Kant: Philosophy of Mathematics

Draft for presentation at the 14th International Kant-Congress, September 2024. -/- Abstract: Kant claims we intuit infinite space. There’s a problem: Kant thinks full awareness of infinite space requires synthesis—the act of putting representations together and comprehending them as one. But our ability to synthesize is finite. Tobias Rosefeldt has argued in a recent paper that Kant’s notion of decomposing synthesis offers a solution. This talk criticizes Rosefeldt’s approach. First, Rosefeldt is committed to nonconceptual yet determinate awareness of (potentially) infinite (...) space, but I argue that such awareness is (1) prima facie implausible, as well as contravened by textual evidence from Kant’s (2a) definitions of ‘infinity’ and (2b) account of geometrical construction. Second, scrutiny of how Kant thinks awareness of potential infinity is afforded by geometrical construction (e.g., extending a line segment ad indefinitum) indicates no essential role for decomposing synthesis. Familiar synthesis from parts to wholes is sufficient. (shrink)

In the Metaphysical Foundations of Natural Science, Kant attempts to argue a priori from the indefinite divisibility of space to the indefinite metaphysical divisibility of matter. This is one type of argument from the continuity of space—purportedly established by Euclidean geometry—to the continuity of matter. I compare Kant's argument to parallel reasoning in Du Châtelet, whose work he knew. Both philosophers appeal to idealism about matter in their reasoning, yet also face difficulties in explaining why continuity, though not some other (...) properties from geometry, applies to matter. Both also risk inconsistency in adopting potentialist accounts of material parts, while also committing to realism about infinitesimals. An important difference between them is that Du Châtelet deploys at least three definitions of continuity; only one of these, amounting to indefinite divisibility, is shared with Kant. (shrink)

For Kant, inquiry into nature properly requires seeking to explain all material wholes merely mechanically, in terms of their parts. There is no consensus on how he justifies this Principle of Mechanism. I argue that Kant seeks to derive this claim about part and wholes neither from his laws or mechanics, nor from the mere discursivity of our understanding (two standard options in the literature), but instead from a priori principles laid out in the first Critique, which govern parts, wholes, (...) and magnitudes. These principles are also fundamental to Kant’s account of mathematics. Therefore, Kant’s Principle of Mechanism and his philosophy of mathematics have common foundations. (shrink)

This book provides the first comprehensive discussion regarding the role that Kant ascribes to systematicity in the sciences. It considers not only what Kant has to say on systematicity in general, but also how the systematicity requirement for science is specified in different fields of knowledge. The chapters are divided into three thematic sections. Part 1 is devoted to historical context. The chapters explore precursors of Kant's account of the systematicity of the sciences. Part 2 addresses the application of systematicity (...) to the special sciences-cosmology, physics, chemistry, logic, mathematics, the life sciences and history. Finally, Part 3 explores the systematicity of philosophy. Kant and the Systematicity of the Sciences will be of interest to scholars and advanced students working on Kant and the history and philosophy of science. (shrink)

Our aim in this chapter is to shed light on Kant’s account of the pure forms of sensibility by focusing on a somewhat neglected issue: Kant’s restriction of his claims about space and time to the case of human sensibility. Kant argues that space and time are the pure forms of sensibility for human cognizers. But he also says that we cannot know whether space and time are likewise the pure forms of sensibility for all discursive cognizers. A great deal (...) of attention has focused on the first of these claims, both on how Kant argues for it and how it relates to transcendental idealism. But a satisfactory interpretation must also account for the second claim, and it must account for the fact that Kant endorses both of them. What we need is an explanation of why Kant thinks our knowledge that space and time are the pure forms of sensibility extends to all human beings but no further. (shrink)

ABSTRACT I suggest how a broadly Kantian critique of classical logic might spring from reflections on constructibility conditions. According to Kant, mathematics is concerned with objects that are given through ‘arbitrary synthesis,’ in the form of ‘constructions of concepts’ in the medium of ‘pure intuition.’ Logic, by contrast, is narrowly constrained – it has no objects of its own and is fixed by the very forms of thought. That is why there is not much room for developments within logic, as (...) compared to the progress in mathematics. Kant’s view of logic remains critical, though – through considerations that are effectively on the scope and limits of classical logic and which play a part in his transcendental idealism. The most important ones are to be found in his critique of the use of reductio ad absurdum proofs in metaphysics and his solutions to the ‘antinomies of pure reason.’ Arguably, these considerations carry over to mathematics as well – by way of ‘analogues’ to the antinomies – in particular in resolving infinity paradoxes. (shrink)

Thorough comparison of Immanuel Kant’s and Christian Wolff’s divergent appraisals of the science of psychology reveals various ways in which Kant fundamentally altered the Wolffian philosophical apparatus that he inherited. Wolff conceived of a thoroughgoing interplay between empirical and rational psychology, of combining different sorts of cognition in psychology, and of a mathematical science of the soul, or psychometrics. Kant however rejected each of these particular theses and deemed psychology to be no natural science, “properly so-called.” This chapter details these (...) departures from Wolff, explains their basis, and concretizes an underlying contrast in Kant’s and Wolff’s respective philosophical approaches. Namely, whereas Wolff’s philosophical system is malleable, allowing for the combination of various kinds of cognition and methods, in Kant’s Critical philosophy, everything has its proper, preordained niche. At bottom, it is the unyielding rigor of Kant’s system that results in his pessimistic evaluation of the prospects for psychology. (shrink)

In Bertrand Russell's 1903 Principles of Mathematics, he offers an apparently devastating criticism of the neo-Kantian Hermann Cohen's Principle of the Infinitesimal Method and its History (PIM). Russell's criticism is motivated by his concern that Cohen's account of the foundations of calculus saddles mathematics with the paradoxes of the infinitesimal and continuum, and thus threatens the very idea of mathematical truth. This paper defends Cohen against that objection of Russell's, and argues that properly understood, Cohen's views of limits and infinitesimals (...) do not entail the paradoxes of the infinitesimal and continuum. Essential to that defense is an interpretation, developed in the paper, of Cohen's positions in the PIM as deeply rationalist. The interest in developing this interpretation is not just that it reveals how Cohen's views in the PIM avoid the paradoxes of the infinitesimal and continuum. It also reveals some of what is at stake, both historically and philosophically, in Russell's criticism of Cohen. (shrink)

Winner of Fichte-Preis für Nachwuchsforscher*innen. In this paper I develop an account of Fichte’s conception of philosophical construction. Following the latter’s definition of philosophy as the ‘science of science’, philosophy is to be understood as a normative theory of what should qualify as science. In order to ground scientific knowledge-production as such, philosophy itself has to acquire a scientific method, through the application of which the constitution of scientific knowledge is secured. In systematic continuity to Kant’s account of geometrical construction, (...) Fichte develops a philosophical method that exploits the special epistemic conditions of performativity. Construction is then defined as an experimental, self-reflexive performance that exemplifies consciousness. Throughout its acts of exemplification this reflexive kind of self-observation yields a particular type of experience, which ultimately satisfies the Science of Knowledge’s demand for certainty, that is intellectual intuition. (shrink)

Name der Zeitschrift: Kant-Studien Jahrgang: 110 Heft: 1 Seiten: 74-125.

Kant’s account of the sublime in the Critique of Judgment has been extremely influential, prompting extensive discussion of the psychology, affect, moral significance, and relevance to artistic representation of the sublime on his provocative view. I focus instead on Kant’s account of the mathematical sublime in connection to his theoretical critical project, namely his attempt to characterize human cognitive powers and to limit human pretensions to knowledge of the supersensible. I argue, first, that his account of the psychology of the (...) sublime is designed to explain not just its affective character, but also to address challenges concerning the coherence of an experience of something as transcending one’s cognitive abilities. Thereby, I argue moreover, Kant provides an alternative, demystifying account of mystical experiences, in which humans might take themselves to intuit that which is beyond human understanding or reason, and thus to claim that they have special cognitive access to the supersensible, transcending the limits Kant claims to establish for human cognition. Kant’s account of the mathematical sublime is not merely so reductive of mystical experience, however; it also, I suggest, describes the aesthetic of Kantian critique itself. (shrink)

Hintikka considers that the “Transcendental Deduction” includes finding the role that concepts in the effort is meant by human activities of acquiring knowledge; and it affirms that the principles governing human activities of knowledge can be objective rules that can become transcendental conditions of experience and no conditions contingent product of nature of human agents involved in the know. In his opinion, intuition as it is used by Kant not be understood in the traditional way, ie as producer of mental (...) images, but rather as that which the mind represents an individual. To support this interpretation refers to the lessons of Kant´s Logics, to individuality space time and the thesis, submitted the winning essay in 1764, characterizing the mathematical method by the use of particular representatives of general concepts. Thus, his exegesis considers the Kantian conceptions of the “Doctrine of Method” not later conceptions, as traditionally interpreted, but prior conceptions to the processing of “Transcendental Aesthetic”. This article reconstructs some of their arguments that eventually will equal the Kantian intuition with the Euclidean ecthesis. (shrink)

Name der Zeitschrift: Kant-Studien Jahrgang: 108 Heft: 1 Seiten: 19-54.

A Estética Transcendental é uma peça chave no programa de pesquisa que Kant desenvolveu e nomeou de filosofia transcendental. Ela se anuncia na Dissertação de 1770 e se configura de forma bem explícita na primeira edição da Crítica da razão pura, de 1781. O modo como Kant a concebeu permitiu-lhe separar radicalmente intelecto e sensibilidade, mas seria importante compreender a raiz dessa separação. Nesse texto procuramos mostrar que o opúsculo “Sobre o primeiro fundamento da distinção de direções no espaço”, de (...) 1768, traz características bastante decisivas para a estética proposta no período crítico. Para evidenciar isso, expusemos ideias marcantes do texto sobre as contrapartidas incongruentes e as cotejamos com as reflexões expostas nos itens 3 e 4 relativos à Exposição Metafísica do conceito de espaço e aquelas apresentadas nos itens 4 e 5 relativos à Exposição Metafísica do conceito de tempo. (shrink)

Name der Zeitschrift: Kant-Studien Jahrgang: 107 Heft: 3 Seiten: 429-450.

In his Metaphysische Anfangsgründe der Naturwissenschaft (MAN), Kant claims that chemistry is an improper, though rational science. The chemistry to which Kant confers this status is the phlogistic chemistry of, for instance, Georg Stahl. In his Opus Postumum (OP), however, Kant espouses a broadly Lavoiserian conception of chemistry. In particular, Kant endorses Antoine Lavoisier's elements, oxygen theory of combustion, and role for the caloric. As Lavoisier's lasting contribution to chemistry, according to some histories of the science, was his emphasis on (...) quantitative methods, Kant's assimilation of the chemical revolution raises the question: did Kant continue to think of chemistry as an improper, though rational, science in OP? In this paper, I answer this question affirmatively. I explain that a proper science requires a priori laws for the mathematical construction of its objects: the mere use of quantitative measurements is insufficient for the satisfaction this requirement. Further, I argue, contra Burkhard Tuschling, that in OP Kant retains a central role for mathematical construction in the foundations of proper natural science. Finally, I contend that the prominent a priori components of chemistry discussed in OP—the aether and the enumeration of the elements—do not make chemistry into a proper science. (shrink)

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.

There is a long tradition, in the history and philosophy of science, of studying Kant’s philosophy of mathematics, but recently philosophers have begun to examine the way in which Kant’s reflections on mathematics play a role in his philosophy more generally, and in its development. For example, in the Critique of Pure Reason , Kant outlines the method of philosophy in general by contrasting it with the method of mathematics; in the Critique of Practical Reason , Kant compares the Formula (...) of Universal Law, central to his theory of moral judgement, to a mathematical postulate; in the Critique of Judgement , where he considers aesthetic judgment, Kant distinguishes the mathematical sublime from the dynamical sublime. This last point rests on the distinction that shapes the Transcendental Analytic of Concepts at the heart of Kant’s Critical philosophy, that between the mathematical and the dynamical categories. These examples make it clear that Kant's transcendental philosophy is strongly influenced by the importance and special status of mathematics. The contributions to this book explore this theme of the centrality of mathematics to Kant’s philosophy as a whole. This book was originally published as a special issue of the Canadian Journal of Philosophy. (shrink)

There is a long tradition, in the history and philosophy of science, of studying Kant’s philosophy of mathematics, but recently philosophers have begun to examine the way in which Kant’s reflections on mathematics play a role in his philosophy more generally, and in its development. For example, in the Critique of Pure Reason , Kant outlines the method of philosophy in general by contrasting it with the method of mathematics; in the Critique of Practical Reason , Kant compares the Formula (...) of Universal Law, central to his theory of moral judgement, to a mathematical postulate; in the Critique of Judgement , where he considers aesthetic judgment, Kant distinguishes the mathematical sublime from the dynamical sublime. This last point rests on the distinction that shapes the Transcendental Analytic of Concepts at the heart of Kant’s Critical philosophy, that between the mathematical and the dynamical categories. These examples make it clear that Kant's transcendental philosophy is strongly influenced by the importance and special status of mathematics. The contributions to this book explore this theme of the centrality of mathematics to Kant’s philosophy as a whole. This book was originally published as a special issue of the Canadian Journal of Philosophy. (shrink)

The question of the dimensionality of space has informed the development of physics since the beginning of the twentieth century in the quest for a unified picture of quantum processes and gravitation. Scientists have worked within various approaches to explain why the universe appears to have a certain number of spatial dimensions. The question of why space has three dimensions has a genuinely philosophical nature that can be shaped as a problem of justifying a contingent necessity of the world. In (...) contrast to explanations of three-dimensionality based on anthropic arguments, we support the search for a theory that provides a justification for the dimensionality of space based on a combination of deductive and inductive reasoning applied to science. In doing so, we argue that Kant correctly approached the question in “Thoughts on the true estimation of living forces” by connecting space dimensionality and the inverse square law. In expounding the strategy of Kant’s argument, we describe the main features of a general Kantian explanation of the dimensionality of space and discuss them with respect to current accounts of explanation in the philosophy of science, such as inference to the best explanation and the deductive-nomological model. (shrink)

Robert Hanna works out a unified contemporary Kantian theory of rational human cognition and knowledge. Along the way, he provides accounts of intentionality and its contents, sense perception and perceptual knowledge, the analytic-synthetic distinction, the nature of logic, and a priori truth and knowledge in mathematics, logic, and philosophy. This book is specifically intended to reach out to two very different audiences: contemporary analytic philosophers of mind and knowledge, and contemporary Kantian philosophers or Kant-scholars. At the same time, it rides (...) the crest of a wave of exciting and revolutionary emerging new trends and new work in the philosophy of mind and epistemology, with a special concentration on the philosophy of perception. What is revolutionary in this new wave are its strong emphases on action, on cognitive phenomenology, on disjunctivist direct realism, on embodiment, and on sense perception as a primitive and proto-rational capacity for cognizing the world. Hanna makes a fundamental contribution to this philosophical revolution by giving it a specifically contemporary Kantian twist, and by pushing these new lines of investigation radically further. (shrink)

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 (...) summarized as follows: mathematical structures that can be defined independently of the objects we experience are necessary for judgments about magnitudes to be generally valid. I suggest that space is conceived by Helmholtz as one such structure. I will analyze his argument in its most detailed version, which is found in Helmholtz (Zählen und Messen, erkenntnistheoretisch betrachtet 1887. In: Schriften zur Erkenntnistheorie. Springer, Berlin, 1921, 70–97). In support of my view, I will consider alternative formulations of the same argument by Ernst Cassirer and Otto Hölder. (shrink)

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 (...) seen to be a form of concept acquisition, one that is related to the modal phenomenology of geometrical judgement. (shrink)

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 (...) Kurt Gödel during Gödel’s later years was Hao Wang” .This book takes Wang’s report about Gödel’s philosophical interests seriously and paves the way “to go beyond the contributions of Plato, Leibniz and Husserl, based on Gödel’s philosophical and technical writings, his comments to Hao Wang, and various items from the Gödel Nachlass” . Tieszen proposes a “type of Gödelian platonic rationalism” about mathematic and logic that makes no pretenc .. (shrink)

A translation of the Introduction to Hermann Cohen's 1883 work The Principle of the Infinitesimal Method and Its History.

A translation of one section of Hermann Cohen's introduction to Friedrich Albert Lange's History of Materialism.

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 (...) applicability of mathematical concepts. (shrink)

This paper shows that Kant's investigation into mathematical purposiveness was central to the development of his understanding of synthetic a priori knowledge. Specifically, it provides a clear historical explanation as to why Kant points to mathematics as an exemplary case of the synthetic a priori, argues that his early analysis of mathematical purposiveness provides a clue to the metaphysical context and motives from which his understanding of synthetic a-priori knowledge emerged, and provides an analysis of the underlying structure of mathematical (...) purposiveness itself, which can be described as unintentional, but also as objective and unlimited. (shrink)

One of a pair of reviews of Michael Strevens’ book, Tychomancy: Inferring Probability from Causal Structure, Harvard University Press, Cambridge, MA, 2013, pp 265, $39.95 hbk, ISBN 978-0674073111. See also Bookstein (2014, this issue).

This paper gives a contextualized reading of Kant's theory of real definitions in geometry. Though Leibniz, Wolff, Lambert and Kant all believe that definitions in geometry must be ‘real’, they disagree about what a real definition is. These disagreements are made vivid by looking at two of Euclid's definitions. I argue that Kant accepted Euclid's definition of circle and rejected his definition of parallel lines because his conception of mathematics placed uniquely stringent requirements on real definitions in geometry. Leibniz, Wolff (...) and Lambert thus accept definitions that Kant rejects because they assign weaker roles to real definitions. (shrink)

I argue that three main interpretations of the aim of Russell’s early logicism in The Principles of Mathematics (1903) are mistaken, and propose a new interpretation. According to this new interpretation, the aim of Russell’s logicism is to show, in opposition to Kant, that mathematical propositions have a certain sort of complete generality which entails that their truth is independent of space and time. I argue that on this interpretation two often-heard objections to Russell’s logicism, deriving from Gödel’s incompleteness theorem (...) and from the non-logical character of some of the axioms of Principia Mathematica respectively, can be seen to be inconclusive. I then proceed to identify two challenges that Russell’s logicism, as presently construed, faces, but argue that these challenges do not appear unanswerable. (shrink)

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 (...) calls the productive imagination. Contrary to what is assumed by many commentators, it is not the case that the Two Stems Doctrine implies that a representation is either sensible or spontaneity-dependent, but not both. Outer sensible intuitions are both sensible and spontaneity-dependent – they are sensible because they exhibit the kind of structure Kant takes to be distinctive of outer sensible intuitions, and they depend on spontaneity because they are cases of sensibly representing a magnitude. (shrink)

This paper tries to make sense of Kant's scattered remarks about conic sections to see what light they shed on his philosophy of mathematics. It proceeds by confronting his remarks with the source that seems to have informed his thinking about conic sections: the Conica of Apollonius. The paper raises questions about Kant's attitude towards mathematics and the way he understood the cognitive resources available to us to do mathematics.

In his Metaphysische Anfangsgründe der Naturwissenschaft, Kant claims that chemistry is a science, but not a proper science (like physics), because it does not adequately allow for the application of mathematics to its objects. This paper argues that the application of mathematics to a proper science is best thought of as depending upon a coordination between mathematically constructible concepts and those of the science. In physics, the proper science that exhausts the a priori knowledge of objects of the outer sense, (...) only motions and concepts reducible to motions can be legitimately coordinated with mathematical constructions. Since chemistry can neither achieve its own a priori principles of coordination nor be reduced to the coordinated doctrine of motion, it is a merely improper science. (shrink)

Marburg Neo-Kantianism has attracted substantial interest among contemporary philosophers drawn by its founding idea that the success of advanced theoretical science is a given fact and it is the task of philosophical inquiry to ground the objectivity of scientific achievement in its a priori sources (Cohen and Natorp 1906, p. i). The Marburg thinkers realized that recent advances and developments in the mathematical sciences had changed the character of Kant’s transcendental project, demanding new methods and approaches to establish the objectivity (...) of the latest insights into physical reality. Hermann Cohen, one of the founders of the Marburg School, attempted to base the objectivity of science on the concepts of .. (shrink)

Geometry is an a priori science. However, its apriority is saddled with problems. The aim of this paper will be to show 1) how Kant understands that the contents of geometry are synthetic a priori judgments in the Critique of Pure Reason, and 2) if it’s still relevant to study Kant’s theory of geometry after the challenges posed by non-Euclidian theories of space. With respect to point 1: Kant understands geometry as the discipline that objectifies the pure intuition of space. (...) Every geometric concept is built upon the pure intuition of space through a synthetic ostensive process. Furthermore, the pure intuition of space is the form of external experiences. Thus, geometry and external phenomena share a common ground – pure space. This common ground is what provides an answer to the question of the possibility of mathematics as a universal and a priori science. With respect to point 2: the relevance of studying Kant’s theory of geometry lies not only in the fact that geometry can serve as an example to philosophy based on the fact that it establishes its propositions a priori, but also because the object-study of geometry – the pure intuition of space– forces the reader to review Kant’s thoughts about sensibility and its relation to space. The analysis of Kant’s theory of geometry then amounts to studying Kant’s theory of sensibility. (shrink)

An integral translation of Kant's 'Über Kästners Abhandlungen' (AA XX: 410-23). This translation is accompanied by an introductory essay on the importance of the Kästner treatise for an understanding of Kant's theory of space as infinite. See Onof & Schulting, "Kant, Kästner and the Distinction between Metaphysical and Geometrical Space".

This paper examines the role of Kant's theory of mathematical cognition in his phoronomy, his pure doctrine of motion. I argue that Kant's account of how we can construct the composition of motion rests on the construction of extended intervals of space and time, and the representation of the identity of the part–whole relations the construction of these intervals allow. Furthermore, the construction of instantaneous velocities and their composition also rests on the representation of extended intervals of space and time, (...) reflecting the general approach to instantaneous velocity in the eighteenth century. (shrink)

The five-volume set Kant and Philosophy in a Cosmopolitan Sense contains the proceedings of the Eleventh International Kant Congress, which took place in Pisa in 2010. The proceedings consist of 25 plenary talks and 341 papers selected by a team of international referees from over 700 submissions. The contributions span 14 sections: Kant’s Concept of Philosophy; Theory of Cognition and Logic; Ontology and Metaphysics; Ethics; Law and Justice; Religion and Theology; Aesthetics; Anthropology and Psychology; Politics and History; Science, Mathematics, and (...) the Philosophy of Nature; Kant and the Leibnizian Tradition; Kant and the Philosophical Tradition; Kant and Schopenhauer; and Kant’s Heritage. Thanks to cooperation from the Schopenhauer Gesellschaft, the 2010 conference was the first in the history of the International Kant Congress to include a session on Kant and Schopenhauer. -/- . (shrink)

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 (...) what, according to Peirce, was right and what was problematic in Lange's account of reasoning. It further seeks to explain how Peirce's theory of deductive reasoning was a combination of Kant's philosophy of mathematics and Lange's philosophy of logic. (shrink)

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 (...) argument pour justifier son conventionnalisme géométrique. Cependant, Hölder montre qu’une stratégie alternative n’est pas exclue: il sait tirer parti des objections kantiennes pour développer un empirisme cohérent. En même temps, surtout dans Die mathematische Methode [Holder 1924], il adopte aussi bien les expressions que les conceptions de Kant. Dans mon article, je considère d’abord les arguments de Hölder pour la méthode déductive en géométrie dans Anschauung und Denken in der Geometrie [Holder 1900], en relation avec sa façon d’aborder la théorie de la quantité [Holder 1901]. Ensuite, j’examine son rapport avec Kant. À mon sens, les considérations méthodologiques de Hölder lui permettent de préfigurer une relativisation de l’a priori. (shrink)

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.