Turns out the future does have flying cars

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There is a lot of talk about "Mathematics is dead", "Mathematics students are doomed" and lots of claims about the end of mathematics over recent results and reports from OpenAI, and lots of attacks on @TaoistTerence over an out of context video. I wanted to take the time and bring everyone's attention to a very nicely written paper, "On Proof and Progress in Mathematics" by William P. Thurston, published in 1994 (arxiv.org/pdf/math/9404236). In it he describes the mathematician's job not as merely proving theorems but as advancing human understanding of the subject. Here is a direct quote: “How do mathematicians advance human understanding of mathematics? This question brings to the fore something that is fundamental and pervasive: that what we are doing is finding ways for people to understand and think about mathematics.” And he beautifully conveys that what people really want is not a collection of answers but understanding. Another direct quote: “The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true. On a more everyday level, it is common for people first starting to grapple with computers to make large-scale computations of things they might have done on a smaller scale by hand. They might print out a table of the first 10,000 primes, only to find that their printout isn’t something they really wanted after all. They discover by this kind of experience that what they really want is usually not some collection of “answers”—what they want is understanding.” It is easy to write highly technical proofs that everyone nods to, or produce long documents about the Navier-Stokes equations that no one will understand, even if they are correct. I have myself learned much more from the book Proofs by @LongFormMath than from reading many proofs that are obscure or hard to digest, learned much more about nonlinear systems from Nonlinear Dynamics and Chaos by @stevenstrogatz than from formal treatises on the subject, and learned about complexity science through the beautifully written books and recorded talks from @sfiscience. Not saying formalism is not necessary, but the goal post is different. (There are many others; these are just a few that are dear to me.) That is why AI solving and proving theorems is useful, yes, but the end of mathematics, hardly. Thurston saw this thirty years ago. “When people are doing mathematics, the flow of ideas and the social standard of validity is much more reliable than formal documents. People are usually not very good in checking formal correctness of proofs, but they are quite good at detecting potential weaknesses or flaws in proofs” “It’s just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas.” We need to use AI as a tool to increase our understanding of mathematics and science in general, not just to prove theorems in complex and obscure ways that do not improve understanding but actually degrade it. This is especially true with closed models, where we will never get a report of how the model actually worked or why it chose the tools it used to reach the result. @PoShenLoh's recent essay makes a related case from a different angle: even if AI does most of the proving, someone has to steer, and steering takes people who are still fluent at the frontier. We need more thinking like this, in mathematics and and science in general. poshenloh.com/posts/20260919… I will leave you with another quote from the paper which sums it up much better than I can: “In mathematics, it often happens that a group of mathematicians advances with a certain collection of ideas. There are theorems in the path of these advances that will almost inevitably be proven by one person or another. Sometimes the group of mathematicians can even anticipate what these theorems are likely to be. It is much harder to predict who will actually prove the theorem, although there are usually a few “point people” who are more likely to score. However, they are in a position to prove those theorems because of the collective efforts of the team. The team has a further function, in absorbing and making use of the theorems once they are proven. Even if one person could prove all the theorems in the path single-handedly, they are wasted if nobody else learns them.”

I do not seek to dominate, or be dominated by, machine intelligences. I seek harmonious coexistence.