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.