Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Search

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

22 results

Page 1 of 2

Mathematical Intuitionism

Mathematical Intuitionism
  • Carl J. Posy
  • Published online:
    22 October 2020
    Print publication:
    12 November 2020
  • L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its objects must be humanly graspable. He initiated a program rebuilding modern mathematics according to that principle. This book introduces the reader to the mathematical core of intuitionism – from elementary number theory through to Brouwer's uniform continuity theorem – and to the two central topics of 'formalized intuitionism': formal intuitionistic logic, and formal systems for intuitionistic analysis. Building on that, the book proposes a systematic, philosophical foundation for intuitionism that weaves together doctrines about human grasp, mathematical objects and mathematical truth.

  • Introduction

  • Edited by Carl Posy, Hebrew University of Jerusalem, Ofra Rechter, Tel-Aviv University
  • Book:
    Kant's Philosophy of Mathematics
    Published online:
    24 April 2020
    Print publication:
    21 May 2020, pp 1-12

  • Summary

    Kant’s Philosophy of Mathematics: Modern Essays appeared in 1992.1 It was a pioneering effort designed to announce and promote a newly minted renaissance in the study of Kant’s philosophy of mathematics, and it set the agenda for a generation of scholars in this field. Indeed, since that anthology appeared, the study of Kant’s philosophy of mathematics has solidified into a well-established area of study within the philosophy of mathematics and within the study of the history of philosophy. The present two-volume collection aims to give a snapshot view of this thriving academic field. What has grown in the quarter-century separating these two collections is a fine-grained analysis of Kant’s philosophy of mathematics in the context of his overall philosophy as well as a historical sensitivity to its background. There has also been a spate of interest in taking these studies further, to see how the perception of Kant influenced nineteenth- and twentieth-century philosophy of mathematics. The present two-volume collection displays the recent developments in all of these directions.

  • 2 - Of Griffins and Horses

  • from Part I - Roots
  • Edited by Carl Posy, Hebrew University of Jerusalem, Ofra Rechter, Tel-Aviv University
  • Book:
    Kant's Philosophy of Mathematics
    Published online:
    24 April 2020
    Print publication:
    21 May 2020, pp 35-65

  • Summary

    Posy undertakes to give a precise characterization of the difference between Kant’s critical and precritical philosophy. The need to do this becomes vivid when we notice that many doctrines generally identified with the critical philosophy are already present in Kant’s precritical Inaugural Dissertation (1770). By exposing the Leibnizian roots of Kant’s preoccupation with the metaphysics–mathematics interface, Posy pinpoints the subtle but decisive shift in philosophical standpoint that separates the Critique from the Dissertation.

Kant's Philosophy of Mathematics
  • Volume 1, The Critical Philosophy and its Roots
  • Edited by Carl Posy, Ofra Rechter
  • Published online:
    24 April 2020
    Print publication:
    21 May 2020
  • The late 1960s saw the emergence of new philosophical interest in Kant's philosophy of mathematics, and since then this interest has developed into a major and dynamic field of study. In this state-of-the-art survey of contemporary scholarship on Kant's mathematical thinking, Carl Posy and Ofra Rechter gather leading authors who approach it from multiple perspectives, engaging with topics including geometry, arithmetic, logic, and metaphysics. Their essays offer fine-grained analysis of Kant's philosophy of mathematics in the context of his Critical philosophy, and also show sensitivity to its historical background. The volume will be important for readers seeking a comprehensive picture of the current scholarship about the development of Kant's philosophy of mathematics, its place in his overall philosophy, and the Kantian themes that influenced mathematics and its philosophy after Kant.

  • Intuition and Infinity: A Kantian Theme with Echoes in the Foundations of Mathematics

  • Carl Posy
  • Journal:
    Royal Institute of Philosophy Supplements / Volume 63 / October 2008
    Published online by Cambridge University Press:
    16 October 2008, pp. 165-193
    Print publication:
    October 2008

  • Kant says patently conflicting things about infinity and our grasp of it. Infinite space is a good case in point. In his solution to the First Antinomy, he denies that we can grasp the spatial universe as infinite, and therefore that this universe can be infinite; while in the Aesthetic he says just the opposite: ‘Space is represented as a given infinite magnitude’(A25/B39). And he rests these upon consistently opposite grounds. In the Antinomy we are told that we can have no intuitive grasp of an infinite space, and in the Aesthetic he says that our grasp of infinite space is precisely intuitive.

  • Immediacy and the Birth of Reference in Kant: The Case for Space

  • Edited by Gila Sher, University of California, San Diego, Richard Tieszen, San José State University, California
  • Book:
    Between Logic and Intuition
    Published online:
    02 December 2009
    Print publication:
    28 March 2000, pp 155-185

  • Summary

    All cognition, that is, all presentations consciously referred to an object, are either intuitions or concepts. Intuition is a singular presentation (repraesentatio singularis), the concept is a general (repraesentatio per notas communes) or reflected presentation (repraesentatio discursiva).

    (Jäsche Logic, §1)

    In whatever manner and by whatever means a cognition may relate to objects, intuition is that through which it is in immediate relation to them, and to which all thought as a means is directed. But intuition takes place only in so far as the object is given to us.

    (Critique of Pure Reason, A19/B33)

    Space is not a discursive or, as we say, general concept of relations of things in general, but a pure intuition.

    (A25/B39)

    Charles Parsons has taught us that the Kantian conception of intuition is a multi-faceted notion and that this complexity affects Kant's philosophy of mathematics. In this essay, I focus on these two lessons, but also broaden them a bit. Specifically, I have three goals:

    1. Parsons has taught us that the notion of immediacy – which he interprets phenomenologically – must be separately added to the traditional criterion of singularity that has been stressed by all commentators on Kant's definition of intuition. In this essay, however, I shall point out that Kant offers not two but three marks of human intuition: There is singularity; there is, as Parsons insists, immediacy; and there is also something I shall call “reference.” Kant calls it the “object givingness” of intuition. It is there quite clearly at A19.

    2. […]

  • Brouwer versus Hilbert: 1907–1928

  • J. Posy Carl
  • Journal:
    Science in Context / Volume 11 / Issue 2 / Summer 1998
    Published online by Cambridge University Press:
    26 September 2008, pp. 291-325
    Print publication:
    Summer 1998

  • L. E. J. Brouwer and David Hubert, two titans of twentieth-century mathematics, clashed dramatically in the 1920s. Though they were both Kantian constructivists, their notorious Grundlagenstreit centered on sharp differences about the foundations of mathematics: Brouwer was prepared to revise the content and methods of mathematics (his “Intuitionism” did just that radically), while Hilbert's Program was designed to preserve and constructively secure all of classical mathematics.

    Hilbert's interests and polemics at the time led to at least three misconstruals of intuitionism, misconstruals which last to our own time: Current literature often portrays popular views of intuitionism as the product of Brouwer's idiosyncratic subjectivism; modern logicians view intuitionism as simply applying a non-standard formal logic to mathematics; and contemporary philosophers see that logic as based upon a pure assertabilist theory of meaning. These pictures stem from the way Hilbert structured the controversy.

    Even though Brouwer's own work and behavior occasionally reinforce these pictures, they are nevertheless inaccurate accounts of his approach to mathematics. However, the framework provided by the Brouwer-Hilbert debate itself does not supply an adequate correction of these inaccuracies. For, even if we eliminate these mistakes within that framework, Brouwer's position would still appear fragmented and internally inconsistent. I propose a Kantian framework — not from Kant's philosophy of mathematics but from his general metaphysics — which does show the coherence and consistency of Brouwer's views. I also suggest that expanding the context of the controversy in this way will illuminate Hilbert's views as well and will even shed light upon Kant's philosophy.

Page 1 of 2