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.
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.
daxfohl · · focus · HN ↗
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.
unsupp0rted · · focus · HN ↗
The universe is bounded by rules, as far as we can tell, and not a lot of them
logicchains · · focus · HN ↗
That's physics. Not all math is physics.
unsupp0rted · · focus · HN ↗
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.
mathgeek · · focus · HN ↗
When you have a moment, please reference some proofs supporting this.
unsupp0rted · · focus · HN ↗
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.
sthomer2 · · focus · HN ↗
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.
unsupp0rted · · focus · HN ↗
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.
solomonb · · focus · HN ↗