Applicability of Maths to the Universe

[polymath257]

And I would say that this is directly contradicted by Godel's results. There are *always* questions that cannot be answered. it isn't simply a matter of not knowing, but that the issue cannot be resolved without making an arbitrary choice in our axioms.

Are there?  To whom or what?  Godel?  Math?  It doesn't seem like we've exhausted the full list of knowers or knowledge.  This would not be enough to reject platonism as a metaphysical stance.  If we assumed that there were some mathematical objects that demonstrably proved mathematical platonism false (and much more from doing it purely by axiom, as we address below) then a platonist can simply concede that mathematical objects are mental objects (and this isn't the only concession that can be made).  It may be the case that there are not always cognitive answers for mental objects.  This would surprise no one.  Satisfied expressions of taste, like "yum" are not cognitive objects, they cannot be true or false, though we've been known to mistake them as such - on account of being satisfied pattern seekers.  

However, in an effort to maintain the position, it will always be posited first and foremost, that we've got something wrong, perhaps as a consequence of our axioms...and what we've got wrong may be godels results.  

Godel proved that *any* axiom system strong enough to talk about the natural numbers has sentences that cannot be resolved. So there will *always* be unanswerable questions.

Well, we *know* that set theory isn't complete in the sense that all meaningful questions have answers that can be proven. And, in any supplemental system, there will ALWAYS be new questions that cannot be answered.

And the *only* real way to resolve such questions is to *arbitrarily* choose which way we want the axioms to go.

Now, we can make such choices based on things like aesthetics, but there is no way to determine the truth or falsity without making additional assumptions.

Which is a leveler for platonism.  If it's just a disagreement over axioms, and if the things that are taken to be indicative that platonism might be false are merely products of differing axioms or aesthetics.....

No, that is NOT the point. No matter what axiom system you choose, there will be questions that cannot be answered. The mathematical system is *defined* by the axioms, so the basic definition guarantees there will be unanswerable questions.

Furthermore, you can always *add* to the axioms either way, defining two *new* systems of mathematics.

It's sort of like geometry. Once we found that Euclidean geometry is NOT automatic, that said that the Platonic ideals for geometry simply don't exist in the way that Plato imagined.

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I wanted to add that we could go through an endless list of maths that might seem to show that this platonist conjecture or that one might be false - but they'll all be as easy to dismiss as the last.  It's because we're arguing from a purely axiomatic standpoint, unless we're willing to bring it back to something in the OP conjecture.  Unless we want to present some reason to believe that one set of axioms is better than another, then no two positions based on divergent sets of axioms can touch each other.

Exactly the point which shows that Platonism isn't correct.

It isn't that one set of axioms is 'correct' and another isn't. It is that *no* axiom system can manage to answer all of the questions and ALL axiom systems produced by adding on undecidable questions are equally legitimate logically.

Platonists like to talk like the notions like 'set' are intuitive and obvious. But, when push came to shove, the intuitive version was self-contradictory.