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Comment by thayne | Hacker News Reader

But solving math problems is not the same as increasing understanding of math concepts.

Applications don't care whether the math was proven and understood by humans or computers. Your algorithm will get faster no matter where the insight came from.

In math, the journey is often more important than the destination. The process of developing a proof may uncover new mathematical techniques, some of which may have practical applications in other fields. Even an attempt that ends up as a dead end towards the intended proof could produce something useful in a difderent area. But if you just get the proof directly, you miss other discoveries you could have made along the way.

Take the Navier-Stoke problem for example. Knowing that there are solutions that "blow up" probably doesn't have a lot of practical applications. Such solutions couldn't happen in a real system. But the process of finding that proof could result in increased understanding of how turbulence works, or new techniques for solving non-linear partial differential equations (which has a lot of applications in science and engineering).

> In math, the journey is often more important than the destination. The process of developing a proof may uncover new mathematical techniques, some of which may have practical applications in other fields.

Sure. And AIs can use ideas from AI published proofs in one domain to inspire other domains just fine. Nothing changes here.

It isn't just the proof. It's everything that leads up to that, including the interchange of ideas with other mathematicians.

Applications do not care about 99.999% of theoretical math production anyway. And especially most of the big results in theoretical math nowadays are really inconsequential in applications.

Applications don't care about Navier Stokes, yes. But they care about eg proving crytographics secure, or proving that your algorithm doesn't blow up under adversarial input.

Formal verification, cryptography and the like is far from what the vast majority of theoretical mathematicians are doing (if those who do them even see themselves as that vs computer scientists or applied mathematicians) especially when it has to do with specific, production systems, and there are not many other examples like this in general outside compsci and statistics. Moreover, I can imagine that these fields will actually flourish more now that AI can make verification and proofs more viable in scale. But even much theoretical work related to cryptography etc is often not very applicable in itself.

There's lots of math in eg operations research.

Yeah, that's usually classified as applied math, not theoretical, though.

to be honest it is difficult to discuss with someone who doesn't even try to understand the basics of basic science (and how it compares with _applied_ sicence), yet talks with so much confidence. even the solution to navier stokes won't have an immediate practical effect...

Huh? The resolution of Navier Stokes won't have much of an effect, yes.

There's lots of problems like that. Eg if we prove P != NP, that won't have much of an immediate effect either.

However, there's also plenty of problems whose solutions will have practical effects, some even immediate.

> there's also plenty of problems whose solutions will have practical effects, some even immediate.

Sure. But do you know which ones they are? Or do we discover later that they were valuable?

Your argument would be 100x more convincing if you gave an example.

I will try: a super-compressor that made my 100Mb web app into a 5 kb binary bundle would immediately speed up my work. Can/will AI move human understanding or machine capabilities on this front?

A browser without security vulnerabilities would be wonderful. I think LLMs are already helping with this a lot, but a lot of complexity remains.

A right to privacy in society would be amazing (see the UN Declararion of Human Rights). AI is eroding this.

So I tried but I’m not very impressed with my list. Do you have one?

> A right to privacy in society would be amazing (see the UN Declararion of Human Rights). AI is eroding this.

This has nothing to do with mathematics.

> So I tried but I’m not very impressed with my list. Do you have one?

Look into operations research. Or narrower, you can look at improvements in linear programming solvers and mixed integer linear programming.

(These are examples of areas that have seen mathematical improvements in applications recently. I don't think good AI has been around for long enough to contribute much to progress there, yet.)

And this is exactly the point of the Statement. The process is more important than the solution itself. Most problems in mathematics don’t have immediate value or applications to the real world.

AI’s solutions are like the answers section to practice problems at the back of a textbook. Answer is 42, so what?You have to attempt the problem yourself, that’s the whole point of the exercise.

As an engineer I’m happy to use AI for math. If I publish a paper that way, very common these days, I think it’s still problematic. My paper would include something I didn’t come up with and I don’t really understand.

I think this is a good time to properly discuss these things because AI is coming for everything. Mathematics and Software were just the first two.

Then instead of publishing a paper, just upload your AI result to github. Same advancement of the state of the art. Papers are overrated here.

You don't seem to have grasped Terry Tao's (and others') criticisms. Basically they are saying that the advancement you get is illusory. Most of the time, it doesn't give you any new capabilities or deep understanding, instead you get an inhibiting of human exploration and ensuing expansion of our real understanding in that particular (sub)field. It's non-intuitive, since from a purely logical standpoint you've only added another set of known truths to the ones we already knew about before. The issue only becomes apparent when one considers the larger context of human collective truth and meaning making.

Of course, you get new capabilities. Why wouldn't you?

Yes, you might not get new human capabilities. But your applications still work better.

All you get is a new, likely to be useless fact, together with the opaque proof of that fact. There are no known or forseeable applications to the finding that there are singularities in the idealized flow. What you lose OTOH, are the many deep mathematical insights humans motivated by the search would have stumbled upon on the way to that fact and which are much more likely to lead to real-world applications. It's these insights that are truly productive, not settling mathematical points, however iconic these might be.

I think the error you're making is that you're assuming these systems have the same mathematical capability as humans (or better). But that's not the case, nor would an informed prediction be that they surely will get there soon enough if technological evolution keeps apace. That would be akin to believing a hiker will reach the moon if they will just keep ascending the mountain. "But look, they are making such good progress!"

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