Humanism about abstract objects

Abstract

In fall 2015 and spring 2016, I was lucky enough to be on sabbatical. My project was to complete a manuscript generalizing my institutional account of mathematics to cover other paradigmatically abstract objects. Ten years earlier, while finishing my dissertation, I had read Reuben’s What is Mathematics, Really? At the time, I remember thinking I’m not really sure whether we agree about the nature of mathematical objects. On the one hand, Reuben’s book was the first one in which I encountered an explicit comparison between mathematical objects (e.g., numbers, circles, and ordered fields) and institutional objects (e.g., marriages, wars, corporations, and the US Supreme Court), a comparison that is central to my institutional account of mathematics. On the other hand, while Reuben seemed to be committed to there being mathematical and (other) institutional objects, many of the things that he said about them suggested that he viewed them more like fictions than genuine existents. I was intrigued, but didn’t really follow up on that intrigue until my sabbatical offered me the opportunity to visit some of the folks who I took to endorse similar accounts of mathematics to my own. Two such individuals were Reuben and Sol Feferman. For me, our meetings revealed something interesting: while we were all three natural allies in taking humans to be responsible for mathematics-we were humanists, to use Reuben’s terminology-we did not agree on the underlying nature of mathematical objects. At least as I interpreted them, Reuben and Sol held fairly similar views about the underlying nature of such objects: they are something like intersubjective mental objects. Sol-see (Feferman 2009, 2014)-expressed his view in this way, “the basic objects of mathematical thought exist only as mental conceptions,” where, according to Sol, these mental conceptions are highly constrained by social interactions concerning them. Reuben (2014, p. 13), on the other hand, expresses his view in this way: “A mathematical entity is a concept, a shared thought” and “The concept … is nothing other than the collection of the mutually congruent … ‘mental models’ … possessed by those participating in the mathematical culture.” To summarize Reuben’s view as he did in personal communications that followed our November 2015 meetings, “mathematical objects are ‘equivalence classes’ of mutually congruent ‘mental models’ of those objects.”.