FilMat international conference

The new-born FilMat italian network for the philosophy of mathematics announces its first international conference on the topic “Philosophy of Mathematics: objectivity, cognition, and proof”. The conference will take place at San Raffaele University, Milan, on 29-31 May 2014.

The FilMat — of which I am proud to be a member — was created few months ago by a group of Italian scholars in philosophy of mathematics , originally met at the Scuola Normale Superiore in Pisa at the 2012 conference “Philosophy of Mathematics: from Logic to Practice”, and aims to foster the gathering of scholars working either in Italy or abroad on the philosophy of mathematics and strictly related fields, with special attention to those at early stages of their careers.

Here you can find the FilMat’s website. Here is the link to the call for abstracts for the 2014 international conference.

Are numbers sets?

One of the milestones of contemporary philosophy of mathematics is Benacerraf‘s 1965 article “What Numbers Could Not Be“.1 There he offers a compelling argument according to which numbers cannot be considered as sets — namely, they cannot be metaphysically identified with sets.

The argument is quite simple: if numbers were sets, we should be able to find a unique progression of sets with which numbers can be identified. But this is apparently impossible: there is a lot of ω-series that can serve as well for the aim. For example, we can adopt von Neumann’s series, and say that 0=\emptyset, 1=\{\emptyset\}, 2=\{\emptyset,\{\emptyset\}\}, and so on, where the successor function S is defined by S(x)=x \cup \{x\}. Or we can adopt Zermelo’s series, and say that 0=\emptyset, 1=\{\emptyset\}, 2=\{\{\emptyset\}\}, and so on, where the successor function S is defined by S(x)=\{x\}. Now, the problem is: is 3=\{\{\{\emptyset\}\}\} or is 3=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}? Benacerraf presents then the example of two children, Ernie and John. The first learned that von Neumann’s ordinals are the natural numbers, while the latter that Zermelo’s ordinals are the natural numbers. Now, they will be easily able to learn arithmetic set theoretically via the above constructions, and they will agree on any arithmetical theorem, except that for Ernie it is true that 3 \in 17, while for John it is false!

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