This quarter I am taking a seminar purportedly on “new wave” philosophy of mathematics. While we have talked about a lot of interesting things, not much of it has seemed particularly “new wave”. We have read some Friedman on Carnap, and a book by Parsons that, while it is inarguably new, seems to be very much concerned with classical issues of mathematical ontology.

As a result, I have periodically taken up reading things on my own that seem more modern. In particular, last night I read a chapter from The Architecture of Modern Mathematics by Jean-Pierre Marquis entitled “A Path to the Epistemology of Mathematics: Homotopy Theory”. This, similarly to the mathematical explanation stuff I wrote about (in a way) yesterday, is fundamentally different than the classical concerns in the philosophy of mathematics.

Although philosophy of mathematics has, in the past, been very occupied with mathematical epistemology, most of the focus has been on issues of mathematical intuition. This chapter, however, is not concerned with that. The focus is trying to draw a parallel between general scientific technologies and their emergence and systematic mathematical technologies. To illustrate the point, the paper takes a look at algebraic topology generally and homotopy theory specifically.

Homotopy theory is a part of algebraic topology concerned with equivalences of paths and path generalizations. By a path in a topological space , all that is meant is a continuous function . We can then consider two paths and to be equivalent when there is a continuous map such that , . This generalizes, allowing the replacement of by and by .

The details are largely irrelevant to the main point except as an example. The main thing I took away from the paper is the importance “for many mathematical notions, to understand that notion, it is irrelevant to specify what it is ‘made of’, or its underlying ‘ontology’ but rather what it is used for: when, why and how.” (245-6) The case of homotopy turns out to be an excellent example of this. Things like fibrations and spectral sequences, while perhaps interesting in their own right, are really only understood when one grasps how, when, and why to employ them as tools for calculating homotopy. Furthermore, homotopy theory itself, while interesting, is best understood in terms of the goal of classifying topological spaces. That is, while the “axioms of homotopy theory” are general enough to apply to many categories, and that makes them intrinsically interesting, their use for developing theories for the classification of spaces is far more central to really understanding them.

In general, it seems like if I am going to continue in the philosophy of mathematics, it will be with regard for things like this. Focusing on mathematical practice with deep examples, answering interesting questions and probably not being concerned with ontology or other seemingly dead-end classical questions.