Some philosophy of mathematics. And of philosophy.
The question I will cover is: Is mathematics purely formal manipulation of strings of symbols?
First, the trivial answers: Yes, since all results, definitions and theorems could in principle be reduced to such. No, because doings mathematics takes imagination, creativity, and whatever other positive-sounding things I would like to mention. (It also requires manipulation of formal systems, occasionally known as abstract nonsense or computation or calculation.)
There was this philosopher named Immanuel Kant. He proved (by very nonrigorous standards and with many hidden assumptions) a number of seemingly contradictory results called antinomies (of pure reason), such that the world is infinite and finite both. I stated something similar above, but it is fairly easy to see that I cheated, since my two statements are actually about different things.
Mathematics is formalistic in that the mathematical results: theorems, definitions and proofs – can be stated formalistically. Mathematics as in the activity that mathematicians do is not so, as it involves building and understanding (often complicated) arguments, seeing analogies, seeing past analogies, and other quite non-formalistic activities.
There is more to the question. Namely, is mathematics something discovered by or created by people? (One could ask this of natural sciences, I think.) Mathematics is discovered in that all the theorems and results are fixed, given a particular set of axioms (which means assumptions): Given proposition is true or false (or undeterminable due to Gödel’s incompleteness or even meaningless in the same sense that not every string of letters is a word or has meaning) based on the axioms in use.
Mathematics is created in that the particular results achieved, axiomatics considered, and so on, are determined by the questions mathematicians ask and answer. Physics, statistics, and various other fields of inquiry motivate plenty of mathematics, for example.
What is our conclusion? It is that the answer one finds is not important, but what philosophy does is highlight concepts that are hard to separate otherwise. This is what I find useful in philosophy. This is what I wish more philosophers recognised.
(Post inspired by this one.)