Erdős was proper. By the point Tao was 24, he had made sufficient new discoveries to have his alternative of everlasting school positions; he finally determined to settle on the College of California, Los Angeles. Round that point, he met a younger English quantity theorist named Ben Inexperienced. The 2 started collaborating on a proof that sure sorts of patterns referred to as arithmetic progressions — during which the numbers in a set improve by a hard and fast interval, like 7, 10, 13, 16 — inevitably seem in giant collections of prime numbers, although primes look like scattered randomly alongside the quantity line. Their proof would change into the signature results of Tao’s early profession, contributing to his profitable the Fields Medal in 2006, and propelling him to the higher echelons of arithmetic.
Tao may have constructed a profitable profession with out collaborating with anybody, however that’s not the best way he appreciated to work. He considered working with different researchers as a main technique to uncover new concepts — take what you understand, pair it with what I do know, and see what occurs.
This strategy led Tao’s mathematical analysis to vary over an unusually broad set of matters, from analytic quantity concept, together with the Inexperienced-Tao theorem about prime numbers, to evaluation, the place he studied properties of the Navier-Stokes equations that describe the conduct of fluids, to algorithms for setting up MRI pictures from digital knowledge. (The MRI collaboration developed throughout conversations Tao had with Emmanuel Candès, a statistician then on the California Institute of Know-how, whereas they had been each dropping off their children at preschool.) This thirst for collaborative discovery additionally led Tao to do numerous his work in public. In 2007, he began a weblog, the place he started publishing common updates about his analysis. By that time, Tao was one of the vital well-known mathematicians not solely in his subject however on the earth. His posts acquired numerous consideration and typically led to lengthy exchanges within the feedback part, the place Tao enthusiastically participated. He did it as a result of he discovered it enjoyable, and since he hoped the dialog may generate new concepts.
Round that point, one other early math blogger had an identical thought. Like Tao, Timothy Gowers was a distinguished analysis mathematician with a style for public alternate. However reasonably than trusting serendipity to strike in his weblog’s remark part, Gowers needed to channel public vitality in a centered method. In January 2009, he printed a weblog publish asserting his want to facilitate a brand new form of “massively collaborative arithmetic.” He would suggest an issue in an open on-line discussion board, and “anyone who had something in any respect to say about the issue may chip in.” He named it the Polymath Venture.
Tao considered working with different researchers as a main technique to uncover new concepts — take what you understand, pair it with what I do know, and see what occurs.
Tao jumped in. Like Gowers, he understood that some math issues had been extra amenable than others to being solved by way of large-scale collaboration. The important thing, as Tao wrote in a touch upon Gowers’ preliminary publish, was to seek out issues that would “generate a lot of less complicated sub-problems … which might largely be labored on in parallel.” By breaking massive issues into particular person circumstances, completely different groups or people may work on their very own after which assemble their outcomes as items of a much bigger entire. On the similar time, Tao knew that maybe the most important problem with the Polymath mannequin can be organizing: moderating contributions and checking to be sure that all of the contributions had been appropriate.
For the primary Polymath mission, Gowers proposed bettering a outcome referred to as the Hales-Jewett theorem, which was about patterns that seem whenever you shade cells in a grid with one among two completely different colours. After a couple of months of labor, coordinated by way of hundreds of feedback by dozens of mathematicians, the group had proved a extra actual assertion about how these coloring patterns emerge. That fall, they launched the work as a first-of-its-kind math paper with the pseudonymous byline “D.H.J. Polymath.” Gowers’ experiment had been successful. It allowed many mathematicians — skilled and beginner alike — to work collectively and yielded a proof ultimately.
Over the following decade, there have been 15 extra Polymath initiatives, a few of which Tao led, and the initiative attracted mainstream consideration. On October 29, 2011, The Wall Road Journal ran an article referred to as “The New Einsteins Will Be Scientists Who Share” and reported that the Polymath Venture had “pioneered a brand new strategy to problem-solving.”
Samuel Velasco/Quanta Journal
But in different methods, the Polymath Venture was an thought earlier than its time. Tao discovered it exhilarating to be on the heart of a frenzy of mathematical exercise, however he acknowledged that the feedback part of a weblog was a restricted platform for collaboration. Huge open collaboration elevated the chance of a sure form of serendipitous discovery, however on the similar time it heightened the chances that any one of many many contributors would contribute a mistake. The one technique to guard towards error was for a moderator to rigorously verify all of the work. However that form of moderation bottleneck undermined the Polymath imaginative and prescient.
What Tao was actually after was an environment friendly new type of scientific discovery. And after some time, he got here to grasp that the Polymath mannequin was not it. To make it actual, he thought, some form of laptop verification can be wanted — a technique to verify contributions robotically, reasonably than by hand. However given the state of know-how within the 2010s, he may as properly have wished for passenger service to Mars.
Tao had been conscious of computer-verified arithmetic for years. He knew about a couple of success tales, however he additionally knew that formal math was nonetheless impractical, requiring much more effort than it was price generally. However, Tao was intrigued by its potential. Virtually uniquely among the many world’s elite mathematicians, he noticed the potential in new strategies for doing arithmetic.
In July 2022, partly to fulfill his curiosity, he started to prepare a workshop on all of the other ways computer systems had been helping mathematical analysis. He introduced on a staff of co-organizers, together with Kevin Buzzard, who was on the time the world’s most seen evangelist for formal arithmetic.
Going into the convention, Tao regarded Lean, software program that permits mathematical proofs to be written and checked as laptop code, as an advanced program that will take months to study. Buzzard satisfied him to offer it a attempt. Together with that encouragement, Tao felt a robust duty to steer by instance — if he was going to proceed selling machine-assisted proof, he wanted to begin making an attempt it himself.
On October 9, 2023, Tao posted on social media, “I’ve determined to lastly get acquainted with the #Lean4 interactive proof system (utilizing AI help as crucial to assist me use it).”
Tao felt a robust duty to steer by instance — if he was going to proceed selling machine-assisted proof, he wanted to begin making an attempt it himself.
On MathOverflow, a preferred on-line dialogue discussion board for mathematicians, Tao discovered a query about one thing referred to as Maclaurin’s inequality. He determined to reply it as an experiment in formalization. First, he wrote up the proof as a typical math paper. It was quick, solely 10 pages lengthy. Then he turned his consideration to his actual objective: seeing if he may formalize the straightforward proof in Lean.
Initially, Tao thought he may be capable to do it in every week, however he was rapidly confronted by the variations between writing math by hand and typing it in Lean. Tao noticed that the exhausting components of the proof had been simple to formalize in Lean, whereas the straightforward components took a shocking quantity of labor.
Within the common paper, Tao spent no time in any respect asserting that when you’ve got three numbers, all of that are larger than 1, their sum is essentially not less than 3. However Lean doesn’t abide assertions, and Tao needed to spend time digging up a lemma in Mathlib — a digital library of already-formalized arithmetic that Lean customers draw on when writing proofs — that proved the self-evident relationship. Equally, in casual math, it’s not essential to all the time specify which quantity system you’re working in. The quantity 3, for instance, is concurrently an integer, a pure quantity, and an actual quantity. In his unique paper, Tao may merely write “3” with out specifying what sort of 3 he had in thoughts. Nonetheless, in Lean he needed to spell it out. Tao discovered his proof stored failing to compile as a result of he had uncared for to specify the proper kind at completely different factors within the formalization. It wasn’t till practically a month later, on November 6, that Tao posted in a touch upon his weblog, “Only a comment that I’ve managed to formalize the outcomes of this paper in Lean4.” The outcome was minor, and the Lean code he had written to formalize it was horrible. But Tao was now formally a contributing member of the Lean neighborhood.
At the identical time he was studying Lean, Tao continued work on a number of different analysis initiatives. These included one together with his longtime collaborators Ben Inexperienced and Tim Gowers, in addition to Freddie Manners, a former pupil of Inexperienced’s and now a professor on the College of California, San Diego. It was an elite set of collaborators — Gowers, like Tao, had received a Fields Medal, whereas Inexperienced was among the many most adorned quantity theorists within the subject.
The group had a selected downside of their sights, one which revolved round a mathematical object referred to as a sumset. You probably have a set of numbers, it may be used to type one other, associated assortment: its sumset. The sumset is made by taking the sum of each distinctive pair of numbers within the first set. All these sums collectively type the sumset of the unique set.
If the unique set is filled with random numbers, then its sumset can be comparatively giant. A set of 10 random numbers has a sumset of about 50 numbers (and a set of 1,000 numbers has a sumset of about 500,000 numbers). But when, as an alternative of containing random numbers, the unique set follows some form of sample, its sumset can be a lot smaller as a result of many sums will seem a number of occasions (and also you solely embody every sum as soon as within the sumset). The set of the numbers 1 to 10 is one instance — its sumset solely comprises 17 numbers (not 50, as you’d count on if it had been a random assortment of 10 numbers), as a result of most of the sums repeat (1 + 6, 2 + 5, and three + 4 all equal 7, and also you solely enter 7 as soon as within the sumset).
Along with having a small sumset, the numbers 1 by way of 10 are an instance of an arithmetic development as a result of they improve by a relentless interval. A conjecture with roots within the Sixties by the pc scientist Katalin Marton asserts that this isn’t a coincidence. She predicted that units that produce small sumsets should additionally embody lengthy arithmetic progressions. Gowers, Inexperienced, and Tao had made headway on a refined model of this downside referred to as the polynomial Freiman-Ruzsa conjecture within the early 2000s however ultimately received caught. Then, in 2023, Tao, Inexperienced, and Manners picked it up once more with a watch towards introducing methods from likelihood concept that Manners had developed.
They realized that by combining these methods with Gowers’ earlier concepts, they could be capable to clear up the entire thing. They introduced Gowers into the collaboration, and the quartet made regular progress by way of the summer time of 2023. By late fall, they’d it. On November 9 — simply three days after Tao uploaded his first formal Lean proof to GitHub — they uploaded their proof to arxiv.org.
With Lean on his thoughts, Tao urged to his three co-authors that they may attempt formalizing their paper. The work appeared like a very good candidate for formalization each as a result of it was an vital outcome and since it relied on comparatively easy methods. They wouldn’t should spend months including prerequisite materials to Mathlib — many of the crucial definitions had been already there.
Nonetheless, Inexperienced, Gowers, and Manners weren’t particularly keen on taking the time to study Lean. So Tao set off on his personal — although he knew he doubtless wouldn’t be alone for lengthy. Any mission he led was doubtless to attract consideration.
On November 13, Tao kicked off a brand new channel in a Lean-focused chat group. “Hello everybody. I’m considering of beginning a mission to formalize in Lean4 the current proof of Timothy Gowers, Ben Inexperienced, Freddie Manners, and myself of the polynomial Freiman-Ruzsa (PFR) conjecture,” he wrote. He would use the channel to coordinate exercise on the mission and can be “glad to simply accept volunteers to contribute to this mission in no matter capability they really feel ready.” It was a reboot of the Polymath Venture, solely this time they had been formalizing an current outcome reasonably than making an attempt to show a brand new one — and all of the work can be verified by Lean, which means Tao wouldn’t should verify it himself.
Equational Theories, in Tao’s view, was the opening act of what he hoped would change into a brand new period of “experimental” arithmetic.
Inside a day, Yaël Dillies, a Ph.D. pupil at Stockholm College, had arrange a tough blueprint for the mission that divided the proof into 13 sections. Inside every part, Tao recognized the sequence of lemmas and definitions that wanted to be formalized. In a typical math paper, lemmas — less complicated outcomes that assist construct towards the proof of a bigger theorem — is perhaps about 20 traces lengthy, however for the PFR formalization, Tao broke the proof down into five-line lemmas. His objective was to make the proof as modular as doable, permitting many individuals to make small contributions.
For the primary week, many of the exercise on the thread was about formalizing fundamental ideas from likelihood concept that the proof required however weren’t but in Mathlib. Specifically, they needed to formalize Shannon entropy — a measure of the uncertainty or dysfunction in a knowledge supply, like a set of numbers. However together with formalizing math, Tao and the others spent that first week determining learn how to work collectively. Initially, the dialog was free-form, with Tao posting about what he thought wanted to be achieved and others chiming in with concepts about learn how to do it, a lot because the Polymath initiatives had unfolded in weblog feedback.
On November 22, Tao posted a listing of twenty-two excellent lemmas and wrote, “If you wish to declare a number of of those lemmas, please accomplish that by replying on this thread.” The replies flooded in: “I’d like to say the entropy of a uniform random variable :),” wrote Paul Lezeau, a Ph.D. pupil on the London Faculty of Geometry and Quantity Principle. “I’m gonna take a stab on the basic fibring identification,” replied Aaron Anderson, a Ph.D. pupil at UCLA.
Drawn by phrase of mouth, increasingly mathematicians joined the trouble. By the tip of November, Tao, like a harried volunteer coordinator, was writing little Lean code himself, as an alternative specializing in discovering duties for others to do. On November 28, he wrote, “Provided that we could briefly have a surplus of volunteers for the PFR mission because it nears completion, I considered one further small job that somebody is perhaps keen to work on.” Forty-six minutes later, Kim Morrison replied that they’d accomplished the duty. “Wow, that was fast! Thanks!” Tao answered.
Even earlier than the formalization was full, the Lean neighborhood started discussing what it meant. Specifically, they debated whether or not the effectivity of the mission signaled a brand new period of quick formalization, or whether or not it mirrored the singular affect of Terry Tao. In a wrap-up publish within the group thread, Tao mirrored that he had not written a lot of the code himself. “That is truly fairly encouraging to me, because it suggests to me that it is going to be doable for mathematicians to steer Lean formalization initiatives with out requiring in depth Lean programming abilities (although one may have not less than sufficient experience to have the ability to state lemmas, if not show them).” Eight minutes later, Johan Commelin, a mathematician at Utrecht College and the director of the Mathlib initiative, replied, “I don’t need to instantly hijack this thread,” earlier than happening to query whether or not the teachings Tao had discovered throughout the mission had been broadly relevant. “In fact you bought numerous assist with this mission due to its high-profile nature,” he wrote.
Commelin additionally famous that whereas initiatives like PFR had been enjoyable and thrilling to participate in, contributing to them was not the form of factor younger mathematicians had been acknowledged for after they utilized for tutorial jobs. “For the time being, it’s nonetheless not clear how formalizers (for lack of a greater job description) can be credited by the mathematical neighborhood, and the way these actions can be valued on the job market.” Tao replied, “For what it’s price, I’m very happy to say contributions to this mission in letters of reference as acceptable.”
By 2024, Tao had change into probably the most distinguished public voice touting the potential of machine-assisted arithmetic. He was three years into his tenure on President Joe Biden’s President’s Council of Advisors on Science and Know-how and had change into co-chair of a working group on generative AI. In a pair of high-profile speeches in 2024, he expressed his imaginative and prescient for a brand new form of mathematical collaboration: one which mixed human perception, the creativity of enormous language fashions, and the ensures of correctness supplied by formal verification methods.
He got here to this view partly as a result of he noticed clear limits on what present AI instruments may do. They excelled at fixing simple issues or duties with loads of prior knowledge, however on the frontier of arithmetic — the place there have been few printed outcomes and little knowledge to coach on — AI faltered. In his early experiments with LLMs, he noticed that they behaved like overconfident undergraduates, providing ideas with out the experience to inform the distinction between good and unhealthy concepts.
But Tao had a method ahead in thoughts. He didn’t suppose AI would exchange human mathematicians anytime quickly, however he did think about it significantly properly suited to serving to clear up sure forms of complicated mathematical issues: ones that may very well be damaged into hundreds of small, manageable subproblems — basically the identical class of issues that labored properly for Polymath initiatives. At that scale, mathematicians may make use of AI to unravel giant swaths of the best subproblems, with its outcomes outputted as formal proofs that Lean may verify, and step in to deal with probably the most troublesome remaining questions themselves. In 2024, Tao was selling this imaginative and prescient to anybody who would hear, and following the PFR mission, he had realized that if he actually believed within the work, he wanted to step up and lead it himself. He additionally knew instantly which downside he would begin with.
It was a query that he had stumbled upon a yr earlier. In July 2023, a person on MathOverflow posed a seemingly easy puzzle. Think about an operation like addition, the person wrote. It would observe sure basic algebraic legal guidelines just like the commutative legislation, which says x + y = y + x, or the associative legislation, which states that (x + y) + z = x + (y + z). In lots of circumstances, there’s no relationship between one legislation and one other — the commutative legislation doesn’t suggest the associative legislation, for instance.
The MathOverflow query involved the connection between two explicit legal guidelines, and one other person answered it rapidly.
However the query of how legal guidelines relate to one another on the whole caught Tao’s curiosity. Moderately than fixing puzzles one after the other, Tao started sketching out a tough diagram that confirmed how completely different doable algebraic legal guidelines relate to 1 one other. It grew to become clear the image is perhaps fairly sophisticated.
He noticed that if he restricted his examine to algebraic legal guidelines involving operations utilized precisely 4 occasions, there have been about 4,694 legal guidelines he needed to account for. Every legislation may doubtlessly suggest or fail to suggest another legislation, creating 22 million logical implications to verify. As soon as he had checked all of them — both by proving they held or by discovering a counterexample during which they failed — he would have a whole image of how all 4,694 of these legal guidelines relate to one another. It felt like precisely the suitable scale for the brand new model of arithmetic he was proposing.
Tao referred to as his new endeavor “Equational Theories” and introduced its formation in a publish on his private weblog on September 25, 2024. He opened by ticking by way of the principle causes large-scale public math collaborations had been exhausting up to now after which wrote, “Proof assistant languages, comparable to Lean, present a possible technique to overcome these obstacles.”
To start out, Tao and the rising variety of volunteers who joined him examined the greater than 4,000 legal guidelines towards easy mathematical buildings often called magmas. Magmas are stripped-down variations of arithmetic that made for a helpful start line as a result of any legislation that failed to carry for magmas couldn’t presumably suggest different, extra complicated legal guidelines. The contributors rapidly examined tens of millions of those simplified methods utilizing fundamental Python scripts and inside days had resolved greater than 99% of the 22 million potential implications. Tao posted on day 2 — September 27 — that he was astonished at how quickly the mission was advancing: “This mission has moved far, far faster, and scaled up a lot a lot sooner, than I had anticipated — solely 48 hours in and already a big fraction of the implications are more likely to be resolved quickly! I believed the 3-week PFR mission was quick, however that is an insane further degree of pace.”
As soon as the only implications had been resolved, the Equational Theories volunteers moved in a decentralized technique to automated theorem provers that would seek for options to issues all on their very own, with out interactive assist. These provers, together with old school human ingenuity, knocked down the open questions one after the other.
Like a scientist watching his personal creation come to life, Tao admired the work because it unfolded. “The mission appears to be efficiently decentralizing; specifically, there may be now numerous exercise happening now that I’m not totally conscious of,” he wrote.
To many mathematicians, Tao’s mission was intriguing however odd. Buzzard adopted alongside, fascinated by the social experiment although bored by the mathematical content material of Equational Theories. He thought it was each elementary and peculiar, although he admired Tao’s inventiveness. John Baez, one other distinguished mathematician, was extra blunt. He remarked, “This looks like a colossal waste of time to me,” earlier than acknowledging that he felt the identical method about school soccer, and loads of individuals appreciated that, too.
Inside a month, the Equational Theories group had narrowed 22 million questions right down to 238. By late November, they had been right down to 138. As they chipped away on the remaining circumstances, progress slowed. When the brand new yr started, about 30 implications had been left unresolved, and the speed of progress slowed even additional. By the tip of March, they’d been caught for a number of weeks on simply 4. Contributors tried their hand on the the rest, however with so few implications left, many individuals drifted away; Tao’s updates slackened from their near-daily cadence to as soon as each few weeks.
However settling each single one of many 22 million implications had by no means actually been the objective of Equational Theories. Out of sheer curiosity Tao had needed a map of the entire panorama, and now he had that, minus a couple of particulars. Extra importantly, he considered Equational Theories as a pilot mission for a essentially new method of doing arithmetic — and in that regard it was an unqualified success.
Equational Theories, in Tao’s view, was the opening act of what he hoped would change into a brand new period of “experimental” arithmetic. He had in thoughts the form of transformation that had already come to fields like physics. Physics had as soon as been a largely theoretical self-discipline the place solitary thinkers or small teams of collaborators tackled one or two issues at a time — in different phrases, it used to look so much like math nonetheless did. However with technological developments got here a brand new, experimental department of the sector — large collaborations at laboratories like CERN’s Giant Hadron Collider, the place lots of and even hundreds of researchers with specialised abilities labored collectively and generated large volumes of knowledge. These experiments didn’t exchange concept however complemented it, with new outcomes washing between the 2 modes of investigation.
Tao imagined an identical evolution taking place in arithmetic. He trusted that novel types of inquiry would inevitably result in novel insights, as they all the time had earlier than. Because the Equational Theories staff methodically crossed implications off their immense desk, they stumbled onto genuinely new mathematical constructions — like “magma cohomology,” an alien extension of the idea of group cohomology, a deep and well-studied subject describing when teams can or can’t be enlarged in sure methods. Tao reached out to John Baez — the Equational Theories naysayer and an professional in cohomology — to ask if this building had been seen earlier than. Baez admitted he had by no means encountered it.
To Tao, that was precisely the purpose. The mission had proven that arithmetic may very well be performed otherwise, experimentally — and in doing so, it had turned up one thing genuinely new. Tao had by no means anticipated Equational Theories to unearth a revelation; he needed it to display the efficacy of a brand new form of mathematical machine. And in that sense it labored. Terry Tao had discovered a brand new method of doing arithmetic, and he confirmed no indicators of going again.
Excerpted from The Proof within the Code: How a Reality Machine Is Remodeling Math and AI by Kevin Hartnett. Copyright © 2026 by Kevin Hartnett. To be printed on June 9, 2026, by Quanta Books in partnership with Farrar, Straus and Giroux. All rights reserved.
