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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 2016-06-28
    Giuseppe Iurato (2012). On the History of Differentiable Manifolds. International Mathematical Forum 7 (10):477-514.
    We discuss central aspects of history of the concept of an affine differentiable manifold, as a proposal confirming the need for using some quantitative methods (drawn from elementary Model Theory) in Mathematical Historiography. In particular, we prove that this geometric structure is a syntactic rigid designator in the sense of Kripke-Putnam.
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  2. added 2016-06-23
    Greg Fried (2016). A Challenge to Divine Psychologism. Theology and Science 14 (2):175-189.
    Alvin Plantinga proposes that mathematical objects and propositions are divine thoughts. This position, which I call divine psychologism, resonates with some remarks by contemporary thinkers. Plantinga claims several advantages for his position, and I add another: it helps to explain the glory of mathematics. But my main purpose is to issue a challenge to divine psychologism. I argue that it has an implausible consequence: it identifies an entity with God’s relation to that entity. I consider and rebut several ways in (...)
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  3. added 2016-06-21
    Jessica Carter (2016). John W. Dawson, Jr. Why Prove It Again: Alternative Proofs in Mathematical Practice. Philosophia Mathematica 24 (2):256-263.
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  4. added 2016-06-21
    Nigel Vinckier & Jean Paul Van Bendegem (2016). Feng Ye. Strict Finitism and the Logic of Mathematical Applications. Philosophia Mathematica 24 (2):247-256.
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  5. added 2016-06-21
    Audrey Yap (2016). Stephen Pollard. A Mathematical Prelude to the Philosophy of Mathematics. Philosophia Mathematica 24 (2):275-277.
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  6. added 2016-06-21
    Burt C. Hopkins (2016). Claire Ortiz Hill and Jairo José da Silva. The Road Not Taken: On Husserl's Philosophy of Logic and Mathematics. Philosophia Mathematica 24 (2):263-275.
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  7. added 2016-06-12
    Michael Starks, What Do Paraconsistent, Undecidable, Random, Computable and Incomplete Mean? A Review of Godel's Way: Exploits Into an Undecidable World by Gregory Chaitin, Francisco A Doria , Newton C.A. Da Costa 160p (2012).
    In ‘Godel’s Way’ three eminent scientists discuss issues such as undecidability, incompleteness, randomness, computability and paraconsistency. I approach these issues from the Wittgensteinian viewpoint that there are two basic issues which have completely different solutions. There are the scientific or empirical issues, which are facts about the world that need to be investigated observationally and philosophical issues as to how language can be used intelligibly (which include certain questions in mathematics and logic), which need to be decided by looking at (...)
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  8. added 2016-06-11
    Malcolm Macleod, A Mathematical Universe Hypothesis.
    In the “Trialogue on the number of fundamental physical constants”, Okun, Veneziano and M. Duff debated the number, from 1 to 3, of dimensionful units required. In this essay the physical constants are defined as geometrical forms constructed from 2 dimensionless constants; the Sommerfeld fine structure constant $\alpha$ and a proposed $\Omega$ and from 2 dimensionful units. The dual-state electron oscillates between a dimensionful magnetic monopole ($\sigma_e$, units = $s^{-1/3}$) state that when combined with Planck time ($T$, units = $s$), (...)
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  9. added 2016-06-03
    Marc Champagne (2016). Diagrams of the Past: How Timelines Can Aid the Growth of Historical Knowledge. Cognitive Semiotics 9 (1):11-44.
    Historians occasionally use timelines, but many seem to regard such signs merely as ways of visually summarizing results that are presumably better expressed in prose. Challenging this language-centered view, I suggest that timelines might assist the generation of novel historical insights. To show this, I begin by looking at studies confirming the cognitive benefits of diagrams like timelines. I then try to survey the remarkable diversity of timelines by analyzing actual examples. Finally, having conveyed this (mostly untapped) potential, I argue (...)
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  10. added 2016-05-29
    Desmond Sander (2014). Mindful Physics — A New Account of What Happens. AENESIDEMUS PRESS.
    A physics that fails to take account of minds, or account for them, cannot be quite right; a physics that accounts so beautifully and so powerfully for so much of what we observe cannot be quite wrong. This book had that conundrum as its starting point, and resolves it. The mindful physics we need is complementary to the compelling and successful but mind-ignoring physics of today. It is the physics that life, especially human life, has made and is making here (...)
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  11. added 2016-05-22
    Kurt Grelling (1910). Die Axiome der Arithmetik mit besonderer Berücksichtigung der Beziehungen zur Mengenlehre. Dissertation, Georg-Augusts-Universität Göttingen
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  12. added 2016-05-18
    Valérie Lynn Therrien (2012). INVENTING LOGIC: THE LÖWENHEIM-SKOLEM THEOREM AND FIRST- AND SECOND-ORDER LOGIC. Pensées Canadiennes 10.
  13. added 2016-05-13
    Michael J. Shaffer (forthcoming). Lakatos’ Quasi-Empiricism in the Philosophy of Mathematics. Polish Journal of Philosophy.
    Imre Lakatos' views on the philosophy of mathematics are important and they have often been underappreciated. The most obvious lacuna in this respect is the lack of detailed discussion and analysis of his 1976a paper and its implications for the methodology of mathematics, particularly its implications with respect to argumentation and the matter of how truths are established in mathematics. The most important themes that run through his work on the philosophy of mathematics and which culminate in the 1976a paper (...)
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  14. added 2016-05-07
    Stephen Puryear (forthcoming). Finitism, Divisibility, and the Beginning of the Universe: Replies to Loke and Dumsday. Australasian Journal of Philosophy:1-6.
    Some philosophers contend that the past must be finite in duration, because otherwise reaching the present would have involved the sequential occurrence of an actual infinity of events, which they regard as impossible. I recently developed a new objection to this finitist argument, to which Andrew Ter Ern Loke and Travis Dumsday have replied. Here I respond to the three main points raised in their replies.
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  15. added 2016-05-05
    Sorin Bangu (2016). On The Unreasonable Effectiveness of Mathematics in the Natural Sciences. In Ippoliti, Sterpetti & Nickles (eds.), Models and Inferences in Science. Springer 11-29.
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  16. added 2016-05-04
    Lior Rabi (2016). Ortega y Gasset on Georg Cantor's Theory of Transfinite Numbers. Kairos. Journal of Philosophy and Science (15):46-70.
    Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and articulated a (...)
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  17. added 2016-04-27
    Luca Incurvati (forthcoming). Maximality Principles in Set Theory. Philosophia Mathematica.
    In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...)
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  18. added 2016-04-23
    S. Hewitt, A Note on Gabriel Uzquiano's 'Varieties of Indefinite Extensibility'.
  19. added 2016-04-21
    Chris Daly & David Liggins (forthcoming). Dorr on the Language of Ontology. Philosophical Studies:1-15.
    In the ‘ordinary business of life’, everyone makes claims about what there is. For instance, we say things like: ‘There are some beautiful chairs in my favourite furniture shop’. Within the context of philosophical debate, some philosophers also make claims about what there is. For instance, some ontologists claim that there are chairs; other ontologists claim that there are no chairs. What is the relation between ontologists’ philosophical claims about what there is and ordinary claims about what there is? According (...)
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  20. added 2016-04-21
    Araceli Ramirez-Cardenas, Maria Moskaleva & Andreas Nieder (2016). Neuronal Representation of Numerosity Zero in the Primate Parieto-Frontal Number Network. Current Biology 26.
    Neurons in the primate parieto-frontal network represent the number of visual items in a collection, but it is unknown whether this system encodes empty sets as conveying null quantity. We recorded from the ventral intraparietal area (VIP) and the prefrontal cortex (PFC) of monkeys performing a matching task including empty sets and countable numerosities as stimuli. VIP neurons encoded empty sets predominantly as a distinct category from numerosities. In contrast, PFC neurons represented empty sets more similarly to numerosity one than (...)
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  21. added 2016-04-13
    Thomas Forster (2016). Mathematical Objects Arising From Equivalence Relations and Their Implementation in Quine's NF. Philosophia Mathematica 24 (1):50-59.
    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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  22. added 2016-04-13
    Jaykov Foukzon, Relevant First-Order Logic LP# and Curry’s Paradox Resolution. Pure and Applied Mathematics Journal Volume 4, Issue 1-1, January 2015 DOI: 10.11648/J.Pamj.S.2015040101.12.
    In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei' sparadox without rejection any contraction postulate is proposed. In additional relevant paraconsistent logic C ̌_n^#,1≤n<ω, in fact,provide an effective way of circumventing triviality of da Costa’s paraconsistent Set Theories〖NF〗n^C.
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  23. added 2016-04-04
    Mohammad Husein Saleh, Doaa Mohammad Shokry & Saada A. Rahman Abu Shammala, A Numerical Approach for Solving Classes of Linear and Nonlinear Volterra Integral Equations by Chebyshev Polynomial.
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  24. added 2016-04-02
    Andrew Arana (2016). Imagination in Mathematics. In Amy Kind (ed.), The Routledge Handbook of Philosophy of Imagination. Routledge 463-477.