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Forthcoming articles
  1. Andrew Arana (forthcoming). On the Depth of Szemerédi's Theorem. Philosophia Mathematica:nku036.
    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 (...)
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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. Juliet Floyd (forthcoming). Depth and ClarityFelix Mühlhölzer. Braucht Die Mathematik Eine Grundlegung? Eine Kommentar des Teils III von Wittgensteins Bemerkungen Über Die Grundlagen der Mathematik [Does Mathematics Need a Foundation? A Commentary on Part III of Wittgenstein's Remarks on the Foundations of Mathematics]. Frankfurt: Vittorio Klostermann, 2010. ISBN: 978-3-465-03667-8. Pp. Xiv + 602. [REVIEW] Philosophia Mathematica:nku037.
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  4. 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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  5. Jeremy Gray (forthcoming). Depth — A Gaussian Tradition in Mathematics. Philosophia Mathematica:nku035.
    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 (...)
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  6. Marc Lange (forthcoming). Depth and Explanation in Mathematics. Philosophia Mathematica:nku022.
    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 (...)
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  7. B. Larvor (forthcoming). The Growth of Mathematical Knowledge. Philosophia Mathematica.
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  8. 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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  9. 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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  10. 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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  11. John Stillwell (forthcoming). What Does ‘Depth’ Mean in Mathematics? Philosophia Mathematica:nku033.
    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 (...)
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  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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  13. 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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  14. Alasdair Urquhart (forthcoming). Mathematical Depth. Philosophia Mathematica:nkv004.
    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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  15. 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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