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Forthcoming articles
  1. Christopher Menzel (forthcoming). Logic, Essence, and Modality — A Critical Review of Hale's Necessary Beings. Philosophia Mathematica 23.
    Bob Hale’s distinguished record of research places him among the most important and influential contemporary analytic metaphysicians. In his deep, wide ranging, yet highly readable book Necessary Beings, Hale draws upon, but substantially integrates and extends, a good deal his past research to produce a sustained and richly textured essay on — as promised in the subtitle — ontology, modality, and the relations between them. I’ve set myself two tasks in this review: first, to provide a reasonably thorough (if not (...)
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  2. Santos Gon?alo (forthcoming). Numbers and Everything. Philosophia Mathematica.
    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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  3. 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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  4. F. Tanswell (forthcoming). A Problem with the Dependence of Informal Proofs on Formal Proofs. Philosophia Mathematica:nkv008.
    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 (...)
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  5. 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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  6. Thomas Forster (forthcoming). Mathematical Objects Arising From Equivalence Relations and Their Implementation in Quine's NF. Philosophia Mathematica:nku005.
    Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for “low” sets, and thus, for example, a set of all ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; so no (...)
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  7. 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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  8. Simon Hewitt (forthcoming). When Do Some Things Form a Set? Philosophia Mathematica:nkv010.
    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 (...)
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  9. 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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  10. B. Larvor (forthcoming). The Growth of Mathematical Knowledge. Philosophia Mathematica.
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  11. Gianluigi Oliveri (forthcoming). Book Review.'I Fondamenti della Matematica nel Logicismo di Bertrand Russell'. Stefano Donati. Firenze (Firenze Atheneum). 2003. ISBN: 88-7255-204-4. 988 pages.€ 39.00. [REVIEW] Philosophia Mathematica.
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  12. Michael Rescorla (forthcoming). The Representational Foundations of Computation. Philosophia Mathematica:nkv009.
    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 (...)
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  13. 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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  14. Jan Sebestik (forthcoming). Bolzano's Theory of Science Disclosed in EnglishBernard 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. [REVIEW] Philosophia Mathematica:nkv001.
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  15. Wilfried Sieg & Dirk Schlimm (forthcoming). Dedekind's Abstract Concepts: Models and Mappings. Philosophia Mathematica:nku021.
    Dedekind's mathematical work is integral to the transformation of mathematics in the nineteenth century and crucial for the emergence of structuralist mathematics in the twentieth century. We investigate the essential components of what Emmy Noether called, his ‘axiomatic standpoint’: abstract concepts, models, and mappings.
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  16. Daniël F. M. Strauss (forthcoming). The On to Log I Cal Sta Tus of the Prin Ci Ple of the Ex Cluded Mid Dle. Philosophia Mathematica.
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  17. Alasdair Urquhart (forthcoming). Pavel Pudlák. Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction. Springer Monographs in Mathematics. Springer, 2013. ISBN: 978-3-319-00118-0 ; 978-3-319-00119-7 . Pp. Xiv + 695. [REVIEW] Philosophia Mathematica:nkv006.
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  18. Audrey Yap (forthcoming). Dedekind and Cassirer on Mathematical Concept Formation. Philosophia Mathematica:nku029.
    Dedekind's major work on the foundations of arithmetic employs several techniques that have left him open to charges of psychologism, and through this, to worries about the objectivity of the natural-number concept he defines. While I accept that Dedekind takes the foundation for arithmetic to lie in certain mental powers, I will also argue that, given an appropriate philosophical background, this need not make numbers into subjective mental objects. Even though Dedekind himself did not provide that background, one can nevertheless (...)
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