Mathematics and artificial intelligence

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.

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. LLMs can also be used to disprove theorems by finding counterexamples. Since mathematics is verifiable, once a proof is written it can be verified by checking that every line of the proof follows from the assumptions, for example by using a proof assistant such as Lean. This makes mathematics very suitable as training data for AI.

History

After the computer revolution, computers have made huge strides in mathematics, for example by helping to solve large numerical calculations 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 Russel and Whitehead's Principia Mathematica. The Logic Theorist, as it became known, has been dubbed "the first artificial intelligence program".

During the 1960s, key strides were made in the theoretical foundations of artificial intelligence. A decade later, computers began to be used in computer assisted proofs.

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

Present

Main article: List of mathematical discoveries by artificial intelligence

Up until roughly 2024, large language models (LLMs) were poor at doing mathematics. Since then, LLMs from major companies have made significant strides in mathematics, heralding an "AI Revolution in Math". 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 $g(n)=O(n^{1+\epsilon }),$ 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.	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. The claim has not currently been verified by external mathematicians.	Internal OpenAI frontier model

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

Peak LLM performance on FrontierMath AI benchmark

Discussion

Meme about the role of linear algebra in AI

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. There is also the question of attribution when AI makes a mathematical discovery. Furthermore, the prohibitive price of the best AI models can potentially restrict mathematical research to elite institutions. 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. 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.

See also

• Artificial intelligence
• Automated theorem proving
• Computer-assisted proof
• List of mathematical discoveries by artificial intelligence
• Numerical Analysis
• Proof assistant