> There is no “one” mathematics because mathematics is the human exploration of idealized objects using a variety of human logical systems.
But isn't the whole point to to do our best to bypass human-centric systems of understanding, and arrive at the "core truth" of the matter? Whether that's possible is another matter, but even if it's not possible, surely it's something that can be theoretically approached, and I would wager is precisely what PG means by "one mathematics."
> And then there’s the possibility that our mathematics and overall perception of reality is shaped by our biology in far deeper ways than we imagine and currently understand.
Yes, but also no. Consider some first principles:
-We have every reason to believe that any and all life would not live forever, or if the life in question is "intelligent" (a nebulous/human-centric term, for sure) would at the very least conceive of other things not lasting forever (such as stars, or even the universe itself [or, if you want to be really generous, "this current iteration of the universe"]).
-Therefore we can reasonably assume that all "intelligent" life in the universe would understand the concept of scarcity (either via finite lifespans/time, food/energy sources, both, or something else), non-infinity. I'd go so far as to say that any life form that doesn't understand its own mortality or other such limits should be not be considered "intelligent," at least for the reasons of this discussion.
-Therefore we can reasonably assume that said intelligent life would somehow conceptualize a binary state (you're either alive or you're not, you either have access to an energy source or you don't, etc), and consequently would somehow or another understand the concept of "zero," "nothing," etc, as well as its opposite, "something." And from there, would necessarily discern the differences between two states of "somethings" (the state of "something" that is "one" is different than the state of "something" that is "two").
I know I'm using a lot of loaded terms here -- "reasonably," "assume," "discerning" -- but just like we look for life by looking for the markers of life that we know were necessary for Earth (carbon, water, etc), we can look for intelligence that exhibits the properties that we understand it to have. We need some sort of frame of reference, after all, if we are to do anything other than simply flail. If that frame of reference is to be proven wrong, that's wonderful, but until that's the case, I don't think it's unreasonable to assume that the universe's "primitives" would be perceived in any truly, truly different way such that the species' interpretation would cause humans to rethink our own understanding of the universe's "primitives" from the ground up.
After all, conceiving a difference between hydrogen and helium requires being able to tell the difference between one and two (electrons, as well as separate elements themselves). And considering we have every reason to believe that those make up the majority of the mass in the universe, any "intelligent" life (there's that human-centric term again) can be expected to somehow conceptualize that difference, and thus, do something like counting, and thus, approach the same primitives of mathematics that we do. The approach might be different, but what they're approaching -- the very fabric of reality, hopefully as objectively as possible -- must be assumed to be the same (again, that is, until we're given compelling evidence to believe otherwise).
That said, I've never studied the philosophy of mathematics, so I could be talking out of my ass here, this is just the reasoning of a layman after all. If anyone reads this and goes "no you're way off base," I'd love to hear it!