Tuesday, September 15, 2009

Cole's Practice-Dependent Realism and Creativity in Mathematics

Julian Cole's "Creativity, Freedom and Authority: A New Perspective on the Metaphysics of Mathematics" is now available via the Australasian Journal of Philosophy. Cole develops what seems to me to be the most careful version of a social constructivist metaphysics for mathematics. Basically the idea is that the activities of mathematics constitute the mathematical entities as abstract entities. This makes it coherent for Cole to insist that the entities have many of the traditional features of abstract objects such as being outside space and time and lacking causal relations. Crucially for the causal point, even though the mathematicians constitute the mathematical entities, they do not cause them to exist.

One consideration in favor of his view that Cole emphasizes is the creativity that mathematicians have to posit new entities. Qua mathematician, he notes "the freedom I felt I had to introduce a new mathematical theory whose variables ranged over any mathematical entities I wished, provided it served a legitimate mathematical purpose" (p. 589). Other mathematicians have of course said similar things, from Cantor's claim that "the essence of mathematics lies precisely in its freedom" (noted by Linnebo in his essay in this volume) and Hilbert's conception of axioms in his debate with Frege.

I have two worries with this starting point. First, is it so clear that mathematicians really have this freedom? The history of mathematics seems filled with controversies about new objects or new mathematical techniques that seem to presuppose the existence problematic objects. Second, even if mathematicians have a certain kind of freedom to posit new objects, how do we determine that this freedom is independent of prior metaphysical commitments? One option for the traditional platonist or the ante rem structuralist is to insist that mathematicians are now free to posit new objects only because it is highly likely that these new objects can find a place in their background set theory or theory of structures. This of course would not settle the issue against practice-dependent realism, but it gives the realist a strategy to accommodate the same data.

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