Summary

• Using AI for mathematics research is a much less hyped field than GenAI in general.
• The pre-print, “The Shape of Math To Come”, from Alex Kontorovich gives an excellent recent, readable, and accessible discussion of the potential for using AI to scale-up and automate formal mathematics research.

Introduction

Like the rest of the universe I’ve been drawn into experimenting with GenAI for general productivity tasks. However, one of my other interests is how GenAI gets used in scientific discovery. Even more interesting is how GenAI can be used in mathematical discovery. If you want to know where the combination of mathematics and AI is going or could go, then I would strongly recommend reading this arXiv pre-print from Alex Kontorovich, titled “The Shape of Math To Come”. The pre-print was uploaded to the archive in October of this year. With some free evenings last week, I finally got round to finish reading and digesting it.

GenAI for rigorous mathematics? Really?

It’s important to point out that I’m not talking about the mathematics behind GenAI, but about how GenAI is used or could be used as a tool to support the formal development of rigorous mathematical proofs, derivations and statements. Because of that rigour, there is actually less hype (compared to mainstream applications of GenAI) around the use of AI in this way. In the pre-print Kontorovich outlines a credible and believable road ahead for using GenAI in formal mathematics. For me the roadmap is credible because of i)Kontorovich’s credentials as Distinguished Professor of mathematics at Rutger’s University, ii) the significant progress that has been made in this area over the last couple of years, particularly the AlphaProof system from DeepMind, iii) and the existence of powerful automated theorem provers such as Lean.

The pre-print is very readable and even if you don’t have a formal mathematical background but have, say an undergraduate science or engineering degree, you will be able to follow the majority of the discussion – there is only one place where specialized knowledge of analysis is used and even then it is not essential for following Kontorovich’s argument.

The main potential sticking point that Kontorovich highlights in using AI to do maths is the tension between the stochastic nature of Large Language Models (LLMs) and the deterministic nature of mathematical proof. Kontorovich’s argument is that the use of automated theorem checkers such as Lean, in combination with a large and increasing library (MathLib) that formally encodes our current rigorous mathematical knowledge base, will provide a checking mechanism to mitigate the stochastic nature of LLM outputs. In this scenario, human mathematicians are elevated to the role of orchestrating and guiding the LLMs in their attempts to construct new proofs and derivations (using Lean and aided by the knowledge base encoded in MathLib). Whether human mathematicians choose to adopt this approach will depend on how fruitful and easy to use the new GenAI-based mathematics tools are. By analogy with the take-up of the LaTeX typesetting language in the early 1990s, Kontorovich says that this will be when “the ratio of time required to discover and formalize a mathematical result (from initial idea through verified formal proof) to the time required to discover and write up the same result in natural language mathematics” falls below one. When this happens there will also be accelerated ‘network’ effects – increased number of verified proofs encoded into MathLib will increase the ability of computer + GenAI assisted workflows to construct further verified proofs.

Less formal mathematics research

What interests me as a Data Scientist is, if such approaches are successful in advancing how mathematicians do formal mathematics, how much of those approaches can be adopted or modified to advance how non-mathematicians do non-formal mathematics. I’m not talking about the “closed-loop” GenAI-based “robot scientist” approaches that are being discussed in drug-discovery and materials development fields. I’m talking about the iterative, exploratory process of developing a new model training algorithm for a new type of data, or new application domain, or new type of problem. Data Science is typically about approximation. Yes, there is formal derivation in places, and rigorous proofs in others, but there is also a lot of heuristics. Those heuristics can be expressed mathematically, but they are still heuristics. This is not a criticism of using such heuristics – sometimes they are used out of ignorance that better approaches exist, but often they are used out of necessity arising from constraints imposed by the scale of the problem or runtime and storage constraints. What I’m driving at is that the Data Science development process is a messy workflow, with some steps being highly mathematical and imposing a high degree of rigour, some less so. Having access to proven tools that accelerate the exploration of the mathematical steps of that workflow can obviously accelerate the overall Data Science development process. At the moment, switching between mathematical and non-mathematical steps introduces context-switching and hence friction, so my tool use for the mathematical steps is limited. I might use Mathematica for an occasional tedious expansion and where tracking and collating expansion terms is error-prone. For my day-to-day work my mathematical steps will be limited to, say, 40 pages of A4, and are more typically between 5 – 15 pages of A4, and occasionally 1-2 pages. At such scales, I can just stick to pen-and-paper. If however, mathematical support tools developed from doing formal mathematics research became more seamlessly integrated into the Data Science development process, I would change my way of working. However, there is another problem. And that is the messy nature of the Data Science development process I outlined. Not all the steps in the process are rigorous. That means the Data Science development workflow doesn’t have access to the self-correcting process that formal mathematics does. In Alex Kontorovich’s paper he makes an optimistic case for how the use of formal theorem checkers, in combination with GenAI, can genuinely advance formal mathematics. I think of this as akin to how physics-based engines are used to provide rigorous environments for reinforcement-learning. The messy Data Science workflow typically has no such luxury. Yes, sometimes we have access to the end-point ground-truth, or we can simulate such truths under specified assumptions. More often than not, our check on the validity of a specific Data Science development workflow is an ad-hoc combination of data-driven checks, commercial vertical domain expertise, technical vertical expertise, and hard-won experience of ‘what works’. That needs to get better and faster. I would love for some of the tools and processes outlined by Kontorovich to be adapted to less formal but still mathematically-intensive fields such as the algorithm development side of Data Science. Will it happen? I don’t know. Can it be done? I don’t know, as yet.

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