Terry Tao: Math Needs to Celebrate More Than Just Proof
If math is more than proof, we need to better celebrate the rest of it
Terry Tao hosts a guest post by Grant Sanderson (3Blue1Brown) arguing that as AI makes proof generation easier, the math community must elevate 'motivated explanations'—work that builds intuition and understanding—to the same status as proofs. Sanderson defines these explanations, gives historical exemplars like Thurston and the Princeton Companion, and proposes treating open exposition problems as seriously as open research problems.
If outsiders believe that proof-generating machines render mathematicians obsolete, while insiders see that as a misconception of what researchers add, it's incumbent on this community to better project its true values through the kind of work that it rewards.
- ForgotMyUUID
I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part. 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.
- sweezyjeezy
The math field is confronting something that coders have been dealing with for a few years now, only far more violently. Today's moat for software seems to be that AI can automate tasks but not a full job (yet). But for a large proportion of mathematicians, doing these tasks really was _the_ job. It's the bit they wanted to do, and if they completed a sufficiently difficult set of tasks, they got tenure. Now this model is failing, they frantically need to pivot the role of humans to save their profession from funding cuts.
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.
- c7b
> we might imagine what it could look like to have an analog of the Millennium Prize Problems for open exposition problems
The core idea seems to me that we should shift the standards for professional evaluation from generating proofs to generating explanations. Makes sense that such a proposal would come from the 3B1B guy, and I actually agree with it, irrespective of AI. But what eludes me is how that could be a defensive mechanism against AI automating humans out of mathematics. AI is likely no less good at producing natural language explanations as it is at generating rigorous proofs. It's telling that even Terrence Tao turned to AI to understand AI-generated results [0]. It seems that the essay doesn't address that issue at all.
- youoy
Part of the controversy here is that now the skill advantage that some Field Medalist had is much narrower. The fact that fields medals have an age limit implies that it favors brain power over understanding. And that was the guiding light award of the community. So i find it "funny" (and natural) when they are offended by AI.
That is the main "crisis" of mathematics.
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.
- pcfwik
If this suggestion were to come to pass, I wonder how new math PhDs would think about choosing between a 'normal' R1 faculty job vs. the "teaching route" (teaching professorships, lectureships, community college professorships, or SLAC professorships).
It's been my understanding that traditionally the ones who care about "motivated explanations" in this sense go for the latter, but if the research community has now decided they care about teaching and understanding, it might "even the playing field" and make the jobs more similar.
- kurthr
This goes in a necessary direction, from my personal take away of Gower's recent post on the subject.
Mathematics is suffering from Goodhart's Law:
"When a measure becomes a target, it ceases to be a good measure."
- alkyon
> It was a short film called Outside In, perhaps the earliest example of a viral video about substantive math, visualizing the key idea of Thurston’s own construction for sphere eversion.
This is really interesting and available here: https://www.youtube.com/watch?v=IbGNZQvobkc
- random3
While I understand and emphatically with Tao's concern I'm afraid it's missing the forest from the trees. Unless you can make a claim that AI will never be able to perform intellectually at the same level as any human at a much lower cost, there's an outstanding utility problem that remains unaddressed.
Sure enough, the AI may not have taste or goals, or many human traits, but that's irrelevant to the much thornier (and much broader than mathematics or even academia) question related to who's getting paid how much and for what.