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Alexander Paseau
Oxford University
  1.  33
    Fitch's Argument and Typing Knowledge.Alexander Paseau - 2008 - Notre Dame Journal of Formal Logic 49 (2):153-176.
    Fitch's argument purports to show that if all truths are knowable then all truths are known. The argument exploits the fact that the knowledge predicate or operator is untyped and may thus apply to sentences containing itself. This article outlines a response to Fitch's argument based on the idea that knowledge is typed. The first part of the article outlines the philosophical motivation for the view, comparing it to the motivation behind typing truth. The second, formal part presents a logic (...)
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  2. Mathematical Knowledge.Mary Leng, Alexander Paseau & Michael Potter (eds.) - 2007 - Oxford University Press.
    What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field. Contents 1. (...)
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  3. Resemblance Theories of Properties.Alexander Paseau - 2012 - Philosophical Studies 157 (3):361-382.
    The paper aims to develop a resemblance theory of properties that technically improves on past versions. The theory is based on a comparative resemblance predicate. In combination with other resources, it solves the various technical problems besetting resemblance nominalism. The paper’s second main aim is to indicate that previously proposed resemblance theories that solve the technical problems, including the comparative theory, are nominalistically unacceptable and have controversial philosophical commitments.
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  4. Knowledge of Mathematics Without Proof.Alexander Paseau - 2015 - British Journal for the Philosophy of Science 66 (4):775-799.
    Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We also show that (...)
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  5. Defining Ultimate Ontological Basis and the Fundamental Layer.Alexander Paseau - 2010 - Philosophical Quarterly 60 (238):169-175.
    I explain why Ross Cameron's definition of ultimate ontological basis is incorrect, and propose a different definition in terms of ontological dependence, as well as a definition of reality's fundamental layer. These new definitions cover the conceptual possibility that self-dependent entities exist. They also apply to different conceptions of the relation of ontological dependence.
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  6.  57
    Why the Subtraction Argument Does Not Add Up.A. Paseau - 2002 - Analysis 62 (1):73-75.
    Gonzalo Rodriguez-Pereyra (1997) has refined an argument due to Thomas Baldwin (1996), which claims to prove nihilism, the thesis that there could have been no concrete objects, and which apparently does so without reliance on any heavy-duty metaphysics of modality. This note will show that on either reading of its key premiss, the subtraction argument Rodriguez-Pereyra proposes is invalid.
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  7. Naturalism in Mathematics and the Authority of Philosophy.Alexander Paseau - 2005 - British Journal for the Philosophy of Science 56 (2):377-396.
    Naturalism in the philosophy of mathematics is the view that philosophy cannot legitimately gainsay mathematics. I distinguish between reinterpretation and reconstruction naturalism: the former states that philosophy cannot legitimately sanction a reinterpretation of mathematics (i.e. an interpretation different from the standard one); the latter that philosophy cannot legitimately change standard mathematics (as opposed to its interpretation). I begin by showing that neither form of naturalism is self-refuting. I then focus on reinterpretation naturalism, which comes in two forms, and examine the (...)
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  8.  79
    Boolos on the Justification of Set Theory.Alexander Paseau - 2007 - Philosophia Mathematica 15 (1):30-53.
    George Boolos has argued that the iterative conception of set justifies most, but not all, the ZFC axioms, and that a second conception of set, the Frege-von Neumann conception (FN), justifies the remaining axioms. This article challenges Boolos's claim that FN does better than the iterative conception at justifying the axioms in question.
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  9.  84
    An Exact Measure of Paradox.A. C. Paseau - 2013 - Analysis 73 (1):17-26.
    We take seriously the idea that paradoxes come in quantifiable degree by offering an exact measure of paradox. We consider three factors relevant to the degree of paradox, which are a function of the degree of belief in each of the individual propositions in the paradox set and the degree of belief in the set as a whole. We illustrate the proposal with a particular measure, and conclude the discussion with some critical remarks.
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  10.  45
    What’s the Point of Complete Rigour?A. C. Paseau - 2016 - Mind 125 (497):177-207.
    Complete inferential rigour is achieved by breaking down arguments into steps that are as small as possible: inferential ‘atoms’. For example, a mathematical or philosophical argument may be made completely inferentially rigorous by decomposing its inferential steps into the type of step found in a natural deduction system. It is commonly thought that atomization, paradigmatically in mathematics but also more generally, is pro tanto epistemically valuable. The paper considers some plausible candidates for the epistemic value arising from atomization and finds (...)
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  11.  50
    Fairness and Aggregation.A. C. Paseau & Ben Saunders - 2015 - Utilitas 27 (4):460-469.
    Sometimes, two unfair distributions cancel out in aggregate. Paradoxically, two distributions each of which is fair in isolation may give rise to aggregate unfairness. When assessing the fairness of distributions, it therefore matters whether we assess transactions piecemeal or focus only on the overall result. This piece illustrates these difficulties for two leading theories of fairness before offering a formal proof that no non-trivial theory guarantees aggregativity. This is not intended as a criticism of any particular theory, but as a (...)
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  12.  33
    Did Frege Commit a Cardinal Sin?A. C. Paseau - 2015 - Analysis 75 (3):379-386.
    Frege’s _Basic Law V_ is inconsistent. The reason often given is that it posits the existence of an injection from the larger collection of first-order concepts to the smaller collection of objects. This article explains what is right and what is wrong with this diagnosis.
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  13.  92
    Review: Logical Pluralism. [REVIEW]A. Paseau - 2007 - Mind 116 (462):391-396.
  14.  65
    Mathematical Instrumentalism, Gödel’s Theorem, and Inductive Evidence.Alexander Paseau - 2011 - Studies in History and Philosophy of Science Part A 42 (1):140-149.
    Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical instrumentalism (...)
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  15. Science and Mathematics: The Scope and Limits of Mathematical Fictionalism. [REVIEW]Christopher Pincock, Alan Baker, Alexander Paseau & Mary Leng - 2012 - Metascience 21 (2):269-294.
    Science and mathematics: the scope and limits of mathematical fictionalism Content Type Journal Article Category Book Symposium Pages 1-26 DOI 10.1007/s11016-011-9640-3 Authors Christopher Pincock, University of Missouri, 438 Strickland Hall, Columbia, MO 65211-4160, USA Alan Baker, Department of Philosophy, Swarthmore College, Swarthmore, PA 19081, USA Alexander Paseau, Wadham College, Oxford, OX1 3PN UK Mary Leng, Department of Philosophy, University of York, Heslington, York, YO10 5DD UK Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.
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  16.  36
    The Overgeneration Argument(S): A Succinct Refutation.A. C. Paseau - 2014 - Analysis 74 (1):40-47.
    The overgeneration argument attempts to show that accepting second-order validity as a sound formal counterpart of logical truth has the unacceptable consequence that the Continuum Hypothesis is either a logical truth or a logical falsehood. The argument was presented and vigorously defended in John Etchemendy’s The Concept of Logical Consequence and it has many proponents to this day. Yet it is nothing but a seductive fallacy. I demonstrate this by considering five versions of the argument; as I show, each is (...)
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  17. David Papineau. Philosophical Devices: Proofs, Probabilities, Possibilities, and Sets. Oxford: Oxford University Press, 2012. ISBN 978-0-19965173-3. Pp. Xix + 224. [REVIEW]A. C. Paseau - 2013 - Philosophia Mathematica (1):nkt006.
  18.  31
    The Overgeneration Argument(S): A Succinct Refutation.A. C. Paseau - 2014 - Analysis 74 (1):ant097.
    The overgeneration argument attempts to show that accepting second-order validity as a sound formal counterpart of logical truth has the unacceptable consequence that the Continuum Hypothesis is either a logical truth or a logical falsehood. The argument was presented and vigorously defended in John Etchemendy’s The Concept of Logical Consequence and it has many proponents to this day. Yet it is nothing but a seductive fallacy. I demonstrate this by considering five versions of the argument; as I show, each is (...)
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  19. Proofs of the Compactness Theorem.Alexander Paseau - 2010 - History and Philosophy of Logic 31 (1):73-98.
    In this study, several proofs of the compactness theorem for propositional logic with countably many atomic sentences are compared. Thereby some steps are taken towards a systematic philosophical study of the compactness theorem. In addition, some related data and morals for the theory of mathematical explanation are presented.
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  20.  39
    The Subtraction Argument(S).Alexander Paseau - 2006 - Dialectica 60 (2):145–156.
    The subtraction argument aims to show that there is an empty world, in the sense of a possible world with no concrete objects. The argument has been endorsed by several philosophers. I show that there are currently two versions of the argument around, and that only one of them is valid. I then sketch the main problem for the valid version of the argument.
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  21.  35
    The Open-Endedness of the Set Concept and the Semantics of Set Theory.A. Paseau - 2003 - Synthese 135 (3):379 - 399.
    Some philosophers have argued that the open-endedness of the set concept has revisionary consequences for the semantics and logic of set theory. I consider (several variants of) an argument for this claim, premissed on the view that quantification in mathematics cannot outrun our conceptual abilities. The argument urges a non-standard semantics for set theory that allegedly sanctions a non-classical logic. I show that the views about quantification the argument relies on turn out to sanction a classical semantics and logic after (...)
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  22.  58
    Genuine Modal Realism and Completeness.Alexander Paseau - 2006 - Mind 115 (459):721-730.
    John Divers and Joseph Melia have argued that Lewis's modal realism is extensionally inadequate. This paper explains why their argument does not succeed.
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  23.  11
    Isomorphism Invariance and Overgeneration – Corrigendum.O. Griffiths & A. C. Paseau - 2017 - Bulletin of Symbolic Logic 23 (4):546.
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  24.  30
    Isomorphism Invariance and Overgeneration.Owen Griffiths & A. C. Paseau - 2016 - Bulletin of Symbolic Logic 22 (4):482-503.
    The isomorphism invariance criterion of logical nature has much to commend it. It can be philosophically motivated by the thought that logic is distinctively general or topic neutral. It is capable of precise set-theoretic formulation. And it delivers an extension of ‘logical constant’ which respects the intuitively clear cases. Despite its attractions, the criterion has recently come under attack. Critics such as Feferman, MacFarlane and Bonnay argue that the criterion overgenerates by incorrectly judging mathematical notions as logical. We consider five (...)
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  25.  73
    Naturalism in the Philosophy of Mathematics.Alexander Paseau - 2008 - In Stanford Encyclopedia of Philosophy.
    Contemporary philosophy’s three main naturalisms are methodological, ontological and epistemological. Methodological naturalism states that the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural. In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. Ontological and epistemological naturalism in the philosophy of mathematics are discussed more briefly in section (...)
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  26.  59
    Proving Induction.Alexander Paseau - 2011 - Australasian Journal of Logic 10:1-17.
    The hard problem of induction is to argue without begging the question that inductive inference, applied properly in the proper circumstances, is conducive to truth. A recent theorem seems to show that the hard problem has a deductive solution. The theorem, provable in zfc, states that a predictive function M exists with the following property: whatever world we live in, M correctly predicts the world’s present state given its previous states at all times apart from a well-ordered subset. On the (...)
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  27.  54
    How to Type: Reply to Halbach.Alexander Paseau - 2009 - Analysis 69 (2):280-286.
    In my paper , I noted that Fitch's argument, which purports to show that if all truths are knowable then all truths are known, can be blocked by typing knowledge. If there is not one knowledge predicate, ‘ K’, but infinitely many, ‘ K 1’, ‘ K 2’, … , then the type rules prevent application of the predicate ‘ K i’ to sentences containing ‘ K i’ such as ‘ p ∧¬ K i⌜ p⌝’. This provides a motivated response (...)
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  28.  60
    Motivating Reductionism About Sets.Alexander Paseau - 2008 - Australasian Journal of Philosophy 86 (2):295 – 307.
    The paper raises some difficulties for the typical motivations behind set reductionism, the view that sets are reducible to entities identified independently of set theory.
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  29.  1
    Proofs of the Compactness Theorem.Alexander Paseau - 2011 - History and Philosophy of Logic 32 (4):407-407.
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  30.  44
    A Measure of Inferential-Role Preservation.A. C. Paseau - forthcoming - Synthese:1-22.
    The point of formalisation is to model various aspects of natural language. Perhaps the main use to which formalisation is put is to model and explain inferential relations between different sentences. Judged solely by this objective, a formalisation is successful in modelling the inferential network of natural language sentences to the extent that it mirrors this network. There is surprisingly little literature on the criteria of good formalisation, and even less on the question of what it is for a formalisation (...)
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  31.  57
    Should the Logic of Set Theory Be Intuitionistic?Alexander Paseau - 2001 - Proceedings of the Aristotelian Society 101 (3):369–378.
    It is commonly assumed that classical logic is the embodiment of a realist ontology. In “Sets and Semantics”, however, Jonathan Lear challenged this assumption in the particular case of set theory, arguing that even if one is a set-theoretic Platonist, due attention to a special feature of set theory leads to the conclusion that the correct logic for it is intuitionistic. The feature of set theory Lear appeals to is the open-endedness of the concept of set. This article advances reasons (...)
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  32.  35
    JOHN P. BURGESS Rigor and Structure.A. C. Paseau - 2016 - British Journal for the Philosophy of Science 67 (4):1185-1187.
  33. Justifying Induction Mathematically: Strategies and Functions.Alexander Paseau - 2008 - Logique Et Analyse 51 (203):263.
    If the total state of the universe is encodable by a real number, Hardin and Taylor have proved that there is a solution to one version of the problem of induction, or at least a solution to a closely related epistemological problem. Is this philosophical application of the Hardin-Taylor result modest enough? I advance grounds for doubt.
     
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  34.  74
    Pure Second-Order Logic with Second-Order Identity.Alexander Paseau - 2010 - Notre Dame Journal of Formal Logic 51 (3):351-360.
    Pure second-order logic is second-order logic without functional or first-order variables. In "Pure Second-Order Logic," Denyer shows that pure second-order logic is compact and that its notion of logical truth is decidable. However, his argument does not extend to pure second-order logic with second-order identity. We give a more general argument, based on elimination of quantifiers, which shows that any formula of pure second-order logic with second-order identity is equivalent to a member of a circumscribed class of formulas. As a (...)
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  35.  59
    A Puzzle About Naturalism.Alexander Paseau - 2010 - Metaphilosophy 41 (5):642-648.
    Abstract: This article presents and solves a puzzle about methodological naturalism. Trumping naturalism is the thesis that we must accept p if science sanctions p, and biconditional naturalism the apparently stronger thesis that we must accept p if and only if science sanctions p. The puzzle is generated by an apparently cogent argument to the effect that trumping naturalism is equivalent to biconditional naturalism. It turns out that the argument for this equivalence is subtly question-begging. The article explains this and (...)
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  36.  14
    Philosophy of the Matrix.A. C. Paseau - 2017 - Philosophia Mathematica 25 (2):246-267.
    A mathematical matrix is usually defined as a two-dimensional array of scalars. And yet, as I explain, matrices are not in fact two-dimensional arrays. So are we to conclude that matrices do not exist? I show how to resolve the puzzle, for both contemporary and older mathematics. The solution generalises to the interpretation of all mathematical discourse. The paper as a whole attempts to reinforce mathematical structuralism by reflecting on how best to interpret mathematics.
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  37.  15
    The Laws of Belief: Ranking Theory & its Philosophical Applications, by Wolfgang Spohn.A. C. Paseau - 2017 - Mind 126 (501):273-278.
    The Laws of Belief: Ranking Theory & its Philosophical Applications, by SpohnWolfgang. New York: Oxford University Press, 2012. Pp. xv + 598.
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  38.  37
    Against the Judgment-Dependence of Mathematics and Logic.Alexander Paseau - 2012 - Erkenntnis 76 (1):23-40.
    Although the case for the judgment-dependence of many other domains has been pored over, surprisingly little attention has been paid to mathematics and logic. This paper presents two dilemmas for a judgment-dependent account of these areas. First, the extensionality-substantiality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the substantiality condition (roughly: non-vacuous specification). Second, the extensionality-extremality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the extremality condition (...)
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  39.  27
    James Robert Brown. Platonism, Naturalism, and Mathematical Knowledge. New York and London: Routledge, 2012. Isbn 978-0-415-87266-9. Pp. X + 182. [REVIEW]A. C. Paseau - 2012 - Philosophia Mathematica 20 (3):359-364.
  40.  12
    Erratum To: A Measure of Inferential-Role Preservation.A. C. Paseau - 2017 - Synthese 194 (4):1425-1425.
    Erratum to: Synthese DOI 10.1007/s11229-015-0705-5In line 3 of footnote 8 on page 4, ‘allow’ should be ‘disallow’.In line 8 of page 5, \ should be \ and \ should be \. Similarly for lines 1, 2, 3, 7, 8, 13 and 14 of page 6.The entry in row 20 column 6 of the table on page 5 should be 1 rather than 0.The entry \ in row 30 column 5 of the table on page 5 should be \.In line 27 (...)
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  41.  24
    What the Foundationalist Filter Kept Out.Alexander Paseau - 2005 - Studies in History and Philosophy of Science Part A 36 (1):191-201.
    From title to back cover, a polemic runs through David Corfield's "Towards a Philosophy of Real Mathematics". Corfield repeatedly complains that philosophers of mathematics have ignored the interesting and important mathematical developments of the past seventy years, ‘filtering’ the details of mathematical practice out of philosophical discussion. His aim is to remedy the discipline’s long-sightedness and, by precept and example, to redirect philosophical attention towards current developments in mathematics. This review discusses some strands of Corfield’s philosophy of real mathematics and (...)
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  42.  24
    On an Application of Categoricity.Alexander Paseau - 2005 - Proceedings of the Aristotelian Society 105 (3):411–415.
    James Walmsley in “Categoricity and Indefinite Extensibility” argues that a realist about some branch of mathematics X (e.g. arithmetic) apparently cannot use the categoricity of an axiomatisation of X to justify her belief that every sentence of the language of X has a truth-value. My note corrects Walmsley’s formulation of his claim, and shows that his argument for it hinges on the implausible idea that grasping that there is some model of the axioms amounts to grasping that there is a (...)
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  43.  7
    The Open-Endedness of the Set Concept and the Semantics of Set Theory.A. Paseau - 2003 - Synthese 135 (3):379-399.
    Some philosophers have argued that the open-endedness of the set concept has revisionary consequences for the semantics and logic of set theory. I consider an argument for this claim, premissed on the view that quantification in mathematics cannot outrun our conceptual abilities. The argument urges a non-standard semantics for set theory that allegedly sanctions a non-classical logic. I show that the views about quantification the argument relies on turn out to sanction a classical semantics and logic after all. More generally, (...)
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  44.  3
    The Subtraction Argument.Alexander Paseau - 2006 - Dialectica 60 (2):145-156.
    The subtraction argument aims to show that there is an empty world, in the sense of a possible world with no concrete objects. The argument has been endorsed by several philosophers. I show that there are currently two versions of the argument around, and that only one of them is valid. I then sketch the main problem for the valid version of the argument.
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  45.  1
    Should the Logic of Set Theory Be Intuitionistic?Alexander Paseau - 2001 - Proceedings of the Aristotelian Society 101 (3):369-378.
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  46. Mathematical Knowledge.Mary Leng, Alexander Paseau & Michael Potter (eds.) - 2007 - Oxford University Press.
    What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions.
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  47. David Papineau. Philosophical Devices: Proofs, Probabilities, Possibilities, and Sets. Oxford: Oxford University Press, 2012. ISBN 978-0-19965173-3. Pp. Xix &Plus; 224: Critical Studies/Book Reviews. [REVIEW]A. Paseau - 2014 - Philosophia Mathematica 22 (1):121-123.
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  48. David Papineau. Philosophical Devices: Proofs, Probabilities, Possibilities, and Sets. Oxford: Oxford University Press, 2012. Isbn 978-0-19965173-3. Pp. XIX + 224. [REVIEW]A. C. Paseau - 2014 - Philosophia Mathematica 22 (1):121-123.
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  49. On an Application of Categoricity.Alexander Paseau - 2005 - Proceedings of the Aristotelian Society 105 (1):395-399.
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  50. Philosophy of Mathematics.A. C. Paseau (ed.) - 2017 - Routledge.
    Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras’ Theorem or Fermat’s Last Theorem? The metaphysical question of what numbers are and the (...)
     
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