Terence Tao – How the world’s top mathematician uses AI
In a Nutshell
Terence Tao likens Kepler's empirical discovery process to high-temperature LLMs generating and verifying wild hypotheses against data, arguing AI now floods science with cheap ideas, shifting the bottleneck to verification, evaluation, and distinguishing progress amid abundance. AI excels at breadth—solving low-hanging Erdős problems via existing techniques and enabling experimental math at scale—but lacks human depth for cumulative insight, partial progress, and novel breakthroughs, making human-AI complementarity key to future science. Tao boosts personal productivity 2x on auxiliary tasks like literature and visuals, predicts AI as co-author by 2026, and advises embracing serendipity and new paradigms as AI lowers barriers for frontier math contributions.
These notes were generated by AI and may contain inaccuracies.
Terence Tao retells the story of Kepler discovering the laws of planetary motion. Kepler built on Copernicus, who proposed the heliocentric model with the Sun at the center and planets orbiting in perfect circles. This fit observations from Greeks, Arabs, and Indians. Kepler noted ratios of Copernicus's predicted orbit sizes had geometric meaning, proposing enclosing Earth's orbit in a cube where the outer sphere matched Mars's orbit, and so forth. With six known planets and five gaps, matching five Platonic solids: cube, tetrahedron, icosahedron, octahedron, dodecahedron. Kepler theorized inscribing these solids between planetary spheres, seeing it as God's mathematical design.
He needed data from Tycho Brahe, a wealthy eccentric Danish astronomer who funded an observatory on an entire island. Brahe took decades of naked-eye observations of planets like Mars and Jupiter on clear nights, the last naked-eye astronomer. Kepler worked with Tycho, who jealously gave data in bits. Kepler stole/copied it, fighting Brahe's descendants to use it. His Platonic solid theory failed, off by 10%. After years of fudges, he analyzed data to find actual orbits were ellipses, not circles—shocking. He derived two laws: orbits are ellipses; equal areas sweep equal times. Ten years later, with more data (Saturn and Jupiter hardest), third law: orbital period proportional to some power of Sun distance. No explanation; all empirical. Newton explained all three a century later.
Kepler like a high-temperature LLM trying random relationships for 20 years, some nonsensical, verified against Brahe's dataset. In his book Harmonics of the World, an aside amid astrology (Earth's note mi-fa-mi causes famine) contains the third law (period cubed proportional to distance squared). Adding Newton's F=ma and centripetal acceleration yields inverse-square law.
LLMs can try random geometries, musical notes, Platonic solids with bias toward geometry, verify empirical regularities to drive progress. Idea generation (prestige part) involves identifying problems, collecting data, hypothesizing, validating. Kepler cycled ideas, many unpublished failures. Needs verification or it's slop. Celebrate Brahe's precise data (10x better), Kepler's Euclidean geometry matching.
Hypothesis generation no longer bottleneck. Paradigms: theory/experiment, then 20th-century simulations, late 20th-century big data. Now collect massive datasets first, deduce patterns—reversed from hypothesis-then-test. Kepler early data scientist but preconceived theories first; didn't start with Brahe's data.
20th-century science like Kepler: 1595-96 Platonic ideas wrong; 20 years random tries post-Brahe data yield regularities. Data like massive simulation bank enables verification, else just harmonics/Platonic books.
Kepler's third law like regression on six data points (orbit length, Sun distance per planet), fitting square-cube law—lucky. Johann Bode fit shifted geometric progression to same data, predicting missing planet in Mars-Jupiter gap. Uranus (Herschel) fit, Ceres (asteroid belt) fit—excited as new law. Neptune way off—numerical fluke. Kepler tentative on third law, sensing six points unreliable sans modern stats.
AI drives idea generation cost to zero like internet did communication—not abundance alone. Now thousands of theories per problem; bottleneck verification/evaluation. Structures must adapt. Past peer review filtered amateur theories; now AI-generated submissions overwhelm journals.
Need scale for thousands daily. Human science solved abundance somehow (e.g., 1940s Bell Labs: pulse-code modulation, digitizing signals over analog wires; one paper invents bit, implications across fields).
Test of time: great ideas often poorly received initially, fruit after others extend (deep learning niche long, controversial data-over-first-principles; trits vs. bits; transformer not inevitable). Depends on future, culture, adoption (base-10 useful due to standardization/inertia vs. Roman numerals).
Can't objectively grade in isolation sans past/future context; not reinforcement-learnable like localized problems.
Correct theories initially imply wrong/implausible things resolved later: Aristarchus heliocentrism (3rd c. BC) implied huge star distances (parallax); Leibniz vs. Newton action-at-distance; Newton stunned inertial=gravitational mass (Einstein resolved). Even falsifiable parts: progress despite worse fit initially (Copernicus less accurate than millennium-tuned Ptolemy; Kepler made it better).
Science work-in-progress; partial solutions look worse than completed incorrect ones (Newton's mysteries resolved centuries later by Einstein). Progress often deletes assumptions (Aristotelian rest → Newton's motion persists; geocentrism held as Earth motion feels absent).
Darwin: species not static (unobvious, no lifetime evolution). Now AI shows non-human intelligences reorder task intelligence assessments.
Origin of Species (1859) two centuries after Principia (1687). Darwin conceptually simpler; Huxley: "How stupid not to have thought of that." Nobody says re Newton. Evidence cumulative/retrospective vs. Newton's moon orbit test. Lucretius (1st c. BC) had adaptation idea, ignored sans experiments.
Progress faster in tight data loops despite conceptual difficulty.
Darwin masterful communicator: plain English, no equations, synthesized facts, compelling vision despite gaps (no DNA/heredity). Newton: Latin, invented math, secretive/competitive (held insights), unpleasant; simplified decades later.
Exposition/narrative crucial to convince, invest in learning theory. Hard to RL-score persuasiveness (marketing departments try). Social: data + narrative of gaps (Darwin predicted transitional forms, inheritance mechanism—found).
Cosmic distance ladder (recommend 3Blue1Brown series): deductive overhang huge; right insight unlocks much from incident Earth data.
Astronomy embraces data analysis, squeezes info as data bottleneck (still is). Astronomers Sherlock-like; quant hedge funds hire astronomy PhDs for signal extraction.
Shawn solved Jane Street puzzle for audience, posted walkthrough on X.
Jane Street trained a ResNet, shuffled all 96 layers, and challenged people to put them back in the right order using only the model's outputs and training data. You can't brute force this—there's more possible orderings than atoms in the universe. Shawn broke the problem into two parts: first, pair the layers into 48 different blocks; second, put those blocks in the right order. For pairing, Shawn realized that in a well-trained ResNet, the product of two weight matrices in a residual block should have a distinctive negative diagonal pattern. This arises as a way for the model to keep the residual stream from growing out of control. From this insight, he was able to recover the right pairings. For ordering, Shawn noticed that the model seemed to improve if he sorted the blocks by the size of their residual contributions. Starting with that rough approximation, he combined a clever ranking heuristic with local swaps to recover the exact right order. His full walkthrough is linked in the description. There's still one puzzle up about backdoored LLMs that even Jane Street doesn't know how to solve, at janestreet.com/dwarkesh.
We under-explore how to extract extra information from various signals. Study on how often scientists read cited papers: measured by how often typos (wrong number or punctuation) get copied from one reference to the next, inferring if authors copy-paste without checking. This gives a measure of attention paid. Clever tricks like this can extract insights. For assessing scientific developments as fruitful, interesting, or real progress, useful metrics or footprints might exist in data: citations, conference mentions. Sociology of science research could detect these.
Over the last few months, AI programs solved 50 out of 1100+ Erdős problems. As of a month ago, pause because low-hanging fruit picked. Still the case: 50-odd solved with AI assistance (great, but 600+ to go). People chipping away at one or two. Fewer pure AI one-shot solutions now—no month of that recently, despite attempts. Three separate attempts to get frontier model AIs to attack every problem simultaneously: they pick minor observations or find already-solved in literature, but no further purely AI-powered solutions.
People use AI a lot: generate proof strategy, critique/rewrite it, generate numerical data, literature survey. Some problems solved by ongoing conversation between lots of humans and AI tools. Analogy: mountain range with cliffs/walls of varying heights (3ft, 6ft, 15ft, mile-high) in the dark. Light candles, make maps, figure out climbable ones, partial tracks. AI tools like jumping machines (2m high, higher than humans): sometimes wrong direction/crash, but reach lowest walls previously unreachable. Exciting period where they found/reached all low ones. Next model advance might breach more.
Different style from normal math (hill climb, markers, partial progress). AIs bad at partial progress or identifying intermediate stages. No way to evaluate partial progress like one-shot success/failure.
Bearish view: AIs only reach certain wall height, not as high as humans. Bullish: once at waterline, fill every problem at that level (humans can't: can't make million copies with million dollars inference compute for 100 years subjective time on million problems). At Terence Tao-level or intermediate, do that version. Human-level AI intelligence qualitatively wider/more powerful than ours due to breadth.
Agree: AIs excel at breadth, humans (experts) at depth—complementary. Current math/science focused on depth (human strength), humans can't do breadth. Redesign science to leverage AI breadth: more effort on broad classes of problems, not just 1-2 deep ones. Keep deep problems for humans. Explore new fields: broad moderately competent AIs map it, make easy observations, identify islands of difficulty for human experts. Future of complementary science. Eventually both breadth and depth. Need practice with breadth—too new, no paradigms yet. Science will be unrecognizable.
Programmers more productive with AI. In software, understanding/abstraction instrumental to world effect. In research (e.g., Millennium Prize Problems), proof instrumental to discovering new objects/techniques advancing math understanding.
In math, process often more important than problem—proxy for progress. Software tasks vary: boilerplate offload to AI (no skill learned). Maintenance/upgrading/compatibility ongoing—even if AI prototypes, meshing with real world requires skills from writing code.
Math uses problems to build intuition, train on true/expect/provable/difficult. Getting answers inhibits that.
Distinction between theory and experiment: most sciences equal division, math almost entirely theoretical (premium on coherent clean theories). Little experimentation on which solution methods more effective—no large-scale studies on 1000 problems.
AI will revolutionize experimental math: gather large-scale data on what works/doesn't, not individual processes. Like software companies scaling workflows, not handcrafting. Math at scale is infancy—AI will revolutionize.
Crux: AIs use/modify existing techniques. How much progress from applying existing techniques? In top math journals: many papers use existing on new problems; some solve 80%, invent new for resistant 20%. Rare to solve with no past literature reliance—math mature.
First step on new problem: try standard past techniques one by one (sometimes works, publishable; sometimes almost, add wrinkle). AIs good at this: try standards, fewer application mistakes than humans (about tie now on error-catching).
AIs not yet at next step: holes where nothing works. Suggestions often waste time chasing. Some hard problems will fall, especially neglected ones. Of 50 AI-solved Erdős problems: almost no literature (Erdős posed once/twice, casual tries). Solution: combine obscure technique + literature result (median AI accomplishment).
Systematic studies: AI success rate 1-2% per problem, but scale picks winners (looks amazing on social media). Same for prestigious problems: some lucky solves via missed backdoor (publicity), but 1-2% rate on favorites. Need standardized datasets/challenge problems, not just AI companies publishing wins.
Progress amazing/disappointing: models apply unwritten techniques to problems. Strange feeling, but acclimatize quickly (e.g., Google search blew others away 20 years ago, now taken for granted; 2026 AI stunning in 2021, now face recognition/speech/college math routine).
2023 prediction: by 2026, AI like colleague/trustworthy co-author if used correctly—looking good.
Personally 2x productive? Productivity multidimensional. Style changing: papers have more code/pictures (easy to generate). Plot taking hours now minutes—or wouldn't include, just words. Papers today without AI: 5x longer. But wouldn't write same way.
Auxiliary: deeper literature, more numerics enrich. Core (hardest part): still pen/paper, unchanged. AI for reformatting (parentheses). Speeds secondary tasks, adds to papers. Rewriting 2020 paper same functionality: not much time saved—richer/broader, not deeper.
Intelligence vs cleverness: intelligence hard to define, know when see. Collaborative math solving: neither knows solution initially; idea → prototype → test/fail → modify; adaptivity, continual improvement; map what works/doesn't, path forward evolving.
AIs mimic somewhat, but like jumping robots: jump/fail repeatedly, no cumulative buildup (jump bit, handhold, stay/pull others up/jump from there). Trial/error, brute force repetition—scales, works in contexts. No interactive cumulative partial progress.
If Gemini 3/Claude 4.5 solves: no own math understanding progress. New session forgets, no new skills for related. Maybe 0.001% training data for next gen.
Plausible: keep training AIs better at Lean → solve impressive problems → surprised by little insight (e.g., Riemann hypothesis as assembly gobbledygook)? Or necessary: Lean constructions/definitions advance understanding?
Some solved pure brute force (e.g., four color theorem—no elegant proof yet, maybe never; enormous cases). Riemann: expect new math type or connection between areas—not exhaustive cases. Unlikely false (compute zero off line, massive verification—disappointing).
Fully autonomous one-shot not right for these. More mileage: humans + powerful AI interplay. Solve via new collaboration type: million zeta variants + AI data analysis → new pattern → transform problem. Latent new construction in Lean applicable widely—how recognize?
If equivalent of Descartes' coordinate system unifying algebra
Formalizing a proof in Lean allows studying any piece atomically. In a paper solving a difficult problem, there's often a big sequence of lemmas and theorems. Authors may not reveal which steps are important versus boilerplate or standard. In Lean, each lemma can be studied in isolation. Some look standard and familiar, with nothing interesting. Others are novel and key to the main result. Lean facilitates assessing whether a step is key, with individual steps identified precisely.
In the future, professions of mathematicians might take giant Lean-generated proofs and perform ablation, removing parts to find more elegant ways. Other AIs could do reinforcement learning to make proofs more elegant, with additional AIs grading if the proof looks better.
Changes in Paper Writing
Writing papers was previously the most time-consuming and expensive part, done rarely after all arguments were checked, as rewriting was painful. Modern AI tools make this easier. Instead of one version, hundreds can be generated. A giant messy Lean proof may not be meaningful alone, but others can refactor it. The Erdős problem website example: AI generates a proof with 3,000 lines of verifying code; other AIs summarize it; people write their own proofs with post-processing. Once one proof exists, tools deconstruct and interpret it. This is nascent but promising.
Some people are concerned about what happens if the Riemann hypothesis is proven with a completely incomprehensible proof. I think once you have the artifact of a proof, we can do a lot of analysis on it.
A formal or semi-formal language for mathematical strategies (beyond proofs, which Lean specializes in) would be helpful. Mathematics formalized laws of logic and axioms (ZFC, first-order logic) only in the early 20th century, started by Euclid. This enabled automation and formal languages like Lean for deductive proofs.
Assessing conjecture plausibility (e.g., testing examples) lacks formalization. Mathematical models like Bayesian probability exist but require subjective base assumptions. A semi-formal framework could assess plausibility semi-automatically without easy hacks. Formal proof assistants must avoid backdoors, as reinforcement learning finds exploits.
A framework mimicking scientists' semi-formal talk (data, argument, narratives) could insert AI usefully, capturing subjective aspects. Efforts exist for automated conjectures with benchmarks, but it's new science.
Intuition and Paradox Resolution
Example of unformalizable scientist communication: conjectures like Gauss's prime number theorem. Gauss computed first 100,000 primes, found statistical pattern: primes up to X are about X / ln(X). Data-driven, no proof, revolutionary as first major statistical conjecture in math. Started analytic number theory. Led to viewing primes as pseudo-random (not truly random, but behaving like random sets with density).
Twin prime conjecture: infinitely many twin primes (e.g., 11 and 13), unproven but believed via random model—random chance produces them infinitely often. Heuristic random model of primes is accurate; proven results match it. Underpins belief in Riemann hypothesis and prime-based cryptography security. Riemann hypothesis failure would reveal secret patterns, threatening crypto.
Conviction from experimental evidence and aligned theoretical results. Consensus could be wrong, as in past paradigm shifts, but hard to measure without data (one historical timeline, ~100 turning points). With million alien civilizations' histories, progress and strategies could be formalized.
Create mini-universes: simulations of AI solving basic arithmetic, evolving strategies. Investigate smallest neural network for 10-digit multiplication to learn from small AIs on simple problems.
Mercury sponsors; user opened account in 2023. Insights feature summarizes money in/out, biggest transactions, flags issues (e.g., revenue drop, uncategorized purchase). Low-friction for business monitoring. Example: track 2025 monthly spending to plan cash needs and invest excess. Mercury (fintech, not FDIC-insured bank; services via Choice Financial Group, Column N.A., FDIC members) adds features constantly.
Process combines depth/breadth (fox vs. hedgehog; collaborates with hedgehogs, can be one if needed). Obsessive streak: bugs when unfamiliar math achieves result one could do otherwise. Weans off computer games due to completionism.
- Collaborate: befriend mathematicians in other areas; they teach basics, known/unknown.
- Write blog posts to retain knowledge (past forgetting frustrated; now records cool tricks).
Blog time: half hour to several hours, done voluntarily when avoiding unpleasant tasks (e.g., referee reports). Feels creative/fun; time flies vs. drudgery. AI helps administrative tasks now.
From first principles, civilization might eliminate podcasts, committees. But senior academia adds responsibilities; reluctant events yield interactions outside comfort zone, networking, learning (e.g., this podcast). Believes in serendipity: optimize some day portions carefully, leave others unstructured—often positive unplanned experiences.
Danger in over-optimization (AI exacerbates). COVID remote meetings kept busy but lost casual interactions (hallway knocks, coffee chats). Library browsing accidentally found interesting articles; now search/AI gives exact matches, no serendipity.
Institute for Advanced Study: great initially (undistracted research), but prolonged stays cause boredom, internet surfing. Needs distraction for randomness/high temperature. No optimal life schedule known; current works.
AIs already do frontier math humans can't (e.g., number crunching like calculators). Within a decade, AI handles much of math students' time/papers. But that won't be most important part.
Historical shifts:
- 19th-century mathematicians solved DEs for physicists (now Mathematica/AI in minutes).
- Computers (once humans) outsourced log tables/primes (Gauss).
- Genome sequencing: once PhD project, now $1,000 sequencer; field moved to ecosystems.
Mathematical progress hybrid human-AI longer; needs breakthroughs. AIs good at some things, terrible at others; frameworks reduce errors but incomplete. Complementary now, accelerates science. Risk: destroys serendipity, inhibits progress. World unpredictable.
Embrace change; era unpredictable, centuries-old assumptions may fail. Prefer boring stability, but adapt. Some studies obsolete, others retained; eye new opportunities.
AI lowers barriers: high schoolers can contribute to frontier via AI/Lean. Non-traditional learning paths. Still need credentials/traditional education, but open to new science ways (some not existing yet). Scary but exciting.
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