Year:

Forthcoming articles
  1.  22
    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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  2.  6
    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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  3.  12
    Neil Barton (forthcoming). Richness and Reflection. Philosophia Mathematica:nkv036.
    A pervasive thought in contemporary philosophy of mathematics is that in order to justify reflection principles, one must hold universism: the view that there is a single universe of pure sets. I challenge this kind of reasoning by contrasting universism with a Zermelian form of multiversism. I argue that if extant justifications of reflection principles using notions of richness are acceptable for the universist, then the Zermelian can use similar justifications. However, I note that for some forms of richness argument, (...)
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  4.  7
    Patricia Blanchette (forthcoming). The Breadth of the Paradox. Philosophia Mathematica:nkv038.
    This essay examines Frege's reaction to Russell's Paradox and his views about the grounding of existence claims in mathematics. It is argued that Frege's strict requirements on existential proofs would rule out the attempt to ground arithmetic in. It is hoped that this discussion will help to clarify the ways in which Frege's position is both coherent and significantly different from the neo-logicist position on the issues of: what's required for proofs of existence; the connection between models, consistency, and existence; (...)
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  5.  14
    Roy T. Cook (forthcoming). Frege's Cardinals and Neo-Logicism. Philosophia Mathematica:nkv029.
    Gottlob Frege defined cardinal numbers in terms of value-ranges governed by the inconsistent Basic Law V. Neo-logicists have revived something like Frege's original project by introducing cardinal numbers as primitive objects, governed by Hume's Principle. A neo-logicist foundation for set theory, however, requires a consistent theory of value-ranges of some sort. Thus, it is natural to ask whether we can reconstruct the cardinal numbers by retaining Frege's definition and adopting an alternative consistent principle governing value-ranges. Given some natural assumptions regarding (...)
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  6.  17
    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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  7.  9
    Philip A. Ebert (forthcoming). Frege on Sense Identity, Basic Law V, and Analysis. Philosophia Mathematica:nkv032.
    The paper challenges a widely held interpretation of Frege's conception of logic on which the constituent clauses of basic law V have the same sense. I argue against this interpretation by first carefully looking at the development of Frege's thoughts in Grundlagen with respect to the status of abstraction principles. In doing so, I put forth a new interpretation of Grundlagen §64 and Frege's idea of ‘recarving of content’. I then argue that there is strong evidence in Grundgesetze that Frege (...)
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  8.  17
    Michèle Friend & Daniele Molinini (forthcoming). Using Mathematics to Explain a Scientific Theory. Philosophia Mathematica:nkv022.
    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 (...)
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  9.  6
    Sébastien Gandon (forthcoming). Rota's Philosophy in its Mathematical Context. Philosophia Mathematica:nkv023.
    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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  10.  5
    Casper Storm Hansen (forthcoming). Brouwer's Conception of Truth. Philosophia Mathematica:nkv025.
    In this paper it is argued that the understanding of Brouwer as replacing truth conditions with assertability or proof conditions, in particular as codified in the so-called Brouwer-Heyting-Kolmogorov Interpretation, is misleading and conflates a weak and a strong notion of truth that have to be kept apart to understand Brouwer properly: truth-as-anticipation and truth- in-content. These notions are explained, exegetical documentation provided, and semi-formal recursive definitions are given.
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  11.  10
    Kevin C. Klement (forthcoming). A Generic Russellian Elimination of Abstract Objects. Philosophia Mathematica:nkv031.
    In this paper I explore a position on which it is possible to eliminate the need for postulating abstract objects through abstraction principles by treating terms for abstracta as ‘incomplete symbols’, using Russell's no-classes theory as a template from which to generalize. I defend views of this stripe against objections, most notably Richard Heck's charge that syntactic forms of nominalism cannot correctly deal with non-first-orderizable quantifcation over apparent abstracta. I further discuss how number theory may be developed in a system (...)
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  12.  5
    Ansten Mørch Klev (forthcoming). Dedekind's Logicism. Philosophia Mathematica:nkv027.
    A detailed argument is provided for the thesis that Dedekind was a logicist about arithmetic. The rules of inference employed in Dedekind's construction of arithmetic are, by his lights, all purely logical in character, and the definitions are all explicit; even the definition of the natural numbers as the abstract type of simply infinite systems can be seen to be explicit. The primitive concepts of the construction are logical in their being intrinsically tied to the functioning of the understanding.
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  13.  5
    Teresa Kouri (forthcoming). Ante Rem Structuralism and the No-Naming Constraint. Philosophia Mathematica:nkv041.
    Tim Räz has presented what he takes to be a new objection to Stewart Shapiro's ante rem structuralism. Räz claims that ARS conflicts with mathematical practice. I will explain why this is similar to an old problem, posed originally by John Burgess in 1999 and Jukka Keränen in 2001, and show that Shapiro can use the solution to the original problem in Räz's case. Additionally, I will suggest that Räz's proposed treatment of the situation does not provide an argument for (...)
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  14.  20
    B. Larvor (forthcoming). The Growth of Mathematical Knowledge. Philosophia Mathematica.
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  15.  11
    Graham Leach-Krouse (forthcoming). Structural-Abstraction Principles. Philosophia Mathematica:nkv033.
    In this paper, I present a class of ‘structural’ abstraction principles, and describe how they are suggested by some features of Cantor's and Dedekind's approach to abstraction. Structural abstraction is a promising source of mathematically tractable new axioms for the neo-logicist. I illustrate this by showing, first, how a theorem of Shelah gives a sufficient condition for consistency in the structural setting, solving what neo-logicists call the ‘bad company’ problem for structural abstraction. Second, I show how, in the structural setting, (...)
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  16.  43
    Øystein Linnebo, Stewart Shapiro & Geoffrey Hellman (forthcoming). Aristotelian Continua. Philosophia Mathematica:nkv024.
    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 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, (...)
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  17. 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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  18.  4
    Richard Pettigrew (forthcoming). John P. Burgess. Rigor and Structure. Oxford: Oxford University Press, 2015. ISBN: 978-0-19-872222-9 ; 978-0-19-103360-5 . Pp. Xii + 215. [REVIEW] Philosophia Mathematica:nkv034.
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  19.  8
    Erich H. Reck & Roy T. Cook (forthcoming). Introduction to Special Issue Reconsidering Frege's Conception of Number Dedicated to the Memory of Aldo Antonelli. Philosophia Mathematica:nkv042.
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  20.  10
    Fred Richman (forthcoming). 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 . Pp. Xvi + 270. [REVIEW] Philosophia Mathematica:nkv035.
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  21.  4
    Peter Roeper (forthcoming). A Vindication of Logicism. Philosophia Mathematica:nkv026.
    Frege regarded Hume's Principle as insufficient for a logicist account of arithmetic, as it does not identify the numbers; it does not tell us which objects the numbers are. His solution, generally regarded as a failure, was to propose certain sets as the referents of numerical terms. I suggest instead that numbers are properties of pluralities, where these properties are treated as objects. Given this identification, the truth-conditions of the statements of arithmetic can be obtained from logical principles with the (...)
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  22.  2
    Dirk Schlimm (forthcoming). Metaphors for Mathematics From Pasch to Hilbert. Philosophia Mathematica:nkv039.
    How mathematicians conceive of the nature of mathematics is reflected in the metaphors they use to talk about it. In this paper I investigate a change in the use of metaphors in the late nineteenth and early twentieth centuries. In particular, I argue that the metaphor of mathematics as a tree was used systematically by Pasch and some of his contemporaries, while that of mathematics as a building was deliberately chosen by Hilbert to reflect a different view of mathematics. By (...)
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  23.  37
    Stewart Shapiro & Geoffrey Hellman (forthcoming). Frege Meets Aristotle: Points as Abstracts. Philosophia Mathematica:nkv021.
    There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake (...)
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  24.  12
    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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  25.  4
    Audrey Yap (forthcoming). Stephen Pollard. A Mathematical Prelude to the Philosophy of Mathematics. Springer, 2014. ISBN: 978-3-319-05815-3 ; 978-3-319-05816-0 . Pp. Xi + 202. [REVIEW] Philosophia Mathematica:nkw001.
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