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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 2014-07-28
    Stephen Puryear (forthcoming). Finitism and the Beginning of the Universe. Australasian Journal of Philosophy.
    Many philosophers have argued that the past must be finite in duration because otherwise reaching the present moment would have involved something impossible, namely, the sequential occurrence of an actual infinity of events. In reply, some philosophers have objected that there can be nothing amiss in such an occurrence, since actually infinite sequences are ‘traversed’ all the time in nature, for example, whenever an object moves from one location in space to another. This essay focuses on one of the two (...)
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  2. added 2014-07-28
    Émilie du Châtelet & Lydia Patton (2014). On the Divisibility and Subtlety of Matter. In L. Patton (ed.), Philosophy, Science, and History. Routledge. 332-42.
    Translation for this volume by Lydia Patton of Chapter 9 (pages 179-200) of Émilie du Châtelet's Institutions de Physique (Foundations of Physics). Original publication date 1750. Paris: Chez Prault Fils.
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  3. added 2014-07-23
    Steven M. Duncan, Platonism by the Numbers.
    In this paper, I defend traditional Platonic mathematical realism from its contemporary detractors, arguing that numbers, understood as abstract, non-physical objects of rational intuition, are indispensable for the act of counting.
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  4. added 2014-07-23
    Alejandro Pérez Carballo (2014). Structuring Logical Space. Philosophy and Phenomenological Research 89 (1).
  5. added 2014-07-19
    Sean Walsh, Eleanor Knox & Adam Caulton (2014). Critical Review of Mathematics and Scientific Representation. [REVIEW] Philosophy of Science 81 (3):460-469.
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  6. added 2014-06-21
    Justin Clarke-Doane (forthcoming). Justification and Explanation in Mathematics and Morality. In Russ Shafer-Landau (ed.), Oxford Studies in Metaethics. Oxford University Press.
    In an influential book, Harman writes, "In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles [1977, 9 – 10]." What is the epistemological relevance of this contrast? In this article, I argue that ethicists and philosophers of mathematics have misunderstood it. They have confused what I shall call the justificatory challenge for realism about an area, D – (...)
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  7. added 2014-06-20
    Carlo Ierna (2011). Brentano and Mathematics. Revue Roumaine de Philosophie 55 (1):149-167.
    Franz Brentano is not usually associated with mathematics. Generally, only Brentano’s discussion of the continuum and his critique of the mathematical accounts of it is treated in the literature. It is this detailed critique which suggests that Brentano had more than a superficial familiarity with mathematics. Indeed, considering the authors and works quoted in his lectures, Brentano appears well-informed and quite interested in the mathematical research of his time. I specifically address his lectures here as there is much less to (...)
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  8. added 2014-06-11
    N. Tennant (forthcoming). Logic, Mathematics, and the A Priori, Part II: Core Logic as Analytic, and as the Basis for Natural Logicism. Philosophia Mathematica:nku009.
    We examine the sense in which logic is a priori , and explain how mathematical theories can be dichotomized non-trivially into analytic and synthetic portions. We argue that Core Logic contains exactly the a-priori -because-analytically-valid deductive principles. We introduce the reader to Core Logic by explaining its relationship to other logical systems, and stating its rules of inference. Important metatheorems about Core Logic are reported, and its important features noted. Core Logic can serve as the basis for a foundational program (...)
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  9. added 2014-06-10
    Jonah Goldwater (2014). Sider's Third Realm. Metaphysica 15 (1):99-112.
    Sider (2011; Writing the Book of the World. Oxford: Oxford University Press) argues it is not only predicates that carve reality at its joints, but expressions of any logical or grammatical category – including quantifiers, operators, and sentential connectives. Even so, he denies these expressions pick out entities in the world; instead, they only represent the world’s “structure”. I argue that this distinction is not viable, and that Sider’s ambitious programme requires an exotic ontology – and even a Fregean “third (...)
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  10. added 2014-06-09
    T. Parent (forthcoming). Theory Dualism and the Metalogic of Mind-Body Problems. In Christopher Daly (ed.), The Palgrave Handbook to Philosophical Method. Palgrave.
    The paper defends the philosophical method of "regimentation" by example, especially in relation to the theory of mind. The starting point is the Place-Smart after-image argument: A green after-image will not be located outside the skull, but if we cracked open your skull, we won't find anything green in there either. (If we did, you'd have some disturbing medical news.) So the after-image seems not to be in physical space, suggesting that it is non-physical. In response, I argue that the (...)
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  11. added 2014-06-05
    David Liggins (forthcoming). Grounding and the Indispensability Argument. Synthese:1-18.
    There has been much discussion of the indispensability argument for the existence of mathematical objects. In this paper I reconsider the debate by using the notion of grounding, or non-causal dependence. First of all, I investigate what proponents of the indispensability argument should say about the grounding of relations between physical objects and mathematical ones. This reveals some resources which nominalists are entitled to use. Making use of these resources, I present a neglected but promising response to the indispensability argument—a (...)
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  12. added 2014-06-03
    David Hyder (forthcoming). Review of Michael Friedman, Kant’s Construction of Nature. [REVIEW] Isis 105 (2).
  13. added 2014-06-02
    R. Trueman (forthcoming). A Dilemma for Neo-Fregeanism. Philosophia Mathematica:nku012.
    Neo-Fregeans need their stipulation of Hume's Principle — $NxFx=NxGx \leftrightarrow \exists R (Fx \,1\hbox {-}1_R\, Gx)$ — to do two things. First, it must implicitly define the term-forming operator ‘ Nx ... x ...’, and second it must guarantee that Hume's Principle as a whole is true. I distinguish two senses in which the neo-Fregeans might ‘stipulate’ Hume's Principle, and argue that while one sort of stipulation fixes a meaning for ‘ Nx ... x ...’ and the other guarantees the (...)
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  14. added 2014-06-02
    James Franklin (2014). Aristotelian Realist Philosophy of Mathematics. Palgrave MacMillan.
    An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...)
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  15. added 2014-05-30
    Catherine Legg (forthcoming). “Things Unreasonably Compulsory”: A Peircean Challenge to a Humean Theory of Perception, Particularly With Respect to Perceiving Necessary Truths. Cognitio.
    Much mainstream analytic epistemology is built around a sceptical treatment of modality which descends from Hume. The roots of this scepticism are argued to lie in Hume’s (nominalist) theory of perception, which is excavated, studied and compared with the very different (realist) theory of perception developed by Peirce. It is argued that Peirce’s theory not only enables a considerably more nuanced and effective epistemology, it also (unlike Hume’s theory) does justice to what happens when we appreciate a proof in mathematics.
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  16. added 2014-05-23
    Cesare Cozzo (2011). Matematica e retorica. Paradigmi (3):59-72.
    The traditional opposition between mathematical proof and rhetorical argument is based on a non-contextual picture of proof, against which historical and theoretical objections have been raised. The author advocates a different opposition, between epistemic rhetoric and instrumental rhetoric. Instrumental rhetoric aims at persuasion without caring for truth. Epistemic rhetoric is a practice aimed at both persuasion and truth. Aiming at truth is a way of acting, which can be characterized in terms of epistemically virtuous behavioural traits. In this sense epistemic (...)
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  17. added 2014-05-20
    Lizhi Xu (2008). Xu Lizhi Tan Shu Xue Fang Fa Lun =. Dalian Li Gong da Xue Chu Ban She.
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  18. added 2014-05-20
    Carlo Cellucci (2007). La Filosofia Della Matematica Del Novecento. Laterza.
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  19. added 2014-05-15
    Renren Liu (2010). Duo Zhi Luo Ji Han Shu Jie Gou Li Lun Yan Jiu. Ke Xue Chu Ban She.
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  20. added 2014-05-12
    N. Tennant (forthcoming). Logic, Mathematics, and the A Priori, Part I: A Problem for Realism. Philosophia Mathematica:nku006.
    This is Part I of a two-part study of the foundations of mathematics through the lenses of (i) apriority and analyticity, and (ii) the resources supplied by Core Logic. Here we explain what is meant by apriority, as the notion applies to knowledge and possibly also to truths in general. We distinguish grounds for knowledge from grounds of truth, in light of our recent work on truthmakers. We then examine the role of apriority in the realism/anti-realism debate. We raise a (...)
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  21. added 2014-05-11
    T. 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 ( e.g ., NF and Church's CUS) 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 (low) ordinals. However, that set has an ordinal in turn which is not (...)
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  22. added 2014-05-10
    Carlos Montemayor & Rasmus Grønfeldt Winther (forthcoming). Review of Space, Time, and Number in the Brain. [REVIEW] Mathematical Intelligencer.
    Albert Einstein once made the following remark about “the world of our sense experiences”: “the fact that it is comprehensible is a miracle.” (1936, p. 351) A few decades later, another physicist, Eugene Wigner, wondered about the unreasonable effectiveness of mathematics in the natural sciences, concluding his classic article thus: “the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve” (1960, p. 14). (...)
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  23. added 2014-05-09
    T. Parent, Self-Reference is Sufficient for Paradox.
    This is a much less technical argument for the same conclusion from my “Paradox with just Self-Reference,” viz., that if self-reference is unconstrained, paradox will result. I first show that in classical logic, expressions must be seen as linguistic types rather than tokens. (Otherwise, ‘this very term = this very term’ is a false instance of the Law of Identity.) But then, one can derive a contradiction from the premise ‘This sentence is not derived’, or from the premise ‘ ‘this (...)
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