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# Francesca Boccuni & Andrea Sereni's talk on Frege's philosophy of mathematics

Francesca Boccuni (San Raffaele, Milano) and Andrea Sereni (IUSS, Pavia) will give two talks on Frege's philosophy of mathematics.

Venue: sala rossa, via Azzo Gardino 23, Bologna

**Francesca Boccuni (joint work with Jack Woods)** : What Should we Render unto Caesar?

**Andrea Sereni** : The philosophical significance of Frege's constraint

**Abstracts:**

**Francesca Boccuni (University San Raffaele, Milan) – Jack Woods (Bilkent University, Ankara)** : What Should we Render unto Caesar?

We tackle the so-called Julius Caesar (JC) problem for Neo-Fregeanism. The problem is that Hume's Principle (HP), Frege's way of carving out the cardinal numbers, does not distinguish between objects of different sorts. HP says that the number of Fs is the same as the number of Gs if and only if F and G are equinumerous – i.e. the Fs and the Gs can be put into 1-1 mapping. This causes the JC problem: in Frege's provocative example, how can we know that the reference of “the number of the rooms in the Philosophy Department building of Bologna University”, as specified by HP, picks out the cardinal number n instead of Julius Caesar? We will suggest a solution to this vexed problem in terms of the arbitrary reference. We argue that introducing the tool of arbitrary reference does not undermine the logical character of Hume's Principle since arbitrary reference satisfies the correct formal criterion of logicality---``invariance'', in the right sense, under isomorphism. In effect, we argue that we should render unto Julius Caesar only what is Caesar's...and unto #(x = Julius Caesar) only what is 1's.

**Andrea Sereni** (IUSS Pavia) : The philosophical significance of Frege's constraint

I discuss the role that Frege's Constraint - the claim, roughly, that the principle regulating actual and potential applications of a given mathematical theory should be built into the central definitions of that theory - has been though to play in the recent debate between neo-logicist and structuralist philosophies of mathematics. I stress some serious shortcomings of most common formulations of the constraint - mainly, their being so strong as to make it question-begging against views rival to neo-logicism - and suggest an alternative modest formulation. By reviewing the discussion between Wright and Shapiro on the one side, and Hale and Batitsky on the other, I then defend such modest formulation from the charge of being "toothless": this is done by showing that the modest constraint remains faithful to Frege's insights by ruling out Frege's original foes, and that it also allows rejecting those definitions neo-logicists classify as "arrogant". At the same time, I suggest that this more plausible formulation makes the constraint satisfiable also by structuralism and other non-platonist views, thus undermining the role it is meant to play by neo-logicists against structuralism and other rival views.