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Philosophia Mathematica (235)
Philosophia Mathematica. 1964; s1-1(1): 1-4. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.1. (1)
Philosophia Mathematica. 1964; s1-1(1): 33-37. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.33. (1)
Philosophia Mathematica. 1964; s1-1(1): 38-44. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.38. (1)
Philosophia Mathematica. 1964; s1-1(1): 45-50. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.45. (1)
Philosophia Mathematica. 1964; s1-1(1): 5-32. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.5. (1)
Philosophia Mathematica. 1964; s1-1(1): 50-52. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.50. (1)
Philosophia Mathematica. 1964; s1-1(1): 53-65. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.53. (1)
Philosophia Mathematica. 1964; s1-1(2): 107-111. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.107. (1)
Philosophia Mathematica. 1964; s1-1(2): 111-113. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.111. (1)
Philosophia Mathematica. 1964; s1-1(2): 113-117. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.113. (1)
Philosophia Mathematica. 1964; s1-1(2): 118-119. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.118. (1)
Philosophia Mathematica. 1964; s1-1(2): 120-121. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.120. (1)
Philosophia Mathematica. 1964; s1-1(2): 67-88. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.67. (1)
Philosophia Mathematica. 1964; s1-1(2): 89-95. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.89. (1)
Philosophia Mathematica. 1964; s1-1(2): 97-106. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.97. (1)
Philosophia Mathematica. 1965; Published 1965 Dec 01. doi: 10.1093/oxfordjournals.philmat.a037281. [early online release] (1)
Philosophia Mathematica. 1965; Published 1965 Dec 01. doi: 10.1093/oxfordjournals.philmat.a037282. [early online release] (1)
Philosophia Mathematica. 1965; Published 1965 Dec 01. doi: 10.1093/oxfordjournals.philmat.a037283. [early online release] (1)
Philosophia Mathematica. 1965; Published 1965 Dec 01. doi: 10.1093/oxfordjournals.philmat.a037284. [early online release] (1)
53289
Philosophy (1,431)

Journal

Philosophia Mathematica (235)
Philosophia Mathematica. 1964; s1-1(1): 1-4. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.1. (1)
Philosophia Mathematica. 1964; s1-1(1): 33-37. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.33. (1)
Philosophia Mathematica. 1964; s1-1(1): 38-44. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.38. (1)
Philosophia Mathematica. 1964; s1-1(1): 45-50. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.45. (1)

Philosophia Mathematica. 1964; s1-1(1): 5-32. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.5. (1)

Philosophia Mathematica. 1964; s1-1(1): 50-52. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.50. (1)

Philosophia Mathematica. 1964; s1-1(1): 53-65. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.1.53. (1)

Philosophia Mathematica. 1964; s1-1(2): 107-111. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.107. (1)

Philosophia Mathematica. 1964; s1-1(2): 111-113. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.111. (1)

Philosophia Mathematica. 1964; s1-1(2): 113-117. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.113. (1)

Philosophia Mathematica. 1964; s1-1(2): 118-119. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.118. (1)

Philosophia Mathematica. 1964; s1-1(2): 120-121. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.120. (1)

Philosophia Mathematica. 1964; s1-1(2): 67-88. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.67. (1)

Philosophia Mathematica. 1964; s1-1(2): 89-95. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.89. (1)

Philosophia Mathematica. 1964; s1-1(2): 97-106. Published 1964 Jan 01. doi: 10.1093/philmat/s1-1.2.97. (1)

Philosophia Mathematica. 1965; Published 1965 Dec 01. doi: 10.1093/oxfordjournals.philmat.a037281. [early online release] (1)

Philosophia Mathematica. 1965; Published 1965 Dec 01. doi: 10.1093/oxfordjournals.philmat.a037282. [early online release] (1)

Philosophia Mathematica. 1965; Published 1965 Dec 01. doi: 10.1093/oxfordjournals.philmat.a037283. [early online release] (1)

Philosophia Mathematica. 1965; Published 1965 Dec 01. doi: 10.1093/oxfordjournals.philmat.a037284. [early online release] (1)

53289

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A Problem with the Dependence of Informal Proofs on Formal Proofs (2015)

Tanswell, F.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2015-09-22

Description: Derivationists, those wishing to explain the correctness and rigour of informal proofs in terms of associated formal proofs, are generally held to be supported by the success of the project of translating informal proofs into computer-checkable formal counterparts. I argue, however, that this project is a false friend for the derivationists because there are too many different associated formal proofs for each informal proof, leading to a serious worry of overgeneration. I press this worry primarily against Azzouni's derivation-indicator account, but conclude that overgeneration is a major obstacle to a successful account of informal proofs in this direction.

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The Representational Foundations of Computation (2015)

Rescorla, M.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2015-09-22

Description: Turing computation over a non-linguistic domain presupposes a notation for the domain. Accordingly, computability theory studies notations for various non-linguistic domains. It illuminates how different ways of representing a domain support different finite mechanical procedures over that domain. Formal definitions and theorems yield a principled classification of notations based upon their computational properties. To understand computability theory, we must recognize that representation is a key target of mathematical inquiry. We must also recognize that computability theory is an intensional enterprise: it studies entities as represented in certain ways, rather than entities detached from any means of representing them.

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Mathematical Structuralism, Modal Nominalism, and the Coherence Principle (2015)

Schwartz, J. S. J.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2015-09-22

Description: According to Stewart Shapiro's coherence principle, structures exist whenever they can be coherently described. I argue that Shapiro's attempts to justify this principle (along with his position, ante rem structuralism) are circular, as he relies on criticisms of modal nominalism which presuppose the coherence principle. I argue further that when the coherence principle is not presupposed, his reasoning more strongly supports modal nominalism than ante rem structuralism.

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Identity in Homotopy Type Theory, Part I: The Justification of Path Induction (2015)

Ladyman, J., Presnell, S.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2015-09-22

Description: Homotopy Type Theory (HoTT) is a proposed new language and foundation for mathematics, combining algebraic topology with logic. An important rule for the treatment of identity in HoTT is path induction , which is commonly explained by appeal to the homotopy interpretation of the theory's types, tokens, and identities as (respectively) spaces, points, and paths. However, if HoTT is to be an autonomous foundation then such an interpretation cannot play a fundamental role. In this paper we give a derivation of path induction, motivated from pre-mathematical considerations, without recourse to homotopy theory.

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Logic, Essence, and Modality Bob Hale. Necessary Beings: An Essay on Ontology, Modality, & the Relations Between Them. Oxford University Press, 2013. ISBN: 978-0-19-966262-3 (hbk); 978-0-19-874803-8 (pbk); 978-0-19-164834-2 (e-book). Pp. ix + 298 (2015)

Menzel, C.

Oxford University Press

In: Philosophia Mathematica

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Bernard Bolzano. Theory of Science. Volumes I-IV. Paul Rusnock and Rolf George, trans. Oxford: Oxford University Press, 2014. ISBN: 978-0-19-968438-0. Pp. 2044. (2015)

Sebestik, J.

Oxford University Press

In: Philosophia Mathematica

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Pavel Pudlak. Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction. Springer Monographs in Mathematics. Springer, 2013. ISBN: 978-3-319-00118-0 (hbk); 978-3-319-00119-7 (ebook). Pp. xiv + 695 (2015)

Urquhart, A.

Oxford University Press

In: Philosophia Mathematica

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Mark van Atten. Essays on Godel's Reception of Leibniz, Husserl, and Brouwer. Logic, Epistemology, and the Unity of Science; 35. Dordrecht: Springer, 2015. ISBN 978-3-319-10030-2 (hbk); 978-3-319-10031-9 (e-book). Pp. xiv + 328 (2015)

Oxford University Press

In: Philosophia Mathematica

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Gabriele Lolli, Marco Panza, and Giorgio Venturi, eds. From Logic to Practice: Italian Studies in the Philosophy of Mathematics. Boston Studies in the Philosophy and History of Science; 308. Springer, 2015. ISBN 978-3-319-10433-1 (hbk); 978-3-319-10434-8 (2015)

Oxford University Press

In: Philosophia Mathematica

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When Do Some Things Form a Set? (2015)

Hewitt, S.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2015-09-22

Description: This paper raises the question under what circumstances a plurality forms a set, parallel to the Special Composition Question for mereology. The range of answers that have been proposed in the literature are surveyed and criticised. I argue that there is good reason to reject both the view that pluralities never form sets and the view that pluralities always form sets. Instead, we need to affirm restricted set formation. Casting doubt on the availability of any informative principle which will settle which pluralities form sets, the paper concludes by affirming a naturalistic approach to the philosophy of set theory.

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Aristotelian Continua (2016)

Linnebo, O., Shapiro, S., Hellman, G.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2016-06-21

Description: In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems (such as the existence of bisections) that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, and we show that the two approaches are equivalent.

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Using Mathematics to Explain a Scientific Theory (2016)

Friend, M., Molinini, D.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2016-06-21

Description: We answer three questions: 1. Can we give a wholly mathematical explanation of a physical phenomenon? 2. Can we give a wholly mathematical explanation for a whole physical theory ? 3. What is gained or lost in giving a wholly, or partially, mathematical explanation of a phenomenon or a scientific theory? To answer these questions we look at a project developed by Hajnal Andréka, Judit Madarász, István Németi and Gergely Székely. They, together with collaborators, present special relativity theory in a three-sorted first-order formal language.

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Stephen Pollard. A Mathematical Prelude to the Philosophy of Mathematics (2016)

Yap, A.

Oxford University Press

In: Philosophia Mathematica

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Nick Haverkamp. Intuitionism vs. Classicism: A Mathematical Attack on Classical Logic. Studies in Theoretical Philosophy, Vol. 2. Frankfurt: Klostermann, 2015. ISBN 978-3-465-03906-8 (pbk). Pp. xvi + 270 (2016)

Richman, F.

Oxford University Press

In: Philosophia Mathematica

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Sorin Costreie, ed. Early Analytic Philosophy: New Perpectives on the Tradition (2016)

Oxford University Press

In: Philosophia Mathematica

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Ernest Davis and Philip J. Davis eds. Mathematics, Substance and Surmise: Views on the Meaning and Ontology of Mathematics (2016)

Oxford University Press

In: Philosophia Mathematica

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Rota's Philosophy in its Mathematical Context (2016)

Gandon, S.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2016-06-21

Description: The goal of this paper is to connect Rota's discussion of the Husserlian notion of Fundierung with Rota's project of giving combinatorics a foundation in his 1964 paper ‘On the foundations of combinatorial theory I’. Section 2 gives the basic tenets of this seminal paper. Sections 3 and 4 spell out the connections made there between Rota's philosophical writings and his mathematical achievements. Section 5 shows how these two developments fit into Rota's analysis of the place of combinatorics in mathematics.

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Claire Ortiz Hill and Jairo Jose da Silva. The Road Not Taken: On Husserl's Philosophy of Logic and Mathematics (2016)

Hopkins, B. C.

Oxford University Press

In: Philosophia Mathematica

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Feng Ye. Strict Finitism and the Logic of Mathematical Applications (2016)

Vinckier, N., Van Bendegem, J. P.

Oxford University Press

In: Philosophia Mathematica

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Reinhard Kahle and Michael Rathjen, eds. Gentzen's Centenary: The Quest for Consistency (2016)

Oxford University Press

In: Philosophia Mathematica

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John W. Dawson, Jr. Why Prove it Again: Alternative Proofs in Mathematical Practice (2016)

Carter, J.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2016-06-21

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Numbers and Everything (2013)

Santos, G.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2013-10-02

Description: I begin by drawing a parallel between the intuitionistic understanding of quantification over all natural numbers and the generality relativist understanding of quantification over absolutely everything. I then argue that adoption of an intuitionistic reading of relativism not only provides an immediate reply to the absolutist's charge of incoherence but it also throws a new light on the debates surrounding absolute generality.

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Dianoia Left and Right (2013)

Pollard, S.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2013-10-02

Description: In Plato's Phaedrus , Socrates offers two speeches, the first portraying madness as mere disease, the second celebrating madness as divine inspiration. Each speech is correct, says Socrates, though neither is complete. The two kinds of madness are like the left and right sides of a living body: no account that focuses on just one half can be adequate. In a recent paper, Hugh Benson gives a left-handed speech about a psychic condition endemic among mathematicians: dianoia . Benson acknowledges that his account is one-sided, but only hints at the virtues of right-handed dianoia . This note sketches a somewhat fuller picture.

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On the Application of the Honeycomb Conjecture to the Bee's Honeycomb (2013)

Raz, T.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2013-10-02

Description: In a recent paper, Aidan Lyon and Mark Colyvan have proposed an explanation of the structure of the bee's honeycomb based on the mathematical Honeycomb Conjecture. This explanation has instantly become one of the standard examples in the philosophical debate on mathematical explanations of physical phenomena. In this critical note, I argue that the explanation is not scientifically adequate. The reason for this is that the explanation fails to do justice to the essentially three-dimensional structure of the bee's honeycomb.

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Hilary Putnam's Consistency Objection against Wittgenstein's Conventionalism in Mathematics (2013)

Garavaso, P.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2013-10-02

Description: Hilary Putnam first published the consistency objection against Ludwig Wittgenstein’s account of mathematics in 1979. In 1983, Putnam and Benacerraf raised this objection against all conventionalist accounts of mathematics. I discuss the 1979 version and the scenario argument, which supports the key premise of the objection. The wide applicability of this objection is not apparent; I thus raise it against an imaginary axiomatic theory T similar to Peano arithmetic in all relevant aspects. I argue that a conventionalist can explain the consistency of T and suggest that an analogous explanation can be provided for the consistency of Peano arithmetic.

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Deconstructing a Topological Sorites (2013)

Rizza, D.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2013-10-02

Description: In this paper I examine some difficulties with the argument presented as a topological sorites in Z. Weber and M. Colyvan, ‘A topological sorites’, Journal of Philosophy 107, 311–325. In particular, I suggest that the argument may be used to support the claim that sorites-type paradoxes cannot arise in a cohesive environment.

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PENELOPE MADDY. Defending the Axioms: On the Philosophical Foundations of Set Theory. Oxford: Oxford University Press, 2011. ISBN 978-0-19-959618-8 (hbk); 978-0-19-967148-9 (pbk). Pp. x + 150 (2013)

Mc; Larty, C.

Oxford University Press

In: Philosophia Mathematica

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SEBASTIEN GANDON. Russell's Unknown Logicism: A Study in the History and Philosophy of Mathematics. New York: Palgrave Macmillan, 2012. ISBN 978-0-230-57699-5. Pp. xiv + 266 (2013)

Urquhart, A.

Oxford University Press

In: Philosophia Mathematica

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BERNARD LINSKY. The Evolution of Principia Mathematica: Bertrand Russell's Manuscripts and Notes for the Second Edition. Cambridge: Cambridge University Press, 2011. ISBN 978-1-107-00327-9. Pp. vii + 407 (2013)

Griffin, N.

Oxford University Press

In: Philosophia Mathematica

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That We See That Some Diagrammatic Proofs Are Perfectly Rigorous (2013)

Azzouni, J.

Oxford University Press

In: Philosophia Mathematica

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Description: Mistaken reasons for thinking diagrammatic proofs aren't rigorous are explored. The main result is that a confusion between the contents of a proof procedure (what's expressed by the referential elements in a proof procedure) and the unarticulated mathematical aspects of a proof procedure (how that proof procedure is enabled) gives the impression that diagrammatic proofs are less rigorous than language proofs. An additional (and independent) factor is treating the impossibility of naturally generalizing a diagrammatic proof procedure as an indication of lack of rigor.

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GORDON BELOT. Geometric Possibility. Oxford and New York: Oxford University Press, 2011. ISBN 978-0-19-959532-7. Pp. x + 219 (2013)

Smeenk, C.

Oxford University Press

In: Philosophia Mathematica

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ALBERT LAUTMAN. Mathematics, Ideas and the Physical Real. Simon B. Duffy, trans. London and New York: Continuum, 2011. 978-1-4411-2344-2 (pbk); 978-1-44114656-4 (hbk); 978-1-44114433-1 (pdf e-bk); 978-1-44114654-0 (epub e-bk). Pp. xlii + 310 (2013)

Cassou-Nogues, P.

Oxford University Press

In: Philosophia Mathematica

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Second-Order Arithmetic Sans Sets (2013)

Berk, L.

Oxford University Press

In: Philosophia Mathematica

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Description: This paper examines the ontological commitments of the second-order language of arithmetic and argues that they do not extend beyond the first-order language. Then, building on an argument by George Boolos, we develop a Tarski-style definition of a truth predicate for the second-order language of arithmetic that does not involve the assignment of sets to second-order variables but rather uses the same class of assignments standardly used in a definition for the first-order language.

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MARK VAN ATTEN, PASCAL BOLDINI, MICHEL BOURDEAU, and GERHARD HEINZMANN, eds., One Hundred Years of Intuitionism (1907-2007): The Cerisy Conference. Basel, Boston, Berlin: Birkhauser, 2008. ISBN 978-3-7643-8652-8. Pp. xiii + 422 (2013)

Bell, J. L.

Oxford University Press

In: Philosophia Mathematica

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APOSTOLOS DOXIADIS and BARRY MAZUR, eds. Circles Disturbed: The Interplay of Mathematics and Narrative (2013)

Oxford University Press

In: Philosophia Mathematica

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CHRISTOPHER PINCOCK. Mathematics and Scientific Representation. Oxford University Press, 2012. ISBN 978-0-19-975710-7. Pp. xv + 330 (2013)

Liston, M.

Oxford University Press

In: Philosophia Mathematica

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GUILLERMO E. ROSADO HADDOCK, Against the Current: Selected Philosophical Papers (2013)

Oxford University Press

In: Philosophia Mathematica

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WILLIAM DEMOPOULOS. Logicism and its Philosophical Legacy (2013)

Oxford University Press

In: Philosophia Mathematica

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In Defense of Bertrand: The Non-Restrictiveness of Reasoning by Example (2013)

Klyve, D.

Oxford University Press

In: Philosophia Mathematica

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Description: This note has three goals. First, we discuss a presentation of Bertrand's paradox in a recent issue of Philosophia Mathematica , which we believe to be a subtle but important misinterpretation of the problem. We compare claims made about Bertrand with his 1889 Calcul des Probabilités . Second, we use this source to understand Bertrand's true intention in describing what we now call his paradox, comparing it both to another problem he describes in the same section and to a modern treatment. Finally, we briefly consider the importance of knowing when a specific example represents a general case.

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MIRCEA PITICI, ed. The Best Writing on Mathematics 2012 (2013)

Oxford University Press

In: Philosophia Mathematica

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Properties and the Interpretation of Second-Order Logic (2013)

Hale, B.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2013-06-11

Description: This paper defends a deflationary conception of properties, according to which a property exists if and only if there could be a predicate with appropriate satisfaction conditions. I argue that purely general properties and relations necessarily exist and discuss the bearing of this conception of properties on the interpretation of higher-order logic and on Quine's charge that higher-order logic is ‘set theory in sheep's clothing’. On my approach, the usual semantics involves a false assimilation of the logic to set theory. I conclude with remarks about implications for the programme of founding mathematical theories in higher-order logic plus abstraction principles.

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Introduction: Logicism Today (2013)

Gandon, S., Halimi, B.

Oxford University Press

In: Philosophia Mathematica

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Zermelo and Russell's Paradox: Is There a Universal set? (2013)

Landini, G.

Oxford University Press

In: Philosophia Mathematica

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Description: Zermelo once wrote that he had anticipated Russell's contradiction of the set of all sets that are not members of themselves. Is this sufficient for having anticipated Russell's Paradox — the paradox that revealed the untenability of the logical notion of a set as an extension? This paper argues that it is not sufficient and offers criteria that are necessary and sufficient for having discovered Russell's Paradox. It is shown that there is ample evidence that Russell satisfied the criteria and that Zermelo did not.

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Variable, Structure, and Restricted Generality (2013)

Gandon, S.

Oxford University Press

In: Philosophia Mathematica

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Description: From 1905–1908 onward, Russell thought that his new ‘substitutional theory’ provided him with the right framework to resolve the set-theoretic paradoxes. Even if he did not finally retain this resolution, the substitutional strategy was instrumental in the development of his thought. The aim of this paper is not historical, however. It is to show that Russell's substitutional insight can shed new light on current issues in philosophy of mathematics. After having briefly expounded Russell's key notion of a ‘structured variable’, I connect it to two ongoing discussions: the Caesar problem, and absolute generality.

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MIRCEA PITICI, ed. The Best Writing on Mathematics 2011 (2013)

Oxford University Press

In: Philosophia Mathematica

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DAVID HILBERT. David Hilbert's lectures on the foundations of geometry, 1891-1902. Michael Hallett and Ulrich Majer, eds. David Hilbert's Foundational Lectures; 1. Berlin: Springer-Verlag, 2004. ISBN 3-540-64373-7. Pp. xxviii + 661 (2013)

Pambuccian, V.

Oxford University Press

In: Philosophia Mathematica

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Enlarging One's Stall or How Did All of These Sets Get in Here? (2013)

Wilson, M.

Oxford University Press

In: Philosophia Mathematica

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Description: Following historical developments, this article traces two basic motives for employing sets within a physical setting and discusses whether they truly pose a problem for ‘mathematical naturalism’.

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Structured Variables (2013)

Halimi, B.

Oxford University Press

In: Philosophia Mathematica

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Description: Drawing on Russell's substitutional theory, this paper examines the notion of ‘structured variable’, in order to compare Russell's and Tarski's conceptions of variables. The framework of syntactic fibrations, coming from categorical logic, is used as a common setting. The main objective of this paper is to make sense of the notion of structured variable beyond the context of Russell's theory, to question the Tarskian way of understanding what it is to be a possible value for a variable, and to bring out semantically structured variables as a renewed and fruitful way of conceiving of logical structure.

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The Relationship of Derivations in Artificial Languages to Ordinary Rigorous Mathematical Proof (2013)

Azzouni, J.

Oxford University Press

In: Philosophia Mathematica

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Description: The relationship is explored between formal derivations, which occur in artificial languages, and mathematical proof, which occurs in natural languages. The suggestion that ordinary mathematical proofs are abbreviations or sketches of formal derivations is presumed false. The alternative suggestion that the existence of appropriate derivations in formal logical languages is a norm for ordinary rigorous mathematical proof is explored and rejected.

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Depth and Clarity * Felix Muhlholzer. Braucht die Mathematik eine Grundlegung? Eine Kommentar des Teils III von Wittgensteins Bemerkungen uber die Grundlagen der Mathematik [Does Mathematics need a Foundation? A Commentary on Part III of Wittgenstein's Re (2015)

Floyd, J.

Oxford University Press

In: Philosophia Mathematica

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Mathematical Depth (2015)

Urquhart, A.

Oxford University Press

In: Philosophia Mathematica

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Description: The first part of the paper is devoted to surveying the remarks that philosophers and mathematicians such as Maddy, Hardy, Gowers, and Zeilberger have made about mathematical depth. The second part is devoted to the question of whether we can make the notion precise by a more formal proof-theoretical approach. The idea of measuring depth by the depth and bushiness of the proof is considered, and compared to the related notion of the depth of a chess combination.

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Depth and Explanation in Mathematics (2015)

Lange, M.

Oxford University Press

In: Philosophia Mathematica

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Description: This paper argues that in at least some cases, one proof of a given theorem is deeper than another by virtue of supplying a deeper explanation of the theorem — that is, a deeper account of why the theorem holds. There are cases of scientific depth that also involve a common abstract structure explaining a similarity between two otherwise unrelated phenomena, making their similarity no coincidence and purchasing depth by answering why questions that separate, dissimilar explanations of the two phenomena cannot correctly answer. The connections between explanation, depth, unification, power, and coincidence in mathematics and science are compared.

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Andrew Aberdein and Ian J. Dove, eds. The Argument of Mathematics. Logic, Epistemology, and the Unity of Science; 30. Dordrecht: Springer, 2013. ISBN: 978-94-007-6533-7 (hbk); 978-94-007-6534-4 (ebook). Pp. x + 393 (2015)

De; Vidi, D.

Oxford University Press

In: Philosophia Mathematica

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Afterword to Special Issue on Mathematical Depth (2015)

Oxford University Press

In: Philosophia Mathematica

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Depth -- A Gaussian Tradition in Mathematics (2015)

Gray, J.

Oxford University Press

In: Philosophia Mathematica

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Description: Mathematicians use the word ‘deep’ to convey a high appreciation of a concept, theorem, or proof. This paper investigates the extent to which the term can be said to have an objective character by examining its first use in mathematics. It was a consequence of Gauss's work on number theory and the agreement among his successors that specific parts of Gauss's work were deep, on grounds that indicate that depth was a structural feature of mathematics for them. In contrast, French mathematicians had a less structural, more problem-oriented approach to mathematics and did not speak of depth so readily.

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On the Depth of Szemeredi's Theorem (2015)

Arana, A.

Oxford University Press

In: Philosophia Mathematica

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Description: Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which to focus in analyzing mathematical depth. After introducing the theorem, four accounts of mathematical depth will be considered.

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What Does 'Depth' Mean in Mathematics? (2015)

Stillwell, J.

Oxford University Press

In: Philosophia Mathematica

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Description: This paper explores different interpretations of the word ‘deep’ as it is used by mathematicians, with a large number of examples illustrating various criteria for depth. Most of the examples are theorems with ‘historical depth’, in the sense that many generations of mathematicians contributed to their proof. Some also have ‘foundational depth’, in the sense that they support large mathematical theories. Finally, concepts from mathematical logic suggest that it may be possible to order certain theorems or problems according to ‘logical depth’.

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Dagfinn Follesdal, guest ed. Mathematical Evidence. Special issue of Inquiry; An Interdisciplinary Journal of Philosophy 58 (2015) 1-98 (2015)

Oxford University Press

In: Philosophia Mathematica

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James Franklin. An Aristotelian Realist Philosophy of Mathematics. New York: Palgrave Macmillan, 2014. ISBN: 978-1-137-40072-7 (hbk); 978-1-137-40073-4 (pdf); 978-1-137-40074-1 (e-book). Pp. x + 308 (2015)

Jones, M.

Oxford University Press

In: Philosophia Mathematica

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Richard G. Heck Jr. Reading Frege's Grundgesetze. Oxford: Oxford University Press, 2012. ISBN: 978-0-19-923370-0 (hbk); 978-0-19-874437-5 (pbk); 978-0-19-165535-7 (pdf). Pp. xvii + 296 (2015)

Ebert, P. A.

Oxford University Press

In: Philosophia Mathematica

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To Diagram, to Demonstrate: To Do, To See, and To Judge in Greek Geometry (2012)

Catton, P., Montelle, C.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2012-02-14

Description: Not simply set out in accompaniment of the Greek geometrical text, the diagram also is coaxed into existence manually (using straightedge and compasses) by commands in the text. The marks that a diligent reader thus sequentially produces typically sum, however, to a figure more complex than the provided one and also not (as it is) artful for being synoptically instructive. To provide a figure artfully is to balance multiple desiderata, interlocking the timelessness of insight with the temporality of construction. Our account of the diagram complements those of Manders and Macbeth by more strongly emphasizing practical synthesis.

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The Philosophical Significance of Tennenbaum's Theorem (2012)

Button, T., Smith, P.

Oxford University Press

In: Philosophia Mathematica

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Description: Tennenbaum's Theorem yields an elegant characterisation of the standard model of arithmetic. Several authors have recently claimed that this result has important philosophical consequences: in particular, it offers us a way of responding to model-theoretic worries about how we manage to grasp the standard model. We disagree. If there ever was such a problem about how we come to grasp the standard model, then Tennenbaum's Theorem does not help. We show this by examining a parallel argument, from a simpler model-theoretic result.

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Husserl's Pluralistic Phenomenology of Mathematics (2012)

Hartimo, M.

Oxford University Press

In: Philosophia Mathematica

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Description: The paper discusses Husserl's phenomenology of mathematics in his Formal and Transcendental Logic (1929). In it Husserl seeks to provide descriptive foundations for mathematics. As sciences and mathematics are normative activities Husserl's attempt is also to describe the norms at work in these disciplines. The description shows that mathematics can be given in several different ways. The phenomenologist's task is to examine whether a given part of mathematics is genuine according to the norms that pertain to the approach in question. The paper will then examine the intuitionistic, formalistic, and structural features of Husserl's philosophy of mathematics.

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CARLO CELLUCCI, EMILY GROSHOLZ and EMILIANO IPPOLITI, eds. Logic and Knowledge (2012)

Oxford University Press

In: Philosophia Mathematica

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HANS HALVORSON, ed. Deep Beauty: Understanding the Quantum World through Mathematical Innovation (2012)

Oxford University Press

In: Philosophia Mathematica

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Categorical Foundations and Mathematical Practice (2012)

Mc; Larty, C.

Oxford University Press

In: Philosophia Mathematica

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Description: Linnebo and Pettigrew's critique in this journal of categorical foundations well emphasizes that the particulars of various categorical foundations matter, and that mathematical practice must be a major consideration. But several categorists named by the authors as proposing categorical foundations do not propose foundations, notably Awodey, and the article's description of current textbook practice seems inaccurate. They say that categorical foundations have justificatory autonomy if and only if mathematics can be justified simply by its practice. Do they seriously believe philosophers may be able to justify some currently unpractised mathematics better than current practice justifies itself?

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BHARATH SRIRAMAN and SIMON GOODCHILD, eds. Relatively and Philosophically Earnest: Festschrift in Honor of Paul Ernest's 65th Birthday (2012)

Oxford University Press

In: Philosophia Mathematica

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MICHEL SERFATI, ed. De la methode: Recherches en histoire et philosophie des mathematiques (2012)

Oxford University Press

In: Philosophia Mathematica

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Platonism and the 'Epistemic Role Puzzle' (2012)

Mc; Evoy, M.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2012-10-20

Description: Jody Azzouni has offered the following argument against the existence of mathematical entities: if, as it seems, mathematical entities play no role in mathematical practice, we therefore have no reason to believe in them. I consider this argument as it applies to mathematical platonism, and argue that it does not present a legitimate novel challenge to platonism. I also assess Azzouni's use of the ‘epistemic role puzzle’ (ERP) to undermine the platonist's alleged parallel between skepticism about mathematical entities and external-world skepticism. I conclude that ERP fails to undermine this parallel.

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On the Indispensability of the Distinctively Mathematical (2012)

Juhl, C.

Oxford University Press

In: Philosophia Mathematica

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Description: Indispensability arguments purport to show that empirical data provide evidence for the existence of mathematical entities. In this paper we argue that indispensability arguments fail to show that empirical data bears on the mathematical. In order to show this we attempt to clarify what it is to be mathematical , and separate the question whether abstracta in general exist from whether distinctively mathematical abstracta exist. We introduce the notion of an ‘agnostified’ empirical theory as a heuristic device for clarifying the role of mathematical terminology in empirical theories. The special case of spaces is then briefly considered.

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Why the Weasel Fails (2012)

Raley, Y.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2012-10-20

Description: In his paper ‘On what there’s not’, Joseph Melia disavows commitment to the existence of objects like average mothers, possibilities, numbers, etc . Since quantification over such objects is at times unavoidable, Melia tries to argue that we can deny the existence of such objects despite the fact that our (true) theories of the world quantify over them. Melia calls this ‘weaseling’. In this paper, I argue that these assumptions of Melia’s render his position incoherent.

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RICHARD G. HECK, Jr. Frege's Theorem. Oxford: Clarendon Press, 2011. ISBN 978-0-19-969564-5. Pp. xiv + 307 (2012)

Cook, R. T.

Oxford University Press

In: Philosophia Mathematica

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Higher-Order Logic or Set Theory: A False Dilemma (2012)

Shapiro, S.

Oxford University Press

In: Philosophia Mathematica

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Description: The purpose of this article is show that second-order logic, as understood through standard semantics, is intimately bound up with set theory, or some other general theory of interpretations, structures, or whatever. Contra Quine, this does not disqualify second-order logic from its role in foundational studies. To wax Quinean, why should there be a sharp border separating mathematics from logic, especially the logic of mathematics?

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CRAIG SMORYNSKI. Adventures in Formalism. London: College Publications, 2012. ISBN 978-1-84890-060-8. Pp. xii + 606 (2012)

Jones, R.

Oxford University Press

In: Philosophia Mathematica

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HILMI DEMIR, ed. Luciano Floridi's Philosophy of Technology: Critical Reflections (2012)

Oxford University Press

In: Philosophia Mathematica

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MICHAEL POTTER TOM RICKETTS, eds. The Cambridge Companion to Frege. Cambridge: Cambridge University Press, 2010. ISBN 978-0-521-62479-4. Pp. xvii+639 (2012)

Landini, G.

Oxford University Press

In: Philosophia Mathematica

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CHRIS MORTENSEN. Inconsistent Geometry. Studies in Logic; 27. London: College Publications, 2010. ISBN 978-1-84890-022-6. Pp. ii+162. (2012)

Van Bendegem, J. P.

Oxford University Press

In: Philosophia Mathematica

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CURTIS FRANKS. The Autonomy of Mathematical Knowledge: Hilbert's Program Revisted. Cambridge: Cambridge University Press, 2009. ISBN 978-0-521-51437-8. Pp. xiii+213{dagger} (2012)

Feferman, S.

Oxford University Press

In: Philosophia Mathematica

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Towards an Institutional Account of the Objectivity, Necessity, and Atemporality of Mathematics (2013)

Cole, J.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2013-02-06

Description: I contend that mathematical domains are freestanding institutional entities that, at least typically, are introduced to serve representational functions. In this paper, I outline an account of institutional reality and a supporting metaontological perspective that clarify the content of this thesis. I also argue that a philosophy of mathematics that has this thesis as its central tenet can account for the objectivity, necessity, and atemporality of mathematics.

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FABRIZIO PALOMBI, The Star & The Whole: Gian-Carlo Rota on Mathematics and Phenomenology. Boca Raton: CRC Press, 2011. ISBN 978-1-56881-583-1 (pbk). Pp. xiv + 124. English translation of La stella e l'intero: La ricerca di Gian-Carlo Rota tra matemat (2013)

van Atten, M.

Oxford University Press

In: Philosophia Mathematica

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DAVID DEVIDI, MICHAEL HALLETT, and PETER CLARK, eds. Logic, Mathematics, Philosophy: Vintage Enthusiasms: Essays in honour of John L. Bell (2013)

Oxford University Press

In: Philosophia Mathematica

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EMILY R. GROSHOLZ. Representation and Productive Ambiguity in Mathematics and the Sciences. Oxford: Oxford University Press, 2007. ISBN 978-0-19-929973-7. Pp. viii + 313. (2012)

Larvor, B. P.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2012-06-26

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Recollection and the Mathematician's Method in Plato's Meno (2012)

Landry, E.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2012-06-26

Description: I argue that recollection, in Plato's Meno , should not be taken as a method, and, if it is taken as a myth, it should not be taken as a mere myth. Neither should it be taken as a truth, a priori or metaphorical. In contrast to such views, I argue that recollection ought to be taken as an hypothesis for learning. Thus, the only methods demonstrated in the Meno are the elenchus and the hypothetical, or mathematical, method. What Plato's Meno demonstrates, then, is that we cannot be philosophers if we fail to make use of the mathematician's hypothetical method.

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GIOVANNI SOMMARUGA, ed. Foundational Theories of Classical and Constructive Mathematics. Dordrecht: Springer, 2011. ISBN 978-94-007-0430-5. Pp. xi + 314. (2012)

Koss, M. R.

Oxford University Press

In: Philosophia Mathematica

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Platonic and Fregean Numbers (2012)

White, N.

Oxford University Press

In: Philosophia Mathematica

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Description: Rather than reading Plato's philosophy of arithmetic ‘charitably’, it is better to try to explain its failure to generate any fruitful ideas. Prominent in the explanation is Plato's focus on predicates assigning cardinalities and on ‘groups’ falling under them. This focus left Plato unable to envisage the possibility, emerging in Dedekind and Frege but which arithmetic in Plato's time would not easily have suggested, of regarding numbers as objects essentially ranged in the structure of a progression.

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PAOLO MANCOSU. The Adventure of Reason: Interplay between Philosophy of Mathematics and Mathematical Logic, 1900-1940. Oxford: Oxford University Press, 2010. ISBN 978-0-19-954653-4. Pp. xii + 618. (2012)

Potter, M.

Oxford University Press

In: Philosophia Mathematica

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BERND BULDT, BENEDIKT LOWE and THOMAS MULLER eds. Special issue Towards a new epistemology of mathematics. Erkenntnis 68 (2008) No. 3. Pp. 305-435. * BENEDIKT LOWE and THOMAS MULLER, eds. PhiMSAMP: Philosophy of Mathematics: Sociological Aspects and (2012)

Thomas, R.

Oxford University Press

In: Philosophia Mathematica

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Category Theory as an Autonomous Foundation (2011)

Linnebo, O., Pettigrew, R.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2011-11-24

Description: Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy : logical, conceptual, and justificatory. We argue that, while a strong case can be made for its logical and conceptual autonomy, its justificatory autonomy turns on whether or not mathematical theories can be justified by appeal to mathematical practice. If they can, a category-theoretical approach will be fully autonomous; if not, the most natural route to justificatory autonomy is blocked.

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MICHAEL HELLER AND W. HUGH WOODIN, eds. Infinity: New Research Frontiers (2011)

Oxford University Press

In: Philosophia Mathematica

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MIRJA HARTIMO, ed. Phenomenology and Mathematics (2011)

Oxford University Press

In: Philosophia Mathematica

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GILA HANNA, HANS NIELS JAHNKE, and HELMUT PULTE, eds Explanation and Proof in Mathematics: Philosophical and educational perspectives (2011)

Oxford University Press

In: Philosophia Mathematica

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Reflections on Concrete Incompleteness (2011)

Longo, G.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2011-11-24

Description: How do we prove true but unprovable propositions? Gödel produced a statement whose undecidability derives from its ad hoc construction. Concrete or mathematical incompleteness results are interesting unprovable statements of formal arithmetic. We point out where exactly the unprovability lies in the ordinary ‘mathematical’ proofs of two interesting formally unprovable propositions, the Kruskal-Friedman theorem on trees and Girard's normalization theorem in type theory. Their validity is based on robust cognitive performances, which ground mathematics in our relation to space and time, such as symmetries and order, or on the generality of Herbrand's notion of ‘prototype proof’.

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Indexing and Mathematical Explanation (2011)

Baker, A., Colyvan, M.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2011-11-24

Description: We discuss a recent attempt by Chris Daly and Simon Langford to do away with mathematical explanations of physical phenomena. Daly and Langford suggest that mathematics merely indexes parts of the physical world, and on this understanding of the role of mathematics in science, there is no need to countenance mathematical explanation of physical facts. We argue that their strategy is at best a sketch and only looks plausible in simple cases. We also draw attention to how frequently Daly and Langford find themselves in conflict with mathematical and scientific practice.

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The Company Kept by Cut Abstraction (and its Relatives) (2011)

Shapiro, S.

Oxford University Press

In: Philosophia Mathematica

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Kant's Philosophy of Geometry--On the Road to a Final Assessment (2011)

Kvasz, L.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2011-11-24

Description: The paper attempts to summarize the debate on Kant’s philosophy of geometry and to offer a restricted area of mathematical practice for which Kant’s philosophy would be a reasonable account. Geometrical theories can be characterized using Wittgenstein’s notion of pictorial form . Kant’s philosophy of geometry can be interpreted as a reconstruction of geometry based on one of these forms — the projective form . If this is correct, Kant’s philosophy is a reasonable reconstruction of such theories as projective geometry; and not only as they were practiced in Kant’s time, but also as architects use them today.

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Alan Weir. Truth through Proof: A Formalist Foundation for Mathematics. Oxford: Clarendon Press, 2010. ISBN 978-0-19-954149-2. Pp. xiv+281 (2011)

Burgess, J. P.

Oxford University Press

In: Philosophia Mathematica

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JOSEPH KOUNEIHER, DOMINIQUE FLAMENT, PHILIPPE NABONNAND, and JEAN-JACQUES SZCZECINIARZ, eds. Geometrie au XXe siecle, 1930-2000: Histoire et horizons (2011)

Oxford University Press

In: Philosophia Mathematica

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ROMAN MURAWSKI. Essays in the Philosophy and History of Logic and Mathematics (2011)

Oxford University Press

In: Philosophia Mathematica

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The Implausibility of Hermeneutic Non-Assertivism (2011)

Armour-Garb, B.

Oxford University Press

In: Philosophia Mathematica

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Description: In a recent paper, Mark Balaguer has responded to the argument that I launched against Hermeneutic Non-Assertivism, claiming that, as a matter of empirical fact, ‘when typical mathematicians utter mathematical sentences, they are doing something that differs from asserting in a pretty subtle way, so that the difference between [asserting] and this other kind of speech act is not obvious’. In this paper, I show the implausibility of this empirical hypothesis.

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Logicism and the Problem of Infinity: The Number of Numbers (2011)

Landini, G.

Oxford University Press

In: Philosophia Mathematica

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Publication Date: 2011-11-24

Description: Simple-type theory is widely regarded as inadequate to capture the metaphysics of mathematics. The problem, however, is not that some kinds of structure cannot be studied within simple-type theory. Even structures that violate simple-types are isomorphic to structures that can be studied in simple-type theory. In disputes over the logicist foundations of mathematics, the central issue concerns the problem that simple-type theory fails to assure an infinity of natural numbers as objects . This paper argues that the problem of infinity is based on a metaphysical prejudice in favor of numbers as objects — a prejudice that mathematics can get along without.

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