Terence Tao: Math 2.0 [pdf]

Math 2.0 Terence Tao October 2026 Terence Tao Math 2.0 AI disruption of mathematics The advent of modern AI tools is disrupting many long-established professions. My own field of mathematical research – which has been largely stable for almost a century – has proven to be particularly susceptible to this type of disruption. Terence Tao Math 2.0 The historical era of proof scarcity Traditional mathematics (or “Math 1.0”) operated under the assumption of proof scarcity. Finding a solution to an open math problem (and proving it to be correct) was difficult and rare, and often required significant human expertise to obtain. Many of the institutions, traditions, incentives, and stated objectives of Math 1.0 have been organized under this proof scarcity assumption. Terence Tao Math 2.0 Objective verifiability On the other hand, proofs have the property — unusual in most other disciplines — of being objectively verifiable. Different mathematicians may disagree on what types of mathematics are interesting or fruitful. But there is broad consensus on what constitutes a correct and complete proof of a given problem. In fact, it is even possible to mechanically check proofs by computer, if they are written in an appropriate proof assistant language (e.g., Lean). Terence Tao Math 2.0 Digitizability and high-quality data Proofs also have the property being highly digitizable; they do not require a physical instantiation, any real world data as input, or any real world experiment to verify the output. In principle, a proof can be generated and verified by pen and paper, using only standard mathematical textbooks or libraries to supply definitions. The solution of a complex math problem often also requires knowledge of the appropriate research literature; but an enormous amount of this literature is now available digitally. Terence Tao Math 2.0 The era of proof abundance This combination of objective verifiability, digitizability, and large, high-quality data sets have made the task of solving mathematical problems particularly amenable to the machine learning techniques that power modern AI. Frontier AI labs have thus prioritized advances in mathematical problem-solving capability, ahead of other applications which lack one or more of the above three features. As a consequence of this massive investment, we are now entering an era of proof abundance in mathematics. Terence Tao Math 2.0 The jagged capability of AI Given sufficient compute and access to frontier AI models, one can now obtain solutions to many open problems that were previously thought difficult, in relatively short amounts of time. Increasingly often, no special expertise from the user prompting the AI is required to obtain such solutions. This is in stark contrast to current AI performance on tasks which are subjective, dependent on real world interactions, or for which data is scarce. AI performance is thus extremely jagged: astounding in some directions, while inadequate in others. This is true both within mathematics, and more broadly. Terence Tao Math 2.0 The new frontiers of mathematics Despite the impression given by press releases and social media, most of the problems at the current frontier of mathematics remain unsolved, even with massive computational expenditure. Indeed, the space of mathematical problems remains infinite; and the difficulty level of such problems is unbounded. Because of this, the future frontiers of mathematics will always continue to expand. Terence Tao Math 2.0 Aligning goals with the long-term health of math Furthermore, problem solving, while historically one of the most visible goals of research mathematics, is only a proxy for more valuable long-term goals. This proxy was largely accurate in the era of proof scarcity, but has become misaligned in the era of proof abundance. Further blind optimization of problem-solving alone is now actively harmful to the long-term health of mathematics. The future of mathematics — “Math 2.0” — will require both expanding the research frontier, while simultaneously decentering the traditional role of problem solving. Terence Tao Math 2.0 Reporting bias Wow, all the math problems I am seeing on social media are being solved by AI! Terence Tao Math 2.0 More transparent and scientific asssessments of AI capability, that fully report key statistics such as resource consumption and negative results, are urgently needed. First Proof (https://1stproof.org/) is one effort in this direction. Terence Tao Math 2.0 The frontier of mathematics All mathematical tasks · · · · · · · · · · · · · · · ... ... Humans + AI Humans AI Current press release and social media activity More imaginative and creative uses of AI Math 1.0 research frontier Math 2.0 research frontier Unreported failed AI attempts Once we adopt AI more imaginatively, the research frontier of mathematics will expand significantly. Terence Tao Math 2.0 A new era of experimental mathematics We now have the capability to collect large-scale mathematical data sets that were impossible or hugely impractical to generate in the past. Early examples include the Equational Theories Project and the SAIR/LMFDB Inverse Galois Problem challenge. With such data sets, one can perform new types of experimental mathematics: “population surveys” rather than “case studies”. This will help map out the new “difficulty landscape” of the research frontier, which has a significantly different shape now that powerful AI tools are available. Terence Tao Math 2.0 Equational Theories Project (2024–2025) The ETP crowdsourced both human and automated arguments to settle over 22 million true/false statements in universal algebra. Terence Tao Math 2.0 Inverse Galois Problem challenge (2026) The first stage of IGP24 was a successful crowdsourced competition to locate degree 24 polynomials with each of the 25, 000 possible Galois groups at this degree. The second stage will be a collaborative effort to extend up to degree 31. Terence Tao Math 2.0 Beyond problem-solving Contrary to popular opinion (or some of our own marketing), mathematicians are not singularly focused on solving open problems. Particularly in pure mathematics, open problems serve as “lighthouses”: not destinations to be reached in and of themselves, but as useful guides to explore the mathematical landscape around these problems. The lessons learned while attempting to solve these problems — whether they succeed, fail, or achieve partial progress — are often more valuable than the final solution to the problem itself. Reaching these lighthouses prematurely by automated tools can disrupt the exploration of the paths not taken, and sterilize the surrounding field. Terence Tao Math 2.0 A naive view of problem-solving Before AI: Math problems Math solutions After AI: Math problems Math solutions AI Surely this is a good thing? Terence Tao Math 2.0 A more accurate model Applied math problems Aligned applied math solutions Misaligned applied math solutions Human understanding Pure math problems Pure math solutions (objectively verifiable) (requires subjective judgement) A simplified view of the “Math 1.0” problem-solving “ecosystem”. Open problems are solved in a renewable cycle; and the insights gained from solving pure math problems can be safely transferred to applied problems. Terence Tao Math 2.0 Applied math problems Aligned applied math solutions Misaligned applied math solutions AI Human understanding Pure math problems Pure math solutions (objectively verifiable) (requires subjective judgement) AI Indiscriminate use of AI to solve problems in a non-renewable fashion damages the long-term health and progress of the field, as well as safe transfer to messier, real-world applications. Terence Tao Math 2.0 A thought experiment on alignment and understanding Suppose an advanced AI is prompted to “find a cure for cancer that passes a stage 3 clinical trial”. After a large amount of compute, it produces a cocktail of previously unknown chemicals which its mathematical model predicts, when mixed and injected into a patient, will kill all their cancer cells. While nobody truly knows how this cocktail was found, this model prediction is confirmed in Lean, and the cocktail indeed passes a stage 3 trial. Could the AI solution to the prompt be somehow misaligned by exploiting a weakness in the trial process? Before injecting this cocktail into your bloodstream, would you want to know that there is at least one human cancer expert who understands the mechanism behind this cure? Or a human mathematician who understands the mathematical model used to locate the cocktail? Terence Tao Math 2.0 Applied math problems Aligned applied math solutions Misaligned applied math solutions Human understanding Pure math problems Pure math solutions (objectively verifiable) (requires subjective judgement) +AI +AI +AI +AI The more sustainable “Math 2.0” use of AI would be to strengthen all the ways in which human understanding interacts with mathematics. Terence Tao Math 2.0 Examples of “Math 2.0” usage of AI “Open exposition problems” as opposed to open research problems: finding motivated explanations of results whose known proofs are currently opaque. (The newly revised Caltech Mathathon is an interesting experiment in this direction.) Collating many research papers into a larger living network. (The Integrated Explicit Analytic Number Theory Network is an initial attempt at such a network.) Creation of large, high-quality libraries of formal mathematics, such as Mathlib. Using AI to find and highlight new principles, methods, or insights, rather than merely new proofs. Terence Tao Math 2.0 Maththon (Nov 13-15) is a student-run event to use open-source AI tools to find new and improved proofs to existing results for which the current proofs are unsatisfactory in one or more aspects. Terence Tao Math 2.0 The integrated explicit analytic number theory network (IEANTN) The IEANTN is a living “spreadsheet” combining a large number