On Benacerraf's Dilemma, Again

International audienceIn spite of its enormous influence, Benacerraf’s dilemma admits no standard unanimously accepted formulation. This mainly depends on Benacerraf’s having originally presented it in a quite colloquial way, by avoiding any compact, somehow codified,but purportedly comprehensive formulation. But it alsodepends on Benacerraf’s appealing, while expounding the dilemma, to so many conceptual ingredients so as to spontaneously generate the feeling that most of them might in fact be inessential to it. Apart from the almost unanimous agreement on the fact that, despite Benacerraf’s appeal to a causal conception of knowledge throughout his exposition, the dilemma does not rely on it, there is still no agreement about which of these many ingredients is essential and which should be left aside, in agreement with an Ockhamist policy, so as to obtain a minimal version of the dilemma. I will firstly offer a discussion of this matter (Sect. 4.1), with a particular attention to Field’s reformulation of the problem (especially in Field 1989a) in order to identify two converging and fundamental challenges addressed by Benacerraf’s dilemma, respectively to a platonist and to a combinatorialist philosophy of mathematics, in Benacerraf’s own sense of these terms (Sects. 4.2 and 4.3, respectively). What I mean by saying that these challenges are convergent is that they share a common core which embeds a crucial puzzle for any plausible philosophy of mathematics,1 and that they suggest a way out along similar lines. Roughing these lines out is the purpose of the last two sections of the paper (Sects. 4.4 and 4.5)