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, as far as we can tell, and not a lot of them
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.
> 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.
That's physics. Not all math is physics.
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.
When you have a moment, please reference some proofs supporting this.
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.
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.
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.
In our lifetimes computers have made a lot of combinatorial and graph questions meaningful that otherwise would not be interesting.