Here again is one of my very first blog-posts, musing about what philosophers of mathematics with my cast of mind might usefully get up to …
Tired of ontology? (May 13, 2006)
It requires a certain kind of philosophical temperament — which I do seem to lack — to get worked up by the question “But do numbers really exist?” and excitedly debate whether to be a fictionalist or a modal structuralist or some other -ist. As younger colleagues gambol around cheerfully chattering about these things, wondering whether to be hermeneutic or revolutionary, I find myself sitting on the side-lines, slightly grumpily muttering under my breath ‘And who cares?’.
To exaggerate a bit, I guess there’s a basic divide here between two camps. One camp is primarily interested in analytical metaphysics, or epistemology, or the philosophy of language, and sees mathematics as a test case for their preferred Quinean naturalist line or their Kantian framework (or whatever). The other camp is puzzled by some internal features of the practice of real mathematics and would like to have a satisfying story to tell about them .
Well, if you’re tired of playing the ontology game with the first camp, then there’s actually quite a bit of fun to be had in the second camp, and maybe more prospect of making some real progress. In the broadest brush terms, here are just a few of the questions that bug me (leaving aside Gödelian matters):
- How should we develop/improve/augment/replace Lakatos’s model of how mathematics develops in his Proofs and Refutations?
- What makes a mathematical proof illuminating/explanatory? (And what are we to make of unsurveyable computer proofs?)
- Is there a single conceptual grounding for the standard axioms of set theory? (And what are we to make of the standing of various large cardinal axioms?)
- What is the significance of the reverse mathematics project? (Is it just a technical “accident” that RCA_0 is used a base theory in that project? Can some kind of conceptual grounding be given for that theory? Would it be more principled to pursue Feferman’s predicative project?)
- Is there any sense in which category theory provides new foundations/suggests a new philosophical understanding for mathematics?
There’s even a possibility that your local friendly mathematicians might be interested in talking about such things!
That still strikes me as quite a good list of questions that still interest me (particularly, at the moment, the last!). But what really good has been published on these in the intervening ten years? Suggestions please!
One issue that interests me, related to your 2. above, is the idea of mathematical depth. What makes a theorem deep? There was an issue from Philosophia Mathematica on this issue (though I remember not finding it too illuminating).
Another issue which I found interesting is about how putting a problem in the right context can make its solution seems much more natural. I think Kenneth Manders has some writings about this (“Logic and Conceptual Relationships in Mathematics”).
On depth, Luca Incurvati and I wondered how much work the notion of depth can do for us at the end of our review of Maddy’s book Defending The Axioms. And yes, I too didn’t get a great deal out of the pieces in that issue of PM.
According to Eugenia Cheng in her book Cakes, Custard and Category Theory: Easy recipes for understanding complex maths, proof isn’t illuminating and category theory is.
(I found this unconvincing, and it reminded me of John Burgess’s distinction, in Rigor and Structure, n 9, p 172, between fans of category theory and fanatics. A fan thinks that translating a problem into category-theoretic terms can facilitate its solution, by revealing analogies with other areas of mathematics; a fanatic regards mere translation of a theorem or proof into category-theoretic terms as already, in itself, a major contribution.)
Yes, I think Eugenia Cheng counts as something of a fanatic — though in a very engaging way!
More often lately I approach mathematical inquiry and inquiry in general from a Peircean angle, so, yes, there are many interesting questions that might be asked.
I’d be genuinely interested to know what distinctive questions, then, a Peircean approach suggests.