Mathematics itself is about structures and operations, properties of mathematical objects (like closedness and completeness) a.s.o.
Numbers and proofs are just the tools of mathematics to work with those things.
This description defines abstract algebra and some topology well, but I am an applied mathematician and for me mathematics is about number crunching and stability of this number crunching. Without numbers or numerical structures like polynomials and matrices you'll only have set theory, some parts of algebra and some mathematical logic.
Philosophers of mathematics spent the 20th century what mathematics is all about and they did not settled on any potential definition.
Philosophers of mathematics spent the 20th century what mathematics is all about and they did not settled on any potential definition.
Ok, I have to disagree with you here. What Godel established is that there's no complete axiom system. Every mathematical formalism will have statements P for which neither P nor ~P can be proven. This was not an unexpected result. What makes Godel's Incompleteness Theorem awesome is that it proved incompleteness, which few people thought possible.
Same with Turing and the Halting Problem. Almost no one actually believed that such a program (that could determine, algorithmically, if a program halted) existed. If one did, it would blow open all of mathematics. Turing proved, in a very elegant way, that it didn't.
What 20th-century mathematicians agree upon is that mathematical statements aren't "true" or "false" in an absolute, platonic sense, but that they are products of the axiom systems that generate them.
Whether the Axiom of Choice or Continuum Hypothesis are "true" is meaningless. These aren't mathematical "controversies" that have people yelling at each other saying that the other is wrong. They're axioms about infinity (specifically, uncountable infinity) that, although they have no physical correlates (you can't actually Banach-Tarski an orange) are logically independent of the "obvious" axioms. What logically independent means is that neither L nor ~L will generate a contradiction, and therefore neither has any absolute high ground.
Most mathematicians use AoC and CH for typical mathematics, but there are alternative mathematical worlds in which they don't hold, and those are interesting in their own right.
I know about that, AoC, CH, Gödel, Turing, Cantor, Hilbert and such, I had a course on mathematical logic and one on set theory when I was an undergrad and the lectures about Gödel were the high point in both of them, although in set theory there were some high points with the idea of "set of all sets" and some other things.
AoC is used indirectly in almost every place of pure mathematics, but I researched numerical methods for PDEs and Stochastic PDEs which is not pure mathematics, maybe someone who is much more intelligent than me will say where you will use AoC directly in this area outside of some theorems of Analysis that are used but well, I've never ever touched this same axiom again in my life and if I chose to spend my life as a researcher I doubt I will ever need it again to work and publish, and yet I was able to get a Ph.d in Applied Mathematics. The theorem that I know uses the AoC that I cited once but not exactly used is Banach–Alaoglu theorem.
As sbi put if you go to a department of mathematics that includes mathematicians, statisticians and applied mathematicians chances are that almost no one will know too much of set theory excluding some pure mathematicians, this was true on almost every department that I saw in my entire life. So there are mathematicians whose life are not spent trying to use abstractions everywhere. That was the point.
I don't want to put words in hazov's mouth here but Gödel is a total nonsequitor here. This thread isn't about formal set theory. The question is not whether mathematics can be axiomatized in a satisfactory way in first-order logic, but what it is mathematicians study. Since not everything in mathematics is obviously related to geometry or numbers, it is hard to write a satisfactory definition; just look at the difficulty that, say, Wikipedians have had trying to cook up a canonical one. But saying that mathematics is just about rigor or abstract structures makes just about everything mathematical. Not all scholars are mathematicians. And if mathematicians just study abstract structures, what's so special about, say, elliptic PDE, or Fourier analysis, or commutative algebra?
