Philosophia Mathematica

ISSN: 0031-8019

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  1.  22
    Breaking the Tie: Benacerraf’s Identification Argument Revisited.Arnon Avron & Balthasar Grabmayr - 2023 - Philosophia Mathematica 31 (1):81-103.
    Most philosophers take Benacerraf’s argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf’s argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions of (...)
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  2.  4
    Jens Erik Fenstad.*Structures and Algorithms: Mathematics and the Nature of Knowledge.Julian C. Cole - 2023 - Philosophia Mathematica 31 (1):125-131.
    This book collects eight essays — written over multiple decades, for a general audience — that address Fenstad’s thoughts on the topics of what there is and how.
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  3.  19
    Joan Weiner.*Taking Frege at His Word.Roy T. Cook - 2023 - Philosophia Mathematica 31 (1):120-124.
    In Taking Frege at his Word, Joan Weiner proposes a wide-ranging re-thinking of Frege’s philosophical and mathematical projects. She begins by outlining what sh.
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  4.  14
    Jan von Plato.* Can Mathematics be Proved Consistent?John W. Dawson - 2023 - Philosophia Mathematica 31 (1):104-111.
    The papers of Kurt Gödel were donated to the Institute for Advanced Study by his widow Adele shortly after his death in 1978. They were catalogued by the review.
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  5.  13
    Negation in Negationless Intuitionistic Mathematics.Thomas Macaulay Ferguson - 2023 - Philosophia Mathematica 31 (1):29-55.
    The mathematician G.F.C. Griss is known for his program of negationless intuitionistic mathematics. Although Griss’s rejection of negation is regarded as characteristic of his philosophy, this is a consequence of an executability requirement that mental constructions presuppose agents’ executing corresponding mental activity. Restoring Griss’s executability requirement to a central role permits a more subtle characterization of the rejection of negation, according to which D. Nelson’s strong constructible negation is compatible with Griss’s principles. This exposes a ‘holographic’ theory of negation in (...)
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  6.  2
    Frédéric Patras.*The Essence of Numbers.Bonnie Gold - 2023 - Philosophia Mathematica 31 (1):132-142.
    This book approaches the issue of the essence of numbers from a very broad perspective: historical, philosophical, and mathematical. The author is a practicing.
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  7.  6
    Maria Hämeen-Anttila* and Jan von Plato,** eds, Kurt Gödel: The Princeton Lectures on Intuitionism.Ulrich Kohlenbach - 2023 - Philosophia Mathematica 31 (1):112-119.
    This book publishes for the first time notes from two notebooks of Gödel which formed the basis of a course on intuitionism Gödel delivered at Princeton in the.
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  8.  15
    Mathematics: Method Without Metaphysics.Elaine Landry - 2023 - Philosophia Mathematica 31 (1):56-80.
    I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the purpose of solving mathematical problems. I then extend this view to modern mathematics showing that when we shift our focus from the method of philosophy to the method of mathematics, we see that an as-if methodological interpretation of mathematical structuralism can be used to provide an account of (...)
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  9.  14
    Mathematical Progress — On Maddy and Beyond.Simon Weisgerber - 2023 - Philosophia Mathematica 31 (1):1-28.
    A key question of the ‘maverick’ tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions ‘of some mathematical importance’ as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry, a few issues with her proposal are identified. Taking these issues into consideration, an alternative account of ‘mathematical importance’, broadly (...)
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