AI and mathematical discovery

Alexander Pilakoutas

As a mathematician, I have always enjoyed empirical methods. I like being able to explore something, find patterns in its behaviour, and try to work out what is going on. This has been a large part of how I have approached research in the past, and I feel that my use of AI has mostly been a continuation of it. Which is why some of the recent discussion about AI-generated mathematics feels at odds with my own experience.

During the summer of my second year as an undergraduate, I worked with Simon Myerson on a project called ‘Quaternion Gauss Sums’. We were defining a new notion of quaternion Gauss sum and exploring its properties. I began by working on theory and developing the core definitions of the project, this enabled me to develop algorithms for evaluating the sums. Running these algorithms then allowed me to make conjectures about their behaviour. Over the ten-week project, I wrote code, ran experiments, and tried to prove the conjectures I was forming. After the project, I found that many of my results had counterparts in the literature for a related, but not identical, notion of matrix Gauss sums over finite fields. The project and report are available through Warwick’s URSS Showcase. [1]

What I enjoyed was the interaction between the theory and the computations. Developing the theory allowed us to define the concept and the calculations gave me new things to try to explain. Initially my computations were not efficient enough to calculate interesting examples but by trying this gave me targets to optimise my algorithms towards using theory. Once my conjectures were established (much more successfully than my proofs in this case) I began working on proving special cases of them hoping to work towards a more general theory. This has played a part in much more significant mathematical discoveries than this one. Gauss and Legendre’s numerical investigations into the distribution of primes helped them formulate the conjectures that preceded the prime number theorem. [2] The Birch–Swinnerton-Dyer conjecture arose from computational experiments on elliptic curves using an early EDSAC computer at Cambridge. [3] In both cases, calculations helped give unique insight into some of the key problems of these historically important fields.

My view of mathematics has been strongly influenced by Hilary Putnam’s ‘What Is Mathematical Truth?’. [4] I believe mathematics is an empirical subject this means we access truth through investigating examples, testing conjectures, and revising our ideas, while even our confidence that a proof is correct depends on fallible practices of checking and criticism. This does not make numerical evidence and proof interchangeable. It means that both discovering mathematical claims and establishing grounds for accepting them involve investigation. I see my use of AI as a continuation of that practice.

Using AI has let me explore mathematics that I may have never been able to discover alone and certainly within my lifetime with increase of production of knowledge and accelerating rate of progress this will be true if it is not right now (perhaps I could've made these experiments over a longer period of time and solved the theoretical questions completely independently). That experience is why I find some of the recent discussion about AI in mathematics difficult to agree with. I think finding out what is true is valuable, even when making sense of the answer takes further work. And if these tools help us do that work, I think we have a responsibility to consider what we lose by declining to use them, as well as what we risk by using them.

What AI has changed in my research

By the time I started my PhD, large language models were already useful for the computational side of my work. I have never particularly enjoyed the manual task of translating ideas for algorithms and optimisations into well-formed code, so I was happy to incorporate them into my research. By reinspecting ideas from the literature and my collaborators, I could explore my topic at a speed and scale that I could not have managed alone. Examining the resulting calculations helped me formulate and test conjectures, including some I would not have known to make without seeing the data. This extended the way I had worked on quaternion Gauss sums: less of my time had to be spent implementing each experiments and I could calculate massive amounts of data to support new conjectures. In addition to calculating novel representations of non-equilibrium steady states (my new area of study), I was also able to calculate asymptotics for bounding integrals by using the AI to generate python code to plot certain regions of integrals I was interested in. At this time I also made use of their skill at formatting latex and ability to perform literature review as well as to source citations.

More recently, these systems have improved massively in generating proofs and examples. This has changed how I work further, because I can now start with a suggested argument and try to understand it. Some of those arguments have been very initially ugly, but reading and revising them has helped me improve both the proofs and the statements being proved. Working out where an assumption is used can suggest that it is unnecessary, while a failed estimate can help me understand why a conjecture might be wrong. There is still LOTS of mathematics for me to do here, including checking, correcting, and discarding the output the system produces. Having an argument to investigate has often helped me get further than I ever could have from a blank page even when the contents are not correct in some subtle ways or even completely wrong, and I have learnt a great deal through that process.

Looking back at my undergraduate project, tools of this quality would have made a big difference. Finding the existing work on matrix Gauss sums earlier would have given me results to draw on and methods to try to adapt to my quaternion examples. Working out which results carried over and how the arguments needed to change would itself have been interesting mathematics. With less time spent implementing algorithms, I might have been able to develop these connections and work towards a more complete classification. I cannot know how far I would have got, but I think I would have come away with a deeper understanding by having more time to explore the sums and connect what I was finding with the existing theory.

The value of knowing what is true

The declaration ‘A Severe Misalignment of AI in Mathematics’ describes solving problems as “only a tool and proxy for achieving the primary goal of conceptual understanding and insight”. It also raises concerns about rushed announcements, attribution, and the development of students, while recognising that AI could enhance mathematical study. [5] Henry Cohn’s ‘The technical debt of AI-generated mathematics’ similarly objects to producing poorly explained arguments and leaving others to make sense of them. He welcomes responsible AI-assisted contributions, but argues that obtaining an answer usually has little value apart from the understanding it provides. [6] While I agree that counting solved problems is obviously not a good indicator of process, and that producing an argument does not automatically make it useful or interesting or worthy of attention. My disagreement concerns the value assigned to reliably establishing what is true. A result can itself deepen our understanding of the objects we study, even when we cannot hope to understand all the details. It can also give us something further to investigate, but its value need not depend on eventually finding a more illuminating explanation of why it holds.

Sometimes I care about a question before I know what conceptual insight its answer might provide. Establishing the answer changes what we can legitimately conclude and what further work we can undertake. It can settle an uncertainty, rule out a possibility, or show us something unanticipated and interesting. Those gains need not wait for an explanation that makes the answer feel inevitable. An illuminating account can add enormously to a results' value, but it's not the only way of progressing our mathematical goals of knowledge and understanding. Knowing what a result tells us about the objects we study, understanding why its proof works, and knowing how to verify it are different achievements. Progress in one can help us make progress in the others, without requiring us to achieve all three at once.

Furthemore, many mathematical results hold great importance because of the philosophical questions which they affect through there answers. In the field of nonequilibrium stationary states of kinetic theory within introductions and motivations we talk about things like deriving fourier's law or perhaps deriving some physical phenomena that is empirically observed within these models. Yu Deng's recent fields medal attribution was largely for his work rigorously connecting billiard ball dynamics to the hard sphere boltzmann equation, this was framed as a solution to Hilbert's Sixth Problem of connecting Fluid Dynamics with physics. This is not to diminish the significance of the new ideas and impact that understanding them will have to the field, just to say we value the answer for it's connection to physics.

Hilbert's problem was ambiguously phrased what is not ambiguous is that the question is one about physics and so any comprehensive answer to this requires more than mathematics, and yet Yu Deng's proof through establishing the truth of this mathematical claim has made great headway in our understanding of this problem. Similarly, mathematics purely through the answers it holds can affect other scientific disciplines and generally contribute to our expanding spheres of knowledge in each discipline, empirical verification within the sciences is often as strong a thing as is needed and formal verification for an algorithm would just be another way of connecting theory to nature. These questions and answers to those kinds of questions are what motivate me most, in this way by discovering mathematics I can then search nature empirically for examples of the phenomena and upon finding interesting examples of phenomena I can direct my theory to perhaps expand explain them each informing the other.

On this point, I find myself closer to Tim Gowers. In ‘Why I didn’t sign the Fields medallists’ letter’, he describes different mathematical temperaments: some mathematicians primarily want to solve problems and develop understanding to do so, while others primarily want to understand and use problems to get there. He resists treating one of these as the right way to be a mathematician. [7] In my own work and throughout history, finding results and developing understanding have continually informed each other. The following examples show how much we can learn from a verified result even when we cannot personally hope to master every detail.

Verification beyond manual checking

The classification of finite simple groups is one of the greatest achievements of modern mathematics, yet its proof extends across thousands of pages and the work of many mathematicians, making it doubtful that any one person could understand it in full. [8] Nor should we simply assume that every part is understood by someone in the current mathematical community so as to fill in the gaps. We still have good reason to trust the result through the history of scrutiny its proof has received, including the discovery and repair of gaps. Our judgement that a proof succeeds rests on evidence gathered through fallible practices of checking and criticism. Having a list of groups and knowing that it is complete are different achievements. If we had the same list and strongly suspected its completeness, we would be in a very different theoretical situation. Establishing completeness tells us something fundamental about the possible building blocks of finite groups and allows suitable problems about finite groups to be reduced to questions about this explicit collection. The value of that achievement does not depend on whether any individual, or even the present community collectively, holds a complete understanding of its proof.

The four-colour theorem gives a more direct example of verification extending beyond manual checking. Robertson, Sanders, Seymour and Thomas describe a proof whose exhaustive case analysis makes a complete check by hand impractical. [9] Gonthier’s later formalisation uses Coq to check the mathematical and computational parts of the proof. [10] We can understand the structure of the argument and investigate the reliability of its verification without personally repeating every operation. The correspondence between the formal statement and the intended theorem, and the trustworthiness of the checking system and its supporting software and hardware, remain matters for scrutiny. Developing and examining those grounds for confidence is itself mathematical work.

This possibility matters for proofs whose length or technical complexity makes reliable manual checking impractical, as well as proofs involving extensive computation. In such cases, making a result usable may require an independently reproducible method of verification and a clear explanation of why that method establishes the stated result. When understanding the entire argument directly is beyond us, understanding how and why we can nevertheless verify it might still be useful for us. I see no reason to assume that every interesting mathematical question admits a proof that a person could reasonably work through in full. A simpler explanation may eventually become available, but the value of establishing the result should not depend on that happening.

What we owe the people who support our research

I think the responsibility to use effective methods is clearest in clearly very important research such as cancer treatment. If a tool would reliably help researchers develop effective treatments sooner, I think they would have an obligation to use it unless there were sufficiently strong reasons not to. Mathematics often has less immediate and less predictable benefits, but we too receive public support on the basis that our work produces knowledge, understanding, and capabilities worth pursuing. If AI substantially improves our ability to do that work responsibly, I think using it will become an obligation. That assessment must include the time spent checking its contributions, its costs, and its effects on our ability to conduct future research. While there are still interesting questions to be solved, I think empowering mathematicians with these tools is more likely to strengthen the case for supporting mathematical research than to weaken it by increasing our ability to answer scientific and philosophical questions. The idea AI tools could do all parts of this so as to replace all human mathematicians seems like a science fiction idea but if it were to happen I imagine very many things would have to change.

This also bears on proposals to preserve open problems for students or holding back research from being released.

What responsible use requires

If I put my name to a paper, I take responsibility for the claims I make and the grounds I give for them. Using AI does not remove that responsibility. I should make clear what has been proved or solved, how it has been checked, and where uncertainty remains, and provide enough explanation and supporting material for others to examine and use the work. The preceding examples show why this need not mean personally checking every detail by hand. It does mean being able to explain why the methods of verification justify the claims. An argument I do not yet understand can still be useful in my research and working through it may be how I develop understanding of the objects I care about. But until I have adequate grounds for accepting it, I should not present it as an established result.

Attribution is part of that responsibility. I should acknowledge how AI contributed to the work, check references it suggests, and give proper credit to the people whose results and methods I use. A result being new to me does not make it new to mathematics, something my undergraduate project taught me before I began using these tools. AI can help me find relevant work, but it can also produce an argument without making its relationship to existing literature clear. Establishing that relationship remains part of the research.

As a PhD student, I need to develop my own ability to recognise useful ideas, construct and assess arguments, and explain them to others. Reading, correcting and extending AI-generated arguments has helped me do that, though there are also things I need to practise independently. Working without AI is still essential in many parts of the training process for students, by uncritically looking at the outputs one can risk becoming a spectator in there own research, with the AI left to its own devices spinning elaborate unverifiable webs. But by staying grounded on what we want to achieve through our use we can use it as a tool to enhance our understanding and verify arguments. I want to judge these uses by what I learn and become able to do.

As a mathematician and scientist, I also have a responsibility to advance knowledge. I want to encounter things I would not have known how to guess, find reliable ways of establishing them, and learn as much from them as I can. That was what attracted me to empirical methods before I began using AI, and it remains what motivates me now. Where these tools help me fulfil that responsibility, I have good reason to use them. The limits of what I could discover or check unaided should not determine the limits of what is worth knowing.

References

[1] Alexander Pilakoutas. “Quaternion Gauss Sums.” URSS Showcase, University of Warwick, 2023. Showcase article and accompanying research report.

[2] J. J. O’Connor and E. F. Robertson. “Prime numbers.” MacTutor History of Mathematics, University of St Andrews.

[3] Andrew Wiles. “The Birch and Swinnerton-Dyer Conjecture.” Clay Mathematics Institute, official problem description, §2, “Recent History”, p. 3.

[4] Hilary Putnam. “What Is Mathematical Truth?” In Mathematics, Matter and Method: Philosophical Papers, volume 1, second edition. Cambridge University Press, 1979, pp. 60–78.

[5] “A Severe Misalignment of AI in Mathematics.” Collective declaration, reproduced on Terence Tao’s What’s New, 11 September 2026.

[6] Henry Cohn. “The technical debt of AI-generated mathematics.” Guest post on Terence Tao’s What’s New, 15 September 2026.

[7] Timothy Gowers. “Why I didn’t sign the Fields medallists’ letter.” Gowers’s Weblog, 17 September 2026.

[8] Michael Aschbacher. “The Status of the Classification of the Finite Simple Groups.” Notices of the American Mathematical Society 51(7), 2004, pp. 736–740.

[9] Neil Robertson, Daniel P. Sanders, Paul Seymour and Robin Thomas. “The Four Color Theorem.” Authors’ expository account, particularly “Outline of the proof”, “Discharging rules” and “Discussion”. Accompanies “The four-colour theorem”, Journal of Combinatorial Theory, Series B 70(1), 1997, pp. 2–44.

[10] Georges Gonthier. “A computer-checked proof of the Four Colour Theorem.” Microsoft Research Cambridge, research report, especially pp. 1–4.