Philosophia Mathematica - current issue

Extending Hartry Field's Instrumental Account of Applied Mathematics to Statistical Mechanics

Meyer, G. — Tue, 06 Oct 2009 00:31:25 PDT

A serious flaw in Hartry Field’s instrumental account of applied mathematics, namely that Field must overestimate the extent to which many of the structures of our mathematical theories are reflected in the physical world, underlies much of the criticism of this account. After reviewing some of this criticism, I illustrate through an examination of the prospects for extending Field’s account to classical equilibrium statistical mechanics how this flaw will prevent any significant extension of this account beyond field theories. I note in the conclusion that this diagnosis of Field’s program also points the way to modifications that may work.

Justifying Definitions in Mathematics--Going Beyond Lakatos

Werndl, C. — Tue, 06 Oct 2009 00:31:25 PDT

This paper addresses the actual practice of justifying definitions in mathematics. First, I introduce the main account of this issue, namely Lakatos's proof-generated definitions. Based on a case study of definitions of randomness in ergodic theory, I identify three other common ways of justifying definitions: natural-world justification, condition justification, and redundancy justification. Also, I clarify the interrelationships between the different kinds of justification. Finally, I point out how Lakatos's ideas are limited: they fail to show how various kinds of justification can be found and can be reasonable, and they fail to acknowledge the interplay among the different kinds of justification.

Probabilistic Proofs and Transferability

Easwaran, K. — Tue, 06 Oct 2009 00:31:25 PDT

In a series of papers, Don Fallis points out that although mathematicians are generally unwilling to accept merely probabilistic proofs, they do accept proofs that are incomplete, long and complicated, or partly carried out by computers. He argues that there are no epistemic grounds on which probabilistic proofs can be rejected while these other proofs are accepted. I defend the practice by presenting a property I call ‘transferability’, which probabilistic proofs lack and acceptable proofs have. I also consider what this says about the similarities between mathematics and, on the one hand natural sciences, and on the other hand philosophy.

Randomized Arguments are Transferable

Jackson, J. C. — Tue, 06 Oct 2009 00:31:25 PDT

Easwaran has given a definition of transferability and argued that, under this definition, randomized arguments are not transferable. I show that certain aspects of his definition are not suitable for addressing the underlying question of whether or not there is an epistemic distinction between randomized and deductive arguments. Furthermore, I demonstrate that for any suitable definition, randomized arguments are in fact transferable.

JODY AZZOUNI. Tracking Reason: Proof, Consequence and Truth

Asmus, C. — Tue, 06 Oct 2009 00:31:25 PDT

C.K. RAJU. Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE.

Ferreiros, J. — Tue, 06 Oct 2009 00:31:25 PDT

CHRISTINE REDECKER. Wittgensteins Philosophie der Mathematik: Eine Neubewertung im Ausgang von der Kritik an Cantors Beweis der Uberabzahlbarkeit der reellen Zahlen. [Wittgenstein's Philosophy of Mathematics: A Reassessment Starting from the Critique of Cantor's Proof of the Uncountability of the Real Numbers]

Ramharter, E. — Tue, 06 Oct 2009 00:31:25 PDT

W.V. QUINE. Quintessence: Basic Readings from the Philosophy of W.V. Quine. Roger F. Gibson, ed

Tue, 06 Oct 2009 00:31:25 PDT

HERMANN WEYL. Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics. Peter Pesic, ed

Tue, 06 Oct 2009 00:31:25 PDT