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  1.  13
    Human-Effective Computability†.Marianna Antonutti Marfori & Leon Horsten - 2019 - Philosophia Mathematica 27 (1):61-87.
    We analyse Kreisel’s notion of human-effective computability. Like Kreisel, we relate this notion to a concept of informal provability, but we disagree with Kreisel about the precise way in which this is best done. The resulting two different ways of analysing human-effective computability give rise to two different variants of Church’s thesis. These are both investigated by relating them to transfinite progressions of formal theories in the sense of Feferman.
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  2.  8
    Geoffrey Hellman* and Stewart Shapiro.**Varieties of Continua—From Regions to Points and Back.Richard T. W. Arthur - 2019 - Philosophia Mathematica 27 (1):148-152.
    HellmanGeoffrey* * and ShapiroStewart.** ** Varieties of Continua—From Regions to Points and Back. Oxford University Press, 2018. ISBN: 978-0-19-871274-9. Pp. x + 208.
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  3.  11
    Axiomatizing Changing Conceptions of the Geometric Continuum II: Archimedes-Descartes-Hilbert-Tarski†.John T. Baldwin - 2019 - Philosophia Mathematica 27 (1):33-60.
    In Part I of this paper we argued that the first-order systems HP5 and EG are modest complete descriptive axiomatization of most of Euclidean geometry. In this paper we discuss two further modest complete descriptive axiomatizations: Tarksi’s for Cartesian geometry and new systems for adding $$\pi$$. In contrast we find Hilbert’s full second-order system immodest for geometrical purposes but appropriate as a foundation for mathematical analysis.
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  4.  1
    Philosophy of Mathematical Practice — Motivations, Themes and Prospects.Jessica Carter - 2019 - Philosophia Mathematica 27 (1):1-32.
    ABSTRACT A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.
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  5.  12
    Paolo Mancosu.*Abstraction and Infinity.Roy T. Cook & Michael Calasso - 2019 - Philosophia Mathematica 27 (1):125-152.
    MancosuPaolo.* *ion and Infinity. Oxford University Press, 2016. ISBN: 978-0-19-872462-9. Pp. viii + 222.
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  6.  25
    Hartry Field. Science Without Numbers: A Defense of Nominalism 2nd Ed.Geoffrey Hellman & Mary Leng - 2019 - Philosophia Mathematica 27 (1):139-148.
    FieldHartry. Science Without Numbers: A Defense of Nominalism 2nd ed.Oxford University Press, 2016. ISBN 978-0-19-877792-2. Pp. vi + 56 + vi + 111.
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  7.  13
    Deflationary Nominalism and Puzzle Avoidance†.David Mark Kovacs - 2019 - Philosophia Mathematica 27 (1):88-104.
    In a series of works, Jody Azzouni has defended deflationary nominalism, the view that certain sentences quantifying over mathematical objects are literally true, although such objects do not exist. One alleged attraction of this view is that it avoids various philosophical puzzles about mathematical objects. I argue that this thought is misguided. I first develop an ontologically neutral counterpart of Field’s reliability challenge and argue that deflationary nominalism offers no distinctive answer to it. I then show how this reasoning generalizes (...)
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  8.  17
    Antireductionism and Ordinals.Beau Madison Mount - 2019 - Philosophia Mathematica 27 (1):105-124.
    I develop a novel argument against the claim that ordinals are sets. In contrast to Benacerraf’s antireductionist argument, I make no use of covert epistemic assumptions. Instead, my argument uses considerations of ontological dependence. I draw on the datum that sets depend immediately and asymmetrically on their elements and argue that this datum is incompatible with reductionism, given plausible assumptions about the dependence profile of ordinals. In addition, I show that a structurally similar argument can be made against the claim (...)
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