Unlike chess, there is no “optimal play” for math. And if there is, I think it would be considered slow/boring (if we do have computational constraints).
No, there’s no single “optimal play” for math, but it wouldn’t surprise me if there were a set of fundamental mathematical structures which would be consistently rediscovered by a wide variety of different “math playing” algorithms (of sufficient math-playing skill).
The objective criteria is fuzzy, sure, but I am not convinced that you couldn’t eventually figure out a mathematical function expressing your target. Once we didn’t know how to formalize chess, I think (I’m sure we didn’t know how to formalize a unbounded-compute algorithm for it). Our brains have encoded the desirable goal, and it’s perhaps just us happening to not be able to extract the true concept that the blackboxes implement that prevents us from formalizing the objective.
In other words: am I really different from a suped up chess bot (with different cognitive archinecture ofc) with a different goal that isn’t directly available to my cognition?
I’m not saying that such formalised objective couldn’t exist. My claim is that we (probably) haven’t yet found one. And if there will be one, it wouldn’t be “metaphysically objective”, it will just spit out very insightful theorems very fast.
I agree that we haven’t found one (extremely confident!). I am not sure how metaphysically objective you are looking for: I expect that part of what humans look for is a ‘metaphysically singled concept’ (besides practicality), something like good math abstractions to handle the space of proofs, though it may only work if you add on a ‘terminal’/‘prior’ weighting for interestingness.
Are there really whole swaths of completely uninteresting ZFC string manipulation theories? I just don’t know. Alternatively: would aliens also have group theory? I think: probably!
Oh, I plan to post on the topic of alien math. But in short—aliens are going to be guided by beauty/interestingness/utility for the same reason evolution pushed humans to value them, so a lot of our math could intersect (but you still need aliens or humans to pluck out those valuable math bits, you can’t force math look at itself hard enough and present those parts to you).
And they would have group theory because our universe is just full of symmetries.
Even if our universe was not, I expect groups to be discovered by aliens regardless.
I’m not saying that something with no info to point it at the sorts of things humans would care about would get close (e.g. the aliens get data about the same universe we got data from), only that a core part of what math humans find interesting will end up being something simple and ‘singled out’.
Tangentially I don’t think of group theory as fundamentally being about symmetry, though I see symmetry as being about groups. that is, that symmetry is well modeled by groups, but groups are not fundamentally about them.
Instead I think of it simply as a generalization of grade school algebra, asking when you can do things like xy = z → x=z/y. Thus, rings and fields, which are also not about symmetry, and instead are about generalized grade school algebra.
In a universe where symmetries weren’t so fundamental to physics but you still had arithmetic being useful for world modelling, I would expect life there to invent polynomials, symbolic algebra, and abstract algebra.
Unlike chess, there is no “optimal play” for math. And if there is, I think it would be considered slow/boring (if we do have computational constraints).
No, there’s no single “optimal play” for math, but it wouldn’t surprise me if there were a set of fundamental mathematical structures which would be consistently rediscovered by a wide variety of different “math playing” algorithms (of sufficient math-playing skill).
The objective criteria is fuzzy, sure, but I am not convinced that you couldn’t eventually figure out a mathematical function expressing your target. Once we didn’t know how to formalize chess, I think (I’m sure we didn’t know how to formalize a unbounded-compute algorithm for it). Our brains have encoded the desirable goal, and it’s perhaps just us happening to not be able to extract the true concept that the blackboxes implement that prevents us from formalizing the objective.
In other words: am I really different from a suped up chess bot (with different cognitive archinecture ofc) with a different goal that isn’t directly available to my cognition?
I’m not saying that such formalised objective couldn’t exist. My claim is that we (probably) haven’t yet found one. And if there will be one, it wouldn’t be “metaphysically objective”, it will just spit out very insightful theorems very fast.
I agree that we haven’t found one (extremely confident!). I am not sure how metaphysically objective you are looking for: I expect that part of what humans look for is a ‘metaphysically singled concept’ (besides practicality), something like good math abstractions to handle the space of proofs, though it may only work if you add on a ‘terminal’/‘prior’ weighting for interestingness.
Are there really whole swaths of completely uninteresting ZFC string manipulation theories? I just don’t know. Alternatively: would aliens also have group theory? I think: probably!
Oh, I plan to post on the topic of alien math. But in short—aliens are going to be guided by beauty/interestingness/utility for the same reason evolution pushed humans to value them, so a lot of our math could intersect (but you still need aliens or humans to pluck out those valuable math bits, you can’t force math look at itself hard enough and present those parts to you).
And they would have group theory because our universe is just full of symmetries.
Even if our universe was not, I expect groups to be discovered by aliens regardless.
I’m not saying that something with no info to point it at the sorts of things humans would care about would get close (e.g. the aliens get data about the same universe we got data from), only that a core part of what math humans find interesting will end up being something simple and ‘singled out’.
Tangentially I don’t think of group theory as fundamentally being about symmetry, though I see symmetry as being about groups. that is, that symmetry is well modeled by groups, but groups are not fundamentally about them.
Instead I think of it simply as a generalization of grade school algebra, asking when you can do things like xy = z → x=z/y. Thus, rings and fields, which are also not about symmetry, and instead are about generalized grade school algebra.
In a universe where symmetries weren’t so fundamental to physics but you still had arithmetic being useful for world modelling, I would expect life there to invent polynomials, symbolic algebra, and abstract algebra.