5

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.

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".

Joonas Ilmavirta
  • 25,809
  • 3
    "Philosophical" is used in those quotes in the same way one would say "morally", and there is nothing philosophical intended, really. – Mariano Suárez-Álvarez Aug 01 '17 at 16:07
  • 2
    @MarianoSuárez-Álvarez There indeed is nothing truly philosophical, so I want to figure out what is actually meant. Thanks for reminding me of "morally", which, morally, is the same as "philosophically". :) – Joonas Ilmavirta Aug 01 '17 at 16:10
  • 3
    It simply means "the idea is that…" For example, the Fourier transform is a unitary linear map, but since it acts on an infinite dimensional vector space it is not represented by a matrix — but it does not hurt to pretend it does when thinking about it. I think you are overthinking about this, really... – Mariano Suárez-Álvarez Aug 01 '17 at 16:16
  • @MarianoSuárez-Álvarez True, I may be overthinking. (In my defense, I'm paid to overthink stuff related to mathematics.) I was hoping someone would have some insight, but I'm not sure if there's anything illuminating to say... If nothing else, I hope the question itself can remind people that the mathematical "philosophy" and "moral" are not really philosophical and moral. – Joonas Ilmavirta Aug 01 '17 at 16:21
  • @Joonas: Hold on, there does exist a mathematical philosophy. Or many, if you may; think of platonism vs constructivism. It's just that what you quoted is not mathematical philosophy. – Vincenzo Oliva Aug 01 '17 at 16:26
  • @VincenzoOliva Oh yes, there is mathematical philosophy. By mathematical "philosophy" I mean the kind of thing described in the question. They are quite different, so I use quotes extensively. I'll edit to clarify. – Joonas Ilmavirta Aug 01 '17 at 16:29
  • @JoonasIlmavirta: Oh, well then – Vincenzo Oliva Aug 01 '17 at 16:31
  • @JoonasIlmavirta: Logicians, Engineers and Computer Scientists would definitely think about the philosophy of mathematics in realising abstract mathematics into computable (approximate) solutions or formally verifiable solutions. For example some take infinite sets as a priori as is, some people (like me) think that if some (or even one example) requires a recursive generating function then perhaps they all do. Then perhaps infinite sets cannot be taken as an a priori foundation for mathematics after all. Most mathematicians would probably find this a hugely controversial step to take. Why? – James Arathoon Aug 01 '17 at 16:40
  • I take it as "not precise" or "poetic". – Yes Aug 01 '17 at 17:17
  • @James I think almost everyone (though definitely not everyone) would agree that asserting that all nonnegative integers exist is safe. I think it's closed minded to say that because we will never observe an infinite set in its totality, it must not exist. It's completely consistent with what's known of physics at the moment that the universe is infinite in size, but most of it is not observable. That's our own limitation, and there's no reason to impose it on mathematics in that case, because all nonnegative integers can be realized as a collection of Planck volumes. – Matt Samuel Aug 01 '17 at 17:49
  • 2
    @VincenzoOliva, that has absolutely nothing to do with the usage of the word about which this question is concerned. The same goes for Jamesarathoon's comment. – Mariano Suárez-Álvarez Aug 01 '17 at 18:03
  • 1
    @MarianoSuárez-Álvarez: The OP has already clarified. I addressed the comment in which he wrote that << the mathematical "philosophy" and "moral" are not really philosophical and moral.>> At that point it wasn't clear to me that he acknowledged the difference between this "philosophy" and mathematical philosophy. Cheers. – Vincenzo Oliva Aug 01 '17 at 18:52
  • 1
    What "Philosophical" means and applies to, depends on whom you ask. It happens to be the case that mathematicians have a really poor understanding of what philosophy is, its breadth, etc, and make fun of and chastise what they think the word "philosophical" means. @Mariano exemplifies such ignorance. – amWhy Apr 14 '18 at 23:59

1 Answers1

3

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.

  1. 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".)

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

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

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:

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

What are the philosophical implications of category theory?

Agnishom Chattopadhyay
  • 2,332