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.

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unsupp0rted 3 hours ago | parent | next [-]

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

	▲	logicchains 2 hours ago \| parent [-]
		>The universe is bounded by rules That's physics. Not all math is physics.

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itishappy 19 hours ago | parent | prev [-]

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

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n4r9 18 hours ago | parent | next [-]

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.

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karmakurtisaani 17 hours ago | parent | next [-]

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

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Smaug123 6 hours ago | parent | prev [-]

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

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dcl 5 hours ago | parent [-]

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

	▲	Smaug123 an hour ago \| parent [-]
		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”).

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captainbland 18 hours ago | parent | prev | next [-]

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.

	▲	itishappy 16 hours ago \| parent [-]
		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.

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tim333 14 hours ago | parent | prev | next [-]

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.

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anon291 5 hours ago | parent | prev [-]

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