Philosophia Mathematica - recent issues

Introduction

Tennant, N. — 2008-03-20

Carnap, Formalism, and Informal Rigour{dagger}

Gregory, L. — 2008-03-20

Carnap's position on mathematical truth in The Logical Syntax of Language has been attacked from two sides: Kreisel argues that it is formalistic but should not be, and Friedman argues that it is not formalistic but needs to be. In this paper I argue that the Carnap of Syntax does not eliminate our ordinary notion of mathematical truth in favour of a formal analogue; so Carnap's notion of mathematical truth is not formalistic. I further argue that there is no conflict between Carnap's use of informal notions and his principle of tolerance; so Carnap's definition of mathematical truth need not be formalistic.

Intuition Between the Analytic-Continental Divide: Hermann Weyl's Philosophy of the Continuum

Folina, J. — 2008-03-20

Though logical positivism is part of Kant's complex legacy, positivists rejected both Kant's theory of intuition and his classification of mathematical knowledge as synthetic a priori. This paper considers some lingering defenses of intuition in mathematics during the early part of the twentieth century, as logical positivism was born. In particular, it focuses on the difficult and changing views of Hermann Weyl about the proper role of intuition in mathematics. I argue that it was not intuition in general, but his commitment to twodifferent types of intuition, which explains his rather unusual and tormented philosophy of the mathematical continuum.

Intuitionism and Logical Syntax

McCarty, C. — 2008-03-20

In Logical Syntax of Language, Rudolf Carnap became a chief proponent of the doctrine that the statements of intuitionism carry nonstandard intuitionistic meanings. This doctrine is linked to Carnap's ‘Principle of Tolerance’ and claims he made on behalf of his notion of pure syntax. From premises independent of intuitionism, we argue that the doctrine, the Principle, and the attendant claims are mistaken, especially Carnap's repeated insistence that, in defining languages, logicians are free of commitment to mathematical statements intuitionists would reject.

Logic and Metaphysics: Heinrich Scholz and the Scientific World View{dagger}

Peckhaus, V. — 2008-03-20

The anti-metaphysical attitude of the neo-positivist movement is notorious. It is an essential mark of what its members regarded as the scientific world view. The paper focuses on a metaphysical variation of the scientific world view as proposed by Heinrich Scholz and his Münster group, who can be regarded as a peripheral part of the movement. They used formal ontology for legitimizing the use of logical calculi. Scholz's relation to the neo-positivist movement and his contributions to logic and foundations are discussed. His heuristic background can be drawn from a set of six methodological ‘articles of faith’, formulated in 1942 and published here for the first time.

Carnap, Godel, and the Analyticity of Arithmetic

Tennant, N. — 2008-03-20

Michael Friedman maintains that Carnap did not fully appreciate the impact of Gödel's first incompleteness theorem on the prospect for a purely syntactic definition of analyticity that would render arithmetic analytically true. This paper argues against this claim. It also challenges a common presumption on the part of defenders of Carnap, in their diagnosis of the force of Gödel's own critique of Carnap in his Gibbs Lecture.

LEO CORRY. David Hilbert and the Axiomatization of Physics (1898-1918)

Brading, K. — 2008-03-20

MATTHIAS WILLE. Mathematics and the Synthetic A Priori: Epistemological Investigations into the Status of Mathematical Axioms

Beisbart, C. — 2008-03-20

TATIANA ARRIGONI. What is meant by V?: Reflections on the universe of all sets

Tiles, M. — 2008-03-20

THOMAS MCKAY. Plural Predication

Burgess, J. P. — 2008-03-20

Jesper Lutzen. Mechanistic Images in Geometric Form: Heinrich Hertz's Principles of Mechanics

Pincock, C. — 2008-03-20

ALAN RICHARDSON and THOMAS UEBEL. The Cambridge Companion to Logical Empiricism

2008-03-20

BART VAN KERKHOVE and JEAN PAUL VAN BENDEGEM, eds. Perspectives on Matmatical Practices: Bringing Together Philosophy of Mathematics, Sociology of Mathematics, and Mathematics Education

2008-03-20

AGUSTIN RAYO and GABRIEL UZQUIANO, eds. Absolute Generality

2008-03-20

MICHAEL FRIEDMAN and ALFRED NORDMANN, editors. The Kantian Legacy in Nineteenth-Century Science

2008-03-20

NATHALIE SINCLAIR, DAVID PIMM, and WILLIAM HIGGINSON, eds. Mathematics and the Aesthetic: New Approaches to an Ancient Affinity

2008-03-20

KAREN FRANCOIS and JEAN PAUL VAN BENDEGEM, eds. Philosophical Dimensions in Mathematics Education

2008-03-20

Ontology, Commitment, and Quine's Criterion

Raley, Y. — 2007-09-21

For Quine, the ontological commitments of a discourse are what fall under its (objectual) quantifiers. The recent literature, however, is beginning to move away from this picture. There are direct challenges to Quine's criterion, and there are also attempts to provide alternatives. Azzouni suggests that the ontological commitments of a discourse should be determined by an existence predicate instead. The availability of this alternative forces an adjudication between Qune's criterion and the predicate approach to ontological commitment. I argue that to adjudicate between these criteria for ontological commitment, we need first to adjudicate between criteria for what exists.

A Critique of a Formalist-Mechanist Version of the Justification of Arguments in Mathematicians' Proof Practices

Rav, Y. — 2007-09-21

In a recent article, Azzouni has argued in favor of a version of formalism according to which ordinary mathematical proofs indicate mechanically checkable derivations. This is taken to account for the quasi-universal agreement among mathematicians on the validity of their proofs. Here, the author subjects these claims to a critical examination, recalls the technical details about formalization and mechanical checking of proofs, and illustrates the main argument with aanalysis of examples. In the author's view, much of mathematical reasoning presents genuine meaning-dependent mathematical characteristics that cannot be captured by formal calculi.

‘...there is a conflict between mathematical practice and the formalist doctrine.’ [Kreisel, 1969, p. 39]

Frege on Consistency and Conceptual Analysis

Blanchette, P. A. — 2007-09-21

Gottlob Frege famously rejects the methodology for consistency and independence proofs offered by David Hilbert in the latter's Foundations of Geometry. The present essay defends against recent criticism the view that this rejection turns on Frege's understanding of logical entailment, on which the entailment relation is sensitive to the contents of non-logical terminology. The goals are (a) to clarify further Frege's understanding of logic and of the role of conceptual analysis in logical investigation, and (b) to point out the extent to which his understanding of logic differs importantly from that of the model-theoretic tradition that grows out of Hilbert's work.

The Conceivability of Platonism

Callard, B. — 2007-09-21

It is widely believed that platonists face a formidable problem: that of providing an intelligible account of mathematical knowledge. The problem is that we seem unable, if the platonist is right, to have the causal relationships with the objects of mathematics without which knowledge of these objects seems unintelligible. The standard platonist response to this challenge is either to deny that knowledge without causation is unintelligible, or to make room for causal interactions by softening the platonism at issue. In this essay I argue that the idea of causal relations with fully platonist objects is unproblematic.

Full-Blooded Reference

Rabin, G. — 2007-09-21

In ‘Just what is full-blooded platonism?’ Greg Restall outlines several objections to Mark Balaguer's theory of full-blooded platonism. I reply to these objections by explicating the semantic framework for the reference of mathematical terms that full-blooded platonism requires. Expanding upon these replies, I then explain how the full-blooded platonist, in light of the explicated semantic framework, should treat mathematical terms and statements in order to avoid certain pitfalls.

LISA A. SHABEL. Mathematics in Kant's Critical Philosophy: Reflections on Mathematical Practice. Studies in Philosophy Outstanding Dissertations, Robert Nozick, ed. New York & London: Routledge, 2003. ISBN 0-415-93955-0. Pp. 178 (cloth)

Jagnow, R. — 2007-09-21

SIOBHAN ROBERTS. King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry

Brown, J. R. — 2007-09-21

JON WILLIAMSON. Bayesian Nets and Causality: Philosophical and Computational Foundations

Korb, K. B. — 2007-09-21

GRAHAM OPPY. Philosophical Perspectives on Infinity

Mendelson, E. — 2007-09-21

SAUNDERS MAC LANE. Saunders Mac Lane: A Mathematical Autobiography

McLarty, C. — 2007-09-21

RAYMOND G. AYOUB, editor. Musings of the Masters

2007-09-21

Naturalism, Truth and Beauty in Mathematics{dagger}

Moore, M. E. — 2007-05-29

Can a scientific naturalist be a mathematical realist? I review some arguments, derived largely from the writings of Penelope Maddy, for a negative answer. The rejoinder from the realist side is that the irrealist cannot explain, as well as the realist can, why a naturalist should grant the mathematician the degree of methodological autonomy that the irrealist's own arguments require. Thus a naturalist, as such, has at least as much reason to embrace mathematical realism as to embrace irrealism.

Falsification of Propensity Models by Statistical Tests and the Goodness-of-Fit Paradox

Hennig, C. — 2007-05-29

Gillies introduced a propensity interpretation of probability which is linked to experience by a falsifying rule for probability statements. The present paper argues that general statistical tests should qualify as falsification rules. The ‘goodness-of-fit paradox’ is introduced: the confirmation of a probability model by a test refutes the model's validity.

An example is given in which an independence test introduces dependence. Several possibilities to interpret the paradox and to deal with it are discussed. It is concluded that the propensity interpretation properly reflects statistical practice, but it is not as objective as some adherents claim.

On Godel Sentences and What They Say

Milne, P. — 2007-05-29

Proofs of Gödel's First Incompleteness Theorem are often accompanied by claims such as that the gödel sentence constructed in the course of the proof says of itself that it is unprovable and that it is true. The validity of such claims depends closely on how the sentence is constructed. Only by tightly constraining the means of construction can one obtain gödel sentences of which it is correct, without further ado, to say that they say of themselves that they are unprovable and that they are true; otherwise a false theory can yield false gödel sentences.

Kitcher, Mathematical Intuition, and Experience

McEvoy, M. — 2007-05-29

Mathematical apriorists sometimes hold that our non-derived mathematical beliefs are warranted by mathematical intuition. Against this, Philip Kitcher has argued that if we had the experience of encountering mathematical experts who insisted that an intuition-produced belief was mistaken, this would undermine that belief. Since this would be a case of experience undermining the warrant provided by intuition, such warrant cannot be a priori.

I argue that this leaves untouched a conception of intuition as merely an aspect of our ordinary ability to reason. Thus the apriorist may still hold that some mathematical beliefs are warranted by intuition.

ALI BEHBOUD. Bolzanos Beitrage zur Mathematik und ihrer Philosophie [Bolzano's Contributions to Mathematics and its Philosophy]. Bern: Bern Studies in the History and Philosophy of Science, 2000. Pp. iii + 141. ISBN 3-8311-1026-3.

Rusnock, P. — 2007-05-29

An Austrian Melange * ECKEHART KOLER, PETER WEIBEL, MICHAEL STOLTZNER, BERND BULDT, CARSTEN KLEIN, and WERNER DEPAULI-SCHIMANOVICH-GOTTIG, eds. Kurt Godel. Wahrheit & Beweisbarkeit. Band 1: Dokumente und historische Analysen [Kurt Godel. Truth and Provability. Vol. 1: Documents and Historical Analyses]. Vienna: obv et hpt, 2002. ISBN 3-209-03824-1. Pp. 279. * BERND BULDT, ECKEHART KOHLER, MICHAEL STOLTZNER, PETER WEIBEL, CARSTEN KLEIN, and WERNER DEPAULI-SCHIMANOVICH-GOTTIG, eds. Kurt Godel. Wahrheit & Beweisbarkeit. Band 2: Kompendium zum Werk [Vol. 2: Compendium of Work]. Vienna: obv et hpt, 2002. ISBN 3-209-03835-X. Pp. 447.

Leitgeb, H. — 2007-05-29

ROLAND OMNES. Converging Realities: Towards a Common Philosophy of Physics and Mathematics. Princeton and Oxford: Princeton University Press, 2005. Pp. xvii + 264. ISBN 0-691-11530-3.

Liston, M. — 2007-05-29

GIANDOMENICO SICA, editor. What Is Category Theory? Advanced Studies in Mathematics and Logic; 3. Monza, Italy: Polimetrica, 2006. ISBN-10 88-7699-031-3. Pp. 290

2007-05-29

GIANDOMENICO SICA, editor. What Is Geometry? Advanced Studies in Mathematics and Logic; 4. Monza, Italy: Polimetrica, 2006. ISBN-10 88-7699-030-5. Pp. 268

2007-05-29

EMILY CARSON and RENATE HUBER, eds. Intuition and the Axiomatic Method. Dordrecht: Springer, 2006. ISBN-10 1-4020-4039-3; ISBN-13 978-1-4020-4039-9. Pp. xiii + 324

2007-05-29

ROBIN MACKAY, ed. Collapse: Philosophical research and development. Vol. 1. Oxford: Urbanomic, 2006. ISBN-10 0-9553087-0-4. Pp. iv + 288.

2007-05-29

THOMAS BOLANDER, VINCENT F. HENDRICKS and STIG ANDUR PEDERSEN, eds. Self-Reference. CSLI Lecture notes; No. 178. Stanford, Calif.: CSLI Publications, 2006. ISBN 1-57586-516-4 (pbk); 1-57586-515-7 (cloth). Pp. iv + 190

2007-05-29