AI and the Future of Mathematics Part 1: How Computers Changed Math Forever

An Empirical Science

Mathematics is at an inflection point. LLMs (Large Language Models) improve with each release, and it’s unclear how the field of math will evolve with them. My day job sits at the forefront of AI and cybersecurity, but I’m a lifelong lover of math. In fact, I’m a pure math PhD dropout, and I have plenty of stories from my time in academia. In the first of my three-part series, I’d like to share a story about the future prophesied by the remarkable Nicholas Katz…

As a sophomore at Princeton, my favorite class was the two-semester course on Algebra. The curriculum varied greatly by teacher—I had the blessing and challenge of learning from Nicholas Katz. He is a legend in the field and by all means an incredible mathematician. He handwrote each problem set, and the class often had to confer about whether he had put a t or a +. The homework served as a guide through complex proofs; each exercise was a lemma in disguise, building on the last toward a deeper result.

The first semester alone stormed through a typical year’s worth of abstract algebra (group theory, ring theory, field theory, Galois theory). In the spring, Katz got to teach whatever he wanted (his choice: local field theory and a whole bunch of p-adic-ness). The first class had 45 students. The last had 12.

Before the year was over, Professor Katz ate at the dining hall with the 12 of us. During dinner, one of the students asked Katz what he thought about the future of mathematics. Katz noted that math nowadays has much more data than it used to. 250 years ago, Carl Friedrich Gauss had to count the number of prime numbers under 3,000,000 by hand and recognize by eye their pattern that is now known as The Prime Number Theorem. Mathematicians today can use computers to run tests, analyze the output to make Conjectures, and try to prove those observations to call them Theorems. Katz predicted that research would increasingly rely on finding patterns in data. He predicted that math would become an empirical science.

The Four Color Theorem and Beyond

The rise of Lean and AI forces mathematicians to rethink the process of research, proof verification, exposition, publication, and education. But this is not the first time technology has unsettled the world of math. Fifty years ago, computers challenged the very meaning of proof.

Consider this seemingly simple question: how many colors are needed to color a map such that no two adjacent shapes share the same color? The figure to the left demonstrates that some graphs need at least 4 colors. (If you stare at it long enough, you should be able to convince yourself of this.) But is there a graph that requires 5 colors?

Humanity did not know the answer until 1976, when Kenneth Appel and Wolfgang Haken proved that only finitely many cases need to be analyzed to get the final answer. The problem: there were 1,936 cases, and some required checking hundreds of thousands of colorings. Appel and Haken used a computer to verify each case, but some mathematicians would not accept the proof because no human could check every configuration. Thomas Tymoczko argued that a proof that requires a computer is less like a deduction and more like an experiment.

Eventually, people scrutinized the logic and simplified the arguments until the objections faded, and the result became known as the Four Color Theorem. Proofs no longer had to be arguments a human could follow in full. As long as mathematicians trust machines and the algorithms they run, the output counts as truth.

Today, AI pushes this mentality to the extreme. LLMs from OpenAI recently found a solution to the Navier–Stokes equations that can blow up in finite time, thereby solving a Millennium Prize Problem. The proof was formalized in Lean, meaning that it is demonstrably correct. However, reading the paper containing the proof is like chewing granite. It may take months, if not years, for top mathematicians to distill the arguments into something tangible and useful. Remember: the point of math is not to prove statements in a vacuum; it is to build understanding for humanity.

We have entered an era where machines can produce logically sound proofs that no one understands. Without the proper framework, this trend will accelerate and results will far outpace understanding.

If a tree falls in a forest and no one is around to hear it, does it make a sound?
If a theorem is proven on a server and no one is around to understand it, is it math?

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In the next post, we’ll explore what aspects of math LLMs are good and bad at. Subscribe below so you don’t miss it!