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?
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.
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".