Philosophia Mathematica - recent issues

Fictionalism, Theft, and the Story of Mathematics

Balaguer, M. — 2009-05-23

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. — 2009-05-23

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. — 2009-05-23

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. — 2009-05-23

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. — 2009-05-23

A Scientific Enterprise?: Penelope Maddy's Second Philosophy

Shapiro, S., Reeder, P. — 2009-05-23

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

2009-05-23

Empirical Regularities in Wittgenstein's Philosophy of Mathematics

Steiner, M. — 2009-02-05

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. — 2009-02-05

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. — 2009-02-05

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. — 2009-02-05

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. — 2009-02-05

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

Oliveri, G. — 2009-02-05

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

Thomas, R. — 2009-02-05

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

Rav, Y. — 2009-02-05

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

2009-02-05

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

2009-02-05

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

2009-02-05

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

2009-02-05

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)

2009-02-05

PAOLO MANCOSU, ed. The Philosophy of Mathematical Practice

2009-02-05

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

2009-02-05

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

2009-02-05

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

2009-02-05

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

2009-02-05

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

2009-02-05

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

Shapiro, S. — 2008-09-29

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.

Platonism and Aristotelianism in Mathematics

Pettigrew, R. — 2008-09-29

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.

Epistemic Optimism

Ganea, M. — 2008-09-29

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.

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

De Bruin, B. — 2008-09-29

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’.

The Epistemological Status of Computer-Assisted Proofs

McEvoy, M. — 2008-09-29

Several high-profile mathematical problems have been solved in recent decades by computer-assisted proofs. Some philosophers have argued that such proofs are a posteriori on the grounds that some such proofs are unsurveyable; that our warrant for accepting these proofs involves empirical claims about the reliability of computers; that there might be errors in the computer or program executing the proof; and that appeal to computer introduces into a proof an experimental element. I argue that none of these arguments withstands scrutiny, and so there is no reason to believe that computer-assisted proofs are not a priori.

Criteria of Identity and Structuralist Ontology

Leitgeb, H., Ladyman, J. — 2008-09-29

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.

A Cause for Concern: Standard Abstracta and Causation

Azzouni, J. — 2008-09-29

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?

CHARLES PARSONS. Mathematical Thought and Its Objects

Burgess, J. P. — 2008-09-29

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

Cole, J. C. — 2008-09-29