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  1.  10
    Axiomatizing Changing Conceptions of the Geometric Continuum I: Euclid-Hilbert†.John T. Baldwin - 2018 - Philosophia Mathematica 26 (3):346-374.
    We give a general account of the goals of axiomatization, introducing a variant on Detlefsen’s notion of ‘complete descriptive axiomatization’. We describe how distinctions between the Greek and modern view of number, magnitude, and proportion impact the interpretation of Hilbert’s axiomatization of geometry. We argue, as did Hilbert, that Euclid’s propositions concerning polygons, area, and similar triangles are derivable from Hilbert’s first-order axioms. We argue that Hilbert’s axioms including continuity show much more than the geometrical propositions of Euclid’s theorems and (...)
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  2.  3
    William Lane Craig.*God and Abstract Objects – The Coherence of Theism : AseityWilliam Lane Craig. God Over All : Divine Aseity and the Challenge of Platonism.Simon Hewitt - 2018 - Philosophia Mathematica 26 (3):418-421.
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  3.  22
    A Role for Representation Theorems†.Emiliano Ippoliti - 2018 - Philosophia Mathematica 26 (3):396-412.
    I argue that the construction of representation theorems is a powerful tool for creating novel objects and theories in mathematics, as the construction of a new representation introduces new pieces of information in a very specific way that enables a solution for a problem and a proof of a new theorem. In more detail I show how the work behind the proof of a representation theorem transforms a mathematical problem in a way that makes it tractable and introduces information into (...)
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  4.  30
    What Are Structural Properties?†.Johannes Korbmacher & Georg Schiemer - 2018 - Philosophia Mathematica 26 (3):295-323.
    Informally, structural properties of mathematical objects are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. (...)
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  5.  11
    Wittgenstein and Gödel: An Attempt to Make ‘Wittgenstein’s Objection’ Reasonable†.Timm Lampert - 2018 - Philosophia Mathematica 26 (3):324-345.
    According to some scholars, such as Rodych and Steiner, Wittgenstein objects to Gödel’s undecidability proof of his formula $$G$$, arguing that given a proof of $$G$$, one could relinquish the meta-mathematical interpretation of $$G$$ instead of relinquishing the assumption that Principia Mathematica is correct. Most scholars agree that such an objection, be it Wittgenstein’s or not, rests on an inadequate understanding of Gödel’s proof. In this paper, I argue that there is a possible reading of such an objection that is, (...)
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  6.  13
    Øystein Linnebo*. Philosophy of Mathematics.Gregory Lavers - 2018 - Philosophia Mathematica 26 (3):413-417.
    Øystein Linnebo*. Philosophy of Mathematics. Princeton University Press, 2017. ISBN: 978-0-691-16140-2 ; 978-1-40088524-4. Pp. xviii + 203.
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  7.  13
    A Study of Mathematical Determination Through Bertrand’s Paradox.Davide Rizza - 2018 - Philosophia Mathematica 26 (3):375-395.
    Certain mathematical problems prove very hard to solve because some of their intuitive features have not been assimilated or cannot be assimilated by the available mathematical resources. This state of affairs triggers an interesting dynamic whereby the introduction of novel conceptual resources converts the intuitive features into further mathematical determinations in light of which a solution to the original problem is made accessible. I illustrate this phenomenon through a study of Bertrand’s paradox.
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  8.  45
    Existence, Mathematical Nominalism, and Meta-Ontology: An Objection to Azzouni on Criteria for Existence.Farbod Akhlaghi-Ghaffarokh - 2018 - Philosophia Mathematica 26 (2):251-265.
    Jody Azzouni argues that whilst it is indeterminate what the criteria for existence are, there is a criterion that has been collectively adopted to use ‘exist’ that we can employ to argue for positions in ontology. I raise and defend a novel objection to Azzouni: his view has the counterintuitive consequence that the facts regarding what exists can and will change when users of the word ‘exist’ change what criteria they associate with its usage. Considering three responses, I argue Azzouni (...)
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  9.  11
    Introduction to Special Issue: Aesthetics in Mathematics†.Angela Breitenbach & Davide Rizza - 2018 - Philosophia Mathematica 26 (2):153-160.
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  10.  10
    Robert Lorne Victor Hale FRSE May 4, 1945 – December 12, 2017.Roy T. Cook & Stewart Shapiro - 2018 - Philosophia Mathematica 26 (2):266-274.
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  11.  15
    The Because of Because Without Cause†.Daniele Molinini - 2018 - Philosophia Mathematica 26 (2):275-286.
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  12.  16
    The Beautiful Art of Mathematics†.Adam Rieger - 2018 - Philosophia Mathematica 26 (2):234-250.
    ABSTRACT Mathematicians frequently use aesthetic vocabulary and sometimes even describe themselves as engaged in producing art. Yet aestheticians, in so far as they have discussed this at all, have often downplayed the ascriptions of aesthetic properties as metaphorical. In this paper I argue firstly that the aesthetic talk should be taken literally, and secondly that it is at least reasonable to classify some mathematics as art.
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  13.  7
    Aesthetic Preferences in Mathematics: A Case Study†.Irina Starikova - 2018 - Philosophia Mathematica 26 (2):161-183.
    Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to their visualisations? Using an example from graph theory, this paper argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians’ sensitivity to aesthetics of the abstract.
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  14.  1
    Fitting Feelings and Elegant Proofs: On the Psychology of Aesthetic Evaluation in Mathematics†.Cain Todd - 2018 - Philosophia Mathematica 26 (2):211-233.
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  15.  14
    Gila Sher. Epistemic Friction: An Essay on Knowledge, Truth, and Logic.Julian C. Cole - 2018 - Philosophia Mathematica 26 (1):136-148.
    © The Authors [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.comGila Sher believes that our basic epistemic situation — that we aim to gain knowledge of a highly complex world using our severely limited, yet highly resourceful, cognitive capacities — demands that all epistemic projects be undertaken within two broad constraints: epistemic freedom and epistemic friction. The former permits us to employ our cognitive resourcefulness fully while undertaking epistemic projects, while the latter requires that (...)
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  16.  11
    Roi Wagner. Making and Breaking Mathematical Sense: Histories and Philosophies of Mathematical Practice.José Ferreirós - 2018 - Philosophia Mathematica 26 (1):131-136.
    © The Authors [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.comIs mathematics a reflection of some already-given realm? It would not matter whether we are talking about the empirical world in a Millian way, or the domain of a priori truths in Leibnizian or maybe Kantian style, or some world of analytical truths à la Carnap. Or perhaps — could mathematics be something more, or something less, than such a reflection? Might it be human, (...)
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  17.  21
    Labyrinth of Continua†.Patrick Reeder - 2018 - Philosophia Mathematica 26 (1):1-39.
    This is a survey of the concept of continuity. Efforts to explicate continuity have produced a plurality of philosophical conceptions of continuity that have provably distinct expressions within contemporary mathematics. I claim that there is a divide between the conceptions that treat the whole continuum as prior to its parts, and those conceptions that treat the parts of the continuum as prior to the whole. Along this divide, a tension emerges between those conceptions that favor philosophical idealizations of continuity and (...)
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  18.  36
    Francesca Biagioli. Space, Number, and Geometry From Helmholtz to Cassirer. [REVIEW]Thomas Mormann - 2018 - Philosophia Mathematica (2).
    © The Authors [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.comThe book Space, Number, and Geometry from Helmholtz to Cassirer is a reworked version of Francesca Biagioli’s PhD thesis. It aims ‘[to offer] a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century’. More precisely, Biagioli concentrates on how the Marburg school of neo-Kantianism dealt with what may be (...)
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