Philosophia Mathematica - Advance Access

GIANLUIGI OLIVERI. A Realist Philosophy of Mathematics. Texts in Philosophy; 6

Cole, J. C. — 2008-04-23

A Cause for Concern: Standard Abstracta and Causation

Azzouni, J. — 2008-04-23

Benjamin Callard has recently suggested that causation between Platonic objects—standardly understood as atemporal and non-spatial—and spatio-temporal objects is not ‘a priori’ unintelligible. He considers the reasons some have given for its purported unintelligibility: apparent impossibility of energy transference, absence of physical contact, etc. He suggests that these considerations fail to rule out a priori Platonic-object causation. However, he has overlooked one important issue. Platonic objects must causally affect different objects differently, and different Platonic objects must causally affect the same objects differently. How are Platonic objects—ones outside space and time—supposed to do that?

Toward a Topic-Specific Logicism? Russell's Theory of Geometry in the Principles of Mathematics.{dagger}

Gandon, S. — 2008-04-16

Russell's philosophy is rightly described as a programme of reduction of mathematics to logic. Now the theory of geometry developed in 1903 does not fit this picture well, since it is deeply rooted in the purely synthetic projective approach, which conflicts with all the endeavours to reduce geometry to analytical geometry. The first goal of this paper is to present an overview of this conception. The second aim is more far-reaching. The fact that such a theory of geometry was sustained by Russell compels us to question the meaning of logicism: how is it possible to reconcile Russell's global reductionist standpoint with his local defence of the specificities of geometry?

YURI I. MANIN. Mathematics as Metaphor: Selected Essays of Yuri I. Manin

2008-04-15

STEFANO DONATI. I Fondamenti della Matematica nel Logicismo di Bertrand Russell [The Foundations of Mathematics in the Logicism of Bertrand Russell]

Oliveri, G. — 2008-04-03

PAUL GOCHET and PHILIPPE DE ROUILHAN. Logique Epistemique & Philosophie des Mathematiques

2008-04-02

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

2008-04-01

ROY T. COOK, ed. The Arche Papers on the Mathematics of Abstraction. The Western Ontario Series in Philosophy of Science; 71

2008-04-01

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

2008-04-01

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. — 2008-04-01

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

2008-03-26

Identity, Indiscernibility, and ante rem Structuralism: The Tale of i and -i

Shapiro, S. — 2008-03-26

Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of the other. I suggest that ‘i’ functions like a parameter in natural deduction systems.

A Puzzle About Ontological Commitments

Ebert, P. A. — 2008-03-05

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.

Wittgenstein on Circularity in the Frege-Russell Definition of Cardinal Number

De Bruin, B. — 2008-03-04

Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out to be valid on its own terms, even though it depends on two epistemological principles the logicist may find too ‘constructivist’.

Epistemic Optimism

Ganea, M. — 2008-01-30

Michael Dummett's argument for intuitionism can be criticized for the implicit reliance on the existence of what might be called absolutely undecidable statements. Neil Tennant attacks epistemic optimism, the view that there are no such statements. I expose what seem serious flaws in his attack, and I suggest a way of defending the use of classical logic in arithmetic that circumvents the issue of optimism.

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

Ronzitti, G. — 2008-01-25

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

Miriam, F. — 2008-01-16

Criteria of Identity and Structuralist Ontology

Leitgeb, H., Ladyman, J. — 2007-11-02

In discussions about whether the Principle of the Identity of Indiscernibles is compatible with structuralist ontologies of mathematics, it is usually assumed that individual objects are subject to criteria of identity which somehow account for the identity of the individuals. Much of this debate concerns structures that admit of non-trivial automorphisms. We consider cases from graph theory that violate even weak formulations of PII. We argue that (i) the identity or difference of places in a structure is not to be accounted for by anything other than the structure itself and that (ii) mathematical practice provides evidence for this view.

Idealization in Cassirer's Philosophy of Mathematics

Mormann, T. — 2007-10-10

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. — 2007-10-04

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.

Multiple Reductions Revisited

Clarke-Doane, J. — 2007-09-27

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.

Platonism and Aristotelianism in Mathematics

Pettigrew, R. — 2007-09-20

Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at ‘face value’ is one on which the expressions ‘ ’, ‘0’, ‘1’, ‘+’, and ‘x’ are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any semantic test that can must beg the question. I draw the same conclusion concerning areas of mathematics beyond arithmetic.

The Explanatory Power of Phase Spaces

Lyon, A., Colyvan, M. — 2007-08-08

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.