An unreleased OpenAI model produced 722 mathematics manuscripts overnight, prompting a joint criticism from 25 Fields Medalists

[[**On October 7, 2026, an undisclosed OpenAI model generated 722 mathematics manuscripts (372 result groups) overnight, covering major breakthroughs such as the quasi-Riemann hypothesis and a new lower bound for matrix multiplication, with each result requiring on average only 3 hours of compute. This event is regarded as the prototype of an unprecedented technological explosion in the history of mathematics, and it also prompted 25 Fields Medalists to jointly criticize the AI company's release method.**

**Key Points**

**1. The scale and breakthrough nature of the results shocked the academic community**

Without public disclosure, an internal OpenAI model produced 722 mathematics manuscripts in one go, containing 372 result groups and covering core fields such as number theory and algebraic geometry. Among them, the "quasi-Riemann hypothesis" pushes the zero-free region to 0.875 (7/8), the greatest breakthrough in the direction of the Riemann hypothesis in nearly a century; the exponent for matrix multiplication over the complex field is pushed down to 2.25, and the four-dimensional version of the Kakeya conjecture is solved. Mathematicians believe that if these had been done by humans, they would directly win the Fields Medal.

**2. Model costs plummeted, and iteration speed is astonishing**

The model spent on average only about 3 hours of ChatGPT Pro-level compute per result (roughly a single inference run by one Agent), whereas in September, when it tackled the Navier–Stokes equations, it still required millions of dollars and 88 hours. From the first recognized result in May 2026 to October, within 5 months the number of results jumped from 1 to 372, an increase of two orders of magnitude.

**3. The release method triggered fierce opposition within the mathematics community**

OpenAI did not follow the academic community's suggestions and did not disclose the model name, prompts, or reasoning details, but instead released everything in bulk directly in the form of a repository. 25 Fields Medalists signed a joint letter, "The Serious Misalignment of AI in Mathematics," criticizing AI companies for using "problem solving" as a marketing tool and for disrupting the mathematics community; the advisory group of the Institute for Advanced Study in Princeton demanded that such practices stop, and OpenAI did not respond.

**4. The technological explosion is accelerating, but real-world bottlenecks remain**

This article is from the WeChat public account: Digital Life Kazik, author: Digital Life Kazik

Yesterday morning may have been a watershed moment in human history.

So much so that even though I had done a great deal of preparation in advance and had a certain amount of background knowledge, I still spent a full day writing this article, finishing at 6:15 a.m. today. There is no other reason: the amount of information in this event was too great, and its historical significance too great. That sense of destiny, during the research and writing, still caused countless enormous emotional swings.

Those who follow the news have already seen this event.

That is, yesterday (October 7) morning, without any warning, OpenAI released a massive amount of mathematics research results. They directly posted a new repository on GitHub, with a very plain name: openai/math.

Inside lay 722 mathematics manuscripts, 372 related results, covering number theory, algebraic geometry, real analysis, combinatorics, theoretical computer science, mathematical physics, partial differential equations, and other directions, all produced by an unnamed internal OpenAI model that has not yet been released to the public.

Overnight, 722 papers. What kind of concept is that quantity?

The famous arXiv has upload limits; each person can submit only two papers per month.

722 papers, if done by humans and uploaded manually, would take 30 years just to upload the papers.

And this is mathematics. Mathematics can be said to be the foundation of science, and also the queen of science.

At that moment, you could understand it as: the gears of fate are no longer slowly turning; the gears of fate have directly drilled out sparks and assembled themselves into a Gundam, looking down at you from in front of you, asking you, look, am I not pretty awesome?

The next 24 hours were basically a state of boiling across the whole world, with some joyful and some worried.

This guy's comment is the one I find most relatable.

This is a chaotic era, and also an era of exponential explosion in fundamental science.

The technological explosion experienced in "The Three-Body Problem" is, as things stand now, very likely to happen within our lifetimes.

Although in mathematics I am an absolute layman, I have indeed been cramming all kinds of knowledge for the past two months. With AI, learning is really fast. Family members and friends also work in mathematics and helped me fill in a lot, and now I have some understanding of my own, so I also want to use my pen as much as possible to write a small popular-science article for everyone, so that as many people as possible can understand this shock.

So fasten your seatbelts; this journey officially begins.

1. What happened?

These 722 mathematics manuscripts were classified into 372 related results, or in other words, into 372 result groups.

The term "result group" needs some explanation. Mathematical results are a bit like having children: when a main result comes out, it often brings along a string of corollaries, supplementary arguments, or another proof from a different angle. So OpenAI grouped these related things together and counted them as one group.

So how did these results come about?

OpenAI's explanation is that the existing mathematics evaluations they had on hand had already been maxed out by AI.

In fact, it can already be seen that today's mathematics benchmark evaluations are approaching saturation. In the International Mathematical Olympiad, the competition attended by the smartest high school students in the world, AI reached gold-medal level last year. Various mathematics problem banks have also been more or less exhausted; further increases in scores no longer prove much.

So how can the model's true mathematical ability be evaluated further?

So OpenAI selected about 4,000 unsolved research problems in mathematics—what mathematicians commonly call "open problems"—and fed them all at once to that undisclosed internal model.

In the end, they filtered out the results of some significance, which are these 372 groups.

Here, no one needs to doubt OpenAI's model's ability in scientific research. Although it has been completely crushed by Claude in programming, compared with programming, GPT models have always been SOTA in scientific research capability.

And that undisclosed internal model, on average, spent about 3 hours of ChatGPT Pro-level thinking compute on each result.

This is the model we usually see on ChatGPT.

Three hours—many people can't sit still just seeing this number.

You should know that some of these problems are truly the crown jewels that human mathematicians have pondered for decades but still have not solved.

Then Altman tweeted that day, sharing his feelings.

II. What Is Mathematics?

After the release, mathematicians were in an uproar, and this uproar displayed every facet of human nature.

Alex Kontorovich, chair of the Department of Mathematics at Rutgers University and a number theorist, saw the "quasi-Riemann hypothesis" item and said directly, "Quasi-Riemann???? Are you kidding me?" He then said that if this had been done by human mathematicians, "it would be a Fields Medal, no question."

There were also rather awkward moments.

To understand why mathematicians are so excited, I think we first need to know one premise: how mathematics differs from other disciplines.

My own shallow understanding is two words: proof.

Physics, chemistry, biology, and the like often rely on experiments. But no matter how beautifully you conduct an experiment, the theory may still be overturned—just as Newtonian mechanics dominated for more than two hundred years and was ultimately revised by relativity.

Mathematics is different. Mathematics starts from a few basic axioms, and every step is derived with rigorous precision. Once a proof holds, under the same axioms and rules it will always hold; it will not become outdated, nor will it be overturned.

Just as the conclusions Euclid proved in the Elements twenty-three hundred years ago still hold today.

Perhaps this is still too abstract. Let's look at a real proof—don't worry, elementary school math is enough.

The question is: Is there a largest prime number?

We all know that a prime is a number divisible only by 1 and itself, such as 2, 3, 5, 7, 11, 13. The further along you go, the sparser primes become. So could they completely stop somewhere?

Euclid proved it like this.

Suppose primes are finite. Then we can certainly write all the primes in the world on a list. Next, we multiply all the primes on the list together and add 1. For example, suppose there are only three primes in the world: 2, 3, and 5. Then compute 2×3×5+1=31.

This new number leaves a remainder of 1 when divided by 2, by 3, and by 5—that is, none of the primes on your list divides it evenly.

So either it is itself a prime, or it can be divided by some prime not on the list. Either way, it shows that your list is incomplete.

Therefore, primes cannot be finite; there are infinitely many of them.

This is one of the most famous proofs in the history of mathematics: the theorem that there are infinitely many primes.

Just these few lines, and for more than two thousand years no one has been able to find a flaw. This is the power of mathematics.

So what is a "conjecture"? It is something everyone believes to be true, but no one has yet proved.

The conjecture we are most familiar with is probably Goldbach's conjecture: any even number greater than 2 can be written as the sum of two primes. In 1978, a piece of reportage by Xu Chi made Chen Jingrun a national idol, precisely because he had gone the furthest on this problem.

More than a decade ago, computers had already verified Goldbach's conjecture up to a number with 4 followed by 18 zeros, and in recent years they have continued pushing higher, without finding a single counterexample. But in the eyes of mathematicians, this is still not enough.

Because what mathematicians want is "all."

Mathematics has suffered from this kind of thing before. There is something called Pólya's conjecture. It was verified from 1 all the way up through millions and tens of millions and held true, but then at 906150257, wouldn't you know it, it failed.

So verifying a trillion examples is still not as good as one proof.

But then the problem arises: proofs themselves can also contain errors.

In 1637, Fermat wrote a conjecture in the margin of a book and casually left a sentence, roughly meaning: I have discovered a truly marvelous proof, but this margin is too small to contain it.

And so this sentence tormented mathematicians for more than 350 years.

In 1993, the British mathematician Wiles announced at Cambridge that he had proved Fermat's Last Theorem, and the hall erupted in applause. But during peer review, a gap was found, and he went into seclusion for more than a year to fill it. In the end, that proof ran to more than a hundred pages, and only a handful of people in the world can fully understand it.

This brings up a very practical problem: proofs are getting longer and harder, and human energy is simply not enough. What to do?

So some people began to wonder whether machines could check them.

This is the Lean that appears repeatedly in OpenAI's news this time.

Friends who write code can understand it this way: Lean is the compiler of the mathematics world. You translate a proof line by line into code that Lean can read. If it compiles, it means every step of reasoning is compliant and nothing has been cut corners.

Of course, it also has limitations, mainly two.

First, if you copied the problem down wrong from the start, or secretly swapped in a simpler version, the compilation will still pass, because the machine only cares whether your proof is correct, but it doesn't much care whether what you proved is the original problem.

Second, it can only tell you "right or wrong"; it cannot tell you "new or not," "important or not," or "beautiful or not."

These two limitations are basically the focus of all subsequent discussions among mathematicians.

III. This time's "dizzying collapse into a seated stupor"

Among the 372 result groups this time, the most explosive and most dizzying, collapse-into-a-seated-stupor result is inevitably the "quasi-Riemann hypothesis."

Let's return to the primes we mentioned above and review. We said earlier that 2, 3, 5, 7, 11, 13... these numbers that can only be divided by 1 and themselves are primes.

They are the atoms of the integers, meaning that any integer greater than 1 can be decomposed into a product of several primes, and the decomposition is unique. Much of the encryption in your phone today is also supported by primes.

Primes have a very strange characteristic: viewed individually, they are a complete mess.

More than two hundred years ago, Gauss, then a teenager, discovered that however messy they are, there is an overall trend: the further along you go, the sparser they become. Near 1 million, there is about one prime in every 14 numbers; near 1 trillion, there is only about one in every 28 numbers.

But this is only approximate, so for centuries, what everyone has been trying to understand is this: behind these seemingly chaotic primes, what pattern is hidden?

In 1859, Riemann, then 33, took a completely different approach. He published a paper of fewer than ten pages that offered another line of thought: instead of studying primes, study something called the zeta function. But this function can be written in a form directly related to all primes.

I know this may be a bit convoluted. You can crudely understand it as: the zeta function is like a formula machine into which all prime information is compressed.

Here we need to know a concept called a "zero," and this "zero" is a zero in the mathematical sense.

We all know that a function can be understood as a machine: you feed in a number, and it spits out an answer.

For example, everyone can surely understand this formula.

Feed in 3, and the answer happens to equal 0, so 3 is called a zero of this function.

The zeta function is the same, except it is certainly not this simple, and its input is not ordinary numbers but complex numbers. You don't need to understand complex numbers; you only need to think of them as a coordinate on a plane.

If some coordinate is fed into the zeta function and the result happens to equal 0, then that point is a zero of the zeta function.

Where these zeros are has an extremely deep relationship with how the primes are distributed.

We have long known roughly at what rate the primes become sparser, but how far can the actual number of primes deviate from this average rule? The positions of the zeta function's zeros control this kind of fluctuation.

But Riemann himself could not prove this conjecture, nor could later generations, so in 1900, Hilbert included it among the famous 23 problems, and in 2000, it became one of the seven Millennium Prize Problems with a $1 million bounty.

Since it cannot be proved head-on, mathematicians have pushed in bit by bit from the edges. Mathematicians have long known that the real parts of these important zeros all lie between 0 and 1, with 1/2 exactly in the middle. Since they cannot directly prove that they all lie on 1/2, they retreat one step and, starting from the side of 1, prove piece by piece that "there are no zeros here," to see how far they can push toward 1/2.

The first step was taken in 1896, when two mathematicians, Hadamard and de la Vallée Poussin, independently proved that where the real part is exactly equal to 1, there is not a single zero. Do not underestimate this step. The grand trend that Gauss saw back then, that primes become sparser the further along you go, was only rigorously proved thanks to it, becoming today's "prime number theorem."

But then they got stuck. For more than a hundred years afterward, what mathematicians could prove was only that there are no zeros in a thin sliver pressed close to 1. The best record in this area stopped in 1958, and for nearly seventy years afterward, there was almost no substantive progress.

So mathematicians retreated another step and set a lower-tier goal: not to prove all the way to 1/2 in one go, but only to prove that there exists some number less than 1, say 0.99, such that wherever the real part exceeds it, no matter how high you go, there are no zeros.

This lower-tier version is called the "quasi-Riemann hypothesis," but even for it, no one has managed it for more than a hundred years.

But OpenAI this time claims to have done it, and in one step pushed it to 7/8, that is, 0.875.

Where the real part is greater than 0.875, from bottom to top, there is not a single zero.

That is why the big names said that if humans had done this, it would be a Fields Medal, with no suspense at all.

The quasi-Riemann result is only one of the bombs. Let me give two more heavyweight examples.

1. Matrix multiplication.

English text is being prepared.

The textbook method has a computational cost on the order of the cube of the table's side length. In 1969, the German mathematician Strassen discovered that it could be faster and pushed this exponent down to 2.81.

Since then, top scholars around the world spent more than fifty years pushing it down bit by bit, until it reached around 2.37. At that point, moving the third or fourth digit after the decimal point was basically enough to publish a decent paper.

Then, OpenAI claimed that over the complex field, it achieved 2.25.

2. The Kakeya conjecture.

In 1917, the Japanese mathematician Sōichi Kakeya asked a question that seemed almost childish: if a needle is placed on a table and made to turn around in place, what is the smallest total area it must sweep over?

Intuitively, we all know there should be some minimum value, but later someone proved that the answer is unexpected: it can be made arbitrarily small, so small as to be almost zero.

This problem later gradually evolved into a profound conjecture in higher-dimensional space, involving several branches of mathematics. Its three-dimensional version was solved in 2025 by Wang Hong and Joshua Zahl, and is regarded as one of the most important breakthroughs in recent years.

In July of this year, Wang Hong received the Fields Medal, becoming famous around the world and the third woman in the award's ninety-year history to win it.

This time OpenAI claims that the four-dimensional version has also been solved, and that on the three-dimensional side they have pushed forward an even stronger version...

Besides these, the list also includes major results related to the unique games conjecture, special cases of the Hodge conjecture, faster integer multiplication... Any one of them, if picked out at random, I think would have been front-page news in the past.

If all of these hold up, saying "unprecedented in the history of mathematics" would be no exaggeration at all.

In 1956, the year the Dartmouth conference was held and "artificial intelligence" formally made its debut, a program called the "Logic Theorist" proved dozens of theorems from Russell's monumental work Principia Mathematica. At the time, everyone thought that machine proof of mathematics was just around the corner.

As it turned out, that wait lasted decades.

In 1976, the four color theorem was proved. Many people know this theorem: it says that any map can be colored with only four colors so that neighboring countries always have different colors.

The two mathematicians who proved it had computers run for thousands of hours, checking thousands of cases one by one.

The mathematics community immediately erupted in argument: does a proof that no human could possibly read through to the end really count as a proof?

This was the most famous debate in the history of mathematics over whether machine proofs count.

In 1998, an even more troublesome case came along. The American mathematician Hales proved the Kepler conjecture, that is, the problem of stacking oranges at a fruit stand: how to stack them to take up the least space.

The answer is the very stacking method fruit-stand vendors have known all along, but the proof was extremely complicated and used a large amount of computer calculation.

The referees looked at it for several years, and the gist of their final opinion was: we are 99% confident that it is correct.

99% is a very awkward number in mathematics.

Hales gritted his teeth and simply launched a project to formally verify the entire proof from beginning to end by machine. That verification took more than ten years and was only completed in 2014.

From then on, the path of machine verification finally began to catch on. Later came Lean, which we mentioned above, and Mathlib, an open-source library that mathematicians around the world fill in together.

But at that stage the division of labor was very clear: humans did the thinking, and machines were only responsible for calculation and verification.

The turning point came in the era of large models.

In the summer of 2024, DeepMind's AI reached silver-medal level at the International Mathematical Olympiad. In the summer of 2025, AI systems from several companies reached gold-medal level, including OpenAI's.

Then, at the time, a big joke broke out.

In October 2025, an OpenAI vice president tweeted excitedly that GPT-5 had solved 10 previously unsolved Erdős problems.

Erdős was a legendary Hungarian mathematician. The man never had a fixed residence; he traveled the world with a suitcase, collaborating with whoever he stayed with, and published about 1,500 papers in his lifetime. He also liked to put bounties on hard problems, ranging from tens of dollars to tens of thousands of dollars, and left behind more than a thousand problems.

But it did not take long before people dug up that GPT-5 had not solved these problems on its own at all; it had, damn it, found in the vast ocean of literature answers that someone had solved long ago but that everyone had forgotten...

The real turning point was May 2026.

An OpenAI model overturned a geometric conjecture proposed by Erdős in 1946. This time, nine external mathematicians, including Timothy Gowers of Cambridge University, wrote accompanying papers specifically for it. Gowers is a Fields Medalist, and he said that he would not hesitate to recommend this result to a top journal.

Then, the pace picked up.

May, 1 result; August, 10 released at once; September, more than a hundred announced; October, 372 groups of results.

In 5 months, it increased by two orders of magnitude. I think that would leave anyone dumbfounded.

There is another change that is very easy to overlook, namely cost.

In September, when OpenAI attacked the Navier–Stokes equations, it used about ten thousand agents, ran for 88 hours, and burned millions of dollars in compute.

By October, OpenAI said that almost every result was produced by "one prompt, one agent," averaging 3 hours.

Moreover, on the mathematics track, OpenAI is far from the only player.

The system DeepMind released in May solved 9 of several hundred Erdős problems, and all the results withstood machine checking. Anthropic spent 11 days translating the entire proof of Fermat's Last Theorem into Lean code, a full 13 million lines.

The entire industry has already lost control of the brakes.

V. The mathematicians strike back

Before today, there had in fact been an even more dramatic confrontation between mathematicians and OpenAI.

This is because on September 8, OpenAI announced that it had achieved a major breakthrough on the Navier–Stokes millennium problem.

The Navier–Stokes equations are one of the seven Millennium Prize Problems, with a $1 million prize.

This was originally a tremendous piece of good news, but behind it there was a lot of drama.

OpenAI itself admitted that on September 1 they heard a rumor that two millennium problems had been solved, and only then urgently launched this project. Later they learned that the rumor was connected to two people: Tristan Buckmaster, a professor at New York University, and Levent Alpöge, an employee of their bitter rival Anthropic.

About 12 hours before OpenAI issued its announcement, these two rushed to post their paper and Lean code online. However, what they proved was a result for the Euler equations in the presence of external forces, which is not the same result as OpenAI's Navier–Stokes proof, though the direction is very close.

Immediately afterward, Buckmaster publicly alleged that OpenAI researcher Bubeck, in a phone call, pressured him over the issue of paper authorship, including demanding that Alpöge be excluded because he works at Anthropic, and also said some things that he understood as threats.

Anyway, the two sides started cursing at each other. You should know that this is the normally quiet world of mathematics; this affair has been extremely melodramatic.

Three days later, on September 11, a joint letter was released.

It was signed by 25 Fields Medalists, a list that included Terence Tao, Germany's Scholze, Ukraine's Viazovska, Belgium's Deligne, and the Chinese mathematician Deng Yu.

The gist was that AI companies treat "solving problems" as a yardstick for measuring how strong their models are, which is severely misaligned with the goals of mathematics itself and is harming mathematics and the mathematical community. What is truly precious about mathematics, especially abstract mathematics, has never been merely obtaining an answer; it is about enabling humanity to gain new understanding and insight.

Within a week, more than seven thousand people had signed.

Looking further back, there was an even earlier foreshadowing.

According to Wired magazine, in August OpenAI had convened a closed-door meeting of about 40 mathematicians. Bryna Kra, former president of the American Mathematical Society, recalled that the atmosphere at the meeting was one of extreme excitement and extreme fear at the same time.

The mathematicians present pleaded with OpenAI: "Don't dump everything out all at once, and don't release mathematics like an influencer, relying only on posting tweets."

On September 21, OpenAI claimed that it had solved more than 100 open problems, but gave no details. The independent advisory group based at the Institute for Advanced Study in Princeton, mentioned at the beginning, specifically sought the mathematics community's views on this situation.

On September 29, this advisory group collected more than 600 pieces of feedback from the mathematics community and issued a set of recommendations. Then it said very sternly: they do not approve of AI labs using proprietary models that outsiders cannot access to test frontier mathematics problems, and demanded that this practice stop immediately.

As for those so-called more than 100 results, they demanded:

Put them into an academic archive not controlled by the lab; disclose the model name, prompts, reasoning process, how much time and compute were spent; formalize as much as possible; and report failed attempts as well. They also specifically added one point: don't use mathematical results to advertise your own models.

March 2016, Seoul, AlphaGo versus Lee Sedol, 4–1.

In the second game, AlphaGo played the famous "Move 37." The live commentators initially thought it was a machine error, and only later realized it was a brilliant move no human would have thought of.

In the fourth game, Lee Sedol played the move known as the "Hand of God," Move 78, winning humanity's only game. Three years later, Lee Sedol announced his retirement, saying that AI is an existence that cannot be defeated.

In 2017, AlphaGo beat Ke Jie, then ranked world number one, 3–0 in Wuzhen. Ke Jie shed tears at the board.

After that, things became very quiet.

I think every discipline in the future may follow the same path as Go.

But to this day, I think what humanity has always lacked most is never intelligence, but something else:

Will physics, chemistry, astronomy, and biology all be rapidly conquered field by field, just like mathematics? Will humanity usher in a true technological explosion?

In The Three-Body Problem, it happens that Luo Ji experienced a technological explosion.

All our disciplines will definitely undergo a technological explosion, definitely, except that the speed will differ. That speed depends on only one factor:

The verifier, or rather, how expensive this referee is.

Mathematics' referee is Lean; running it takes a few seconds. Chess's referee is a game, tens of minutes.

Both of these are very fast.

But chemistry's referee is the laboratory; a new material has to be synthesized and tested, taking weeks or months.

New drugs' referee is clinical trials; a drug from discovery to market used to often take ten years.

High-energy physics' referee may be a collider costing billions of dollars.

Astronomy's referee, well, that may require waiting for the next telescope to launch.

Therefore, the technological explosion must also follow this principle: the first to explode will definitely be mathematics and code, because the referee is almost free, followed by fields that can be simulated on computers, such as weather, proteins, and materials screening.

Finally, there is hard science, where experiments must actually be done by hand, and everything that involves people—medicine, for example.

One is an experiment in the real world; the other is people.

People have to verify, to understand, to decide which direction is worth funding and worth taking risks on.

By then, humans themselves will become the slowest link in scientific progress.

In fact, this scene has already played out in mathematics ahead of time.

Mathematics is now standing at the same crossroads: generating proofs has become cheap, but reviewing proofs and understanding proofs have become the scarcest things.

At the International Congress of Mathematicians that summer, the German mathematician Hilbert gave a lecture and proposed a set of mathematical problems he considered the most important of the new century; they were later compiled and published as the famous 23 problems.

These 23 problems supported mathematicians for an entire century; whoever solved one of them could go down in history, and the direction of mathematics in the entire 20th century was, to a large extent, shaped by this list.

In September of that year, Hilbert, already famous and successful, gave a radio address in his hometown.

But few noticed that the day before his speech, at an academic conference in the same city, a 24-year-old young man, Gödel, lightly announced a result.

Later, it became known as the "incompleteness theorem."

"As long as the axioms of a mathematical system can be clearly listed one by one, are strong enough to describe the most basic arithmetic, and are themselves consistent, then there must exist some propositions within it that, relying on this set of axioms alone, you can neither prove true nor prove false."

One of Hilbert's dreams back then was to build an impregnable foundation for mathematics, to axiomatize mathematics, and then to use rigorous mathematical methods to prove that the whole building would not collapse.

Gödel said, however, that as long as the building is complex enough, you cannot stand entirely inside it and use the building itself to prove that it will never collapse.

I tell these two stories because my family members and friends who do mathematics tell me that, in their understanding, mathematics has never been merely "for knowing the answer."

But this craft of mathematics has been practiced by humans for thousands of years—from Euclid to Fermat, from Gauss to Riemann—and its most fascinating aspect has never been the final Q.E.D.

It is the lifelong process of pursuit and understanding.

The British mathematician Hardy said that mathematicians, like painters and poets, are makers of patterns.

First, there is asking questions. Which questions are worth asking, which direction to go in—behind this lie human curiosity and accumulated aesthetic judgment.

Second, there is understanding. A machine gives a hundred-page proof; it is correct, but what exactly does it show? What connections does it have to other things?

Third, there is meaning. You see, with cars, people still run marathons; with cable cars, people still climb mountains. After AlphaGo, Go players did not disappear either; they began to learn Go from AI and to reunderstand this ancient game. Often, we do something never entirely for the result, but also for meaning.

Finally, let me end with Hilbert's sentence.

We must know.

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