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  1.  30
    Justin Clarke-Doane* Morality and Mathematics.Michael Bevan & A. C. Paseau - 2020 - Philosophia Mathematica 28 (3):442-446.
    _Justin Clarke-Doane* * Morality and Mathematics. _ Oxford University Press, 2020. Pp. xx + 208. ISBN: 978-0-19-882366-7 ; 978-0-19-2556806.† †.
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  2.  57
    Structuralist Neologicism†.Francesca Boccuni & Jack Woods - 2020 - Philosophia Mathematica 28 (3):296-316.
    Neofregeanism and structuralism are among the most promising recent approaches to the philosophy of mathematics. Yet both have serious costs. We develop a view, structuralist neologicism, which retains the central advantages of each while avoiding their more serious costs. The key to our approach is using arbitrary reference to explicate how mathematical terms, introduced by abstraction principles, refer. Focusing on numerical terms, this allows us to treat abstraction principles as implicit definitions determining all properties of the numbers, achieving a key (...)
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  3.  12
    Structuralism and Mathematical Practice in Felix Klein’s Work on Non-Euclidean Geometry†.Biagioli Francesca - 2020 - Philosophia Mathematica 28 (3):360-384.
    It is well known that Felix Klein took a decisive step in investigating the invariants of transformation groups. However, less attention has been given to Klein’s considerations on the epistemological implications of his work on geometry. This paper proposes an interpretation of Klein’s view as a form of mathematical structuralism, according to which the study of mathematical structures provides the basis for a better understanding of how mathematical research and practice develop.
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  4.  11
    Tim Button and Sean Walsh* Philosophy and Model Theory.Brice Halimi - 2020 - Philosophia Mathematica 28 (3):404-415.
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  5.  28
    Luca Incurvati* Conceptions of Set and the Foundations of Mathematics.Burgess John - 2020 - Philosophia Mathematica 28 (3):395-403.
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  6.  10
    Gerhard Jäger* and Wilfried Sieg.** Feferman on Foundations: Logic, Mathematics, Philosophy.Reinhard Kahle - 2020 - Philosophia Mathematica 28 (3):421-425.
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  7.  11
    Matthias Wille.* ›Largely unknown‹ Gottlob Frege und der posthume Ruhm ›alles in den Wind geschrieben‹ Gottlob Frege wider den Zeitgeist.Ansten Klev - 2020 - Philosophia Mathematica 28 (3):426-430.
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  8.  26
    On Non-Eliminative Structuralism. Unlabeled Graphs as a Case Study, Part A†.Hannes Leitgeb - 2020 - Philosophia Mathematica 28 (3):317-346.
    This is Part A of an article that defends non-eliminative structuralism about mathematics by means of a concrete case study: a theory of unlabeled graphs. Part A summarizes the general attractions of non-eliminative structuralism. Afterwards, it motivates an understanding of unlabeled graphs as structures sui generis and develops a corresponding axiomatic theory of unlabeled graphs. As the theory demonstrates, graph theory can be developed consistently without eliminating unlabeled graphs in favour of sets; and the usual structuralist criterion of identity can (...)
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  9.  53
    An ‘I’ for an I, a Truth for a Truth†.Mary Leng - 2020 - Philosophia Mathematica 28 (3):347-359.
    Stewart Shapiro’s ante rem structuralism recognizes the structural or ‘algebraic’ aspects of mathematical practice while still offering a face-value semantics. Fictionalism, as a purely ‘algebraic’ approach, is held to be at a disadvantage, as compared with Shapiro’s structuralism, in not interpreting mathematics at face value. However, the face-value reading of mathematical singular terms has difficulty explaining how we can use such terms to pick out a unique referent in cases where the relevant mathematical structures admit non-trivial automorphisms. Shapiro offers a (...)
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  10.  17
    Erich Reck* and Georg Schiemer.** The Prehistory of Mathematical Structuralism.Jean-Pierre Marquis - 2020 - Philosophia Mathematica 28 (3):416-420.
    _Erich Reck* * and Georg Schiemer.** ** The Prehistory of Mathematical Structuralism. _Oxford University Press, 2020. Pp. 454. ISBN: 978-0-19-064122-1 ; 978-0-19-064123-8. doi: 10.1093/oso/9780190641221.001.0001.
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  11.  19
    Intuitive and Regressive Justifications†.Michael Potter - 2020 - Philosophia Mathematica 28 (3):385-394.
    In his recent book, Quine, New Foundations, and the Philosophy of Set Theory, Sean Morris attempts to rehabilitate Quine’s NF as a possible foundation for mathematics. I explain why he does not succeed.
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  12.  11
    Introduction to Special Issue: Foundations of Mathematical Structuralism.Georg Schiemer & John Wigglesworth - 2020 - Philosophia Mathematica 28 (3):291-295.
    Structuralism, the view that mathematics is the science of structures, can be characterized as a philosophical response to a general structural turn in modern mathematics. Structuralists aim to understand the ontological, epistemological, and semantical implications of this structural approach in mathematics. Theories of structuralism began to develop following the publication of Paul Benacerraf’s paper ‘What numbers could not be’ in 1965. These theories include non-eliminative approaches, formulated in a background ontology of sui generis structures, such as Stewart Shapiro’s ante rem (...)
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  13.  10
    Emily Rolfe* Great Circles: The Transits of Mathematics and Poetry.Jean Paul Van Bendegem & Bart Van Kerkhove - 2020 - Philosophia Mathematica 28 (3):431-441.
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  14.  35
    Intuitionism in the Philosophy of Mathematics: Introducing a Phenomenological Account.Philipp Berghofer - 2020 - Philosophia Mathematica 28 (2):204-235.
    ABSTRACT The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such (...)
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  15.  16
    Øystein Linnebo.*Thin Objects.Thomas Donaldson - 2020 - Philosophia Mathematica 28 (2):258-263.
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  16.  26
    Critical Plural Logic.Salvatore Florio & Øystein Linnebo - 2020 - Philosophia Mathematica 28 (2):172-203.
    What is the relation between some things and the set of these things? Mathematical practice does not provide a univocal answer. On the one hand, it relies on ordinary plural talk, which is implicitly committed to a traditional form of plural logic. On the other hand, mathematical practice favors a liberal view of definitions which entails that traditional plural logic must be restricted. We explore this predicament and develop a “critical” alternative to traditional plural logic.
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  17.  14
    Persi Diaconis and Brian Skyrms. Ten Great Ideas About Chance.Christian Hennig - 2020 - Philosophia Mathematica 28 (2):282-285.
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  18.  6
    Philip A. Ebert and Marcus Rossberg, Eds.*Essays on Frege’s Basic Laws of Arithmetic. [REVIEW]Gregory Landini - 2020 - Philosophia Mathematica 28 (2):264-276.
    EbertPhilip A and RossbergMarcus, eds.* * _ Essays on Frege’s Basic Laws of Arithmetic_. Oxford: Oxford University Press, 2019. Pp. xii + 673. ISBN: 978-0-19-871208-4 ; 978-0-19-102005-6, 978-0-19-178024-0. doi: 10.1093/oso/9780198712084.001.0001.
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  19.  14
    Reflections on Frege’s Theory of Real Numbers†.Peter Roeper - 2020 - Philosophia Mathematica 28 (2):236-257.
    ABSTRACT Although Frege’s theory of real numbers in Grundgesetze der Arithmetik, Vol. II, is incomplete, it is possible to provide a logicist justification for the approach he is taking and to construct a plausible completion of his account by an extrapolation which parallels his theory of cardinal numbers.
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  20.  14
    Geoffrey Hellman* and Stewart Shapiro.**Mathematical Structuralism. Cambridge Elements in the Philosophy of Mathematics, Penelope Rush and Stewart Shapiro, Eds.Andrea Sereni - 2020 - Philosophia Mathematica 28 (2):277-281.
    HellmanGeoffrey ** and ShapiroStewart. **** Mathematical Structuralism. Cambridge Elements in the Philosophy of Mathematics, RushPenelope and ShapiroStewart, eds. Cambridge University Press, 2019. Pp. iv + 94. ISBN 978-1-108-45643-2, 978-1-108-69728-6. doi: 10.1017/9781108582933.
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  21.  17
    Does Choice Really Imply Excluded Middle? Part I: Regimentation of the Goodman–Myhill Result, and Its Immediate Reception†.Neil Tennant - 2020 - Philosophia Mathematica 28 (2):139-171.
    ABSTRACT The one-page 1978 informal proof of Goodman and Myhill is regimented in a weak constructive set theory in free logic. The decidability of identities in general is derived; then, of sentences in general. Martin-Löf’s and Bell’s receptions of the latter result are discussed. Regimentation reveals the form of Choice used in deriving Excluded Middle. It also reveals an abstraction principle that the proof employs. It will be argued that the Goodman–Myhill result does not provide the constructive set theorist with (...)
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  22.  68
    A Counterfactual Approach to Explanation in Mathematics.Sam Baron, Mark Colyvan & David Ripley - 2020 - Philosophia Mathematica 28 (1):1-34.
    ABSTRACT Our goal in this paper is to extend counterfactual accounts of scientific explanation to mathematics. Our focus, in particular, is on intra-mathematical explanations: explanations of one mathematical fact in terms of another. We offer a basic counterfactual theory of intra-mathematical explanations, before modelling the explanatory structure of a test case using counterfactual machinery. We finish by considering the application of counterpossibles to mathematical explanation, and explore a second test case along these lines.
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  23.  73
    Elaine Landry,* Ed. Categories for the Working Philosopher. [REVIEW]Neil Barton - 2020 - Philosophia Mathematica 28 (1):95-108.
    LandryElaine, * ed. Categories for the Working Philosopher. Oxford University Press, 2017. ISBN 978-0-19-874899-1 ; 978-0-19-106582-8. Pp. xiv + 471.
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  24.  21
    The Benacerraf Problem as a Challenge for Ontic Structural Realism†.Majid Davoody Beni - 2020 - Philosophia Mathematica 28 (1):35-59.
    ABSTRACT Benacerraf has presented two problems for the philosophy of mathematics. These are the problem of identification and the problem of representation. This paper aims to reconstruct the latter problem and to unpack its undermining bearing on the version of Ontic Structural Realism that frames scientific representations in terms of abstract structures. I argue that the dichotomy between mathematical structures and physical ones cannot be used to address the Benacerraf problem but strengthens it. I conclude by arguing that versions of (...)
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  25.  24
    What Types Should Not Be.Bruno Bentzen - 2020 - Philosophia Mathematica 28 (1):60-76.
    In a series of papers Ladyman and Presnell raise an interesting challenge of providing a pre-mathematical justification for homotopy type theory. In response, they propose what they claim to be an informal semantics for homotopy type theory where types and terms are regarded as mathematical concepts. The aim of this paper is to raise some issues which need to be resolved for the successful development of their types-as-concepts interpretation.
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  26.  21
    John Stillwell.*A Concise History of Mathematics for Philosophers.Emily Carson - 2020 - Philosophia Mathematica 28 (1):128-131.
    StillwellJohn.* * _ A Concise History of Mathematics for Philosophers. _ Cambridge Elements in the Philosophy of Mathematics. Cambridge University Press, 2019. Pp. 69. ISBN: 978-1-108-45623-4, 978-1-108-61012-4. doi.org/10.1017/9781108610124.
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  27.  48
    Review of John Stillwell, Reverse Mathematics: Proofs From the Inside Out. [REVIEW]Benedict Eastaugh - 2020 - Philosophia Mathematica 28 (1):108-116.
    Review of John Stillwell, Reverse Mathematics: Proofs from the Inside Out. Princeton, NJ: Princeton University Press, 2018, pp. 200. ISBN 978-0-69-117717-5 (hbk), 978-0-69-119641-1 (pbk), 978-1-40-088903-7 (e-book).
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  28.  87
    Otávio Bueno* and Steven French.**Applying Mathematics: Immersion, Inference, Interpretation. [REVIEW]Anthony F. Peressini - 2020 - Philosophia Mathematica 28 (1):116-127.
    Otávio Bueno* * and Steven French.** ** Applying Mathematics: Immersion, Inference, Interpretation. Oxford University Press, 2018. ISBN: 978-0-19-881504-4 978-0-19-185286-2. doi:10.1093/oso/9780198815044. 001.0001. Pp. xvii + 257.
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  29.  19
    Leon Horsten*The Metaphysics and Mathematics of Arbitrary Objects. [REVIEW]Eric Snyder - 2020 - Philosophia Mathematica 28 (1):79-95.
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