Thursday, April 23, 2009

'Mavericks' vs 'foundationalists' and the search for a third way into philosophy of mathematics

Paolo Mancosu, in his recent work 'The philosophy of mathematical practice' (OUP, 2008), raises - pp. 3-7 - a stimulating issue about philosophy of mathematics, that is the emergence of two different traditions in philosophy of math, namely the 'maverick' tradition and the 'foundationalist' tradition.
One of the most interesting aspects of Mancosu's analysis is the fact that he seems to aim at constructing a 'third way' into philosophy of math (pp. 18-20), trying to see into its future.
Mancosu characterizes the two traditions in the following way:

- 'Foundationalist'
The analytic philosophy of mathematics

- 'Maverick' (according to Mancosu: Lakatos, Aspray, Kitcher, Gillies, Cellucci, Hersh, Tymoczko, Davis, Kline, Grosholz, Breger, van Kerkhove, van Bengedem, Krieger, Corfield, Ferreiros, Gray):
a) anti-foundationalism, i.e. there is no certain foundation for mathematics; mathematics is a fallible activity;
b) anti-logicism, i.e. mathematical logic cannot provide the tools for an adequate analysis of mathematics and its development;
c) attention to mathematical practice: only detailed analysis and reconstruction of large and significant parts of mathematical practice can provide a philosophy of mathematics worth its name.

On one side, according to Mancosu, "there is no question that the ‘mavericks’ have managed to extend the boundaries of philosophy of mathematics", but is seems that they "were throwing away the baby with the bathwater"(p. 6) and that, in the end, they have "not managed to substantially redirect the course of philosophy of mathematics" (p. 5).
On the other side, "the aspects of mathematical practice we investigate are absolutely vital to an understanding of mathematics" and "having ignored them has drastically impoverished analytic philosophy of mathematicshas", but this does not mean "that we think that the achievements of this tradition should be discarded or ignored as being irrelevant to philosophy of mathematics. [...] We do not dismiss the analytic tradition in philosophy of mathematics but rather seek to extend its tools to a variety of areas that have been, by and large, ignored" (p. 18).
Therefore, Mancosu seems to aim at moving beyond these two traditions, and tries to construct a 'third way' which, as he puts it, is "calling for an extension to a philosophy of mathematics that will be able to address topics that the foundationalist tradition has ignored" (p. 18) .
The starting point of this new direction into philosophy of math suggested by Mancosu is not "to make any invidious comparisons but only to provide a fair account of what the previous traditions have achieved and why we think we have achieved something worth proposing to the reader. We are only too aware that we are making the first steps in a very difficult area and we hope that our efforts might stimulate others to do better" (pp. 19-20).
So, roughly, that's the way the philosophy of mathematics of the future, according to Mancosu, should follow.

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