In an interesting debate in the comments to this post, Steve Esser made an assertion to the effect that, while the tokens we use to represent mathematical objects are arbitrary and conventional, the "logical and mathematical relations" we use are not, and, were it possible to "replay the tape" of (presumably cultural) evolution, those would be replicated. His point in making this hypothetical statement is that it illustrates that such relations are, as he puts it, intrinsic to reality and not merely human inventions or conventions. And that in turn may buttress the case for the existence of abstract objects -- i.e., for platonism, of at least a mathematical variety.
My position is a different one, though, as I said in a response, it may look similar up to a point -- I think that the mathematical entities and relations we use are indeed human inventions, but useful inventions as opposed to mere conventions, and this usefulness would lead to the reappearance, in some form, of at least the simpler and more basic ones -- that is, there would be at least some degree of convergence, based upon practical utility, in any replay of the tape. But it's interesting that the " tape" metaphor was famously used by Stephen Gould in Wonderful Life to make just the opposite assertion -- in his hypothetical, the replay of the evolutionary tape would be very unlikely to replicate the same biological forms we see today (such as ourselves). The point he was concerned to make was that we commonly fail to appreciate just how varied are the options that evolutionary processes have before them -- nature, for Gould, is no platonist.
In any case, with that as setup, here's the fable:
Imagine that we finally encounter technological aliens. I think we underestimate the problems involved in translating the communicative processes of radically dissimilar life forms, but let's say that those problems are mutually ironed out and communications are established. At that point there was some consternation on the human side when we had a hard time even detecting, in the alien culture, anything that looked like mathematics as we understand it, and what we did find seemed to bear little resemblance to the concepts, objects, or relations that we use. Now, platonism had long since won the day within human culture, and so this seemed puzzling to say the least, particularly in understanding how they could have developed a space-warp drive without even discovering complex numbers. But human mathematicians were in any case happy to share such insights into the intrinsic nature of reality with their alien counterparts, and in doing so they even learned from the aliens a few computational tricks, involving some fictitious "entities" and highly dubious "relations", but which nevertheless turned out to greatly simplify certain crucial calculations.
The aliens, as it happened, were as puzzled by our own "mathematics" and our own technological success as we were of theirs, but, since the idea that "abstract objects" might have an actual existence hadn't occurred to them, they took a more pragmatic approach to the problem. Complex numbers, it turned out, weren't really of much use to them, though they were polite about it, but they too found that other human abstractions and relations did work better than their own, and these they simply incorporated into their version of mathematics quite readily, since they didn't feel that they were in any way thumbing their noses (they did have noses) at intrinsic reality in doing so.
Eventually, though, a human-alien team working on the problem of reconciling the two versions announced a fairly comprehensive revision of mathematics/[rough alien equivalent] which brought together the most efficient and useful concepts, objects, and relations from both traditions. Humans, working within their platonist assumptions, were torn -- some were scandalized by the revision, and felt that, in its alien-inspired portions, it really amounted to little more than an assortment of cheap tricks; some even began to have doubts as to whether mathematical abstractions really did exist in nature, rather than being just a handy and practical way of organizing our thoughts about nature. Others, however -- the more progressive platonists -- hailed the revision as a fundamental change in our understanding of the true nature of reality. For these people, our old mathematics, though appearing to provide us with a grasp of the abstract objects inherent in the universe, had actually mislead us in certain subtle but critical ways, which the revision had fixed -- and this now was the new, the really real, intrinsic reality.
And then, just as this progressive view was taking hold, and platonism had regained its old confident self within the airy reaches of human higher learning, another technological alien race was discovered....
The moral: Oh, something like -- "Even platonists need to be elastic" (which is admittedly a bit limp).