## Does mathematics need a philosophy? — 2

*A few more thoughts after the TMS meeting (mainly for non-philosophers) … *

‘Does mathematics need a philosophy?’ The question isn’t exactly transparent. So, to ask one of those really, really annoying questions which philosophers like to ask, what exactly does it *mean*?

Well, here’s *one* more focused question it could mean (and it was in part taken to mean in the TMS discussion): should mathematicians take note of, care about, the philosophy of mathematics as currently typically done by paid-up philosophers of mathematics? Both Imre Leader and Thomas Forster had something to say about this. And they agreed. The answer to this more focused question, they said, is basically “no”. Thomas went as far as saying,

The entirety of “Philosophy of Mathematics” as practised in philosophy departments is — to a first approximation — a waste of time, at least from the point of view of the working mathematician.

Fighting talk, eh?! But is that a reasonable assessment?

Well, I suppose it could have been that much of the philosophy is a waste of time because philosophers just don’t know what the heck they are talking about when it comes to mathematics. But that’s rather unlikely given how many professional philosophers have maths degrees (when I was in the Philosophy Faculty, a third of us had maths degrees, including one with a PhD and another with Part III under their belts). So it probably isn’t going to be just a matter of brute ignorance. What’s going on among the philosophers, then, that enables Imre and Thomas to be quite so sniffy about the philosophy of mathematics as practised?

Here’s my best shot at making a case for their shared view. There’s a lovely quote from the great philosopher Wilfrid Sellars that many modern philosophers in the Anglo-American tradition [apologies to those Down Under and in Scandinavia …] would also take as their motto:

The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term.

Concerning mathematics, then, we might wonder: how do the abstract entities that maths *seems* to talk about fit into our predominantly naturalistic world view (in which empirical science, in the end, gets to call the shots about what is real and what is not)? How do we get to know about these supposed abstract entities (gathering knowledge seems normally to involve some sort of causal interactions with the things we are trying to find out about, but we can’t get a causal grip on the abstract entities of mathematics)? Hmmmm: what maths is about and how we get to know about it — or if you prefer than in Greek, the ontology and epistemology of maths — seems very puzzlingly disconnected from the world, and from our cognitive capacities in getting a grip on the world, as revealed by our best going science. And yet, … And yet maths is intrinsically bound up with, seems to be positively indispensable to, our best going science. That’s odd! How is it that enquiry into the abstract realms of mathematics gets to be so empirically damned useful? A puzzle that prompted the physicist Eugene Wigner to write a famous paper called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”.

Well, perhaps it’s the very idea of mathematics describing an abstract realm sharply marked off from the rest of the universe — roughly, *Platonism* — that gets us into trouble. But in that case, what else is mathematics about? Structures in some sense (where structures can be exemplified in the non-mathematical world too, which is how maths gets applied)? — so, ahah!, maybe we should go for some kind of *Structuralism* about maths? But then, on second thoughts, what are structures if not *very* abstract entities? Hmmmm. Maybe mathematics is really best thought of as not being *about* anything “out there” at all, and we should go for some kind of sophisticated version of *Formalism* after all?

And so we get swept away into esoteric philosophical fights, as the big Isms slug it out (there are more guys than I’ve mentioned waiting on the sidelines to join in too: I’ll come back to them in the next post).

Now: the sorts of questions that ignite the Battle of the Isms do look like perfectly good questions … for philosophers. But they are questions which seem to get a lot of their bite, as I say, from worries about how maths hangs together with other things we tend to believe about the world and our knowledge of it. And the working mathematician is likely to think that, fine questions though they may be, s/he has quite enough nitty-gritty problems to think about *within* mathematics, thank you very much, and is far too busy to pause to worry about how what s/he’s up to relates to *other* areas of enquiry. So it’s division of labour time: let the philosophers get on with their own thing, building broad-brush ontological and epistemological stories about Life, the Universe, and Everything (including the place of maths); and let the mathematicians get on doing *their* more particular things. The philosophers had better know a smidgin about maths so their stories about how it fits into the Big Picture aren’t too unrealistic. But the mathematicians needn’t return the compliment, ’cos Big Picture stuff frankly isn’t their concern.

Right ….

Doesn’t that actually look a pretty sensible view, which would sustain the line that Imre and Thomas took (and indeed, between them, they made a few remarks suggesting this sort of picture)?

However, I’ll suggest in the next instalment of these comments that there is after all some reason to think that mathematicians are inevitably, like it or not, entangled with some Big Picture questions. But first, in the rest of *this* instalment, let’s note a rather different reason why we (qua mathematicians) should hesitate to be *quite* so quick to ignore the philosophers.

The simple truth is that philosophers in fact talk about *much* more than the Big Picture stuff. To be sure, the beginning undergraduate curriculum tends to concentrate in that region: e.g. for an excellent textbook see Stewart Shapiro’s very readable *Thinking about Mathematics* (OUP, 2000). [And by the way, Shapiro has interesting things to say in the very first chapter of the book relevant to the general question whether mathematical practice is underpinned by philosophical assumptions.] But the philosophers also worry about more specific questions like this: Have we any reason to suppose that the Continuum Hypothesis has a determinate truth-value? How do we decide on new axioms for set theory as we beef up ZFC trying to decide the likes of the Continuum Hypothesis? Anyway, what’s so great about ZFC as against other set theories (does it have a privileged motivation)? In what sense if any does set theory serve as a foundation for mathematics? Is there some sense in which topos theory, say, is a rival foundation? What kind of explanations/insights do very abstract theories like category theory give us? What makes for an explanatory proof in mathematics anyway? Is the phenomenon of mathematical depth just in the eye of the beholder, or is there something objective there? What are we to make of the reverse mathematics project (which shows that applicable mathematics can be founded in a very weak system of so-called predicative second-order arithmetic)? Must every genuine proof be formalisable (in the sort of way I talked about in the last post), and if so, using what grade of logical apparatus? Are there irreducibly diagrammatic proofs? …

I could go on. And on. But the point is already made. These questions, standing-back-a-bit and reflecting on our mathematical practice, can still reasonably enough be called philosophical questions (even if they don’t quite fit Sellars’s motto). They are more local than what I was calling the Big Picture questions — they don’t arise from looking over our shoulders and comparing mathematics with some other form of enquiry and wondering how they fit together, but rather the questions are internal to the mathematical enterprise. Yet certainly they are discussed by mathematically-minded people who call themselves philosophers as well as by philosophically-minded people who call themselves mathematicians (sometimes it is difficult to remember who is which, and some people such as Solomon Feferman and Steve Awodey are in both camps — it is worth having a look at their lists of papers to see what they get up to). *And the sort of questions we’ve listed surely are worth some mathematicians thinking about some of the time*. Which, thankfully, they do.

So let’s not, after all, be as dismissive of the philosophers as Thomas was!

*To be continued …*