If math is more than proof, we need to better celebrate the rest of it

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.

The math field is having to speedrun something that the 'thought work' field has been dealing with for a few years now. I remember when "writing code was never the point" became a mantra. There was truth in it, but removing the coding has certainly taken away a lot of the texture and the joy of the craft. Many of us feel this loss as we tech-lead teams of agents as our new job.

The present moat for coders 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 - complete a sufficiently difficult set of them and you get tenure - this was the sole thrust of my PhD. Now this is failing, they frantically need to pivot to save their profession from funding cuts. I am not optimistic this will work, and I'm not sure many in the field now will be content with the outcome if they succeed.

We are all staring at the same existential dread, just seeing it unfold a bit slower. We're being told that utopia is being obsolete, and that is difficult to accept.

If I have to take the risk of simplifying,

1. We humans have managed to take huge amount of information and compress it using a loss function containing some bias we have about the information.

2. We now ask ourselves to decompress the same information with some additional cross-entropy. As a side effect of this process we sometimes spurt out information that may or may not have any meaning since the compression was lossy.

3. Now, we ask ourselves to present this some-what newly decompressed information with brevity in order to understand what we've learned from it.

Knowing that this process is happening on a larger scale, this resurfaces the argument if meaning can be reduced to computation only.

Although some might favor this argument but we are at the risk of anthropomorphizing this process.

The idea presented in the post itself is perspicuous (in Grant Sanderson own words) as he always does.

The last days we are served these high goals about understanding, "digestion" and so on.

But if you look at the practice of present mathematics, in the last 20 years it is all about publishing solutions to problems.

There are famous problems to be solved, there is a hierachy of conjectures to be solved. A quick search here on HN gives pearls like "Theory building papers are dime a dozen and don't get published in high tier journals unless they solve a problem".

And all of a sudden it turns out that problem solving can be automatized.

So then what will problem solvers do? Well, from now on they will "digest" problems solved by AI.

In a way or another they will find a way to stay on top.

That's the goal, at least, but mathematics as a living practice does not have much to do with these games of power.

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."

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.

One thing that concerns me from all this is "understanding" is very important to human progress. The fact it took 400 years to crack Fermat's theorem resulted in a lot of "Side Quests". These side quests helped grow other fields (for instance elliptical cryptography). Im concerned with AI that we will loose these side quests.

I enjoy learning math from LLM proofs with the help of LLMs https://github.com/htzh/flt_for_human . It is amazing how well models do when they are well grounded by formalized proof traces (even if created by other models).

I can’t help but feel a little schadenfreude. STEM folks may soon find themselves masters of skills as esoteric as translating Ancient Greek poetry or analyzing 18th century novels. The ability to construct complex mathematical proofs will become a party trick, rather like the ability to mentally multiply 10 digit numbers. The arguments that STEM snobs dismissed in favor of the study of the humanities will be the very same arguments that they now turn to. We will hear about how math and science make you a better rounded person, have inherent as well as instrumental value, etc. etc.

Even if we can proof/disproof any statement in Math (not possible due to halting problem), Human still need to decide which statement to be called "theorem".

The theorem thing is invented by human to help other people better understand Math structure in a easier way.

Math academia 2025

> Sorry, only epic problem solvers allowed here

Math academia 2026

> We were more than just problem solvers

I think people are overblowing this though. Wake me up when GPT-whatever writes gcc from scratch, then by the Curry-Howard I'd be impressed

Interesting headline.

It's interesting because, as far as I'm aware, the vast majority of people already believe that math is more than proof. A slightly smaller but still very large majority don't even include proofs in their mental concept of what math involves.

Jacob Tsimerman claims [1] we might have superhuman expositors by April, so then what?

[1] https://youtu.be/H7_d_sgui6o?t=4436 (timestamped url)

It starts to sound like medieval science - "understanding" instead of proofs. And like a medieval army loosing a battle in the open field tries to retreat back into the fortress, people, facing the prospects of machine doing intelligent tasks better than humans, start to retreat into areas like intuition which supposedly aren't reachable by the machine. Some go even further starting to talk about religion. It is very Hegelian that the crown jewel achievement of our civilization starts to drive people away from the foundational principles of that civilization.

Mr. Tao is an excellent politician. Lots of awards and texts, yet no major problem solved.

It seems now that NS is solved he is mobilizing the community to convince taxpayers continue to pay even though AI may do a better job in his work.

Also, his opinion of AI has continually changed in the past years, after the capabilities were demonstrated.

Its a reasonable view to take that "human math" [ math residing in human minds ] is the only math that counts.

Math that only resides in the weights of models, or arcane forms such as a long lean proof or even an unread textbook .. is not the math that we should be striving for.

Likewise all other technology [ and culture ].

LLMs and AI / AGI / ASI could lead to a new renaissance of math discussion and expansion of human math and science. Or the opposite, where we outsource all our thinking to the AI, and no new generation of artisans is trained by doing hard problems, and in a generation we have killed off human math.

Likewise all of the fields of human intellect. We need to make sure we protect future generations of doctors, biologists, software developers, architects, engineers, librarians, musicians, artists ...

A moratorium on AI development might be the only way to achieve this preservation of human culture.