Philosophia Mathematica - current issue

Fictionalism, Theft, and the Story of Mathematics

Balaguer, M. — 2009-05-23

This paper develops a novel version of mathematical fictionalism and defends it against three objections or worries, viz., (i) an objection based on the fact that there are obvious disanalogies between mathematics and fiction; (ii) a worry about whether fictionalism is consistent with the fact that certain mathematical sentences are objectively correct whereas others are incorrect; and (iii) a recent objection due to John Burgess concerning "hermeneuticism" and "revolutionism".

Empty de re Attitudes About Numbers

Azzouni, J. — 2009-05-23

I dub a certain central tradition in philosophy of language (and mind) the de re tradition. Compelling thought experiments show that in certain common cases the truth conditions for thoughts and public-language expressions categorically turn on external objects referred to, rather than on linguistic meanings and/or belief assumptions. However, de re phenomena in language and thought occur even when the objects in question don't exist. Call these empty de re phenomena. Empty de re thought with respect to numeration is explored in this paper, and such thought with respect to hallucinations is touched on.

On Formally Measuring and Eliminating Extraneous Notions in Proofs

Arana, A. — 2009-05-23

Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen's cut-elimination theorem. I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures.

The Godel Paradox and Wittgenstein's Reasons

Berto, F. — 2009-05-23

An interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match with some intuitions underlying Wittgenstein's philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question.

WILLIAM TAIT. The Provenance of Pure Reason. Essays on the Philosophy of Mathematics and on its History

Parsons, C. — 2009-05-23

A Scientific Enterprise?: Penelope Maddy's Second Philosophy

Shapiro, S., Reeder, P. — 2009-05-23

STEN LINDSTROM, ERIK PALMGREN, KRISTER SEGERBERG, and VIGGO STOLTENBERG-HANSE, editors. Logicism, Intuitionism, and Formalism: What Has Become of Them?

2009-05-23