Why do some mathematicians feel like mathematical objects are "really out there" in some metaphysically fundamental sense? For example, if you ask mathematicians whether ZFC + not Consistent(ZFC) is consistent, they will say "no, of course not!" But given ZFC is consistent, the statement is in fact consistent due to by Godel's second incompleteness theorem^[1]. Similarly, if we have the Peano axioms without induction, mathematicians will say that induction should be there, but in fact you cannot prove this fact from within Peano, and given induction mathematicians will say transfinite induction should be there.

I argue that an explanation could be from logical induction. In logical induction, fast but possibly wrong sub-processes bet with each other over whether different mathematical facts will be proven true or false by a slow but ground-truth formal system prover. Another example of backstops in learning. But one result of this is that the successful sub-processes are not selected very hard to give null results on unprovable statements, producing spurious generalization and the subjective feeling--as expressed by probabilities for propositions--that some impossible theorems are true.

Of course, the platonist can still claim that this logical induction stuff is very similar to bayesian updating in the sense that both tell you something about the world, even when you can't directly observe the relevant facts. If a photon exists your lightcone, there is no reason to stop believing the photon exists, even though there is no chance for you to ever encounter it again. Similarly, just because a statement is unprovable, doesn't mean its right for you to have no opinion on the subject, insofar as the simplest & best internal logical-induction market traders have strong beliefs on the subject, they may very well be picking up on something metaphysically fundamental. Its simply the simplest explanation consistent with the facts.