I wish I understood how mathematicians conceptualize the notion of 'right' and 'wrong' axioms in light of Gödel. I would've thought that the incompleteness theorems would be taken as evidence against Platonism, but clearly this didn't happen. What do people think is the source of 'ground truth'? Is it more than aesthetic appeal? Are there any articles or books that convey it?
I've talked to a friend with a doctorate in algebraic geometry about this several times, but all I've really gotten out of her is that if I studied a lot more math, I would gradually get the same sense.
(For reference, I have about as much mathematics background as a US student who minored in maths.)
> I would've thought that the incompleteness theorems would be taken as evidence against Platonism, but clearly this didn't happen
I don't think they really are evidence against it. The first incompleteness theorem says (to put it simply) there are truths about the natural numbers you can't prove, and (if we equate proof with knowledge) can't know. I don't know why a Platonist would find that objectionable. I mean, naïve materialism would imply there are lots of facts about the material world we are never going to be able to know (e.g. the particular arrangement of rocks on a lifeless planet in a distant galaxy right now, or as close to right now as relativity permits). If unknowable truths isn't evidence against materialism, why would it be evidence against Platonism?
Really, Gödel's theorems were a much bigger problem for formalism than Platonism. Formalists wanted to identify mathematical truth with provability, and Gödel shattered that dream. Platonists never dreamt the dream, so its destruction didn't discourage them.
> the incompleteness theorems would be taken as evidence against Platonism
I wonder if this is a map/territory thing. Incompleteness theorems apply to models (axiomatisations) of mathematics, e.g. ZFC set theory. Platonism applies to the actual act of doing mathematics, i.e. persuading other mathematicians that certain statements about abstract entities are true or false.
Right. According to Platonism, a mathematician publishing a result about spherical geometry is in fact exploring an objective reality beyond human minds. Contrast that with, say, intuitionism, according to which they are simply describing the mental activity which arises as a result of considering an object of shared understanding known as a "sphere".
I've talked to a friend with a doctorate in algebraic geometry about this several times, but all I've really gotten out of her is that if I studied a lot more math, I would gradually get the same sense.
(For reference, I have about as much mathematics background as a US student who minored in maths.)