Marco Panza (Paris I) gives a talk at the Workshop on Mathematics: Objectivity by Representation (11 November, 2014) titled "What are the challenges of Benacerrafs Dilemma? A Reinterpretation". Abstract: Despite its enormous influence, Benacerraf's dilemma admits no standard, unanimously accepted, version. 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 also depends on Benacerraf's appealing, while expounding the dilemma, to so many conceptual ingredients so as to spontaneously generate the feeling that most of them are in fact inessential for stating it. It is almost unanimously admitted that the dilemma is, as such, independent of the adoption of a causal conception of knowledge, though Benacerraf appealed to it. This apart, there have not been, however, and still there is no agreement about which of these ingredients have to be conserved so as to get a sort of minimal version of the dilemma, and which others can, rather, be left aside (or should be so, in agreement with an Okkamist policy).

My purpose is to come back to the discussion on this matter, with a particular attention to Field's reformulation of the problem, so as to identify two converging and quite basic challenges, addressed by Benacerraf's dilemma to a platonist and to a combinatorialist (in Benacerraf's own sense) philosophy of mathematics, respectively. What I mean by dubbing these challenges 'converging' is both that they share a common kernel, which encompasses a challenge for any plausible philosophy of mathematics, and that they suggest (at least to me) a way-out along similar lines. Roughing these lines out is the purpose of the two last part of the talk.