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Summary Kant's philosophy of mathematics brings together many of the signature doctrines in his theoretical philosophy.  On the one hand, Kant famously distinguishes mathematics from logic, and famously claims, furthermore, that the former requires the use of intuition in order to arrive at its basic concepts and principles, and that distinctively mathematical cognition is synthetic rather than analytic.  On the other hand, Kant equally famously claims that the subject-matter of geometry is something that is ideal rather than real, due to the fact that this subject-matter consists in the form of sensible outer appearances, rather than something that pertains to things besides or outside of appearances.  Both claims have proved to be heavily influential in the shaping of subsequent debates in the philosophy of mathematics.
Key works Key discussions of mathematics are found at the beginning and the end of the Critique of Pure Reason, as well as in Part I of the Prolegomena.  There are also important remarks about the role of mathematics in other sciences at the outset of the Metaphysical Foundations of Natural Science
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  1. Kantova Filosofiia Matematiki Starye I Novye Spory.Lev Abrahamian - 1978 - Izd-Vo.
  2. Kant's View of Mathematical Premisses and Reasonings.Robert Adamson - 1883 - Mind 8 (31):421 - 425.
  3. Acerca de las parejas incongruentes y las figuras simétricas.Carlos Álvarez - 2003 - Critica 35 (104):31-68.
    Kant plantea el problema de las parejas incongruentes en 1768, posteriormente en 1770 y 1783. Este problema, relacionado con su concepción acerca de la naturaleza del espacio, se vincula también con su idea sobre la naturaleza del conocimiento geométrico. Mi objetivo en este texto es analizar las observaciones de Kant sobre este punto--tres de las cuales son, a nuestro juicio, de suma relevancia--a partir de la geometría sólida euclidiana, la que constituye precisamente el marco teórico en el cual él pretende (...)
  4. Metaphysics and Methods in Descartes and Kant.Abraham Anderson - 1994 - Philosophical Quarterly 44 (174).
    This essay is a review of Daniel Garber's "Descartes' Metaphysical Physics" (Chicago U P 1992) and Michael Friedman's "Kant and the Exact Sciences" (Harvard U P 1992). Garber's study of Descartes is scrupulous but his historicist assumptions result in a failure to grasp Descartes' originality or the unity and power of his thought. Friedman, by taking Kant's conception of science seriously, sheds great light on Kant's thought generally and implicitly raises important philosophical problems for the present day.
  5. A Note on the Syntheticity of Mathematical Propositions in Kant'sprolegomena.Daniel E. Anderson - 1979 - Southern Journal of Philosophy 17 (2):149-153.
  6. The Poverty of Conceptual Truth: Kant's Analytic/Synthetic Distinction and the Limits of Metaphysics.R. Lanier Anderson - 2015 - Oxford University Press.
    R. Lanier Anderson presents a new account of Kant's distinction between analytic and synthetic judgments, and provides it with a clear basis within traditional logic. He reconstructs compelling claims about the syntheticity of elementary mathematics, and re-animates Kant's arguments against traditional metaphysics in the Critique of Pure Reason.
  7. The Wolffian Paradigm and its Discontent: Kant's Containment Definition of Analyticity in Historical Context.R. Lanier Anderson - 2005 - Archiv für Geschichte der Philosophie 87 (1):22-74.
    I defend Kant’s definition of analyticity in terms of concept “containment”, which has engendered widespread scepticism. Kant deployed a clear, technical notion of containment based on ideas standard within traditional logic, notably genus/species hierarchies formed via logical division. Kant’s analytic/synthetic distinction thereby undermines the logico-metaphysical system of Christian Wolff, showing that the Wolffian paradigm lacks the expressive power even to represent essential knowledge, including elementary mathematics, and so cannot provide an adequate system of philosophy. The results clarify the extent to (...)
  8. Form of Intuition and Formal Intuition. A Priori and Sensibility in Kant's Philosophy.Anselmo Aportone - 2011 - Rivista di Storia Della Filosofia 66 (3):431-470.
  9. Why Do Informal Proofs Conform to Formal Norms?Jody Azzouni - 2009 - Foundations of Science 14 (1-2):9-26.
    Kant discovered a philosophical problem with mathematical proof. Despite being a priori , its methodology involves more than analytic truth. But what else is involved? This problem is widely taken to have been solved by Frege’s extension of logic beyond its restricted (and largely Aristotelian) form. Nevertheless, a successor problem remains: both traditional and contemporary (classical) mathematical proofs, although conforming to the norms of contemporary (classical) logic, never were, and still aren’t, executed by mathematicians in a way that transparently reveals (...)
  10. L'espace dans ses dimensions transcendantale et pragmatiste.Manuel Bächtold - 2011 - Kant-Studien 102 (2):145-167.
    This article examines the Kantian thesis of the a priori nature of our knowledge of space. Because it makes the representation of objects possible as external to us and all others, and consequently, as distinct and individualized, space (whatever its structure may be) claims the status as necessary condition and as apriori possibility of all knowledge. However, in the light of various physical, psychological and philosophical considerations, it seems that the particular structure allocated by Kant to space (i.e. uniqueness, infinity, (...)
  11. Kant and Whitehead, and the Philosophy of Mathematics.Edward G. Ballard - 1961 - Tulane Studies in Philosophy 10:3-29.
  12. How Wrong Was Kant About Geometry?Stephen F. Barker - 1984 - Topoi 3 (2):133-142.
  13. Erfahrung und Geometrie in ihrem erkenntnistheoretischen Verhältnis.Bruno Bauch - 1907 - Kant-Studien 12 (1-3):213-235.
  14. Kant and Analytic Methodology.Michael Beaney - 2002 - British Journal for the History of Philosophy 10 (3):455 – 466.
  15. Mathematical Method in Kant, Schelling, and Hegel.Frederick Beiser - 2010 - In Michael Friedman, Mary Domski & Michael Dickson (eds.), Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science. Open Court.
  16. The Philosophy of Mathematics.John Bell - manuscript
    THE CLOSE CONNECTION BETWEEN mathematics and philosophy has long been recognized by practitioners of both disciplines. The apparent timelessness of mathematical truth, the exactness and objective nature of its concepts, its applicability to the phenomena of the empirical world—explicating such facts presents philosophy with some of its subtlest problems. We shall discuss some of the attempts made by philosophers and mathematicians to explain the nature of mathematics. We begin with a brief presentation of the views of four major classical philosophers: (...)
  17. Paul Rusnock. Bolzano's Philosophy and the Emergence of Modern Mathematics. Studien Zur Österreichischen Philosophie [Studies in Austrian Philosophy], Vol. 30. Amsterdam & Atlanta: Editions Rodopi, 2000. Isbn 90-420-1501-2. Pp. 218. [REVIEW]John L. Bell - 2006 - Philosophia Mathematica 14 (3):362-364.
    Bernard Bolzano , one of the leading figures of the Bohemian Enlightenment, made important contributions both to mathematics and philosophy which were virtually unknown in his lifetime and are still largely unacknowledged today. As a mathematician, he was a pioneer in the clarification and rigorization of mathematical analysis; as a philosopher, he may be considered a forerunner of the analytic movement later to emerge with Frege and Russell.Rusnock's account of Bolzano's work is laid out in five chapters and two appendices. (...)
  18. Diagrammatic Reasoning: Some Notes on Charles S. Peirce and Friedrich A. Lange.Francesco Bellucci - 2013 - History and Philosophy of Logic 34 (4):293 - 305.
    According to the received view, Charles S. Peirce's theory of diagrammatic reasoning is derived from Kant's philosophy of mathematics. For Kant, only mathematics is constructive/synthetic, logic being instead discursive/analytic, while for Peirce, the entire domain of necessary reasoning, comprising mathematics and deductive logic, is diagrammatic, i.e. constructive in the Kantian sense. This shift was stimulated, as Peirce himself acknowledged, by the doctrines contained in Friedrich Albert Lange's Logische Studien (1877). The present paper reconstructs Peirce's reading of Lange's book, and illustrates (...)
  19. Pure and Applied Geometry in Kant.Marissa Bennett - manuscript
  20. Kant's Conception of Proper Science.Hein Berg - 2011 - Synthese 183 (1):7-26.
    Kant is well known for his restrictive conception of proper science. In the present paper I will try to explain why Kant adopted this conception. I will identify three core conditions which Kant thinks a proper science must satisfy: systematicity, objective grounding, and apodictic certainty. These conditions conform to conditions codified in the Classical Model of Science. Kant’s infamous claim that any proper natural science must be mathematical should be understood on the basis of these conditions. In order to substantiate (...)
  21. What is “Formal Logic”?Jean-Yves Beziau - 2008 - Proceedings of the Xxii World Congress of Philosophy 13:9-22.
    “Formal logic”, an expression created by Kant to characterize Aristotelian logic, has also been used as a name for modern logic, originated by Boole and Frege, which in many aspects differs radically from traditional logic. We shed light on this paradox by distinguishing in this paper five different meanings of the expression “formal logic”: (1) Formal reasoning according to the Aristotelian dichotomy of form and content, (2) Formal logic as a formal science by opposition to an empirical science, (3) Formal (...)
  22. What Does It Mean That “Space Can Be Transcendental Without the Axioms Being So”?Francesca Biagioli - 2014 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 45 (1):1-21.
    In 1870, Hermann von Helmholtz criticized the Kantian conception of geometrical axioms as a priori synthetic judgments grounded in spatial intuition. However, during his dispute with Albrecht Krause (Kant und Helmholtz über den Ursprung und die Bedeutung der Raumanschauung und der geometrischen Axiome. Lahr, Schauenburg, 1878), Helmholtz maintained that space can be transcendental without the axioms being so. In this paper, I will analyze Helmholtz’s claim in connection with his theory of measurement. Helmholtz uses a Kantian argument that can be (...)
  23. Between Kantianism and Empiricism: Otto Hölder's Philosophy of Geometry.Francesca Biagioli - 2013 - Philosophia Scientiæ 17 (17-1):71-92.
    La philosophie de la géométrie de Hölder, si l’on s’en tient à une lecture superficielle, est la part la plus problématique de son épistémologie. Il soutient que la géométrie est fondée sur l’expérience à la manière de Helmholtz, malgré les objections sérieuses de Poincaré. Néanmoins, je pense que la position de Hölder mérite d’être discutée pour deux motifs. Premièrement, ses implications méthodologiques furent importantes pour le développement de son épistémologie. Deuxièmement, Poincaré utilise l’opposition entre le kantisme et l’empirisme comme un (...)
  24. Reply to Edward Kanterian.Graham Bird - 2013 - Kantian Review 18 (2):289-300.
    The reply to Kanterian offers a rebuttal of his central criticisms. It reaffirms the difference between Kant's arguments in the Aesthetic and at B 148-9; it rejects the alleged error of logic in Fischer's (and my) arguments; and it rejects Kanterian's reading of passages in the Preface (A xx-xxii) and of the Amphiboly. Beyond these specific points Kanterian assumes that Kant's project in the first Critique cannot be understood as a and so begs the question at issue.
  25. A Companion to Kant.Graham Bird (ed.) - 2006 - Wiley-Blackwell.
    This _Companion_ provides an authoritative survey of the whole range of Kant’s work, giving readers an idea of its immense scope, its extraordinary achievement, and its continuing ability to generate philosophical interest. Written by an international cast of scholars Covers all the major works of the critical philosophy, as well as the pre-critical works Subjects covered range from mathematics and philosophy of science, through epistemology and metaphysics, to moral and political philosophy.
  26. Kantian Themes in Contemporary Philosophy: Graham Bird.Graham Bird - 1998 - Aristotelian Society Supplementary Volume 72 (1):131–152.
    [Michael Friedman] This paper considers the extent to which Kant's vision of a distinctively 'transcendental' task for philosophy is essentially tied to his views on the foundations of the mathematical and physical sciences. Contemporary philosophers with broadly Kantian sympathies have attempted to reinterpret his project so as to isolate a more general philosophical core not so closely tied to the details of now outmoded mathematical-physical theories (Euclidean geometry and Newtonian physics). I consider two such attempts, those of Strawson and McDowell, (...)
  27. The Completeness of Kant's Metaphysical Exposition of Space.Henny Blomme - 2012 - Kant-Studien 103 (2):139-162.
    In the first edition of his book on the completeness of Kant’s table of judgments, Klaus Reich shortly indicates that the B-version of the metaphysical exposition of space in the Critique of pure reason is structured following the inverse order of the table of categories. In this paper, I develop Reich’s claim and provide further evidence for it. My argumentation is as follows: Through analysis of our actually given representation of space as some kind of object (the formal intuition of (...)
  28. Les géométries non euclidiennes, le problème philosophique de l'espace et la conception transcendantale; Helmholtz et Kant, les néo-kantiens, Einstein, Poincaré et Mach.Luciano Boi - 1996 - Kant-Studien 87 (3):257-289.
  29. Vagueness, Kant and Topology: A Study of Formal Epistemology.Giovanni Boniolo & Silvio Valentini - 2008 - Journal of Philosophical Logic 37 (2):141-168.
    In this paper we propose an approach to vagueness characterised by two features. The first one is philosophical: we move along a Kantian path emphasizing the knowing subject’s conceptual apparatus. The second one is formal: to face vagueness, and our philosophical view on it, we propose to use topology and formal topology. We show that the Kantian and the topological features joined together allow us an atypical, but promising, way of considering vagueness.
  30. Mary Domski and Michael Dickson, Eds. , Discourse on a New Method. Reinvigorating the Marriage of History and Philosophy of Science . Reviewed By. [REVIEW]Giacomo Borbone - 2011 - Philosophy in Review 31 (4):264-266.
  31. From Kant to Hilbert, Volume 1: A Source Book in the Foundations of Mathematics.William Bragg Ewald - 2005 - Oxford University Press UK.
    Immanuel Kant's Critique of Pure Reason is widely taken to be the starting point of the modern period of mathematics while David Hilbert was the last great mainstream mathematician to pursue important nineteenth cnetury ideas. This two-volume work provides an overview of this important era of mathematical research through a carefully chosen selection of articles. They provide an insight into the foundations of each of the main branches of mathematics--algebra, geometry, number theory, analysis, logic and set theory--with narratives to show (...)
  32. Bolzano & Kant.Johannes L. Brandl, Marian David, Maria E. Reicher & Leopold Stubenberg (eds.) - 2012 - Rodopi.
    Inhaltsverzeichnis/Table of ContentsThemenschwerpunkt/Special Topic: Bolzano & KantGastherausgeber/Guest Editor: Sandra LapointeSandra Lapointe: IntroductionSandra Lapointe: Is Logic Formal? Bolzano, Kant and the Kantian LogiciansNicholas F. Stang: A Kantian Reply to Bolzano’s Critique of Kant’s Analytic-Synthetic DistinctionClinton Tolley: Bolzano and Kant on the Place of Subjectivity in a WissenschaftslehreTimothy Rosenkoetter: Kant and Bolzano on the Singularity of IntuitionsWaldemar Rohloff: From Ordinary Language to Definition in Kant and BolzanoWeitere Artikel/Further ArticlesChristian Damböck: Wilhelm Diltheys empirische Philosophie und der rezente Methodenstreit in der analytischen PhilosophieBernd Prien: (...)
  33. Kant's Philosophical Use of Mathematics : Negative Magnitudes.Eva Brann - 2006 - In Stanley Rosen & Nalin Ranasinghe (eds.), Logos and Eros: Essays Honoring Stanley Rosen. St. Augustine's Press.
  34. Arithmetic and Theory of Combination in Kant's Philosophy.Henry Walter Brann - 1974 - Philosophy and History 7 (2):150-152.
  35. Incongruent Counterparts and Modal Relationism.Carolyn Brighouse - 1999 - International Studies in the Philosophy of Science 13 (1):53 – 68.
    Kant's argument from incongruent counterparts for substantival space is examined; it is concluded that the argument has no force against a relationist. The argument does suggest that a relationist cannot give an account of enantiomorphism, incongruent counterparts and orientability. The prospects for a relationist account of these notions are assessed, and it is found that they are good provided the relationist is some kind of modal relationist. An illustration and interpretation of these modal commitments is given.
  36. Gottfried Martin., Arithmetic and Combinatorics: Kant and His Contemporaries.Gordon G. Brittan - 1989 - International Studies in Philosophy 21 (1):100-101.
  37. Kant's Theory of Mathematical and Philosophical Reasoning.C. D. Broad - 1941 - Proceedings of the Aristotelian Society 42:1 - 24.
  38. Book Review:Kant and the Exact Sciences Michael Friedman. [REVIEW]Jill Vance Buroker - 1994 - Philosophy of Science 61 (2):321-.
  39. Rules, Examples and Constructions Kant's Theory of Mathematics.Robert E. Butts - 1981 - Synthese 47 (2):257 - 288.
  40. Kant on the Acquisition of Geometrical Concepts.John J. Callanan - 2014 - Canadian Journal of Philosophy 44 (5-6):580-604.
    It is often maintained that one insight of Kant's Critical philosophy is its recognition of the need to distinguish accounts of knowledge acquisition from knowledge justification. In particular, it is claimed that Kant held that the detailing of a concept's acquisition conditions is insufficient to determine its legitimacy. I argue that this is not the case at least with regard to geometrical concepts. Considered in the light of his pre-Critical writings on the mathematical method, construction in the Critique can be (...)
  41. Mendelssohn and Kant on Mathematics and Metaphysics.John J. Callanan - 2014 - Kant Yearbook 6 (1):1-22.
  42. Topology Change and the Unity of Space.C. Callender & R. Weingard - 2000 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 31 (2):227-246.
    Must space be a unity? This question, which exercised Aristotle, Descartes and Kant, is a specific instance of a more general one; namely, can the topology of physical space change with time? In this paper we show how the discussion of the unity of space has been altered but survives in contemporary research in theoretical physics. With a pedagogical review of the role played by the Euler characteristic in the mathematics of relativistic spacetimes, we explain how classical general relativity (modulo (...)
  43. Answers in Search of a Question: 'Proofs' of the Tri-Dimensionality of Space.Craig Callender - 2005 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 36 (1):113-136.
    From Kant’s first published work to recent articles in the physics literature, philosophers and physicists have long sought an answer to the question, why does space have three dimensions. In this paper, I will flesh out Kant’s claim with a brief detour through Gauss’ law. I then describe Büchel’s version of the common argument that stable orbits are possible only if space is three-dimensional. After examining objections by Russell and van Fraassen, I develop three original criticisms of my own. These (...)
  44. Geometry and Measurement in Otto Hölder's Epistemology.Paola Cantù - 2013 - Philosophia Scientiæ 17 (17-1):131-164.
    L’article a pour but d’analyser la conception de la géométrie et de la mesure présentée dans Intuition et Raisonnement [Hölder 1900], « Les axiomes de la grandeur et la théorie de la mensuration » [Hölder 1901] et La Méthode mathématique [Hölder 1924]. L’article examine les relations entre a) la démarcation introduite par Hölder entre géométrie et arithmétique à partir de la notion de ‘concept donné’, b) sa position philosophique par rapport à l’apriorisme kantien et à l’empirisme et c) le choix (...)
  45. Bolzano Versus Kant: Mathematics as a Scientia Universalis.Paola Cantù - 2011 - Philosophical Papers Dedicated to Kevin Mulligan.
    The paper discusses some changes in Bolzano's definition of mathematics attested in several quotations from the Beyträge, Wissenschaftslehre and Grössenlehre: is mathematics a theory of forms or a theory of quantities? Several issues that are maintained throughout Bolzano's works are distinguished from others that were accepted in the Beyträge and abandoned in the Grössenlehre. Changes are interpreted as a consequence of the new logical theory of truth introduced in the Wissenschaftslehre, but also as a consequence of the overcome of Kant's (...)
  46. Philosophy and Writing: The Philosophical Book According to Kant.Mirella Capozzi - 2011 - Quaestio 11 (1):307-350.
    Moving from the conviction that philosophy differs from mathematics because the signs of philosophy are words, i.e. audible Sprachlaute, and given that the vagueness of natural language cannot be eliminated by adopting a characteristic writing on the model of algebra, Kant poses the problem of how to write a philosophical book with a necessarily only phonetic writing, and yet aspiring to a certainty comparable to that of mathematics. His solution consists in showing, by means of acroamatic proofs, that there are (...)
  47. Iseli, Rebecca. Kants Philosophie der Mathematik.Luigi Caranti - 2002 - Review of Metaphysics 56 (1):179-181.
  48. La construcción esquemática en Kant, y la imaginación como facultad determinante a priori de la sensibilidad.Patricio Lepe Carrión - 2009 - A Parte Rei 61:3.
  49. Arithmetic and Possible Experience.Emily Carson - manuscript
    This paper is part of a larger project about the relation between mathematics and transcendental philosophy that I think is the most interesting feature of Kant’s philosophy of mathematics. This general view is that in the course of arguing independently of mathematical considerations for conditions of experience, Kant also establishes conditions of the possibility of mathematics. My broad aim in this paper is to clarify the sense in which this is an accurate description of Kant’s view of the relation between (...)
  50. Pure Intuition and Kant's Synthetic A Priori.Emily Carson - 2013 - In Stewart Duncan & Antonia LoLordo (eds.), Debates in Modern Philosophy: Essential Readings and Contemporary Responses. Routledge. pp. 307.
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