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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 2016-11-30
    Salvatore Florio & Graham Leach-Krouse (forthcoming). What Russell Should Have Said to Burali-Forti. Review of Symbolic Logic.
    The paradox that appears under Burali-Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be—absurdly—an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali-Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some (...)
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  2. added 2016-11-24
    Jared Warren & Daniel Waxman (forthcoming). A Metasemantic Challenge for Mathematical Determinacy. Synthese:1-19.
    This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics. From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate, motivate two important constraints on attempts to meet our challenge, and then use these constraints to develop an argument against determinacy and discuss a particularly popular approach (...)
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  3. added 2016-11-22
    William Lane Craig (2016). 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 24 (1):136-141.
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  4. added 2016-11-22
    Neil Barton (2016). Multiversism and Concepts of Set: How Much Relativism is Acceptable? In Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. Springer 189-209.
    Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account of (...)
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  5. added 2016-11-15
    Bas Rasmussen, Finding Structure in a Meditative State.
    I have been experimenting with meditation for a long time, but just recently I seem to have come across another being in there. It may just be me looking at me, but whatever it is, it is showing me some really interesting arrangements of colored balls. At first, I thought it was just random colors and shapes, but it became very ordered. It was like this being (me?) was trying to talk to me but couldn’t, so was showing me some (...)
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  6. added 2016-11-15
    Bas Rasmussen, Finding Structure in a Meditative State.
    I have been experimenting with meditation for a long time, but just recently I seem to have come across another being in there. It may just be me looking at me, but whatever it is, it is showing me some really interesting arrangements of colored balls. At first, I thought it was just random colors and shapes, but it became very ordered. It was like this being (me?) was trying to talk to me but couldn’t, so was showing me some (...)
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  7. added 2016-11-15
    Bas Rasmussen, Finding Structure in a Meditative State.
    I have been experimenting with meditation for a long time, but just recently I seem to have come across another being in there. It may just be me looking at me, but whatever it is, it is showing me some really interesting arrangements of colored balls. At first, I thought it was just random colors and shapes, but it became very ordered. It was like this being (me?) was trying to talk to me but couldn’t, so was showing me some (...)
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  8. added 2016-11-15
    Bas Rasmussen, Finding Structure in a Meditative State.
    I have been experimenting with meditation for a long time, but just recently I seem to have come across another being in there. It may just be me looking at me, but whatever it is, it is showing me some really interesting arrangements of colored balls. At first, I thought it was just random colors and shapes, but it became very ordered. It was like this being (me?) was trying to talk to me but couldn’t, so was showing me some (...)
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  9. added 2016-11-15
    Bas Rasmussen, Finding Structure in a Meditative State.
    I have been experimenting with meditation for a long time, but just recently I seem to have come across another being in there. It may just be me looking at me, but whatever it is, it is showing me some really interesting arrangements of colored balls. At first, I thought it was just random colors and shapes, but it became very ordered. It was like this being (me?) was trying to talk to me but couldn’t, so was showing me some (...)
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  10. added 2016-11-11
    Dieter Lohmar & Carlo Ierna (2016). Husserl’s Manuscript A I 35. In Guillermo E. Rosado Haddock (ed.), Husserl and Analytic Philosophy. De Gruyter 289-320.
  11. added 2016-11-11
    Carlo Ierna (2016). The Reception of Russell’s Paradox in Early Phenomenology and the School of Brentano: The Case of Husserl’s Manuscript A I 35α. In Guillermo E. Rosado Haddock (ed.), Husserl and Analytic Philosophy. De Gruyter 119-142.
  12. added 2016-11-07
    Nikolay Milkov (forthcoming). The 1900 Turn in Bertrand Russell’s Logic, the Emergence of His Paradox, and the Way Out. Siegener Beiträge Zur Geschichte Und Philosophie der Mathematik 7.
    Russell’s initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russell’s logic was nothing but mereology. First, his acquaintance with Peano’s ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called ‘denoting phrase’. Unfortunately, (...)
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  13. added 2016-11-04
    Iulian D. Toader (2013). Is Mathematics the Theory of Instantiated Structural Universals? Transylvanian Review 22:132-142.
    The paper argues against defending realism about numbers on the basis of realism about instantiated structural universals. After presenting Armstrong’s theory of structural properties as instantiated universals and Lewis’s devastating criticism of it, I argue that several responses to this criticism are unsuccessful, and that one possible construal of structural universals via non-well-founded sets should be resisted by the mathematical realist.
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  14. added 2016-11-04
    Iulian D. Toader (2012). Fictionalism and Mathematical Objectivity. In Metaphysics and Science. Festschrift for Professor Ilie Pârvu. University of Bucharest Press 137-158.
  15. added 2016-11-01
    Silvia Jonas (forthcoming). Modal Structuralism and Theistic Fictionalism. In Fiona Ellis (ed.), New Models of Religious Understanding. Oxford University Press
    I apply Geoffrey Hellman’s modal structuralism about mathematics to the context of theistic discourse and argue that, for every allegedly a priori domain D whose claims are necessarily true if true at all, there is a fictionalist account F of D that preserves definite truth-values while remaining neutral on the question of ontology.
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  16. added 2016-10-31
    Silvia Jonas (forthcoming). Access Problems and Explanatory Overkill. Philosophical Studies:1-12.
    I argue that recent attempts to deflect Access Problems for realism about a priori domains such as mathematics, logic, morality, and modality using arguments from evolution result in two kinds of explanatory overkill: (1) the Access Problem is eliminated for contentious domains, and (2) realist belief becomes viciously immune to arguments from dispensability, and to non-rebutting counter-arguments more generally.
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  17. added 2016-10-31
    Vladimir Rogozhin, The Formula of Justice: The OntoTopological Basis of Physica and Mathematica*.
    Dialectica: Mathematica and Physica, Truth and Justice, Trick and Life. Mathematica as the Constructive Metaphysica and Ontology. Mathematica as the constructive existential method. Сonsciousness and Mathematica: Dialectica of "eidos" and "logos". Mathematica is the Total Dialectica. The basic maternal Structure - "La Structure mère". Mathematica and Physica: loss of existential certainty. Is effectiveness of Mathematica "unreasonable"? The ontological structure of space. Axiomatization of the ontological basis of knowledge: one axiom, one principle and one mathematical object. The main ideas and concepts (...)
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  18. added 2016-10-29
    Desmond Fearnley-Sander (1989). The Idea of a Diagram. In Hassan Ait-Kaci & Maurice Nivat (eds.), Resolution of Equations in Algebraic Structures. Academic Press
    A detailed axiomatisation of diagrams (in affine geometry) is presented, which supports typing of geometric objects, calculation of geometric quantities and automated proof of theorems.
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  19. added 2016-10-20
    Alan Baker (forthcoming). Mathematics and Explanatory Generality. Philosophia Mathematica:nkw021.
    According to one popular nominalist picture, even when mathematics features indispensably in scientific explanations, this mathematics plays only a purely representational role: physical facts are represented, and these exclusively carry the explanatory load. I think that this view is mistaken, and that there are cases where mathematics itself plays an explanatory role. I distinguish two kinds of explanatory generality: scope generality and topic generality. Using the well-known periodical-cicada example, and also a new case study involving bicycle gears, I argue that (...)
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  20. added 2016-10-20
    Don Berry (forthcoming). Proof and the Virtues of Shared Enquiry. Philosophia Mathematica:nkw022.
    This paper investigates an important aspect of mathematical practice: that proof is required for a finished piece of mathematics. If follows that non-deductive arguments — however convincing — are never sufficient. I explore four aspects of mathematical research that have facilitated the impressive success of the discipline. These I call the Practical Virtues: Permanence, Reliability, Autonomy, and Consensus. I then argue that permitting results to become established on the basis of non-deductive evidence alone would lead to their deterioration. This furnishes (...)
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  21. added 2016-10-15
    Marcelo E. Coniglio, Aldo Figallo-Orellano & Ana C. Golzio (2016). Towards an Hyperalgebraic Theory of Non-Algebraizable Logics. CLE E-Prints 16 (4):1-27.
    Multialgebras (or hyperalgebras) have been very much studied in the literature. In the realm of Logic, they were considered by Avron and his collaborators under the name of non-deterministic matrices (or Nmatrices) as a useful semantics tool for characterizing some logics (in particular, several logics of formal inconsistency or LFIs) which cannot be characterized by a single finite matrix. In particular, these LFIs are not algebraizable by any method, including Blok and Pigozzi general theory. Carnielli and Coniglio introduced a semantics (...)
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  22. added 2016-10-14
    Michael Starks, Wolpert, Chaitin and Wittgenstein on Impossibility, Incompleteness, the Limits of Computation, Theism and the Universe as Computer-the Ultimate Turing Theorem.
    I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv.org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, and even (...)
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  23. added 2016-10-11
    Rafael Duarte Oliveira Venancio (2013). Cyberpunk Entre Literatura E Matemática: Processos Comunicacionais da Literatura Massiva Na Crítica Científica da Realidade. Conexão 12 (23).
    O presente artigo busca definir o movimento literário cyberpunk a partir da sua influência teórica vinda do campo da matemática. Utilizando a teorização interna ao movimento, centrada em Rudy Rucker, o objetivo aqui é entender como os campos da análise e dos fundamentos da matemática criam uma importante distinção entre os cyberpunks e as demais distopias literárias. Com isso, há a pressuposição de um movimento de uma crítica sociomatemática feita pelos cyberpunks cujos conceitos matemáticos tornam possível criticar o tempo presente, (...)
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  24. added 2016-10-09
    Michael Bulmer, Desmond Fearnley-Sander & Tim Stokes (2001). The Kinds of Truth of Geometry Theorems. In Jürgen Richter-Gebert & Dongming Wang (eds.), LNCS: Lecture Notes In Computer Science. Springer-Verlag 129-142.
    Proof by refutation of a geometry theorem that is not universally true produces a Gröbner basis whose elements, called side polynomials, may be used to give inequations that can be added to the hypotheses to give a valid theorem. We show that (in a certain sense) all possible subsidiary conditions are implied by those obtained from the basis; that what we call the kind of truth of the theorem may be derived from the basis; and that the side polynomials may (...)
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  25. added 2016-10-07
    Philip A. Ebert & Marcus Rossberg (eds.) (forthcoming). Essays on Frege's Basic Laws of Arithmetic. Oxford University Press.
  26. added 2016-10-06
    Christopher Menzel (forthcoming). The Argument From Collections. In J. Walls & T. Dougherty (eds.), Two Dozen (or so) Arguments for God: The Plantinga Project. Oxford University Press
    Very broadly, an argument from collections is an argument that purports to show that our beliefs about sets imply — in some sense — the existence of God. Plantinga (2007) first sketched such an argument in “Two Dozen” and filled it out somewhat in his 2011 monograph Where the Conflict Really Lies: Religion, Science, and Naturalism. In this paper I reconstruct what strikes me as the most plausible version of Plantinga’s argument. While it is a good argument in at least (...)
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  27. added 2016-10-05
    Bruno Whittle (forthcoming). Proving Unprovability. Review of Symbolic Logic.
    This paper addresses the question: given some theory T that we accept, is there some natural, generally applicable way of extending T to a theory S that can prove a range of things about what it itself (i.e. S) can prove, including a range of things about what it cannot prove, such as claims to the effect that it cannot prove certain particular sentences (e.g. 0 = 1), or the claim that it is consistent? Typical characterizations of Gödel’s second incompleteness (...)
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  28. added 2016-10-03
    Aberdein Andrew (forthcoming). Mohan Ganesalingam. The Language of Mathematics: A Linguistic and Philosophical Investigation. FoLLI Publications on Logic, Language and Information. [REVIEW] Philosophia Mathematica:nkw020.
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  29. added 2016-10-01
    Cezary Cieśliński (forthcoming). Minimalism and the Generalisation Problem: On Horwich’s Second Solution. Synthese:1-25.
    Disquotational theories of truth are often criticised for being too weak to prove interesting generalisations about truth. In this paper we will propose a certain formal theory to serve as a framework for a solution of the generalisation problem. In contrast with Horwich’s original proposal, our framework will eschew psychological notions altogether, replacing them with the epistemic notion of believability. The aim will be to explain why someone who accepts a given disquotational truth theory Th, should also accept various generalisations (...)
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  30. added 2016-09-30
    Fraser MacBride (forthcoming). NeoFregean Metaontology. In P. Ebert & M. Rossberg (eds.), Abstractionism. Oxford University Press
  31. added 2016-09-27
    David Henley (1992). Mathematical Intuition and Wittgenstein. In Christopher Ormell (ed.), New Thinking About the Nature of Mathematics. 39-43.
    This paper covers some large subjects: as well as intuition and Wittgenstein, it also discusses modern computing. However it only traces one thread through these topics. Basically it proposes that a computational analysis of Wittgenstein's Tractatus can shed light upon processes of discovery in mathematics.
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  32. added 2016-09-23
    Dagfinn Føllesdal (2016). Richard Tieszen. After Gödel. Platonism and Rationalism in Mathematics and Logic. Philosophia Mathematica 24 (3):405-421.
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  33. added 2016-09-21
    T. Scott Dixon & Cody Gilmore (2016). Speaks's Reduction of Propositions to Properties: A Benacerraf Problem. Thought: A Journal of Philosophy 5 (3):275-284.
    Speaks defends the view that propositions are properties: for example, the proposition that grass is green is the property being such that grass is green. We argue that there is no reason to prefer Speaks's theory to analogous but competing theories that identify propositions with, say, 2-adic relations. This style of argument has recently been deployed by many, including Moore and King, against the view that propositions are n-tuples, and by Caplan and Tillman against King's view that propositions are facts (...)
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  34. added 2016-09-21
    David DeVidi (2004). Choice Principles and Constructive Logics. Philosophia Mathematica 12:222-243.
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  35. added 2016-09-15
    R. S. D. Thomas (forthcoming). Beauty Is Not All There Is to Aesthetics in Mathematics. Philosophia Mathematica:nkw019.
    Aesthetics in philosophy of mathematics is too narrowly construed. Beauty is not the only feature in mathematics that is arguably aesthetic. While not the highest aesthetic value, being interesting is a sine qua non for publishability. Of the many ways to be interesting, being explanatory has recently been discussed. The motivational power of what is interesting is important for both directing research and stimulating education. The scientific satisfaction of curiosity and the artistic desire for beautiful results are complementary but both (...)
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  36. added 2016-09-15
    Helen De Cruz (2016). Numerical Cognition and Mathematical Realism. Philosophers' Imprint 16 (16).
    Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...)
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  37. added 2016-09-09
    Lydia Patton (forthcoming). Russell’s Method of Analysis and the Axioms of Mathematics. In Sandra Lapointe Christopher Pincock (ed.), Innovations in the History of Analytical Philosophy. Palgrave-Macmillan
    In the early 1900s, Russell began to recognize that he, and many other mathematicians, had been using assertions like the Axiom of Choice implicitly, and without explicitly proving them. In working with the Axioms of Choice, Infinity, and Reducibility, and his and Whitehead’s Multiplicative Axiom, Russell came to take the position that some axioms are necessary to recovering certain results of mathematics, but may not be proven to be true absolutely. The essay traces historical roots of, and motivations for, Russell’s (...)
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  38. added 2016-09-08
    Annie Selden, John Selden & Kerry McKee (2010). Affect, Behavioural Schemas and the Proving Process. International Journal for Mathematical Education in Science and Technology 41 (2):199-215.
    In this largely theoretical article, we discuss the relation between a kind of affect, behavioural schemas and aspects of the proving process. We begin with affect as described in the mathematics education literature, but soon narrow our focus to a particular kind of affect – nonemotional cognitive feelings. We then mention the position of feelings in consciousness because that bears on the kind of data about feelings that students can be expected to be able to report. Next we introduce the (...)
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