What does a mathematician do?

So I'm not quite up to the level of legitimate mathematician right now, but I feel like I'm on the right path by entering a Ph.D program.  As such, many people have asked, "So... What exactly do you do as a mathematician?" and "Math research?  I thought we had all of that figured out..." In order to give those of you curious enough to ask the question an answer, I'm going to post 3 papers I wrote last semester.  Two of them are about the philosophy of mathematics (which is about as esoteric as one can get), and the last is a final paper I wrote for an undergraduate research project on Hadamard matrices.

The first paper is about logic and it's role in mathematics.  Most people, I believe, confuse "logic" with "rational thought."  Logic is the languageless tool use when performing rational thought.  The paper discusses this distinction a bit, explains how three of the most prominent 20th century philosophies of mathematics view logic, and (if I remember correctly) has my own little take on it.  I wrote this for an  independent study I did with Matt Fahy, and you can find a link to it right here:  Link.  (I just realized I hadn't put my name on this paper, but trust me, it's my own work.)

The next paper is about whether or not this philosopher, Stephen Korner, was (he committed suicide...) a nominalist or a realist about mathematical objects, entities, etc.  This paper might not be quite as interesting since it was sort of a response to a book of Korner's I read, but it should be self-contained for the most part.  This paper was also for the I.S. I did with Matt Fahy.  Find it's link here: Link.

The last paper goes through an introduction to Hadamard matrices, their generalizations, and constructions, then outlines our attempts at generalizing a particular construction.  We also provide a new definition of a Hadamard hypercube that is more inline with the historical definition of a Hadamard matrix.  We didn't have time to do much with this definition, but hopefully this paper will provide someone with the desire to do slay some Hadamard hypercubes.  Here's the link to my last paper:  Link.
(Note:  While this is the paper that I put the most work into, and also the paper that I'm most proud of, it's by far also the hardest to understand, especially if you have little mathematical background.  There are a bunch of proofs and proof like things included, so enjoy it as best you can.)

Lemme know what y'all think!

(I'll be posting a more interesting thing post later, I've been wanting to do this for a bit though)