Software:Mathematics and artificial intelligence

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Mathematics and artificial intelligence (AI) have been interrelated since the initial development of AI. On one hand, mathematics plays a key role in artificial intelligence. For example, many AI algorithms use methods relying on optimization, statistics and linear algebra. On the other hand, researchers have for decades been trying to use AI and computers to help solve mathematics problems. The impact of AI has been pronounced in mathematics, where results are exactly verifiable.

Background

Mathematics in AI

Many AI systems aim to approximate a set of data. This data, represented as numbers, is combined with weights using the techniques of linear algebra. The weighted data is typically non-linearized using for instance a sigmoid function, before defining a loss function to minimize. The loss function is then minimized using an optimization algorithm, for example gradient descent. This produces a set of weights which can be used to approximate the initial data.

File:Gradient descent maximum likelihood.svg
AI relies on mathematical techniques such as gradient descent

AI in mathematics

In mathematics the goal is often to prove a statement or theorem. Large language models (LLMs) can assist mathematicians by writing proofs or helping find mistakes in written work.[1] LLMs can also be used to disprove conjectures or proposed theorems by finding counterexamples.[1] Mathematics is verifiable, because once a formal proof is written, it can be checked that every line of the proof follows from the assumptions or previous lines. Automation can be applied, by using a proof assistant such as Lean.[2]

History

After the introduction of the digital computer, computers made large numerical calculations useful in physics, engineering and military research. In 1956, a significant step was made when mathematicians built a program able to prove many of the theorems from Russell and Whitehead's Principia Mathematica. The Logic Theorist, as it became known, has been dubbed "the first artificial intelligence program".[3]

During the 1960s, theoretical foundations for the field of artificial intelligence were laid.[4] A decade later, computers began to be used in computer-assisted proofs.[5]

In 2017, transformer architecture, a key component in modern LLMs, was developed using mathematics.[6] In 2021, DeepMind's AI enabled researchers to identify patterns in mathematics which had remained elusive to humans.[7] Google then launched AlphaGeometry in 2024, which was able to use artificial intelligence to solve advanced problems in Euclidean geometry.[8]

Present

Up until roughly 2024, large language models (LLMs) were poor at doing mathematics.[9] Since then, LLMs from major companies have made significant strides in mathematics, heralding an "AI Revolution in Math".[10][11] Some of the most significant breakthroughs are summarized in the following table.

Selected Breakthroughs by AI in mathematics
Date solved Problem Comments Model
April 24, 2026 Erdös primitive sets #1196 Internal model, OpenAI
May 20, 2026 Erdös Unit distance conjecture An internal model at OpenAI disproved Erdös' conjecture that g(n)=O(n1+ϵ), where

g(n) is the number of pairs of points in a set of n points which are a unit distance from each other.

Internal model, OpenAI
July 14, 2026 Cycle double cover for any graph without bridges — edges whose removal would disconnect it — can you find a collection of cycles such that each edge is contained in exactly two of those cycles? 50 years open. Internal model, OpenAI
July 20, 2026 Jacobian conjecture Claude Fable 5 found a counterexample to the Jacobian conjecture for n>2. This conjecture had been open for more than 80 years. The result was published in a tweet on X. Fable 5
August 1, 2026 Existence of non-sofic groups OpenAI announced the construction of a non-sofic countable discrete group.[12][13] Internal version of Astra
September 8, 2026 Navier–Stokes existence and smoothness problem (unverified) OpenAI announced an unbounded counterexample to the existence and smoothness of the Navier–Stokes equations.[14] The claim has not currently been verified by external mathematicians.[15] Internal OpenAI frontier model

In 2026 the First Proof [1] project was launched to benchmark the capability of current AI models to perform research level mathematics.

File:FrontierMath Tier 4 (v2).png
Peak LLM performance on FrontierMath AI benchmark

Discussion

Concerns have been raised that the importance of the problems solved by AI in mathematics is overstated, and that the companies publishing these results are using the proofs as advertising for their models.[16] There is also the question of attribution when AI makes a mathematical discovery.[17] Furthermore, the prohibitive price of the best AI models can potentially restrict mathematical research to elite institutions.[18] To address some of these issues, several mathematicians have signed the Leiden Declaration on Artificial Intelligence and Mathematics.

Mathematician Terence Tao has argued that AI will fundamentally change the way mathematics is done in the future.[19] In particular, AI will soon generate proofs that are verifiable by proof assistants yet indigestible to humans. The role of the mathematician will therefore have to involve greater emphasis on digesting and disseminating proofs.[20]

References

  1. ↑ 1.0 1.1 Tsoukalas, George; Kovsharov, Anton; Shirobokov, Sergey; Surina, Anja; Firsching, Moritz; Bérczi, Gergely; Ruiz, Francisco J. R.; Suggala, Arun; Wagner, Adam Zsolt (2026-06-08). "Advancing Mathematics Research with AI-Driven Formal Proof Search". arXiv:2605.22763 [cs.AI].
  2. ↑ "Mathematics and the formal turn" (in en). https://www.ams.org/journals/bull/2024-61-02/S0273-0979-2024-01832-1/viewer/. 
  3. ↑ McCorduck 2004, pp. 123–125, Crevier 1993, pp. 44–46 and Russell & Norvig 2021, p. 17
  4. ↑ Nilsson, Nils J. (2009). The Quest for Artificial Intelligence. Cambridge: Cambridge University Press. ISBN 978-0-521-11639-8. https://www.cambridge.org/core/books/quest-for-artificial-intelligence/32C727961B24223BBB1B3511F44F343E. 
  5. ↑ "Celebrating the Four Color Theorem | College of Liberal Arts & Sciences | Illinois" (in en). https://las.illinois.edu/news/2017-10-18/celebrating-four-color-theorem. 
  6. ↑ Vaswani, Ashish; Shazeer, Noam; Parmar, Niki; Uszkoreit, Jakob; Jones, Llion; Gomez, Aidan N; Kaiser, Łukasz; Polosukhin, Illia (Dec 2017). "Attention is All you Need". in I. Guyon and U. Von Luxburg and S. Bengio and H. Wallach and R. Fergus and S. Vishwanathan and R. Garnett. 30. Curran Associates, Inc.. https://papers.nips.cc/paper_files/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf. 
  7. ↑ Castelvecchi, Davide (2021-12-01). "DeepMind's AI helps untangle the mathematics of knots" (in en). Nature 600 (7888): 202. doi:10.1038/d41586-021-03593-1. PMID 34853424. Bibcode: 2021Natur.600..202C. https://www.nature.com/articles/d41586-021-03593-1. 
  8. ↑ Roberts, Siobhan (17 January 2024). "A.I.'s Latest Challenge: the Math Olympics". https://www.nytimes.com/2024/01/17/science/ai-computers-mathematics-olympiad.html. 
  9. ↑ Werner, John. "AI Is Usually Bad At Math. Here's Why It Matters" (in en). https://www.forbes.com/sites/johnwerner/2024/10/07/ai-is-usually-bad-at-math-heres-what-will-happen-if-it-gets-better/. 
  10. ↑ Kakaes, Konstantin (2026-04-13). "The AI Revolution in Math Has Arrived" (in en). https://www.quantamagazine.org/the-ai-revolution-in-math-has-arrived-20260413/. 
  11. ↑ Kakaes, Konstantin (2026-08-03). "Why the Legendary Erdős Problems Are Falling to AI" (in en). https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/. 
  12. ↑ "Ten advances in mathematics and theoretical computer science" (in en-US). 2026-08-01. https://openai.com/index/ten-advances-in-mathematics/. 
  13. ↑ Sparkes, Matthew (2026-08-03). "OpenAI announces solutions to 10 longstanding maths problems". New Scientist. https://www.newscientist.com/article/2582793-openai-announces-solutions-to-10-longstanding-maths-problems/. 
  14. ↑ "On the Navier–Stokes Millennium Prize Problem" (in en-US). 2026-09-06. https://openai.com/index/navier-stokes-solution/. 
  15. ↑ "OpenAI says it cracked 90-year-old maths problem in 88 hours" (in en-GB). 2026-09-08. https://www.bbc.co.uk/news/articles/cy7zygy3rl2o. 
  16. ↑ Hart, Robert (2026-08-11). "The AI takeover of mathematics has begun" (in en-US). https://www.theverge.com/ai-artificial-intelligence/977273/the-ai-takeover-of-mathematics-has-begun. 
  17. ↑ Lee, Melissa (2026-08-09). "Generative AI has changed mathematics forever. Where to from here?" (in en-US). https://theconversation.com/generative-ai-has-changed-mathematics-forever-where-to-from-here-288954. 
  18. ↑ "'Brutal' math test stumps AI but not human experts" (in en). https://www.science.org/content/article/brutal-math-test-stumps-ai-not-human-experts. 
  19. ↑ "Watch: Fields Medalist Terence Tao on Artificial Intelligence and Why We Do Math" (in en-US). 2026-08-13. https://www.simonsfoundation.org/2026/08/13/fields-medalist-terence-tao-on-artificial-intelligence-and-why-we-do-math/. 
  20. ↑ Tao, Terence (2026-08-17). "Mathematics in the age of AI". arXiv:2608.16753 [math.HO].

See also

Notes



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