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points by niemandhier 4 days ago | 0 comments

We all know Terry Tao, do you know Peter Scholz?

Fields medalist and as Tao listed as one of the most important mathematicians of our age.

Scholz is known to find „precisely the correct and cleanest way to incorporate all the previously done work and find an elegant formulation for that — and then, because he found really the correct framework, go way beyond the known results“

In a way that man is the complement of an AI, he generates understanding rather than just proofs.

Maybe he is the right template for the mathematicians of the future.

https://www.quantamagazine.org/peter-scholze-and-the-future-...

[–] leonidasrup 4 days ago

Even Peter Scholz is not always sure of correctness of extremely complicated mathematical statements and welcomes computer assistance in verification of his results.

See the Liquid tensor experiment:

https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-e...

https://www.ma.imperial.ac.uk/~buzzard/xena/pdfs/liquid_tens...

" I spent much of 2019 obsessed with the proof of this theorem, almost getting crazy over it. In the end, we were able to get an argument pinned down on paper, but I think nobody else has dared to look at the details of this, and so I still have some small lingering doubts. "

In a sense, mathematics is not very different from software engineering.

More precisely: Curry–Howard correspondence

https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...

[–] niemandhier 4 days ago

He is also a member of the association of human mathematicians.

“ 1. Members commit to not working with commercial AI companies by providing technical labor, mathematical knowledge, consultation, or publicity.

2. Members commit to not publishing mathematical texts (including papers, reviews, referee reports, lecture notes, teaching materials, and books) generated by AI models. Members only publish their own writing, as opposed to outputs of AI tools. We affirm that mathematical writing should have human authorship. “

https://www.ahmath.org/members

[–] hardbass 3 days ago

Are they publishing cryptographic proof they never used AI? They should request the AI companies that their IP addresses, etc blacklisted and hard banned from AI services and present some kind of verifiable cryptographic proof they submitted to such a ban.

[–] datsci_est_2015 3 days ago

Or, maybe, they assume their peers within the organization are operating in good faith because they're not drawn from morally bankrupt communities like Silicon Valley.

[–] hardbass 3 days ago

How would you know? Its easy to also accidentally use AI. They are mathematicians, they know mathematics demands proofs. If they pledge they shall not use AI, they should provide cryptographic proof otherwise the statement is considered untrue.

[–] hardbass 3 days ago

How is silicon valley morally bankrupt?

[–] datsci_est_2015 2 days ago

Because they're the only community that requests cryptographic proof that their peers are acting in good faith. (:

[–] bongofizzbuzz 4 days ago

Just like small numbers appear in almost all places in mathematics. Small numbers are limited. Sometimes these may appear as coincidences. Similarly small elegant explanations are limited in numbers. As mathematical universe expands, only way to explain these additional results will be to have complex explanations.

[–] adrianN 3 days ago

That is true in theory, in practice the space of proofs that fit in thirty pages or so is so vast that I find it unlikely that we’ll exhaust it for the foreseeable future.

[–] FuriouslyAdrift 4 days ago

Ramanujan is another one that did much in his head and lacked a lot of formality in his methods. Many intuitive mathematicians are this way (Shinichi Mochizuki is another although his work may just be so hard for others to undertsand that it appears to lack rigor).

[–] specialist 4 days ago

I did not; thank you.

One bit stands out to me: to better understand, Scholz will work the problem backwards. Sometimes that strategy is just so spot on.

To my eternal shame, philosophy (like this OC) makes my head hurt. Pretzels my brain, ouroboros like (aka "head up my own butt").

Caveats covered, here's my cave man cold take on two aspects of this topic.

#1 As we're all trying to paraphrase here: answers are great; understanding is even better. I like the word "legibility" (h/t James Scott). The current models are not very legible.

Which directly leads to:

#2 Understanding enables predictions.

If LLMs are stochastic parrots, then pi.dev (GPT + reasoning + ralph loop) is a drunken sailors walk (thru a solution space). A kind of messy heuristic for optimization. With some kind of gate for declaring "done!" Pretty useful.

However, I don't expect Pi.dev to make predictions. That'd require something else. Maybe that something is being worked on; am noob, so I wouldn't know. (I'm still unclear about whatever Jeff Hawkins is doing, in practical terms.)

Wrapping up: Automated prediction would be cool. That's very different from finding solutions to problems. And the current cohort of AI can't even explain why, much less make predictions.

Assuming predictions requires understanding. Maybe there are other strategies to generate predictions. Like a meta drunken sailor's walk, with drunken sailors all the way down.

(Someone please correct me as needed. TIA.)

[Aside: That article has an ad for their podcast They Joy of ₍₎. Perfect. I have no idea if it lives up to its promise.]

[–] funcDropShadow 4 days ago

> One bit stands out to me: to better understand, Scholz will work the problem backwards. Sometimes that strategy is just so spot on.

In David Deutsch's book The Beginning of Infinity, he argues, that the purpose of science is not to merely predict or to model the world, but to explain it. And he gives in my opinion convincing arguments, why a better explanation is needed to advance science. Therefore, he deduces that pure empiricism has failed. Because no matter how much data you have gathered, an explanation does not follow from it. The explanation is always an insight the scientist has to contribute. Then it is open to scrutiny by criticism or experiments.

I guess he would argue the same is true for mathematicians. Merely constructing proofs to arrive at theorem, is of no intrinsic value, if it doesn't spark an idea of a more elegant formalism that corresponds to a simpler and even more far reaching explanation.

I cannot recommend this book highly enough to anyone who is intellectually curious, not only about epistemology.

[–] threatofrain 3 days ago

Many high stakes problems are blocked by a specific insight, and once known, the people who are highly invested in those problems won't put sufficient care into whether we've built a supporting framework of ergonomic theory for humans.

That's not to say there isn't value in building mathematical insight for humans, just that there is a goldmine and a lot of people know what to do with gold, even if they don't understand the ecosystem surrounding the gold.

[–] funcDropShadow 3 days ago

True, but shortsighted. If we continue to pile proofs and new theorems on top of each other without taking care of making those new abstractions understandable, then we will limit our reach. Even though AI might be able to progress further.

[–] happy-go-lucky 3 days ago

https://en.wikipedia.org/wiki/Peter_Scholze

> Scholze was a student of Rapoport, who was a student of Deligne, who was a student of Grothendieck, who was a student of Schwartz; in this chain, Scholze, Deligne, Grothendieck, and Schwartz are all Fields medallists.