“Math 2.0” will need to value mathematical progress more holistically

Can we please stop reducing human activity to "taste", conferences, talks, "understanding"? I think this is a very unproductive trap.

There is a world where we get to the edge of AI capabilities, and we build on top of that. As humans have always done with every new technology.

There is another more pessimistic view where LLMs just replace every human capability, and our economic overlords dont need us for anything and we just eat the small pieces of bread that are left.

This comes down to the fact of:

is human existence/intelligence just the simbolic representations we make in our brain? Or are they just a tool?

I tend to think of Godels incompleteness theorem as a proof that on the limit LLMs are useless. The real question for me is at what point approaching this limit becomes an issue, and if it has any practical consequences.

Related:

AHM Statement on OpenAI's October 6 Release of Mathematical Documents

https://news.ycombinator.com/item?id=50000421 / https://news.ycombinator.com/item?id=49999159

I would argue the problem is not AI, but putting value on proofs by mathematics community. They're not fully to blame, part of it is need of billionaires (and other elites) to appear educated, and the general status culture that sees mathematics as something that needs to be socially justified.

Grigori Perelman warned about this when he refused the Millenium problem prize. He understood mathematics should be a journey, not a destination.

""Math 1.0" placed a premium on being the first to solve an open problem, even if the solution was not initially well understood"

Arbitrary conclusion. This is the corporate take on "mathematics"

Mathematics were meant to further our understanding of nature and solve people's problem. Not to serve corporate delusional CEOs for their psychopathic purposes.

I think it's disappointing [1] for a group of people (however smart they are, whatever titles and fame they have) to think they are the gatekeepers of mathematics, the core of human knowledge. In this respect "mathematics is what mathematicians do" is no longer true, perhaps it was never a useful thing to say. Mathematics is (downstream) consumed in some shape or form by every human on Earth, and no elite, appointed group gets to decide how new mathematics is created. I feel this is a reaction to the sting of the reality check for these people, who are gifted with amazing genetics, and can think faster and deeper than other humans, that their gifts are relative to humans, and machines can still outdo them.. but this is already true for so many other areas, from physical strength to playing chess. Also, I feel this reaction is just non-sensical, it reminds me of when commercial software company CEOs [2] said open source is communism in the early 2000s, because they felt open source is threating their business model.

Lastly, deep down I don't really get what mathematicians are so upset about. All open problems, once solved, are not solved by 99.9999% of mathematicians, because it's solved by one or a handful of others, and the others just learn of the solution/proof. Mathematicians can now still organize conferences about these proofs, discuss them, digest them, think of new avenues of research, etc. They don't even have to invite OpenAI, in 6 months whatever model is available on chatgpt.com will be this smart anyway, and they can use it in the workshops for explanations, etc.

[1] I was going to write "I'm a bit disappointed by the response of the math community..", but then I remembered, whatever T. Tao writes is not the position of the math community, it's his position. Then I was going to write "I'm a bit disappointed by the response of T. Tao..", but then I remembered, I don't know Tao personally, so why am I disappointed?

[2] Steve Ballmer of Microsoft, I believe

I get bonked for saying this again and again, because people get annoyed with an oversimplified analogy or the use of whateverfallacy,

but can someone please try to set aside their knee-jerk reactions for a while to give a good reason:

  WHY do humans NEED to understand the basics of something?

× You don't know how to mine raw materials — That doesn't prevent you from using computers/phones made with aluminum, copper, glass etc.

× You don't know how to fell trees and shape lumber — That doesn't prevent you from sitting in that comfy chair.

× You don't know assembly language or how to write operating systems — That doesn't prevent you from using Windows or macOS or Linux.

—

EVERYDAY you use hundreds of things made from THOUSANDS of technologies you don't understand, because other people already MASTERED them.

so YOU can go on to go do GREATER things.

(but you CAN still go do farming, mining, logging, writing your own OS, if you ENJOY it — nothing's stopping you — you just won't be as good as the technology that has been specialized for that over centuries, and almost certainly you won't be bringing anything new to those fields, and it'll take time away from doing other things.)

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Maybe we shouldn't be wasting time on "oshit how do we uninvent or slow down this new technology because it makes things easier than what we grew up on"

and focus more on "what other greater things can we move on to?"

There's a whole freakin universe out there and we haven't even stepped off our home planet yet.

"Taste"

This is all reasonable of course, and I think you're have to be pretty cold-hearted to disagree too vociferously. What OpenAI did (opening an announcement by name-dropping the exact institution that told them not to, implying they got buy-in) is just objectively bad-faith, and more of the same from Mr. Altman.

Mathematics is an academy, and academies are human assemblages for producing truth (and the tools therein); they will stop producing if we forget to repair and refine them. It's just undeniable in the abstract.

(Sorry for the length, cut it as much as I could; mod(s) remove if you'd like. Talking to myself in the shadow of giants is how I'm coping with the ennui, I think.) That said, four philosophy nits on paradigms, scope, motivation, and pride:

1. Paradigms | The 'Math 1.0' rhetoric is undeniably powerful, but it makes it seem like he's unaware of his standpoint[1] by lumping all of "traditional mathematics" together. At the very least we've gone through four methodological revolutions in math, each one changing how the field is done on a fundamental level: ??? => Euclidean Certainty => Aristotlean Computation (~800s) => ~Newtonian Calculation (1600s) => ~Gaussian Systems (~1850s), and perhaps one in the 20th c. I lack the expertise to even gesture at. We also have clear analogues from parts of the other two acadamies in the 20th century alone: physics becoming an arcane, inelegant group effort in the ~1920s, and mainstream philosophy adopting a cloud of Kiki ideas vaguely revolving around Wittgeinstein & Chomsky in the ~1960s.

I totally understand this being distressing, especially when it's happening quickly. They, too, had people decrying the future of their fields. But we wouldn't obviously wouldn't change it, in hindsight; much of modern physics would be completely intractable without those strange, boring, unnerving methods, for example. More than intractable: unthinkable.

2. Scope | This all seems overly focused on Autumn 2026. Most egregiously, this is all built on the premise that RSI never happens, and we never acheive ASI. If we do, mathematics is almost assuredly A) the first academy to be completely outmoded, and B) the least of our problems. I cut a long thing about the caveats and effects here; at this point... if you know, you know.

3. Motivation | Ultimately this thread is focusing on human motivation throughout, a fact that would be more forgivable if acknowledged as an intentional tradeoff. Speculating that it'll be harder to have interest in math is just not worth withholding truth; for one thing, knowing that computers could solve a problem but it's banned to try would ruin motivation anyway, and worse. It's up to us to be motivated, and if I know us, we'll have no problem doing so as long as there's any utility there at all.

In more stark terms: trading progress in the fundamental academy for the sake of its current methods of recruitment and motivation seems like something posterity will almost definitely frown upon.

4. Pride | This is the common thread that weaves through all three preceeding points, I think, and is even stated in pretty blatant terms (that's Tao -- always a clear writer!):

  ...promising open directions are now being withheld from the public in fear that this will cause their own research to be "scooped"... "Math 1.0" placed a premium on being the first to solve an open problem, even if the solution was not initially well understood.

Doing science (mathematic or otherwise) in this competitive, individualistic way is just clearly counterintuitive to me, even if it weren't a recent development. Imagine taking it to its conclusion and applying some kind of patent system to mathematics -- or even worse, copyright to combinations of symbols! Perhaps more riches would motivate some mathematicians, but it would so obviously eat away at the democratic principles that have brought us unimaginably far over the past 406 years.

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TL;DR: What worked well for the past ~century is not particularly relevant, and I think Tao is missing the forest here, despite one of the best sylvan trailblazers around. On his side practically-speaking for heuristic and contingent reasons, regardless.

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I think we can say by now that we should not listen to the early nay-sayers and just wait a bit. With every trend, not just "AI". They still have some points (the ethics and environment etc), but the we don't hear from the Stochastic Parrot folks anymore.

Of course it's good to have the discussion... So maybe, we listen to the nay-sayers, but defer judgement on the matter... That's wisdom.

Edit, to be clear, I consider Tao to be the wisdom provider, not an early nay-sayer!

“doing math responsibly” is just a euphemism for “ensuring we can continue doing our paid hobby”.

Take a look at this interview from two days ago: https://m.youtube.com/watch?v=oQypVVv1u1o

The interviewee is worried about the future of math research. He is not strictly worried about being replaced, instead he is worried that he will no longer be able to launder math-as-a-hobby through math-as-something-useful as is the case today. He lays out very clearly that grant proposals claim to have useful outcomes while the proposers know those claims are nonsense.

Business as usual in math, and frankly in all the other sciences, is to do research that furthers the researchers careers or personal interests and pretend that it’s somehow useful. This would be absolutely fine if it were privately funded, but it’s not, this is public money.

In every other endeavour, lying in order to get money is considered fraud.

We have collectively wasted a huge amount of taxpayer money and human time, entire careers, on things not likely to ever matter to anyone.

I look forward to science becoming automated so that we can have real progress instead of the current broken system.

I have this feeling that the frontier of maths is going to accelerate faster than humans can keep up with it, even if machines get exceptionally good at math exposition. There’ll be some event horizon of new discoveries which are so complex we’ll never understand their intermediate steps. Beyond that point, humans will revert to “Math 1.0”, where we’ll need to rediscover proofs that have already been solved by machines, and we’ll have a pair of frontiers each for the humans and the machines.

I feel like there first will be a lot of rationalizations, hamd-wringing, and existential tummy aches, but then the eventual resigned acknowledgment (just like in Chess), that humans do math because they like to do math, not because humans will ever again be as good as computers are at math, then come to make discernments between human math proofs, and the work of an engine, maybe they'll even start start doing proofs on short time controls and stream it on Twitch.

Because the other solutions are to a) quite literally become inhuman, with cyborg integrated TPUs running local models and networked interfaces to propierary models run in data centers, or b) assert dominance of human ignorance by burning civilization down, which doesn't sound pleasant.

I suspect the whole field of mathematics will simply disappear as a career path. It seems obvious that the trajectory is for the machines to be able to provide proof on demand for any solvable problem. Whether or not the proof is understandable by humans is perhaps irrelevant in the larger sense. Doing hard math will simply become another black box tool in the larger AI toolkit for goal optimisation. Is this sad and should we try to prevent it? Is it any less sad than the venerable London cabbie who spent a life time memorising every street to gain "the knowledge" and almost overnight supplanted by machine intelligence.