Since you had restarted this thread after it had had a well deserved rest for two and half years,
I, and others, said two and half years ago, that thinking that axioms are "self evident" is an error.
No, there wasn't. You may be thinking of Goedel's proof. But that was in the 1920's and what he proved was that any system of axioms (large enough to encompass the natural numbers) is either
"incomplete" (there exist statements you can neither prove nor disprove) or "inconsistent" (you can prove both a statement and its negation). This does not mean that nothing can be proved. There exist good reason to believe that all axiom systems we use are consistent so that just says that there will be some things we cannot prove without adding new axioms. No problem with that.
It would be if we stopped there. But we don't. We "define" Euclidean geometry to be a system in which the parallel postulate is true. If we stopped there and refused to consider any other kind of geometry, we would be wrong (or at least limited). But we don't. We also "define" other kinds of geometries in which the parallel postulate is false and see what happens in those. That is what I meant about "if... then...". If
"given a line and a point not on that line, there exist a unique line parallel to the given line through the given point" then
all the results of Euclidean geometry. But also if
"given a line and a point not on that line, there exist more than one line parallel to the given line through the given point" then
all the results of hyperbolic geometry and if
" given a line and a point not on that line, there exist no line parallel to the given line through the given point" then
all the results of elliptic geometry.
While mathematics "assumes" things for one kind of mathematics, in total, it considers all
If you want to say that mathematics alone cannot "prove" statements about nature or, say, physics, then
you would be perfectly correct. That is not what mathematical proofs do. I will say the same thing I said above (and two and a half years ago!)- all statements in mathematics are of the form "If
... then ...". All mathematics does is say "If" the axioms are true, then these are the things that will follow. You cannot argue that mathematics is "wrong" by arguing against the axioms, though you could argue that it is useless because we do not know whether those things are true or not. But history has shown that, indeed, mathematics is very useful! And it is useful in so many different ways specifically because it "assumes" so many different things. For any application, there is bound to be some form of mathematics that "assumes" just what you want!