The few Mathematics I have been studying so far is pure Mathematics. I happen to have some discussions with philosophers of Mathematics, but as they know I totally ignore their subject, we do not usually go through deep questions. However, I often find it very hard to even go beyond the very first steps of our conversations. I want to share this with you, so that you can help me to understand what I miss in their viewpoint (hopefully, this will happen before this question gets closed).
Let us look at just one particular instance. I was told Philosophy of Mathematics is concerned with questions like:
Q. When can we say that a mathematical object exists?
Well, I have an answer for the above question - maybe not a complete one, but a start. I would like someone to help me figure out what is wrong or unsatisfactory.
I would start by saying in Mathematics there is a crucial tool, that of a definition; if one finds an object fitting in the definition, that object exists as a mathematical object. "Mathematical objects" are always defined before one can say anything about them (this is another lovely peculiarity of Mathematics): to introduce a "mathematical object" is to give its definition! Then I say such an object exists if I can explicitly produce one fulfilling the definition.
Let me stress more about this with a stupid example: First, I tell you, say, what a prime number is. After that, any human being should in theory be able to pick any number $n$ and check through the definition whether $n$ is prime or not.
That is my idea of a definition. And it applies with no distinction to both
- mathematical objects one can try to visualize (spheres, triangles...) and
- to more abstract ones (all the remaining, e.g.: functors, Galois groups... whatever).
I believe that what explains the survival of question Q. (as an open question) is that many people believe that, just because they can handle an orange, spheres exist (in the perverse sense that one might, sooner or later, experience one).
They somehow prefer thinking about objects of type 1. Of course, spheres do exist, but not on earth! (The unique way one can get a sphere is, by definition, to write down an equation of the sort $x^2+y^2+z^2=1$.) I feel like there always is some sort of unconfessed attraction towards reality, or the secret hope that a resolution of singularities will eventually help us fixing our leaking sinks. This is what, in my opinion, makes question Q. survive, and my answer not an answer.
Let me end with a small but hopefully funny provocation: are we sure we still want Mathematics to be considered as a part of Science? After all, the main feature of Science is: it deals with the real world; no mathematical object properly exists in the real world. It would not be a shame at all not to be considered "scientific"...