Philosophia Mathematica - recent issues

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

Fictionalism, Theft, and the Story of Mathematics

Balaguer, M. — Sat, 23 May 2009 19:28:32 PDT

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. — Sat, 23 May 2009 19:28:32 PDT

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. — Sat, 23 May 2009 19:28:33 PDT

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. — Sat, 23 May 2009 19:28:33 PDT

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. — Sat, 23 May 2009 19:28:33 PDT

A Scientific Enterprise?: Penelope Maddy's Second Philosophy

Shapiro, S., Reeder, P. — Sat, 23 May 2009 19:28:33 PDT

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

Sat, 23 May 2009 19:28:33 PDT

Empirical Regularities in Wittgenstein's Philosophy of Mathematics

Steiner, M. — Thu, 05 Feb 2009 04:08:17 PST

During the course of about ten years, Wittgenstein revised some of his most basic views in philosophy of mathematics, for example that a mathematical theorem can have only one proof. This essay argues that these changes are rooted in his growing belief that mathematical theorems are ‘internally’ connected to their canonical applications, i.e., that mathematical theorems are ‘hardened’ empirical regularities, upon which the former are supervenient. The central role Wittgenstein increasingly assigns to empirical regularities had profound implications for all of his later philosophy; some of these implications (particularly to rule following) are addressed in the essay.

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

Gandon, S. — Thu, 05 Feb 2009 04:08:17 PST

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?

Deflating Existence Away? A Critique of Azzouni's Nominalism

Raley, Y. — Thu, 05 Feb 2009 04:08:17 PST

In his Deflating Existential Consequence, Azzouni claims to be a nominalist. Yet, he also says that it is philosophically indeterminate which criterion for what exists is correct. Nominalism is the view that certain objects (i.e., abstract objects) do not exist, and not the view that it is philosophically indeterminate whether or not they do. I resolve the dilemma that Azzouni's claims pose: Azzouni is a non-factualist about what exists, but he is a factualist about which criterion for what exists our community of speakers has adopted. It is in the latter sense only that Azzouni can call himself a nominalist.

Why Do We Believe Theorems?

Pelc, A. — Thu, 05 Feb 2009 04:08:17 PST

The formalist point of view maintains that formal derivations underlying proofs, although usually not carried out in practice, contribute to the confidence in mathematical theorems. Opposing this opinion, the main claim of the present paper is that such a gain of confidence obtained from any link between proofs and formal derivations is, even in principle, impossible in the present state of knowledge. Our argument is based on considerations concerning length of formal derivations.

MARCUS GIAQUINTO. Visual Thinking in Mathematics: An Epistemological Study

Avigad, J. — Thu, 05 Feb 2009 04:08:17 PST

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

Oliveri, G. — Thu, 05 Feb 2009 04:08:18 PST

WILLIAM BYERS. How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics

Thomas, R. — Thu, 05 Feb 2009 04:08:18 PST

PIERRE CASSOU-NOGUES. Les Demons de Godel: Logique et Folie. [Godel's Demons: Logic and Craziness]

Rav, Y. — Thu, 05 Feb 2009 04:08:18 PST

ROY T. COOK, ed. The Arche Papers on the Mathematics of Abstraction

Thu, 05 Feb 2009 04:08:18 PST

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

Thu, 05 Feb 2009 04:08:18 PST

PAUL GOCHET and PHILIPPE DE ROUILHAN. Logique Epistemique & Philosophie des Mathematiques. Thierry Martin and Philippe Mongin, eds

Thu, 05 Feb 2009 04:08:18 PST

MICHAEL FRIEDMAN and RICHARD CREATH. eds. The Cambridge Companion to Carnap

Thu, 05 Feb 2009 04:08:18 PST

M. VAN ATTEN, P. BOLDINI, M. BOURDEAU, and G. HEINZMANN, editors. One Hundred Years of Intuitionism (1907-2007): The Cerisy Conference. (Publications of the Henri Poincare Archives. Science around 1900)

Thu, 05 Feb 2009 04:08:18 PST

PAOLO MANCOSU, ed. The Philosophy of Mathematical Practice

Thu, 05 Feb 2009 04:08:18 PST

R. LUPACCHINI and G. CORSI, eds. Deduction, Computation, Experiment: Exploring the Effectiveness of Proof

Thu, 05 Feb 2009 04:08:18 PST

JOHN P. BURGESS. Mathematics, Models, and Modality: Selected Philosophical Essays

Thu, 05 Feb 2009 04:08:18 PST

GERHARD PREYER, editor. Philosophy of Mathematics--Set Theory, Measuring Theories, and Nominalism

Thu, 05 Feb 2009 04:08:18 PST

VINCENT F. HENDRICKS and HANNES LEITGEB, eds. Philosophy of Mathematics: 5 Questions

Thu, 05 Feb 2009 04:08:18 PST

BONNIE GOLD, and ROGER A. SIMONS, eds. Proof and Other Dilemmas: Mathematics and Philosophy

Thu, 05 Feb 2009 04:08:18 PST