Picture Philosophy and Foundations of Mathematics

“How is pure mathematics possible?” is a chapter title of one of Kant’s books. Plato, Descartes, Leibniz, Frege, Husserl, Russell, Wittgenstein, Quine and other great philosophers have accorded a fundamental role to questions in the philosophy of mathematics. We may even say (with J. P. Mayberry) that it is this feature of western philosophy that sharply distinguishes it from the other great philosophical traditions. The interest in the philosophy of mathematics has various reasons. Let me mention two. Firstly, mathematics is the vintage example of a domain of knowledge whose truths are not (or do not seem to be) rooted in experience. How is that possible? Secondly, some of the deepest problems in philosophy find their most perspicuous formulation when they are specialized to mathematics and its foundations.

Working mathematicians have also been interested in the foundations of their subject. This was specially true in the first quarter of the last century, when Poincaré, Hilbert, Weyl, Brouwer et al. have passionately debated the foundations of mathematics. Foundational doctrines are strongly informed by technical results in mathematical logic and, at times, they can even be carved out and made precise, opening the possibility of their mathematical development and/or refutation. Historically, as it is well known, refutation has actually happened in some cases. Take, for instance, the refutation of the Fregean brand of logicism by Russell's paradox, or the downfall of Hilbert's original programme due to Gödel's incompleteness theorems. At other times, like with any philosophical issue, philosophy of mathematics seems prone to irremovable ambiguity. “There is no mathematical substitute for philosophy,” said Saul Kripke; “not even for the philosophy of mathematics,” I would add. It is this sweet combination of mathematical rigour and philosophical reflection that makes the foundations and philosophy of mathematics such a fascinating subject.

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Fernando Ferreira

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