If math is more than proof, we need to better celebrate the rest of it
https://terrytao.wordpress.com/2026/09/18/if-math-is-more-than-proof-we-need-to-better-celebrate-the-rest-of-it/
380 pointsbynum42•1日前•274 コメント
I remember when "writing code was never the point" became a mantra here. There was truth in it, but removing the coding has certainly taken away a lot of the texture of the work and enjoyment of the craft. Many of us feel this loss as we tech-lead teams of agents as our source of income. I am not optimistic the mathematics pivot is going to work, but I'm certain that most will be depressed with the outcome even if they succeed.
We are all staring at the same existential dread, just seeing it unfold slower. We're being told that utopia is to be obsolete, and that is a jarring idea to contend with.
In my opinion there has never been a better time to be a mathematitian, and there has never been a better time to be a software builder.
But there has never been a worst time to have the need to prove your economic value as a mathematitian or software developer alone. Because "understanding" is not something you can prove in one afternoon, its something that you prove with a life.
It occurs to me on reading this that there’s a connection to other computerized mathematical activities.
Occasionally some computer lab in the past would announce that they have computed pi to more digits than ever before; or a new Mersenne prime will be found.
These count as ‘math news’ but they’re of little interest to mathematicians. These computational efforts demonstrate the great power of computers but they do nothing to advance mathematical understanding. Finding a larger Mersenne prime is not surprising to anyone; we’re pretty sure there’s an infinite number of them. Finding the largest Mersenne prime would be the surprise.
So it is with proofs. An LLM might prove some conjecture - Riemann, say or P≠NP. But in general we know that things can be proven and we think those things are probably true, so the existence of a proof doesn’t do much more than producing a new Mersenne prime does.
It’s only if in proving the thing we learned something that there’s actual value in the proof.
As much as I hate it, I don't think we'll ever get a proof of the four color theorem that isn't enumerating cases.
When you have an integral or the sum of an infinite series that comes out to pi, you know there must be some satisfying explanation involving a circle.
Contrast with "Examples of patterns that eventually fail" on math stackexchange[^1]. When a pattern ends at 906150257, you don't really expect the proof for that to be something beautiful. The reason for the exact value of an upper bound is that it isn't smaller and it isn't bigger.
There's a relationship between e, i, pi, and -1 comes from a deeper relationship between complex numbers and rotation.
The relationship between planar graphs, vertex coloring, and 4 might just be because we put planar graphs and vertex coloring in the same room and 4 popped out, instead of 3 or 5.
Others, I think, will be beyond both human and AI. And so what then? Mathematicians just throw in the towel and say it's not worth trying? Of course not. We will continue that pursuit, and as we do, new ideas will arise and new problems will need to be solved. It's math. There is no end.
It's easy to look at the current landscape and see AI ticking off solutions to problems and imagine that soon there will be nothing left. Machines replaced the need for much manual labor, but they also established a basis for an economy that provides the opportunity for more labor. This is the situation with math now. It will take some getting used to. There will be little-to-none pencil-to-paper working out of problems anymore, but there will always be work to do, things to solve, curiosities to unravel. And it will still be professional mathematicians who are the ones most capable of directing that effort. Because, if nothing else, they're the ones whose curiosity is piqued by the problems. Which, let's face it, has been 99% of the motivation for graduate-level math in the first place.
There's the the old question: is math invented or discovered? I think it's both: the problems are invented, and the solutions are discovered. In the age of AI, the discovery part will be greatly affected, but the invention part will remain firmly in the human domain.
Mathematics is suffering from Goodhart's Law:
"When a measure becomes a target, it ceases to be a good measure."
This is really interesting and available here: https://www.youtube.com/watch?v=IbGNZQvobkc
Basically, as long as this is the mechanism by which people earn money to stay alive, the world will always be optimized towards results.
I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them.
Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it.
And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.