philosophy of mathematics - my current position
When choosing some extra subjects for the semester of autumn 2022, I accidently came across the course "Pluralist Philosophy of Mathematics" and immediately knew, I was going to take it since this combination seemed really interesting to me for the last three years or so. Now, right at the beginning of the course, I want to stake my current position in the philosophy of mathematics to have a point of reference when writing about my position and my change of thought during and at the end of the semester. As I am new to discussing this topic with others and learning about what has already been thought of, the following text will probably sound a bit unknowing since I will nearly not use any technical terms, but that's what I simply am in this field at the time. All the more, I'm looking forward to being able to express myself more precisely in this interesting field by the end of the semester.
Coming to my current position, I will argue about three very distinct thoughts I have at the time which will hopefully becoming somewhat more connecting in the upcoming weeks. These are, firstly, my strong believe that number theory and some other fields of mathematics which have in common that they are very intuitive and often done by mathematicians simply for the sake of "playing" with the objects belonging to these theories are to be approached in a rather intuitive and non-abstract manner because we will come to results far faster by doing so and not trying to prove everything starting from some axioms (even though it, of course, is necessary to reduce all of the findings to the axioms these theories are based on in the very end). My reasoning for this approach to these "playing grounds of mathematics" is, first and foremost, that there are many theorems arround which have been discovered due to suggestions and claims stated by mathematicians with a good intuition in this field which then have been profen e.g. using induction. This is a strong argument in my opinion since the induction itself doesn't help us to find the right way to prove a claim, but is really only a tool for checking whether a given formula holds (for integers).
On the contrary, I would at the time argue that keeping oneself close to the axioms when working with all sorts of algebraic structures, like vector spaces, groups, rings and fields or other mathematical objects which are not too easy to grasp, like tensors is pretty important since, at least when beginning to work with these structures/objects one's intuition about them is very little. Hence, I also see the constructivist part of mathematics (hopefully I used the word construtivist in a correct sense here) on the other end of the story and yet don't know how to connect these two very different approaches to mathematics. At the time, I see myself as one or the other depending on which field I'm currently working in.
Lastly, after reading something about foundationalism which is about taking one theory as The Foundation of mathematics and trying to derive everything from it respectively, reduce everything we already found to it, I am really eager to have a word on this view because I am very opposed to this philosophy of mathematics to some very simple reasons: In my opinion mathematics is not about having one theory and proving everything in this specific theory, but rather about coming up with a theory which at best but not necessarily is somewhat relevant to our description of the world and deriving as much as possible out of this theory, only to come up with a new theory later on. Hence, having an ultimate foundation of mathematics would mean to ignore all the theories described by taking the negative of some axiom of The Foundation and turning it into an axiom itself (while discarding the original axiom). As a result, defining a foundation would directly lead to curtail mathematics and I am very much opposed to doing so.
I hope I could make some interesting points here and am looking forward to discussing these points out of a retroperspective in half a year's time!