It is usually said that most mathematicians are mathematical platonists or mathematical realists (those two are often considered to be the same). It should be pointed out, however, that most mathematicians have not studied a lot of philosophy so what they say about their mathematical philosophy is not always well thought out. Perhaps that same comment applies to what I am posting here.
There are good accounts of the philosophy of mathematics in both Wikipedia and the Stanford Encyclopedia. A major issue in the philosophy of mathematics is the question of what are mathematical entities such as numbers. Most mathematicians agree that mathematical entities are abstract, so are not physical. The platonist view is supposedly that numbers exist in a platonic world of ideal forms. Personally, I take numbers and other mathematical entities to be useful fictions. I suppose that makes me a fictionalist, though others might consider me a platonist. But then the platonic world of forms seems like a fictional place. My main disagreement with platonists is over the Continuum Hypothesis. As Gödel and Cohen have shown, the continuum hypothesis is independent of the other axioms of set theory. I see that as settling the issue, while many platonists believe that there is a fact of the matter which we have yet to settle.
Some philosophers seem to think that the status of numbers and other entities has important implications for mathematical knowledge, or for the question of whether there is such a thing as mathematical knowledge. And some think that this also raises questions about why mathematics has turned out to be so useful to science. Yet, to me, it is quite clear that there is mathematical knowledge, and it seems quite apparent as to why mathematics is useful in the sciences. And whether or not numbers are fictions, platonic entities, or anything else has little relevance to those issues.
I intend to discuss mathematical knowledge, and the relation of mathematics to the sciences in future posts. The current post is a warm up exercise.
I tend to see mathematical formalism as a kind of fallback position. That is, mathematics can be described as if it consisted of playing formal games with symbols. But I doubt that many mathematicians actually think about mathematics in that way. Similarly, I suspect that most mathematicians would find logicism (the view that mathematics is a part of logic) to be implausible.
I see intuitionism and the constructivism of Errett Bishop as interesting, and as a bit like doing mathematics with one’s hands tied. I’m sure that Bishop didn’t see it that way. However, I am not at all persuaded by the philosophical arguments often used to support intuitionism and constructivism.
That’s a casual overview of how I see mathematics. I shall be more detailed in some upcoming posts.