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Philosophy of Mathematics

Edited by Øystein Linnebo (Birkbeck College)
Assistant editor: Sam Roberts (Birkbeck College, University of Oslo)
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  1. added 2015-07-31
    Farzad Didehvar, Consistency Problem and “Unexpected Hanging Problem”.
  2. added 2015-07-21
    Hartry Field (1984). Dale Gottlieb, Ontological Economy: Substitutional Quantification and Mathematics. [REVIEW] Noûs 18 (1):160-165.
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  3. added 2015-07-21
    Hans Hahn (1959). Logic, Mathematics, and Knowledge of Nature. In A. J. Ayer (ed.), Logical Positivism. The Free Press 147-161.
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  4. added 2015-07-19
    Richard G. Heck (forthcoming). Is Frege's Definition of the Ancestral Adequate. Philosophia Mathematica:nkv020.
    Why should one think Frege's definition of the ancestral correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that seems to undermine Frege's claim to have justified induction in purely logical terms. I discuss such circularity objections and then offer a new definition of the ancestral intended to be intensionally correct; its extensional correctness then follows without proof. This new definition can be proven equivalent to Frege's without any use of arithmetical induction. This (...)
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  5. added 2015-07-19
    Catherine Legg (2015). Review of Danielle Macbeth, "Realizing Reason: A Narraitve of Truth and Knowing". [REVIEW] Notre Dame Philosophical Reviews:online.
  6. added 2015-07-10
    Justin Clarke-Doane (forthcoming). Debunking and Dispensability. In Uri D. Leibowitz & Neil Sinclair (eds.), Explanation in Ethics and Mathematics. Oxford University Press
    In his précis of a recent book, Richard Joyce writes, “My contention…is that…any epistemological benefit-of-the-doubt that might have been extended to moral beliefs…will be neutralized by the availability of an empirically confirmed moral genealogy that nowhere…presupposes their truth.” Such reasoning – falling under the heading “Genealogical Debunking Arguments” – is now commonplace. But how might “the availability of an empirically confirmed moral genealogy that nowhere… presupposes” the truth of our moral beliefs “neutralize” whatever “epistemological benefit-of-the-doubt that might have been extended (...)
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  7. added 2015-07-04
    William Lane Craig (forthcoming). Bernulf Kanitscheider. Natur Und Zahl: Die Mathematisierbarkeit der Welt [Nature and Number: The Mathematizability of the World]. Berlin: Springer Verlag, 2013. ISBN: 978-3-642-37707-5 ; 978-3-642-37708-2 . Pp. Vii + 385. [REVIEW] Philosophia Mathematica:nkv018.
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  8. added 2015-07-01
    Porto André (2015). Semantical Mutation, Algorithms and Programs. Dissertatio:44-76.
    This article offers an explanation of perhaps Wittgenstein’s strangest and least intuitive thesis – the semantical mutation thesis – according to which one can never answer a mathematical conjecture because the new proof alters the very meanings of the terms involved in the original question. Instead of basing our justification on the distinction between mere calculation and proofs of isolated propositions, characteristic of Wittgenstein’s intermediary period, we generalize it to include conjectures involving effective procedures as well.
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  9. added 2015-06-28
    Christopher Menzel (forthcoming). Logic, Essence, and Modality -- A Critical Review of Bob Hale, Necessary Beings: An Essay on Ontology, Modality, & the Relations Between Them. [REVIEW] Philosophia Mathematica:nkv017.
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  10. added 2015-06-23
    Ceenom, Message Ceenom: The Brief.
    The unabridged Message Ceenom is available at http://ceen.me and http://ceen.in.
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  11. added 2015-06-23
    Robert Schwartzkopff (2015). Singular Terms Revisited. Synthese:1-28.
    Neo-Fregeans take their argument for arithmetical realism to depend on the availability of certain, so-called broadly syntactic tests for whether a given expression functions as a singular term. The broadly syntactic tests proposed in the neo-Fregean tradition are the so-called inferential test and the Aristotelian test. If these tests are to subserve the neo-Fregean argument, they must be at least adequate, in the sense of correctly classifying paradigm cases of singular terms and non-singular terms. In this paper, I pursue two (...)
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  12. added 2015-06-23
    Robert Schwartzkopff (2011). Numbers as Ontologically Dependent Objects - Hume's Principle Revisited. Grazer Philosophische Studien 82:353-373.
    Adherents of Ockham’s fundamental razor contend that considerations of ontological parsimony pertain primarily to fundamental objects. Derivative objects, on the other hand, are thought to be quite unobjectionable. One way to understand the fundamental vs. derivative distinction is in terms of the Aristotelian distinction between ontologically independent and dependent objects. In this paper I will defend the thesis that every natural number greater than 0 is an ontologically dependent object thereby exempting the natural numbers from Ockham’s fundamental razor.
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  13. added 2015-06-18
    David Ellerman, On the Self-Predicative Universals of Category Theory.
    This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having (...)
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  14. added 2015-06-18
    David Ellerman, Mac Lane, Bourbaki, and Adjoints: A Heteromorphic Retrospective.
    Saunders Mac Lane famously remarked that "Bourbaki just missed" formulating adjoints in a 1948 appendix (written no doubt by Pierre Samuel) to an early draft of Algebre--which then had to wait until Daniel Kan's 1958 paper on adjoint functors. But Mac Lane was using the orthodox treatment of adjoints that only contemplates the object-to-object morphisms within a category, i.e., homomorphisms. When Samuel's treatment is reconsidered in view of the treatment of adjoints using heteromorphisms or hets (object-to-object morphisms between objects in (...)
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  15. added 2015-06-17
    David DeVidi (2015). 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 ; 978-94-007-6534-4 . Pp. X + 393. [REVIEW] Philosophia Mathematica 23 (2):276-280.
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  16. added 2015-06-15
    John Corcoran (2014). Formalizing Euclid’s First Axiom. Bulletin of Symbolic Logic 20:404-405.
    Formalizing Euclid’s first axiom. Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný) -/- Euclid [fl. 300 BCE] divides his basic principles into what came to be called ‘postulates’ and ‘axioms’—two words that are synonyms today but which are commonly used to translate Greek words meant by Euclid as contrasting terms. -/- Euclid’s postulates are specifically geometric: they concern geometric magnitudes, shapes, figures, etc.—nothing else. The first: “to draw a line from any point to any point”; the last: the (...)
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  17. added 2015-06-14
    Robert Knowles & David Liggins (forthcoming). Good Weasel Hunting. Synthese.
    The ‘indispensability argument’ for the existence of mathematical objects appeals to the role mathematics plays in science. In a series of publications, Joseph Melia has offered a distinctive reply to the indispensability argument. The purpose of this paper is to clarify Melia’s response to the indispensability argument and to advise Melia and his critics on how best to carry forward the debate. We will begin by presenting Melia’s response and diagnosing some recent misunderstandings of it. Then we will discuss four (...)
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  18. added 2015-06-12
    Daniel Nolan (2015). The Unreasonable Effectiveness of Abstract Metaphysics. Oxford Studies in Metaphysics 9:61-88.
  19. added 2015-06-09
    Sean Walsh (forthcoming). Predicativity, the Russell-Myhill Paradox, and Church’s Intensional Logic. Journal of Philosophical Logic:1-50.
    This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church's intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms (...)
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  20. added 2015-06-09
    James Franklin (2014). Global and Local. Mathematical Intelligencer 36 (4).
    The global/local contrast is ubiquitous in mathematics. This paper explains it with straightforward examples. It is possible to build a circular staircase that is rising at any point (locally) but impossible to build one that rises at all points and comes back to where it started (a global restriction). Differential equations describe the local structure of a process; their solution describes the global structure that results. The interplay between global and local structure is one of the great themes of mathematics, (...)
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  21. added 2015-06-04
    James Ladyman & Stuart Presnell (forthcoming). Identity in Homotopy Type Theory, Part I: The Justification of Path Induction. Philosophia Mathematica:nkv014.
    Homotopy Type Theory 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 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 (...)
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  22. added 2015-06-04
    Catherine Legg (forthcoming). An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure, by Franklin, James. [REVIEW] Australasian Journal of Philosophy:1-1.
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  23. added 2015-06-03
    John Corcoran (2008). Subregular Tetrahedra. Bulletin of Symbolic Logic 14:411-2.
    This largely expository lecture deals with aspects of traditional solid geometry suitable for applications in logic courses. Polygons are plane or two-dimensional; the simplest are triangles. Polyhedra [or polyhedrons] are solid or three-dimensional; the simplest are tetrahedra [or triangular pyramids, made of four triangles]. -/- A regular polygon has equal sides and equal angles. A polyhedron having congruent faces and congruent [polyhedral] angles is not called regular, as some might expect; rather they are said to be subregular—a word coined for (...)
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  24. added 2015-05-27
    Iulian D. Toader, Weyl's Criticism of Dedekind on the Norm of Belief in Mathematics.
    This paper discusses an intriguing, though rather overlooked case of normative disagreement in the history of philosophy of mathematics: Weyl's criticism of Dedekind's famous principle that ``In science, what is provable ought not to be believed without proof.'' This criticism challenges not only a logicist norm of belief in mathematics, but also a realist view about whether there is a fact of the matter as to what norms of belief are correct.
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  25. added 2015-05-15
    Philip A. Ebert (2015). Richard G. Heck Jr. Reading Frege's Grundgesetze. Oxford: Oxford University Press, 2012. ISBN: 978-0-19-923370-0 ; 978-0-19-874437-5 ; 978-0-19-165535-7 . Pp. Xvii + 296. [REVIEW] Philosophia Mathematica 23 (2):289-293.
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  26. added 2015-05-14
    James S. J. Schwartz (forthcoming). Mathematical Structuralism, Modal Nominalism, and the Coherence Principle. Philosophia Mathematica:nkv013.
    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 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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  27. added 2015-05-07
    Uri D. Leibowitz & Neil Sinclair (eds.) (forthcoming). Explanation in Ethics and Mathematics. Oxford University Press.
    Contents: 1.'Introduction: Explanation in Ethics and Mathematics' Neil Sinclair & Uri D. Leibowitz. Part I: Evolutionary Debunking Arguments 2.'Debunking and Dispensability' Justin Clarke-Doane. 3.'Explaining the Reliability of Moral Beliefs' Folke Tersman. 4.'Genealogical Explanations of Chance and Morals' Toby Handfield. 5.'Evolutionary Debunking Arguments in Religion and Morality' Erik J. Wielenberg. 6.‘An Assumption of Extreme Significance’: Moore, Ross and Spencer on Ethics and Evolution' Hallvard Lillehammer. 7.'Reply: Confessions of a Modest Debunker' Richard Joyce. Part II: Indispensability Arguments. 8.'Moral Explanation for Moral Anti-Realism' (...)
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