Josipa Petrunic: Platonism, Cognitive Science and the Learning of Mathematics: a Wittgensteinian response to exclusionary trends in the philosophy of mathematics

Support for this talk has been provided in part by the SSHRC Cluster Grant on Situating Science.The Science Technology and Society Program is pleased to present:

Josipa Petrunic , Postdoctoral research fellow, IHPST, University of Toronto; Visiting research fellow, HPS, University of Cambridge

Platonism, Cognitive Science and the Learning of Mathematics: a Wittgensteinian response to exclusionary trends in the philosophy of mathematics

Wednesday, November 2, 2011 3pm
Tory 8-22

The philosopher of mathematics, James Brown, has argued that mathematical objects exist in an atemporal realm. We gain knowledge of these objects through a special intuition. This unique knowledge is filtered through the conduit of an expert, a philosopher king or a really smart mathematician who can show us the way.

Speaking from the perspective of Kantian cognitive science, the philosopher of mathematics, Marcus Giaquinto, has claimed we necessarily come to “know” mathematical objects over time, because our brains are hardwired to grasp their special intuitive existence. Our brains are equipped with binoculars that can peer into a meta-realm where mathematical objects exist.

Brown and Giaquinto’s respective arguments boil down to much the same thing: mathematical objects exist outside of human life; they can be discovered by humans, but not created by humans. Both Brown and Giaquinto view mathematics as a special domain of thinking (different from political, economic, religious or other social thinking). Both authors believe mathematics requires special people to do the thinking.

Not only are Brown and Giaquinto’s arguments based on a number of unquestioned suppositions, they are stances to be avoided from the perspective of democratic education.

First, neither Brown’s Platonism nor Giaquinto’s neo-Kantianism explains why we “know” mathematical theorems or mathematical objects as we know them today. Neither account offers an explanation as to why we use the mathematical techniques we use. Neither account explains why we pose the mathematical questions that we pose.

A better explanatory account of mathematical activity and mathematical knowing comes from Wittgenstein. A Wittgensteinian would argue that what we know in mathematics and what we use in mathematics is based on what we are taught, what we learn, and what the social realm of possible extension of that knowledge to new case studies permits. By looking at the case studies of failed mathematics, the historian can work to explain why mathematicians “discover” so much mathematics that is so wrong, so erroneous, and eventually so disused (or “failed”) over time. I will look at the case study of William Kingdon Clifford’s biquaternions as a starting point to argue that mathematics serves interests, and interests are historically contingent.

Second, I will argue that we (as historians, teachers and legislators) ought to avoid Platonist accounts of mathematics at all costs, and we ought to approach neo-Kantian cognitive scientific accounts with trepidation or at least a sense of democratic injustice. These accounts tell us only that certain, special people can properly know mathematics, while the rest of us are out of luck. These views feed into the belief that mathematics is elite, special and exclusionary. In practice, however, mathematics can be practiced by anyone and it can be practiced in a heterogeneity of ways. Promoting the image of its elitism suggest to those people who are convinced by the image of mathematics as unique and special (rather than as mundane and human made) that they themselves are (in the words of Augustine), “without grace” and, thus, without the capacity to pursue a career in mathematics.
 
 
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