Chapter 2 uses the superficial form of Bildungsroman (recounting the authors’ moderately successful career) in order to present some elements for a sociological analysis of the mathematical profession, as the author sees it. But it also advances a philosophical thesis: that since mathematics does not rest on conformity to some preexisting reality, it derives its meaning from coherence with a hierarchical system of values, defined in principle by continuity with a historical narrative of “progress toward a telos” as perceived by contemporary practitioners. There is no claim that the contemporary perception of history is accurate, only that it serves simultaneously as motivation and to provide continuity to the tradition. Nor is there a claim that the values have independent validity, although there’s no doubt that sensitivity to the needs of the natural sciences has always played an important (but not primary) role in maintaining the system of values.
I put out the thesis on pp. 18-19 and in several other places in Chapter 2, but I’m not a philosopher and I didn’t choose to defend it in the book; I’m certainly not going to try to defend it here. I think it captures something important about the sociology of the profession that is missing from most philosophies of mathematics, but I don’t want to insist on that now. Instead I want to focus on what appears to be the naked elitism of this conception of mathematics. If the meaning of mathematics is derived from a hierarchical system of values, then how can mathematics be democratic? If the values can be compared to no external standard open to universal evaluation, how do we know that those who preserve these values are not simply a self-perpetuating aristocracy, excluding visions of mathematics that are no less valid but that do not benefit from proximity to power?
My belief is that mathematics is NOT a self-perpetuating aristocracy, but I don’t see how to prove this, and I don’t think philosophy will be of much help, except maybe to help work through comparisons with other models. What other practices share similar systems of values? Improvisational jazz? Poetry? Medieval guilds? The contemporary building trades? Philosophy?
A close reading of my book, and especially Chapters 2 and 10, should make it clear that I am NOT celebrating the hierarchical character of mathematics, much less the competitiveness that keeps the hierarchy in place. Actually, my sense is that mathematics is much LESS competitive than other university disciplines. Once, for example, while exchanging professional notes with a biologist, I mentioned that it was our tradition not only to thank our colleagues in the acknowledgments sections of our papers but also to cite all other work on related questions. The biologist said that her colleagues didn’t do that at all; they cited their own previous work, of course, but only cited competitors in order to prove they were wrong. She may have been exaggerating for effect, of course, but I did feel awfully virtuous after that exchange.
In a later post I will tell the story of how I was — not flamed, exactly, but at least singed by an accusation of elitism — when I attempted to take part in an online discussion of alternatives to the contemporary model of publication.