Philosophia Mathematica - current issue

Idealization in Cassirer's Philosophy of Mathematics

Mormann, T. — 2008-06-02

The notion of idealization has received considerable attention in contemporary philosophy of science but less in philosophy of mathematics. An exception was the ‘critical idealism’ of the neo-Kantian philosopher Ernst Cassirer. According to Cassirer the methodology of idealization plays a central role for mathematics and empirical science. In this paper it is argued that Cassirer's contributions in this area still deserve to be taken into account in the current debates in philosophy of mathematics.

Abstraction and Additional Nature

Hale, B., Wright, C. — 2008-06-02

In ‘What is wrong with abstraction’, Michael Potter and Peter Sullivan explain a further objection to the abstractionist programme in the foundations of mathematics which they first presented in their ‘Hale on Caesar’ and which they believe our discussion in The Reason's Proper Study misunderstood. The aims of the present note are:

To get the character of this objection into sharper focus;

To explore further certain of the assumptions—primarily, about reference-fixing in mathematics, about certain putative limitations of abstractionist set theory, and about the effects of impredicativity in abstraction principles—which underlie it; and

To advance the debate of the issues thereby raised.

A Puzzle About Ontological Commitments

Ebert, P. A. — 2008-06-02

This paper raises and then discusses a puzzle concerning the ontological commitments of mathematical principles. The main focus here is Hume's Principle—a statement that, embedded in second-order logic, allows for a deduction of the second-order Peano axioms. The puzzle aims to put pressure on so-called epistemic rejectionism, a position that rejects the analytic status of Hume's Principle. The upshot will be to elicit a new and very basic disagreement between epistemic rejectionism and the neo-Fregeans, defenders of the analytic status of Hume's Principle, which will provide a new angle from which properly to assess and re-evaluate the current debate.

The Explanatory Power of Phase Spaces

Lyon, A., Colyvan, M. — 2008-06-02

David Malament argued that Hartry Field's nominalisation program is unlikely to be able to deal with non-space-time theories such as phase-space theories. We give a specific example of such a phase-space theory and argue that this presentation of the theory delivers explanations that are not available in the classical presentation of the theory. This suggests that even if phase-space theories can be nominalised, the resulting theory will not have the explanatory power of the original. Phase-space theories thus raise problems for nominalists that go beyond Malament's initial concerns.

Multiple Reductions Revisited

Clarke-Doane, J. — 2008-06-02

Paul Benacerraf's argument from multiple reductions consists of a general argument against realism about the natural numbers (the view that numbers are objects), and a limited argument against reductionism about them (the view that numbers are identical with prima facie distinct entities). There is a widely recognized and severe difficulty with the former argument, but no comparably recognized such difficulty with the latter. Even so, reductionism in mathematics continues to thrive. In this paper I develop a difficulty for Benacerraf's argument against reductionism that is of comparable severity to the now widely recognized difficulty with his general argument against realism.

NATHANIEL MILLER. Euclid and his Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry. CSLI Studies in the Theory and Applications of Diagrams

Mumma, J. — 2008-06-02

RICHARD TIESZEN. Phenomenology, Logic, and the Philosophy of Mathematics

Ronzitti, G. — 2008-06-02

MARK VAN ATTEN. Brouwer meets Husserl: On the Phenomenology of Choice Sequences

Franchella, M. — 2008-06-02

PETR HAJEK, LUIS VALDES-VILLANUEVA, and DAG WESTERSTAHL, eds. Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress [2003]

2008-06-02

MARY LENG, ALEXANDER PASEAU, and MICHAEL POTTER, eds. Mathematical Knowledge

2008-06-02

JOHAN VAN BENTHEM, GERHARD HEINZMANN, MANUEL REBUSCHI, and HENK VISSER, eds. The Age of Alternative Logics: Assessing Philosophy of Logic and Mathematics Today. Logic, Epistemology, and the Unity of Science; 3

2008-06-02