(December 10, 2021 at 2:34 am)vulcanlogician Wrote: Do you think non-Euclidean geometry refutes Euclid's axioms? If so, how? If not, why not?
Refutes? No. They show that Euclid's axioms are one possibility among many non-equivalent geometries.
If you assume Euclidean geometry, you can construct a model of non-Euclidean geometry inside. If you assume non-Euclidean geometry, you can construct a model of Euclidean geometry inside. The internal consistency of either implies that of both.
At that point, the geometry of 'real space' becomes a physics question and not a mathematics question.
Things start getting interesting when we get to the same phenomena in set theory: there are many different alternatives for set theory, many of which can be proved to be mutually consistent.
And, even more, there are multiple alternatives to *logic*, with standard two-valued logic only one of many possibilities (including paraconsistent logic).
All that means is that we get to choose which mathematical system we want to use to describe other things. Internally, we can study all of the alternatives.
BTW, I am not a Platonist. I am a Formalist when it comes to math.
