What does "philosophical" mean in mathematics?

Question

When teaching or otherwise explaining mathematical ideas and concepts, some mathematicians use the word "philosophical"¹, usually in reference to something that's not. It can also be a way to describe a question to be intuitive, soft or general in some way. In my experience "philosophical" often refers to things that are not rigorously defined (in the particular context) or to ways of thinking by analogy. However, the word does not usually refer to philosophy as a field or in a way a philosopher would understand the word. If nothing philosophical is intended, what is?

For example:

Philosophically, you can consider the Fourier transform as a unitary matrix. There are some complicated details to it, but the key properties are the same.

I have read in some book the following "philosophical" statement: "Introducing randomness we can make unstable things stable." (A philosophical question on randomness)

This question might be more philosophical than mathematical. (What does it mean to solve an equation?)

I apologize in advance because this question might be a bit philosophical, but I do think it is probably a genuine question with non-vacuous content. (Why do differential forms have a much richer structure than vector fields?)

What does "philosophical" mean in mathematical context? Is there a good definition out there? Are there some important subclasses of "philosophy"²? How does a mathematician's "philosophical" differ from that of a philosopher's? I have some vague ideas, but I don't feel I fully grasp the meaning. There are undoubtedly personal and other (temporal or local?) variations in the usage, but I believe that there are some typical uses and meanings — and that I should be more consciously aware of them.

The word "morally" is often used in a similar fashion. Commentary on other similarly used words is welcome, but I want to focus my attention to "philosophically" to avoid making the question overly broad.

This question was inspired by a comment to the accepted answer to this recent question.

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¹ Or "philosophically" or "philosophy". All the words starting with "philosoph-" seem to refer to the same kind of thing in mathematics.

² There is also mathematical philosophy, which is a subfield of philosophy. I have used quotes whenever referring "philosophy" in the sense described in this question, in an attempt to keep a clear distinction between real philosophy and something many mathematicians call "philosophy".

Answer 1

What does "philosophical" mean in mathematical context? Is there a good definition out there?

No. Your question is more philosophical than mathematical :)

Here are three broad senses in which the word could be used, with some usage examples:

1. intuition/principle:

Let's try to imitate the definitions we had for discrete probability spaces to continuous probability spaces. The philosophy is the same.

2. Loose/imprecise ideas/Interpretations:

The philosophy here is that whatever can be achieved by probabilistic polynomial time machines can be achieved by non-uniform polynomial time "machines".

(The above is obviously wrong if interpretted in the literal sense of "whatever".)

3. Soft Ideas (ideas that cannot be objectively evaluated):

The philosophy of giving you these assignments is to get you as much practice as possible.

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How does a mathematician's "philosophical" differ from that of a philosopher's?

The mathematician is only using the word philsophical like any other layman would.

I wouldn't imagine that a philosopher would literally call his idea philosophical. It sounds sort of artificial to me. However, I believe a philosopher of mathematics could also occasionally use the word philosophi-* in a sentence:

4. relating to the methodology, assumptions and foundations of mathematics:

What are the philosophical implications of category theory?