Comment by daxfohl | Hacker News Reader

Comment by daxfohl | Hacker News Reader

People keep talking about this like there's a finite amount of math to be done, and then the party's over. But that's never how math has worked, is it? Every problem you solve, ten new ones open up. Like a fractal, the more you zoom in, the more detail emerges. No matter how much better AI is at solving problems, it's not going to generate the "final, complete compendium of mathematics" that that seems to hover over this post.

Math is meaningful because ... some people like to do it. The same as any other human pursuit. It doesn't need a reason beyond that. And AI won't change that. There will continue to be things to explore, things to find out, things that are maybe just at the edge of AI's reach and needs a human to decide whether it's worth continuing to explore or not. (Remember, AI isn't free).

So, IDK, I think for people who enjoy exploring math, there will always be interesting areas to explore. AI just gives us a better flashlight.

BTW I do agree that there's going to be an incident soon, whether intentional, accidental, or paperclip-factory, that leads governments around the world to shut all this down for some time, perhaps even shutting off access to GPUs entirely. It seems unavoidable. But that's just a temporary respite and skirts the core philosophical premise of the post.

> Every problem you solve, ten new ones open up. Like a fractal, the more you zoom in, the more detail emerges. No matter how much better AI is at solving problems, it's not going to generate the "final, complete compendium of mathematics" that that seems to hover over this post.

I guess Terence's main point has been all the time that if we let AI solve all these existing problems, we don't notice the new ones and then there is stagnation.

There probably is a finite amount of math to be done

The universe is bounded by rules, as far as we can tell, and not a lot of them

>The universe is bounded by rules

That's physics. Not all math is physics.

The universe imposes strict limits on the math that can exist within it, and certain broader limits on the math the creatures and well-organized sand within the universe can conceive of in the first place, whether or not it can maybe exist in other universes.

At those levels math and physics are the same bound: the bound of things the universe allows to be conceived of inside it.

It’s possible the math our monkey brains + sand can ever conceive of in this universe is a low and accessible amount.

> The universe imposes strict limits on the math that can exist within it, and certain broader limits on the math the creatures and well-organized sand within the universe can conceive of in the first place, whether or not it can maybe exist in other universes.

When you have a moment, please reference some proofs supporting this.

There are lots of thoughts you’re not biologically capable of thinking. That means the list of thoughts you’re capable of thinking is finite. That means the amount of math you’re capable of discovering, even if you lived forever, is finite.

That’s true of all humans and all constructs.

So however much math there is to discover, that’s the finite subset you’ll ever have access to.

It’s hard to prove what thoughts no human and no construct is capable of generating, but surely there are some, and it’s possible or even likely some of those are math-related.

Even if we accept that there are "lots of thoughts" we're not capable of thinking (though I wonder how you would define a thought if not as something that you can think), it still does not follow that the "amount of math" (as if it's a definite quantity) that one could discover is necessarily finite, if you lived forever.

By analogy, the "amount of math" we can possibly discover could still be countably infinite even if the space of all possible thoughts would be uncountably infinite. Countably infinite is still plenty big, and it is certainly not finite.

To put what I said another way, math may be infinite in principle, but the creatures in our universe can only conceive of or perceive so much of it: a finite amount.

We (or our constructs) can plausibly mine all there is and then there's no more that's physically possible for us or our constructs to mine within the universe in which we exist.

If an ant can't conceive of or perceive trigonometry that doesn't mean trigonometry doesn't exist. But if neither an ant nor a human nor any creature or construct or technology inside the current universe, now or ever, can conceive of or perceive trigonometry, then we might as well call it non-existent. It may exist: but we'd literally never know it and neither would our constructs or space aliens or inter-dimensional aliens, or their constructs, etc.

Say that there's a level of mathematics at which a blerg is a zorg, but our universe includes neither blergs nor zorgs and no intelligences in our universe can conceive of blergs/zorgs because that would require having evolved outside our universe... in that case we can consider our universe's mathematics solved without it needing to work down to the blerg and zorg level.

The natural numbers are countably infinite. For each n ∈ ℕ, there is a proof by reflexivity that n = n. Hence there are countably infinitely many such proofs, one for each natural number.

Please make substantive comments. Going around message boards demanding proofs is not conducive to good discussion.

I didn’t demand anything. I requested some proofs since this thread is quite literally about a claim that requires proofs. As evidenced by the further comments in the thread. If you disapprove of the conversation, please feel free to use the tools and move on. No hard feelings!

Please don’t use pedantry.

Which math is not physics (and chemistry)? Fundamentally, all these are same

The concept of infinity?

That's why it is a concept. We lack sufficient ability to measure physical world, introduce imperfect abstractions, thus need concepts like infinity.

Why does the universe being bounded imply mathematics is finite?

Because the bounded universe creates bounds on what the intelligences within it can conceive of as mathematics.

We can conceive of lots of mathematics that our own universe doesn't necessarily support. But only that much and no more: we're still made of stuff inside the universe, and so are our tools, so there are upper limits on our conception.

Bounds do not imply bounds. There are an infinite number of numbers between zero and one. Heck, with our finite minds in finite time we have created uncountable infinities!

Mathematics that our universe does not support is still mathematics. Also you're viewing this through a human lens. How do you know it doesn't support what you think it doesn't support? Mathematics after all is a very human pursuit. Some alien species may have started off with ternary or even continuous logic. We have no counterexamples to show that that is beyond our universe.

If you study the history of math you’ll find that the interest in certain problems changes based on culture and technology.

In our lifetimes computers have made a lot of combinatorial and graph questions meaningful that otherwise would not be interesting.

The history of math was done on human scales. Same as the history of chess.

> There probably is a finite amount of math to be done

> The universe is bounded by rules, as far as we can tell, and not a lot of them

Respectfully, this is not a useful frame for the discussion. Nobody is expecting to reach the limit you have noted either with or without the assistance of LLMs. So there is always more math that could be done.

It seems likely that there are an infinite number of math problems but only a finite number of interesting ones.

I think it really depends on what the universe looks like as you drill down into it. It seems like the further down into smaller systems you get, the more analytically complex it gets. And then there will always be more value in enhancing the generalisations you have.

I would argue that novel and/or valuable results are not necessarily interesting!

I (a human) am interested in things that are applicable to my realm of understanding, but I see a very plausible future where novel and/or valuable results leave that realm.

I'd further argue that's already the case for most math for most humans. What's interesting to Terrance Tao is rarely of immediate interesting to me.

Trivially false. Let P be the set of maths problems and I be the interesting subset of P. If I is finite, then there exists an element x belonging to P\I whose description is minimal among P\I. Then x is interesting. QED.

Why is x interesting? Just because it has a minimal description in P\I? That makes it interesting in strictly technical sense only.

I think that's a variation on the interesting numbers paradox joke. Statement: All numbers are interesting. Proof: Assume by contradiction that there's a non-empty set of uninteresting numbers. Then that set contains the smallest uninteresting number. That property makes it interesting.

Yeah, I've seen this before as well. I guess I've just become old and grumpy and can't appreciate jokes like these anymore. Also taking jokes seriously is peak HN so..

Edit: actually, forget about the above. I just find it very annoying when people dismiss good conversations with not-so-good jokes.

I mean, there's some truth to that joke in this case, so I wouldn't agree that it's dismissing. The point of it is that what matters is how we define "interesting", because by the joke's definition in particular, there can't be uninteresting problems. That seems to be a very subjective concept that can also change with time. Mathematics is so specialized that there can be 3 people in the world who find one specific problem interesting. If they solve it, they'll move on to something else.

I find it curious that you've protested against that joke but not against the statement that "it seems likely that there's a finite number of interesting math problems". It doesn't seem likely to me personally and I haven't seen proof of that, even in a joke form. There's a finite number of problems at any given time, obviously, because mathematicians are finite, but it would require a very good understanding of the whole of our mathematical knowledge to declare that if we keep expanding it we'll hit some kind of wall, of "interestingness" or whatever else.

Firstly, I'm very happy we're having this conversation. It's so pointless, yet pedantic it warms my heart in the best way possible.

Secondly, I stand by the statement that there is no merit to this joke. This is because the way it defines interesting is very hand-wavy. There are interesting and non-interesting problems, but by a sleigh of hand you can turn the non-interesting problems interesting, thus proving that basically everything in the universe is interesting. At least in the mathematically describable universe. When everything is interesting, nothing is interesting. So we can dismiss the proof as a silly joke.

What mathematicians find interesting is a different story. However, we can almost certainly say there is only a finite number of problems mathematicians as physical beings can solve. If we have 200 mathematical symbols at our disposal, and we consider all strings of these symbols of length 1000,000, we have captured all the descriptions of problems that fit to 1M symbols. But that's a finite number. Going beyond that starts to be difficult for a human to grasp (if 1M is not too much already), so all mathematical problems that are solvable by a physical mathematician are in that set of strings. And that's not even saying anything about whether or not they're interesting..

I still don't quite agree with the approach to mathematics as enumerating problems of a certain size, but to be honest I'm not prepared or motivated to keep this conversation going without turning to handwavy arguments based on my imperfect idea of what mathematics is and how it works. However, it's certainly given me something to think about, so it hasn't been completely pointless. Thank you for the discussion.

No worries, and thank you as well.

By the way, if I'm not completely mistaken, Gödel's argument to show the incompleteness of mathematics relies on encoding all mathematical statements as numbers. So I'm certainly not being very original here.

FWIW I think the concept of "interesting" is so subjective, time-dependent, and nebulous, that to say there's "only a finite number of interesting" mathematics problems feels absurd to me. What's interesting in ten years time will depend on what other interesting things have been discovered in the intervening time. But you're right, I shouldn't have responded with a cheap joke.

Godel's incompleteness theorem technically relates to individual axiomatic theories (i.e. the set of facts that logically derive from a given set of axioms). The numerical encoding you refer to applies to logical statements within that theory. Arguably, the kind of mathematics that humans do is not constrained to a single theory, but is a more general form of reasoning that is often flexible about which axioms may or may not be assumed.

An interesting problem must have a description that fits in a brain, at least for now. Your description-length argument assumes arbitrarily large storage.

the smallest problem that cannot fit in a brain would be pretty interesting

Sorry, I assumed the inductive construction was implied; you can indeed describe properties of that particular interesting problem (though of course you can’t hold its definition in your head), so it goes in the list. Keep going. At some point you’ll hit problems where the process of constructing the problem doesn’t even fit in a brain, etc. There are at least countably many problems, but finitely many problems which any algorithm-which-fits-in-the-brain can describe given finitely many inputs-which-fit-in-the-brain.

This isn’t an enormously important point - the actual question at issue is an empirical one, “in a steady state, can we produce interesting problems at a rate that exceeds our ability to solve them and integrate our understanding” or something like that - but I did rankle at a “trivial” proof which is invalid due to equivocating between multiple definitions of the word “interesting” (which should really take an object, “interesting to me” vs “interesting to something smarter than me”).

As a concept, sure. As a formal proof structure?

Interesting is a bit in the eye of the beholder. Some people probably find maths boring full stop, some probably find all of it interesting.

That's a good question that can be answered by methods in mathematics.

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