Maximality vs. Extendability:
Reflections on Structuralism and
In a recent paper, while discussing the role of the notion of analyticity in Carnap's thought, Howard Stein wrote:
The primitive view--surely that of Kant — was that whatever is trivial is obvious.
We know that this is wrong; and 1 would put it that the nature of mathemati-
cal knowledge appears more deeply mysterious today than it ever did in earlier
centuries—that one of the advances we have made in philosophy has been to
come to an understanding of just how deeply puzzling the epistemology of
mathematics really is. (1993, 283)
Although our principal concern here is not with analyticity but rather with competing visions of the very subject matter of mathematics, the present essay can certainly be read as an extended illustration of this poignant remark.1
There is a recurring emphasis in Stein's writing on the importance of theoretical insights of our leading intellectual forebears in science and mathematics for the healthy practice of philosophy. In this spirit, I can think of no better way to introduce the subject of this essay than by quoting from the concluding remarks of Zermelo's great, yet underappreciated, 1930 paper, “Über Grenzzablen und Mengenbereiche.” Having formulated
I am grateful to audiences at the Steinfest. University of Chicago, May 21–23, 1999,
and at the Philosophy of Mathematics Conference at the University of California,
Santa Barbara, Feb. 4–6, 2000, and especially to Stewart Shapiro and Tony Anderson,
for helpful comments on earlier drafts of this paper.