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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. Robert Adamson (1883). Kant's View of Mathematical Premisses and Reasonings. Mind 8 (31):421 - 425.
  2. Abraham Anderson (1994). Metaphysics and Methods in Descartes and Kant. 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.
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  3. Daniel E. Anderson (1979). A Note on the Syntheticity of Mathematical Propositions in Kant'sprolegomena. Southern Journal of Philosophy 17 (2):149-153.
  4. R. Lanier Anderson (2005). The Wolffian Paradigm and its Discontent: Kant's Containment Definition of Analyticity in Historical Context. 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 (...)
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  5. Anselmo Aportone (2011). Form of Intuition and Formal Intuition. A Priori and Sensibility in Kant's Philosophy. Rivista di Storia Della Filosofia 66 (3):431-470.
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  6. Jody Azzouni (2009). Why Do Informal Proofs Conform to Formal Norms? 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 (...)
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  7. Manuel Bächtold (2011). L'espace dans ses dimensions transcendantale et pragmatiste. 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, (...)
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  8. Edward G. Ballard (1961). Kant and Whitehead, and the Philosophy of Mathematics. Tulane Studies in Philosophy 10:3-29.
  9. Stephen F. Barker (1984). How Wrong Was Kant About Geometry? Topoi 3 (2):133-142.
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  10. Bruno Bauch (1907). Erfahrung und Geometrie in ihrem erkenntnistheoretischen Verhältnis. Kant-Studien 12 (1-3):213-235.
  11. Michael Beaney (2002). Kant and Analytic Methodology. British Journal for the History of Philosophy 10 (3):455 – 466.
  12. Frederick Beiser (2010). Mathematical Method in Kant, Schelling, and Hegel. In Michael Friedman, Mary Domski & Michael Dickson (eds.), Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science. Open Court.
  13. John Bell, The Philosophy of Mathematics.
    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: (...)
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  14. Francesco Bellucci (2013). Diagrammatic Reasoning: Some Notes on Charles S. Peirce and Friedrich A. Lange. 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 (...)
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  15. Hein Berg (2011). Kant's Conception of Proper Science. 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 (...)
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  16. Jean-Yves Beziau (2008). What is “Formal Logic”? 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 (...)
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  17. Francesca Biagioli (2013). Between Kantianism and Empiricism: Otto Hölder's Philosophy of Geometry. 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 (...)
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  18. Graham Bird (2013). Reply to Edward Kanterian. 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.
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  19. Graham Bird (ed.) (2006). A Companion to Kant. Blackwell Pub..
    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.
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  20. Henny Blomme (2012). The Completeness of Kant's Metaphysical Exposition of Space. 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 (...)
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  21. Luciano Boi (1996). Les géométries non euclidiennes, le problème philosophique de l'espace et la conception transcendantale; Helmholtz et Kant, les néo-kantiens, Einstein, Poincaré et Mach. Kant-Studien 87 (3):257-289.
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  22. Giovanni Boniolo & Silvio Valentini (2008). Vagueness, Kant and Topology: A Study of Formal Epistemology. 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.
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  23. Eva Brann (2006). Kant's Philosophical Use of Mathematics : Negative Magnitudes. In Stanley Rosen & Nalin Ranasinghe (eds.), Logos and Eros: Essays Honoring Stanley Rosen. St. Augustine's Press.
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  24. Henry Walter Brann (1974). Arithmetic and Theory of Combination in Kant's Philosophy. Philosophy and History 7 (2):150-152.
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  25. Jill Vance Buroker (1994). Book Review:Kant and the Exact Sciences Michael Friedman. [REVIEW] Philosophy of Science 61 (2):321-.
  26. Robert E. Butts (1981). Rules, Examples and Constructions Kant's Theory of Mathematics. Synthese 47 (2):257 - 288.
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  27. C. Callender & R. Weingard (2000). Topology Change and the Unity of Space. Studies in History and Philosophy of Science Part B 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 (...)
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  28. Craig Callender (2005). Answers in Search of a Question: 'Proofs' of the Tri-Dimensionality of Space. Studies in History and Philosophy of Science Part B 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 (...)
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  29. Paola Cantù, Bolzano Versus Kant: Mathematics as a Scientia Universalis. 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 (...)
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  30. Patricio Lepe Carrión (2009). La construcción esquemática en Kant, y la imaginación como facultad determinante a priori de la sensibilidad. A Parte Rei: Revista de Filosofía 61:3.
  31. Emily Carson (2013). Pure Intuition and Kant's Synthetic A Priori. In Stewart Duncan & Antonia LoLordo (eds.), Debates in Modern Philosophy: Essential Readings and Contemporary Responses. Routledge. 307.
  32. Emily Carson (2009). Hintikka on Kant's mathematical method. Revue Internationale de Philosophie 4 (250):435-449.
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  33. Emily Carson (2006). Locke and Kant on Mathematical Knowledge. In Emily Carson & Renate Huber (eds.), Intuition and the Axiomatic Method. Springer. 3--19.
  34. Emily Carson (2006). Review: Pierobon, Kant Et les Mathématiques: La Conception Kantienne des Mathématiques [Kant and Mathematics: The Kantian Conception of Mathematics]. [REVIEW] Philosophia Mathematica 14 (3):370-378.
  35. Emily Carson (2004). Metaphysics, Mathematics and the Distinction Between the Sensible and the Intelligible in Kant's Inaugural Dissertation. Journal of the History of Philosophy 42 (2):165-194.
    In this paper I argue that Kant's distinction in the Inaugural Dissertation between the sensible and the intelligible arises in part out of certain open questions left open by his comparison between mathematics and metaphysics in the Prize Essay. This distinction provides a philosophical justification for his distinction between the respective methods of mathematics and metaphysics and his claim that mathematics admits of a greater degree of certainty. More generally, this illustrates the importance of Kant's reflections on mathematics for the (...)
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  36. Emily Carson (1999). Kant on the Method of Mathematics. Journal of the History of Philosophy 37 (4):629-652.
  37. Emily Carson (1997). Kant on Intuition in Geometry. Canadian Journal of Philosophy 27 (4):489 - 512.
  38. Emily Carson & Renate Huber (eds.) (2006). Intuition and the Axiomatic Method. Springer.
    By way of these investigations, we hope to understand better the rationale behind Kant's theory of intuition, as well as to grasp many facets of the relations ...
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  39. Ernst Cassirer (1907). Kant und die moderne Mathematik. (Mit Bezug auf Bertrand Russells und Louis Couturats Werke über die Prinzipien der Mathematik.). Kant-Studien 12 (1-3):1-49.
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  40. Hector Neri Castañeda (1960). "7 + 5 = 12" as a Synthetic Proposition. Philosophy and Phenomenological Research 21 (2):141-158.
  41. Albert Casullo, Counterfactuals and Modal Knowledge.
    One infl uential argument in support of the existence of a priori knowledge is due to Kant, who claimed that necessity is a criterion of the a priori—that is, that all knowledge of necessary propositions is a priori. Th at claim, together with two others that Kant took to be evident—we know some mathematical propositions and such propositions are necessary—led directly to the conclusion that some knowledge is a priori. Kripke ( 1971 , 1980 ) challenged Kant’s central claim by (...)
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  42. Albert Casullo, Intuition, Thought Experiments, and the A Priori.
    There has been a significant shift in the discussion of a priori knowledge. The shift is due largely to the influence of Quine. The traditional debate focused on the epistemic status of mathematics and logic. Kant, for example, maintained that arithmetic and geometry provide clear examples of synthetic a priori knowledge and that principles of logic, such as the principle of contradiction, provide the basis for analytic a priori knowledge. Quine’s rejection of the analytic-synthetic distinction and his holistic empiricist account (...)
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  43. Stefania Centrone (2014). Richard Tieszen, After Gödel. Platonism and Rationalism in Mathematics and Logic. [REVIEW] Husserl Studies 30 (2):153-162.
    It is well known that Husserl, together with Plato and Leibniz, counted among Gödel’s favorite philosophers and was, in fact, an important source and reference point for the elaboration of Gödel’s own philosophical thought. Among the scholars who emphasized this connection we find, as Richard Tieszen reminds us, Gian-Carlo Rota, George Kreisel, Charles Parsons, Heinz Pagels and, especially, Hao Wang. Right at the beginning of After Gödel we read: “The logician who conducted and recorded the most extensive philosophical discussions with (...)
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  44. Leon Chernyak & David Kazhdan (1996). Kant and the Aesthetic-Expressive Vision of Mathematics. In. In Alfred I. Tauber (ed.), The Elusive Synthesis: Aesthetics and Science. Kluwer. 203--225.
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  45. Alberto Coffa (1991). The Semantic Tradition From Kant to Carnap: To the Vienna Station. Cambridge University Press.
    This major publication is a history of the semantic tradition in philosophy from the early nineteenth century through its incarnation in the work of the Vienna Circle, the group of logical positivists that emerged in the years 1925-1935 in Vienna who were characterised by a strong commitment to empiricism, a high regard for science, and a conviction that modern logic is the primary tool of analytic philosophy. In the first part of the book, Alberto Coffa traces the roots of logical (...)
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  46. Alberto Coffa (1982). Kant, Bolzano, and the Emergence of Logicism. Journal of Philosophy 79 (11):679-689.
  47. J. Alberto Coffa (1981). Russell and Kant. Synthese 46 (2):247 - 263.
  48. Daniel Cohnitz (2008). Daniel Ørsteds "Gedankenexperiment": eine Kantianische Fundierung der Infinitesimalrechnung? Ein Beitrag zur Begriffsgeschichte von "Gedankenexperiment" und zur Mathematikgeschichte des frühen 19. Jahrhunderts. Kant-Studien 99 (4):407-433.
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  49. Daniel Cohnitz (2008). Ørsteds „Gedankenexperiment“: eine Kantianische Fundierung der Infinitesimalrechnung? Ein Beitrag zur Begriffsgeschichte von ‚Gedankenexperiment' und zur Mathematikgeschichte des frühen 19. Jahrhunderts. Kant-Studien 99 (4):407-433.
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  50. Helen De Cruz (2007). An Enhanced Argument for Innate Elementary Geometric Knowledge and its Philosophical Implications. In Bart Van Kerkhove (ed.), New perspectives on mathematical practices. Essays in philosophy and history of mathematics. World Scientific.
    The idea that formal geometry derives from intuitive notions of space has appeared in many guises, most notably in Kant’s argument from geometry. Kant claimed that an a priori knowledge of spatial relationships both allows and constrains formal geometry: it serves as the actual source of our cognition of principles of geometry and as a basis for its further cultural development. The development of non-Euclidean geometries, however, seemed to definitely undermine the idea that there is some privileged relationship between our (...)
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