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  1. Kant and non-euclidean geometry.Amit Hagar - 2008 - Kant Studien 99 (1):80-98.
    It is occasionally claimed that the important work of philosophers, physicists, and mathematicians in the nineteenth and in the early twentieth centuries made Kant’s critical philosophy of geometry look somewhat unattractive. Indeed, from the wider perspective of the discovery of non-Euclidean geometries, the replacement of Newtonian physics with Einstein’s theories of relativity, and the rise of quantificational logic, Kant’s philosophy seems “quaint at best and silly at worst”.1 While there is no doubt that Kant’s transcendental project involves his own conceptions (...)
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  • The point of Kant's axioms of intuition.Daniel Sutherland - 2005 - Pacific Philosophical Quarterly 86 (1):135–159.
    Kant's Critique of Pure Reason makes important claims about space, time and mathematics in both the Transcendental Aesthetic and the Axioms of Intuition, claims that appear to overlap in some ways and contradict in others. Various interpretations have been offered to resolve these tensions; I argue for an interpretation that accords the Axioms of Intuition a more important role in explaining mathematical cognition than it is usually given. Appreciation for this larger role reveals that magnitudes are central to Kant's philosophy (...)
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  • It Adds Up After All: Kant’s Philosophy of Arithmetic in Light of the Traditional Logic.R. Lanier Anderson - 2004 - Philosophy and Phenomenological Research 69 (3):501–540.
    Officially, for Kant, judgments are analytic iff the predicate is "contained in" the subject. I defend the containment definition against the common charge of obscurity, and argue that arithmetic cannot be analytic, in the resulting sense. My account deploys two traditional logical notions: logical division and concept hierarchies. Division separates a genus concept into exclusive, exhaustive species. Repeated divisions generate a hierarchy, in which lower species are derived from their genus, by adding differentia(e). Hierarchies afford a straightforward sense of containment: (...)
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  • Kant’s Philosophy of Mathematics and the Greek Mathematical Tradition.Daniel Sutherland - 2004 - Philosophical Review 113 (2):157-201.
    The aggregate EIRP of an N-element antenna array is proportional to N 2. This observation illustrates an effective approach for providing deep space networks with very powerful uplinks. The increased aggregate EIRP can be employed in a number of ways, including improved emergency communications, reaching farther into deep space, increased uplink data rates, and the flexibility of simultaneously providing more than one uplink beam with the array. Furthermore, potential for cost savings also exists since the array can be formed using (...)
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  • Not Quite Yet a Hazy Limbo of Mystery: Intuition in Russell’s An Essay on the Foundations of Geometry.Tyke Nunez - forthcoming - Mind.
    I argue that in Bertrand Russell’s An Essay on the Foundations of Geometry (1897), his forms of externality serve the same fundamental role in grounding the possibility of geometry that Immanuel Kant’s forms of intuition serve in grounding geometry in his critical philosophy. Specifically, both provide knowledge of bare numerical difference, where we have no concept of this difference. Because geometry deals with such conceptually homogeneous magnitudes and their composition on both accounts, forms of intuition or externality (respectively) are at (...)
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  • Algunas apreciaciones acerca del concepto crítico de demostración.Luciana Martínez - 2022 - Logos. Anales Del Seminario de Metafísica [Universidad Complutense de Madrid, España] 55 (1):109-124.
    En este artículo se examina la noción kantiana de las demostraciones matemáticas. Esta noción se encuentra desarrollada en el apartado titulado “Disciplina de la razón pura en su uso dogmático” de la _Crítica de la razón pura. _En este texto, Kant explica por qué los procedimientos exitosos en el conocimiento matemático resultan impracticables en metafísica. En primer lugar se estudian dos pasajes en los que el filósofo describe dos demostraciones: la demostración de la congruencia de los ángulos de la base (...)
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  • Manifold, Intuition, and Synthesis in Kant and Husserl.Burt C. Hopkins - 2013 - History of Philosophy & Logical Analysis 16 (1):264-307.
    The problem of ‘collective unity’ in the transcendental philosophies of Kant and Husserl is investigated on the basis of number’s exemplary ‘collective unity’. To this end, the investigation reconstructs the historical context of the conceptuality of the mathematics that informs Kant’s and Husserl’s accounts of manifold, intuition, and synthesis. On the basis of this reconstruction, the argument is advanced that the unity of number – not the unity of the ‘concept’ of number – is presupposed by each transcendental philosopher in (...)
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  • After Non-Euclidean Geometry: Intuition, Truth and the Autonomy of Mathematics.Janet Folina - 2018 - Journal for the History of Analytical Philosophy 6 (3).
    The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses to (...)
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  • Kant's Conception of Number.Daniel Sutherland - 2017 - Philosophical Review Current Issue 126 (2):147-190.
    Despite the importance of Kant's claims about mathematical cognition for his philosophy as a whole and for subsequent philosophy of mathematics, there is still no consensus on his philosophy of arithmetic, and in particular the role he assigns intuition in it. This inquiry sets aside the role of intuition for the nonce to investigate Kant's conception of natural number. Although Kant himself doesn't distinguish between a cardinal and an ordinal conception of number, some of the properties Kant attributes to number (...)
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  • Reflections On Kant’s Concept Of Space.Lisa Shabel - 2003 - Studies in History and Philosophy of Science Part A 34 (1):45-57.
    In this paper, I investigate an important aspect of Kant’s theory of pure sensible intuition. I argue that, according to Kant, a pure concept of space warrants and constrains intuitions of finite regions of space. That is, an a priori conceptual representation of space provides a governing principle for all spatial construction, which is necessary for mathematical demonstration as Kant understood it.Author Keywords: Kant; Space; Pure sensible intuition; Philosophy of mathematics.
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  • The Role of Magnitude in Kant's Critical Philosophy.Daniel Sutherland - 2004 - Canadian Journal of Philosophy 34 (3):411-441.
    In theCritique of Pure Reason,Kant argues for two principles that concern magnitudes. The first is the principle that ‘All intuitions are extensive magnitudes,’ which appears in the Axioms of Intuition (B202); the second is the principle that ‘In all appearances the real, which is an object of sensation, has an intensive magnitude, that is, a degree,’ which appears in the Anticipations of Perception (B207). A circle drawn in geometry and the space occupied by an object such as a book are (...)
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  • Kant's A Priori Intuition of Space Independent of Postulates.Edgar J. Valdez - 2012 - Kantian Review 17 (1):135-160.
    Defences of Kant's foundations of geometry fall short if they are unable to equally ground Euclidean and non-Euclidean geometries. Thus, Kant's account must be separated from geometrical postulates. I argue that characterizing space as the form of outer intuition must be independent of postulates. Geometrical postulates are then expressions of particular spatializing activities made possible by the a priori intuition of space. While Amit Hagar contends that this is to speak of noumena, I argue that a Kantian account of space (...)
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  • Beauty in Proofs: Kant on Aesthetics in Mathematics.Angela Breitenbach - 2013 - European Journal of Philosophy 23 (4):955-977.
    It is a common thought that mathematics can be not only true but also beautiful, and many of the greatest mathematicians have attached central importance to the aesthetic merit of their theorems, proofs and theories. But how, exactly, should we conceive of the character of beauty in mathematics? In this paper I suggest that Kant's philosophy provides the resources for a compelling answer to this question. Focusing on §62 of the ‘Critique of Aesthetic Judgment’, I argue against the common view (...)
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  • Kant on analogy.John J. Callanan - 2008 - British Journal for the History of Philosophy 16 (4):747 – 772.
    The role of analogy appears in surprisingly different areas of the first Critique. On the one hand, Kant considered the concept to have a specific enough meaning to entitle the principle concerned with causation an analogy; on the other hand we can find Kant referring to analogy in various parts of the Transcendental Dialectic in a seemingly different manner. Whereas in the Transcendental Analytic, Kant takes some time to provide a detailed (if not clear) account of the meaning of the (...)
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  • Kant’s Theory of Arithmetic: A Constructive Approach?Kristina Engelhard & Peter Mittelstaedt - 2008 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 39 (2):245-271.
    Kant's theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant's theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant's theory of arithmetic can (...)
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  • Spatial representation, magnitude and the two stems of cognition.Thomas Land - 2014 - Canadian Journal of Philosophy 44 (5-6):524-550.
    The aim of this paper is to show that attention to Kant's philosophy of mathematics sheds light on the doctrine that there are two stems of the cognitive capacity, which are distinct, but equally necessary for cognition. Specifically, I argue for the following four claims: The distinctive structure of outer sensible intuitions must be understood in terms of the concept of magnitude. The act of sensibly representing a magnitude involves a special act of spontaneity Kant ascribes to a capacity he (...)
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  • Kant and real numbers.Mark van Atten - unknown
    Kant held that under the concept of √2 falls a geometrical magnitude, but not a number. In particular, he explicitly distinguished this root from potentially infinite converging sequences of rationals. Like Kant, Brouwer based his foundations of mathematics on the a priori intuition of time, but unlike Kant, Brouwer did identify this root with a potentially infinite sequence. In this paper I discuss the systematical reasons why in Kant's philosophy this identification is impossible.
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  • Kant on geometry and spatial intuition.Michael Friedman - 2012 - Synthese 186 (1):231-255.
    I use recent work on Kant and diagrammatic reasoning to develop a reconsideration of central aspects of Kant’s philosophy of geometry and its relation to spatial intuition. In particular, I reconsider in this light the relations between geometrical concepts and their schemata, and the relationship between pure and empirical intuition. I argue that diagrammatic interpretations of Kant’s theory of geometrical intuition can, at best, capture only part of what Kant’s conception involves and that, for example, they cannot explain why Kant (...)
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  • Kant's Philosophy of Geometry--On the Road to a Final Assessment.L. Kvasz - 2011 - Philosophia Mathematica 19 (2):139-166.
    The paper attempts to summarize the debate on Kant’s philosophy of geometry and to offer a restricted area of mathematical practice for which Kant’s philosophy would be a reasonable account. Geometrical theories can be characterized using Wittgenstein’s notion of pictorial form . Kant’s philosophy of geometry can be interpreted as a reconstruction of geometry based on one of these forms — the projective form . If this is correct, Kant’s philosophy is a reasonable reconstruction of such theories as projective geometry; (...)
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  • Lisa A. Shabel. Mathematics in Kant's critical philosophy: Reflections on mathematical practice. Studies in philosophy outstanding dissertations, Robert Nozick, ed. new York & London: Routledge, 2003. ISBN 0-415-93955-0. Pp. 178 (cloth). [REVIEW]René Jagnow - 2007 - Philosophia Mathematica 15 (3):366-386.
    In this interesting and engaging book, Shabel offers an interpretation of Kant's philosophy of mathematics as expressed in his critical writings. Shabel's analysis is based on the insight that Kant's philosophical standpoint on mathematics cannot be understood without an investigation into his perception of mathematical practice in the seventeenth and eighteenth centuries. She aims to illuminate Kant's theory of the construction of concepts in pure intuition—the basis for his conclusion that mathematical knowledge is synthetic a priori. She does this through (...)
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  • Reading Kant’s doctrine of schematism algebraically.Farhad Alavi - 2020 - Philosophical Forum 51 (3):315-329.
    Kant’s investigations into so‐called a priori judgments of pure mathematics in the Critique of Pure Reason (KrV) are mainly confined to geometry and arithmetic both of which are grounded on our pure forms of intuition, space, and time. Nevertheless, as regards notions such as irrational numbers and continuous magnitudes, such a restricted account is crucially problematic. I argue that algebra can play a transcendental role with respect to the two pure intuitive sciences, arithmetic and geometry, as the condition of their (...)
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  • Síntese e formação de conceitos empíricos na crítica da razão pura.Elliot Santovich Scaramal - 2013 - XV Colóquio Kant da Unicamp: Intuições Sem Conceitos São Cegas (Caderno de Resumos).
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  • Arbitrary combination and the use of signs in mathematics: Kant’s 1763 Prize Essay and its Wolffian background.Katherine Dunlop - 2014 - Canadian Journal of Philosophy 44 (5-6):658-685.
    In his 1763 Prize Essay, Kant is thought to endorse a version of formalism on which mathematical concepts need not apply to extramental objects. Against this reading, I argue that the Prize Essay has sufficient resources to explain how the objective reference of mathematical concepts is secured. This account of mathematical concepts’ objective reference employs material from Wolffian philosophy. On my reading, Kant's 1763 view still falls short of his Critical view in that it does not explain the universal, unconditional (...)
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  • Mathematical method and Newtonian science in the philosophy of Christian Wolff.Katherine Dunlop - 2013 - Studies in History and Philosophy of Science Part A 44 (3):457-469.
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  • Epistemology Versus Ontology: Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf.Peter Dybjer, Sten Lindström, Erik Palmgren & Göran Sundholm (eds.) - 2012 - Dordrecht, Netherland: Springer.
    This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...)
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  • Kant’s Prize Essay and Nineteenth Century Formalism.Richard Lawrence - 2024 - Kant Yearbook 16 (1):31-52.
    Kant’s Prize Essay of 1764 emphasizes the importance for mathematical cognition of manipulating signs according to rules, which has led some recent commentators to ask whether Kant’s position there is a species of mathematical formalism. While most have hesitated to find formalism in the Prize Essay, this hesitation derives from misconceptions about what formalists actually believe. I therefore examine some nineteenth century formalists who were in dialogue with Kant, using their views as a model against which to compare the Prize (...)
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  • The 1860s Kant revival and the Philosophical Society of Berlin.Lauri Kallio - 2021 - Kant E-Prints 15 (3):192-219.
    Neo-Kantianism emerged over the course of the 1860s and it occupied a leading position in the German universities from the 1870s until the First World War. Demands for getting "back to Kant" had become common since the early 1860s, and these demands were discussed in the meetings of the Philosophical Society of Berlin (Philosophische Gesellschaft zu Berlin; PGB), which was the international organization of Hegelians. In this paper I address some reactions among the PGB members to the 1860s Kant revival. (...)
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  • Newton and Hamilton: In defense of truth in algebra.Janet Folina - 2012 - Southern Journal of Philosophy 50 (3):504-527.
    Although it is clear that Sir William Rowan Hamilton supported a Kantian account of algebra, I argue that there is an important sense in which Hamilton's philosophy of mathematics can be situated in the Newtonian tradition. Drawing from both Niccolo Guicciardini's (2009) and Stephen Gaukroger's (2010) readings of the Newton–Leibniz controversy over the calculus, I aim to show that the very epistemic ideals that underpin Newton's argument for the superiority of geometry over algebra also motivate Hamilton's philosophy of algebra. Namely, (...)
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  • What is Kantian Philosophy of Mathematics? An Overview of Contemporary Studies.Maksim D. Evstigneev - 2021 - Kantian Journal 40 (2):151-178.
    This review of contemporary discussions of Kantian philosophy of mathematics is timed for the publication of the essay Kant’s Philosophy of Mathematics. Volume 1: The Critical Philosophy and Its Roots (2020) edited by Carl Posy and Ofra Rechter. The main discussions and comments are based on the texts contained in this collection. I first examine the more general questions which have to do not only with the philosophy of mathematics, but also with related areas of Kant’s philosophy, e. g. the (...)
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  • Los esquemas de los conceptos empíricos y matemáticos como procedimientos de síntesis gobernados por reglas conceptuales.Martín Arias Albisu - 2014 - Studia Kantiana 17:74-103.
    El objetivo del presente artículo es ofrecer una interpretación de la doctrina del esquematismo de los conceptos empíricos y matemáticos presentada por Kant en su Crítica de la razón pura. Mostramos que los esquemas de los conceptos empíricos y matemáticos son procedimientos de síntesis gobernados por reglas conceptuales. Aunque no consideramos que esta doctrina kantiana carece de problemas, nuestro trabajo muestra que: 1) esos esquemas pueden distinguirse rigurosamente de sus correspondientes conceptos; 2) esos esquemas no son entidades superfluas. Estas conclusiones (...)
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  • Kant vergegenwärtigt. Fundamento, postulado, intuição intelectual.Federico Ferraguto - forthcoming - Kant E-Prints:10-34.
    O artigo visa mostrar como o gesto filosófico reinholdiano de construção de uma filosofia elementar e o fichteano de fundamentar toda filosofia como ciência rigorosa em uma intuição intelectual alcançada através de um postulado, mais do que uma traição da impostação fundamental da crítica da razão kantiana, refletem uma tentativa de aprofundar elementos implícitos nela, sobretudo no que diz respeito à sua dimensão metodológica. Isto permite estabelecer uma continuidade entre o criticismo de Kant e o chamado de idealismo alemão que (...)
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