I am still working on papers, so there hasn’t been much time to think about anything interestingly new. However, I would like to mention something that would make life much easier.

I think it would be an awesome addition to the field of philosophy to develop the equivalent of the AMS’s MathSciNet for philosophical publications. Not only does MathSciNet provide reviews of papers and books in a centralized place, it also provides for the exportation of references and abstracts. Seeing as most of my papers use Bibtex for reference management, a centralized service for philosophical papers with the ability to export a Bibtex reference would be far superior to the Google Scholar/manual entry method I have to use at the moment.

Granted, getting something like MathSciNet off the ground for philosophy would be amazingly difficult, but it might be worth it.

So I found the following idea in the book Lila: An Inquiry into Morals by Robert M. Pirsig (specifically at the beginning of chapter 26). The following bit on philosophy vs. philosophology doesn’t have anything particularly to do with the main ideas of the book, but being a philosophy grad student, it got me thinking a bit. Hence, I thought I would share it.

Pirsig makes a distinction between philosophy and philosophology. The former is to the latter, he claims, as art is to art history, music to musicology, or creative writing to literary criticism. I take this distinction to be pointing out something like actual creative philosophy versus historical philosophy. “It’s a derivative, secondary field, a sometimes parasitic growth that likes to think it controls its host by analyzing and intellectualizing its host’s behavior.”(370) Having made this distinction, though, he goes on to note how, unlike artists, musicians, and writers, the ranks of pure philosophers are virtually empty. That is, almost everything that goes around calling itself philosophy is actually philosophology. New ideas that might have come up are often compared with Old-Dead-White-Guys¹ and found inferior.

However, he doesn’t just comment on this, but makes a potentially useful suggestion: figure out what you think first, and then compare it to Old-Dead-White-Guys. By proceeding in this way, you can better see the similarities and differences with other philosophers without simply getting carried away in their quite persuasive rhetoric. Furthermore, if you have strong personal views, then considering classical objections to similar Old-Dead-White-Guys may incite deeper consideration and perhaps revision of your own views instead of just abandoning a “Kant/Hume/Rawls/Quine was right” position. “You’re not limited by any dead-ends of [their] thought and can often see ways of going around [them].”(372)

I’m quite interested by this, and will be keeping it in mind as I progress through the program here. My initial reaction is that both philosophy and philosophology are taught here, but I think I need to pay more careful attention.

¹ By Old-Dead-White-Guys I simply mean already-existing philosophy.

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.

Since the NSF finally got around to releasing the last GRFP decisions today¹, I thought I would post a bit from my research proposal. I haven’t thought about this stuff for some time, but reading it again is starting to make me want to actually undertake some of the work.

Research Proposal Background Sketch

Science can generally be seen as undertaking the task of explaining empirical facts — answering questions concerning why things are the way they are. Although mathematics is similar to the natural sciences in many ways, there are substantial differences that have caused problems when attempting to apply theories of explanation from the philosophy of science to mathematical practice.

Although a number of papers have been published on the notion of explanation in mathematics, nothing has been satisfactorily decided. Suggestions have included the unification theory of Kitcher (1981), the use of characteristic properties that provide opportunities for proof deformation in Steiner (1978a), and van Fraassen’s theory of why-question logic as discussed by Sandborg (1998). The examination of various case studies, however, have led Hafner and Mancosu (2008) to successfully argue against Kitcher (1981), and Resnik and Kushner (1987) have successfully argued against Steiner.

Despite the failure of each proposed theory of mathematical explanation thus far, searching for an overriding theory is still fruitful. In the naturalist program, for example, mathematical explanation could be construed as a theoretical virtue of mathematics. For instance, perhaps it should be considered when attempting to adjudicate the adoption of new axioms in set theory. (see Maddy 1997; 2007; Mancosu 2008) More importantly, it is widely accepted that the study of actual mathematical practice is vital to improving understanding of the nature of mathematics.

The study of mathematical explanation and the nature of mathematics has definite connections to general philosophy, history, and practice of mathematics. For instance, a better understanding of mathematical practice may help explain why, when set theory was inconsistent, things didn’t “explode.” Such answers are intrinsically interesting, and they might also prove useful should similar problems reoccur. Moreover, practicing mathematicians might reconsider the style in which they present results. This, in turn, may allow for quicker and more efficient distribution of applicable developments to other fields.

Additionally, mathematical explanation is connected to scientific explanation in general. Steiner (1978b) considers the relationship between mathematical explanations of scientific facts and mathematical explanations of mathematical facts to be alike in methodology. Although mathematical practice and general scientific practice appear substantially different, a general theory of mathematical explanation may throw light on explanation in other sciences and on explanation in many other corners of academia.

References

Hafner, J. and Mancosu, P. (2008). “Beyond Unification” in The Philosophy of Mathematical Practice 151-178. Oxford University Press.

Kitcher, P. (1981). “Explanatory Unification” in Philosophy of Science 48(4):507-531.

Maddy, P. (1997). Natrualism in Mathematics. Oxford University Press.

______ (2007). Second Philosophy: A Naturalistic Method. Oxford University Press.

Resnik, M. D. and Kushner, D. (1987). “Explanation, independence, and realism in mathematics”. The British Journal for the Philosophy of Science, 38(2):141-158.

Sandborg, D. (1998). “Mathematical explanation and the theory of why-questions”. The British Journal for the Philosophy of Science, 49(4):603-624.

Steiner, M. (1978a). “Mathematical explanation”. Philos. Stud., 34(2):135-151.

______ (1978b). “Mathematical explanation and scientific knowledge”. Nos, 12(1):17-28.

¹ I ended up getting an Honorable Mention, for the record.