Maths is based on axioms. Fundamentally axioms are just things we take for granted for no reason other than we need to have some foundation upon which to build upon. But it's all just stuff we have decided is true because.
There's no problem in maths to shift around the axioms to your hearts content. As long as you make the maths work you're golden. Maths is fundamentally just a game of smarts without application. It's nice if it is applicable. But that's not the point of it. The point is making the patterns work and proving they're true within the same pattern.
I remember when doing logic we had a lecture setting out to prove why 1 is 1 by logical necessity, and why 2 is 2 by logical necessity. By the end of the lecture the board was filled with absurdly convoluted chains of logical deductions that all boiled down to that in one way or another everything is better if we assume this is true.
Logic is a game of deduction, which also doesn't need to be applicable.
It is it's own system. But it sure is nice when logical deductions and scientific proofs come to the same conclusion. If they don't, we use that as evidence for something being wrong.
Sorry, you're not making sense here. You start on maths and then drop it midway in favour of science. Sure maths is it's own system but I would assume mathematicians thinks it's a logical one. What's the evidence it's not? As to "
logical deductions and scientific proofs", I'd need an example to understand what you mean.
EB
"In favour of science"? Science uses both logic and maths. There's no competition here regarding what is better to use. Whatever works works. Both are useful.
Maths can be correct AND illogical. Logic has grown from natural language. It's a tool to understand natural language. Logic is a collection of rules that have grown over time to make this analysis easier. But maths isn't natural language and doesn't always follow the rules of natural language. Maths can be illogical and still be mathematically true.