## Saturday, 17 September 2011

### Roy's Fortnightly Puzzle: Volume 9

OK, so not so much a puzzle as a question this time.

I am currently co-teaching a graduate seminar on philosophy of mathematics this semester (structuralism versus logicism, to be more specific). We did a pretty good job of advertising the seminar, and as a result have a number of mathematicians sitting in the class (both faculty and graduate students).

The issue is this: As we talk about the philosophical questions and their possible solutions (for example, last week we read Benecerraf's "What Sets Could Not Be" and "Mathematical Truth", since these set up the issues at issue between modal structuralism and Scottish logicism quite nicely), the mathematicians kept coming back to the fact that none of these issues seem to have any bearing on what mathematicians actually do.

At one level I agree with this - when actually doing mathematics, mathematicians need not, and probably ought not, be thinking about whether their quantifiers range over abstract objects or something else. Rather, they should be worrying about what follows from what (to put it in an overly simplistic way).

There might be an exception to the above paragraph in moments of mathematical crisis - for example, if one were a nineteenth-century mathematician working in real analysis. But in general the point seems, on a certain level, right.

On the other hand, however, it seems obvious to me that mathematicians will benefit from thinking about philosophical issues (and benefit qua mathematician). But it is somewhat difficult to articulate why they would benefit.

So, any thoughts? In short, what should we say to mathematicians regarding why they ought to care about what philosophers say?

#### 5 comments:

1. Hi Roy,

I'm tempted by the view that /in general/ it's misguided to expect philosophy of maths to have any bearing on what mathematicians actually do". The problems we're interested in are interesting on their own terms. It's just as much a mistake to justify interest in philosophy of maths on the basis of its consequences for what mathematicians actually do" as it is to justify interest in maths on the basis of its consequences for what non-mathematicians actually do".

Having said that, there are still cases, I think, where philosophy of maths does have a direct bearing on what mathematicians actually do -- assuming that we don't read this too narrowly. For instance, Hamkins' philosophical views on the multiverse have stimulated his (and others) research into very particular mathematical questions about generic extensions of V. And I'm sure there are many other examples of a similar kind.

In addition, there seems to be examples where philosophy of maths gives weight to answers to purely mathematical questions. For instance, if one is able to provide a good argument that V is ineffable, and that this is to be cashed out in terms of reflection principles, one would thereby lend weight to the hypothesis that there are inaccessibles, Mahlos, weakly compact cardinals etc. In these cases, I can see no good reason (other than division of labour) for mathematicians not to be interested in the associated philosophy.

Even if some mathematician aren't happy to trust philosophical answers to mathematical questions (even for those questions which are not answerable via standard mathematical means), such methodological questions like why not trust philosophical answers to mathematical questions?" will inevitably turn on philosophical issues.

Best,
Sam

2. To be clearer: There are two questions at work here:

(1) Does philosophical work have bearing on how one ought to do mathematics?

(2) Should philosophical work be or interest to mathematicians (and, especially: even if the answer to (1) were negative)?

As noted in my original post (and as illustrated by Sam's useful comments), the answer to (1) is, I think, 'sometimes'. But I am actually at present a bit more interested in question (2). That is, even if philosophical work never had any bearing on how mathematics should proceed, might it not nevertheless be the case that philosophical work should be of interest to mathematicians?

My intuition is that the answer to this question is "yes", but I am having a hard time articulating why the answer is "yes".

3. Eugenia Cheng, a category theorist, has done a nice paper in which she answers 'yes - but the philosophical questions relevant to day-to-day mathematics are not the ones of most interest to philosophers of mathematics. But please note that, due to an unfortunate evolution of word usage in English, you have to ignore all meanings of the word morality as it pertains to ethics while reading http://www.cheng.staff.shef.ac.uk/morality/morality.pdf

4. I can think of 4 possible approaches to convincing a mathematician that they should (sometimes) care about what philosophers say:

1) In the recent past, attempts to address philosophical issues have led to new mathematics. It seems to me that set theory and computability theory are plausible examples.

2) (Some particular) philosophers and (some particular) mathematicians may have a common enemy who is best attacked on two fronts. The enemy will usually be other philosophers and/or mathematicians, but I think that string theorists and economists might also be fair game to some mathematicians.

3) If you want to know more about the history of mathematics, or the life and work of one particular mathematician, it will often be necessary to learn about the philosophy that was relevant at the time.

4) Other disciplines, especially physics, already have an influence on what direction mathematical research will take, even in pure maths. Exposure to the doubts of philosophers of mathematics might loosen the grip of such influences even where they are hard to recognise, and allow new ideas to be explored. I have heard that the (fairly unrelated, I think) philosophies of Brouwer and Whitehead inspired the study of pointless topology, involving a release from powerful intuitions associated with the real numbers, which loom large in mathematical imaginations due to their position in physics and the historical development of maths. Pointless topology may turn out to be, well, pointless, but for some people that’s actually a plus. (I know next to nothing about the subject of that example, and can’t provide any evidence for its philosophical origin at short notice, but I’m sure there are other examples to hand.)

5. Very good points by anonymous.
On point (1), maybe add many mathematical developments within logic, during the Great Foundational Period, 1879-1938. Predicate logic, theory of truth, model theory, proof theory, type theory, variety of paradoxes, modal logics and many-valued logics. Much
of this work has now been absorbed into mathematics and computer science, usually becoming somewhat autonomous from the philosophical questions and conceptual analysis (about ontology, semantics and epistemology) that stimulated the developments.