most? I mean,

5:43what kind of strategy can you use to place those dots on your infinitely large

5:48paper so that the number of dot pairs that are 1 cm apart increases as

5:53fast as possible? And when you do that,

5:56does the number of those dots 1 cm apart increase at the same rate as the total

6:02number of dots you put down, or does it increase exponentially, or does it increase

6:08but slow down, or is it something else entirely? The question is simple, but

6:13even though mathematicians have been developing strategies to maximize

6:18this growth rate for decades, they haven't managed to offer very

6:22impressive solutions. Right now, the best available solution consists of

6:27very slightly improved versions of the grid approach, which positions

6:32all dots one unit apart. It's a boring and pretty uninspiring solution.

6:37What if there are better solutions? For 80 years, no one, including Erdős,

6:43was able to prove that the near-linear growth rate could be meaningfully exceeded.

6:49That is, until a few months ago. A new AI model developed by OpenAI, completely

6:55autonomously, found that Erdős's prediction of a nearly linear upper bound was wrong

7:02by discovering a brand new family of examples. Can you imagine that?

7:06A machine that just a few years ago couldn't even answer the simplest questions can

7:13now solve a problem that the smartest people in the world have been pondering for

7:19almost a century. But the really mind-blowing part isn't that this AI reached the

7:25goal; it's the actual path it took to get there. To solve this problem, the model

7:31used highly sophisticated tools known as algebraic number theory,

7:36which seemingly have almost nothing to do with this dot geometry problem,

7:42in incredibly creative ways that no mathematician had ever thought of before. It's

7:49like using, say, quantum mechanics equations to decipher the rhyme scheme of a poem.

7:55While limited human lifespans and the strict specialization in academia

8:01often make these giant interdisciplinary leaps impossible,

8:06AI can build bridges in seconds between disciplines that diverged decades ago.

8:12It's a beautiful example, but it comes with a risk. As you know, even though this

8:18problem has decreased quite a bit lately, AIs continue to hallucinate, or make

8:23things up, especially when dealing with long and complex tasks. And when it comes

8:29to a subject like math, which even humans have a hard time doing and understanding,

8:35how are we going to verify the AI's supposed proofs? How can we trust it?

8:40By using other AIs trained specifically for this job, of course. Jokes aside,

8:46expert mathematicians do check these AI results themselves,

8:51but the beauty of math is that it's a language built on explicitly stated axioms.

8:58Meaning it's a very formal language.

9:01In fact, that's why we categorize math under the formal sciences.

9:05And right here, there are languages like Lean

9:08and proof assistants like Aristotle that take advantage of this exact formality.

9:14Here's how they work: A base model like ChatGPT comes up with an intuitive idea,

9:19just like a human does, and writes a mathematical proof outline in natural language.

9:25Then, another system named Aristotle translates this draft into Lean code,

9:31which is a formal proof language that a computer can verify step by step.

9:36If any step of the proof translated into Lean doesn't comply with definitions

9:41or axioms, Lean can't compile it

9:44and throws an error. The AI then takes this feedback and tries to fix the error

9:49preventing Lean from compiling. This self-advancing dialogue, at least in some

9:54cases, turns into a formally verified mathematical result that experts can

9:59read and evaluate. And even Terence Tao, one of the greatest mathematicians of our

10:05time who I've told you a bit about before, believes this robotic back-and-forth

10:11is revolutionary for math. In fact, Tao makes a much more fascinating point. Once

10:17the AI solves the problem, it can rewrite that complex proof from top to bottom

10:23for different levels of expertise in mere seconds.

10:27Because normally in the academic world, rewriting a paper is seen as a months-long

10:34agony avoided out of fear of making new logical errors. Yet, it looks like

10:41in the future, a theory will have a standard and very strictly written main

10:47paper, but alongside it, there will be hundreds of secondary versions

10:52instantly generated by AI, tailored to the current reader's needs,

10:57their intelligence, and their skill level, complete with adapted narratives

11:02and references. The fact that scientific communication is becoming

11:07this dynamic truly shows that an incredible scientific revolution

11:12is knocking on our door. Of course,

11:15AI doesn't always solve these math problems on its own. Symbiotic collaborations

11:20where humans are in the loop can yield some truly fascinating results. A great

11:26example of this is when applied mathematician Ernest Ryu from UCLA and his colleag

11:31ues solved the pointwise convergence question for Nesterov's accelerated gradient

11:37method, which had been open since 1983. I know that name sounds intimidating,

11:42but again, the idea behind it is super simple.

11:45I've told you before about the gradient descent algorithm,

11:49which is also the backbone of AIs. Basically, thanks to this method,

11:53the error systems make decreases over time by following a path just like a

11:58ball rolling down a hilly surface, and settles in a specific valley.

12:03And to date, tens of thousands of methods have been developed to accelerate and

12:08improve that ball, or the margin of error,

12:12reaching a certain minimum. Nesterov optimization is a really

12:18clever method used to optimize the momentum of this descent.

12:23Normally, as you know, momentum is an object's mass times its velocity.

12:28For example, as a rock rolls down a hill,

12:31since it accelerates in the direction it's going, its momentum in that

12:36direction—the majesty of its movement—increases too. Nesterov momentum considers

12:41the momentum at a future point in the ongoing direction and looks at the

12:46slope there to prevent the path from veering away from the desired direction,

12:52making corrections in a very smart optimization. So

12:55it sort of gives intelligence to the rock's fall.

12:59Think of it like that. And one unknown thing about this optimization method

13:04was whether the falling motion into that valley ultimately stopped at a

13:09single point, or if it constantly wobbled slightly around the valley point.

13:14So Ryu sat down with ChatGPT and chatted for days to solve this

13:19problem that had gone unsolved for over 42 years. Only,

13:23when he gave the system the problem, the AI kept giving him flawed proofs.

13:29But inside the AI's flawed constructs, there were such brilliantly

13:33discovered partial results, such original intermediate steps that the human

13:39mind, with its conventional way of thinking,

13:42might have never taken. Ryu extracted the precious crumbs from those

13:46wrong proofs the AI generated and guided the system step by step, saying,

13:51"This part is wrong, but that idea is great. Keep going from here."

13:55guiding the system step by step.

13:58That way, this problem, which normally could have taken much longer,

14:02was successfully solved a few months ago. The funny thing is,

14:06AI can sometimes give us answers to questions we never even asked. For example,

14:12you remember permutations from high school. You know, the branch of math that

14:17studies how many different ways objects can be ordered. It might surprise you

14:22that there's a whole branch of math just for this. But remember my short

14:26video about playing cards that people still have a hard time believing.

14:30Just the number of different combinations of a mere 52 playing cards is greater than

14:37the number of atoms in the Milky Way galaxy. So, no matter how many different card

14:43games have been played on Earth with how many different full decks, the probability

14:49of two people getting the exact same sequence of cards is practically zero, and it

14:55probably will never happen from now on. And well, in the field of permutations,

15:00there's a topic we call Bruhat order. You can think of it as a measure

15:05of how simple or complex the ordering of playing cards is. Sort of

15:09like the entropy of the cards. And the structure formed by all the

15:13intermediate permutations that fall in this Bruhat order between

15:17two permutations is called a Bruhat interval. Well,

15:21researchers were getting help from Google DeepMind's AlphaEvolve model—which

15:26is powered by evolutionary algorithms, which was also my PhD topic—to

15:31research large values of a technical quantity of these Bruhat intervals

15:36called the D-invariant. But while AlphaEvolve was thinking about what

15:41they asked, it discovered a hidden pattern in these intervals that no

15:46one had ever noticed before. If the number of elements is a power of 2,

15:52some of those Bruhat intervals organize themselves into giant

15:57hypercubes. No one had seen this. But the even more interesting part is,

16:03humans didn't ask the AI about hypercubes or a pattern like this.

16:08The AI managed to uncover this hidden architecture purely because it knew how to

16:14look there and lacked the shackles of the human mind. Now, if you're studying math

16:20or want to, what I've told you might have caused you to experience the same fear

16:26that developers, translators, and other experts have been feeling for months. Well,

16:32if AI is going to do math, what do we need mathematicians for, right?

16:36First off, I'm sorry if I made you feel that way, but no, don't think like that.

16:42First of all, despite everything I've told you,

16:45AI still hasn't completely taken the jobs of even developers and translators.

16:50I mean, even that promise hasn't materialized yet, at least for now.

16:54In fact, the need for high-quality developers

16:57and high-quality translators seems to have increased even more with AI.

17:02We'll all see how things change as models develop further. One of

17:06the reasons for this is a very interesting topic that I'll detail in

17:10another video. It's becoming increasingly certain that we won't be

17:14able to fully model human intelligence using the large language model

17:19approach that today's models rely on. Especially when it comes to

17:23interaction with the environment and physical intuition,

17:27it seems these models need something extra.

17:30So, language alone isn't enough to create a complete model of a mind. For instance,

17:35think of the Navier-Stokes equations, the bane of physics and engineering.

17:40These are an amazing group of equations that can very successfully model the motion

17:46of many fluids, from gases in the atmosphere to blood in our veins, from rocket

17:51fuels to the flow of a river, acting as the backbone for many modern technologies.

17:57Actually, there's no problem with the equations. We know perfectly

18:01well what they say, and we use them in engineering every day.

18:04The problem is that we can't mathematically solve certain behaviors of this set of

18:10equations. For example, if there's a reasonable and smooth flow in a three-dimensio

18:16nal space, can the Navier-Stokes equations continue to predict this flow within smoo

18:22th and physically reasonable bounds forever? Or does the speed or vorticity, I mean

18:28swirliness, blow up at a certain point

18:30and approach infinity? No one has ever been able to solve this. That includes

18:36AI. Even though there's a $1 million prize for whoever solves the question.

18:41Actually, AI and physics-informed neural networks managed to find some

18:46possible singularity candidates using simpler fluid equations on the path

18:51to Navier-Stokes. But in the three-dimensional Navier-Stokes problem,

18:56especially when viscosity, or flow resistance, is factored in,

19:00no meaningful proof of singularity that could solve this problem has

19:05been produced yet. One of the biggest reasons for this is that AI's

19:10perceptions of the physical world are incredibly limited, and this is a big

19:15problem. Right at this point, the story of a mathematician named Steve

19:20Shkoller holds a tremendous lesson,

19:22in my opinion. Shkoller isn't just a theorist who lives staring at a computer

19:28screen. He's also been surfing since childhood. He's internalized that magnificent

19:33energy on the ocean and the constantly changing, chaotic nature of fluids

19:38through a bodily experience. One day, unable to surf due to an injury,

19:43Shkoller was lying on his couch imagining a time he surfed giant waves.

19:48That's when he realized that the splash moment of fluids shouldn't be

19:53modeled like a static photograph that AI looks for, but rather like a

19:58dynamic filmstrip where new water droplets constantly enter the scene

20:02and players are constantly changing. Simply using this intuition

20:07gifted to him by his evolutionary heritage and physical experience,

20:12he proved that fluids in motion with a free surface could create a

20:17singularity he called a splash singularity. This, of course,

20:21isn't that sought-after solution to Navier-Stokes I told you about, but it's

20:26a discovery that AI still hasn't noticed,

20:30even though it ponders these topics too. This makes sense.

20:34Because AI has never felt the ocean spray, wind resistance, or water's acceleration.

20:40For it, the world is nothing more than a model built on data, equations,

20:45simulations, and formal representations, rather than a lived bodily experience.

20:51The bodily intuition we gain from feeling gravity

20:55and fluidity in our cells as biological beings is something machines can't

21:00easily mimic, even with trillions of calculations. For now,

21:04because it's not impossible. As models develop and physical perception starts taking

21:10root in the minds of these language models, they might start dominating these areas

21:15too. And right at that point, a strange question arises:

21:19How will we understand them? I mean, as American philosopher

21:23and mathematician Reuben Hersh emphasized years ago, "Math isn't just a set

21:28of absolute truths in some platonic universe; it's also a sociological

21:33phenomenon. It requires people to actively communicate with each other and reach

21:39a shared comprehension. That's how it progresses." But if things go this way, and a

21:45computer does a proof of tens of thousands of steps that's too complex for any human

21:51to understand, and another program verifies it as 100% correct, yet not a single

21:57human soul on Earth can grasp why that proof is correct, does this still count as

22:03math? Because an incomprehensible truth, even if it might be technologically

22:09useful to us, cannot enlighten us. Moreover, just like how number theory

22:14suddenly became the foundation of modern digital security, or cryptography,

22:20in the 1970s, what if tomorrow we blindly trust these dark proofs generated by AI

22:26and build our global encryption systems

22:28or medical algorithms on these incomprehensible foundations, and there happens

22:34to be a contextual disconnect deep within the proof that we failed to notice?

22:39We could make massive, irreversible mistakes. Therefore,

22:43we have to understand exactly how the black-box minds of AIs work,

22:48so that we can improve ourselves using what we learn from them.

22:53>> Consider this to see how fast things are moving in the field. Between

22:58the time I shot this video and published it,

23:01two more long-standing problems were solved using AI. One is an 87-year-old

23:07problem. Finding a counterexample to the Jacobian conjecture, Levent Alpöge

23:12from Harvard University, during the World Cup final, gave Claude a simple

23:18prompt that led to the first-ever discovery of a counterexample to this

23:22conjecture. It allowed such an example to be found. The other is another

23:2630-year-old problem. It was solved in just four prompts thanks to

23:30ChatGPT's new model, o1. Both of these happened just in the last week.

23:35Think about that. So mathematicians have a really interesting tool

23:39in their hands. We'll all see what they'll do with it.

23:43Anyway, back to the video. Ah, speaking of improving ourselves, the revolution AI

23:48is starting to create in math will require us to reinvent education, too.

23:53Because today, many math departments in universities are experiencing a serious ass

23:58essment and evaluation crisis due to AI being able to solve assignments in seconds.

24:03When students are prevented from racking their brains over a problem

24:08for hours to build their mental muscles,

24:11how will we train the experts of the future who will verify machine-generated

24:17proofs and make those intuitive leaps? This isn't just a problem for math

24:22departments, of course, but this revolution will definitely affect them, too.

24:27So there it is. I don't know if you can notice this amidst the daily hustle

24:32and bustle, but we are truly living through extraordinary times. I

24:37don't know where AI will take us either. I'm very curious. On the one hand,

24:42yes, it's the most effective tool we've ever seen, but on the other hand,

24:46it feels like it's gradually eliminating the need for humans

24:50in every field. But well, maybe we should listen to Terence Tao on this issue,

24:56too. Tao says mathematical discovery is like exploring a landscape

25:01made of massive mountain ranges. We humans make long-term,

25:05strategic plans to climb Everest, but we can only advance toward the goal

25:10step by step. AIs, meanwhile, are currently like highly athletic robots

25:15that can easily vault over 6 to 10-foot walls doing parkour. They

25:20can handle calculations in seconds that would take us months

25:24and plant their flags on intermediate peaks we thought were unreachable.

25:29But you see, it's still the human mind that will chart that long,

25:34strategic, exhausting route, full of a search for meaning,

25:38to the summit of Everest. At least we hope so. And if we look at it from this angle,

25:44AI isn't actually pushing us out of math. On the contrary,

25:48it gives us an incredibly powerful flashlight to illuminate the dark corners

25:54of this extraordinary cathedral we've built brick by brick over centuries

25:59and gotten lost in its corridors—math, the queen of sciences.

26:04Sure, this flashlight flickers sometimes. Sometimes it shows us hallucinations

26:10and shadow plays. But as we learn how to hold it,

26:13we'll get the chance to understand the flawless fabric of the universe much more

26:19deeply. And in that quest, it falls on us to continue that most human of acts,

26:24which we can never hand off to machines: to keep asking questions and dreaming.

26:29If you liked this video, I highly recommend watching our math video that

26:36also spawned the "are we idiots?" meme. See you in the next video. Take care.