Striving for Truth in the Practice of Mathematics: Kant and Frege (original) (raw)
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The sensible foundation for mathematics: A defense of Kant's view
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of mathematics is sometimes regarded as the weakest of his philosophical contributions. Commentators of stature, such as Frege, Russell, Strawson, Parsons, and Kitcher, have pointed out deep difficulties. The criticism would be less troubling if the philosophy of mathematics were a peripheral part of Kant's project. But it is not. The mathematical examples which appear and reappear throughout the Critique of Pure Reason' are symptoms of Kant's concern with mathematics. There are passages which are inextricably bound up with his view of mathematics, for instance the argument that space and time are transcendentally ideal but empirically real,2 "Schematism of the Pure Concepts of Understanding",3 and the "Axioms of Intuition".4 Moreover, since mathematical judgements are synthetic a priori for Kant, they are among the kinds of judgements he is concerned to account for. If Kant's philosophy of mathematics is deeply flawed, then the entire edifice of the Critique is deeply flawed as well. Criticisms of Kant's philosophy of mathematics may be divided into two kinds. Some question the internal coherence of the view and its ability to capture the mathematical practice of Kant's day. Others arise from contemporary developments in science and mathematics. In this essay I will marshal responses on Kant's behalf to both kinds of objection. In the fist section I will begin by presenting a simplified account of Kant's philosophy of mathematics. This presentation will allow us to see the force of the internal criticisms of Kant's views brought by Frege, Strawson, and Kitcher. I will attempt to defend Kant's view by formulating a comprehensive reading of his remarks on mathematics which is consonant with the rest of his epistemology. My reading of the texts is not particularly controversial, and I will not be concerned to defend it against the possible alternative readings. The interesting point is that
KANT’S PHILOSOPHY OF MATHEMATICS
Kant's 1 philosophy of mathematics plays a crucial role in his critical philosophy, and a clear understanding of his notion of mathematical construction would do much to elucidate his general epistemology. Friedman M. in Shabel L. insists that Kant's philosophical achievement consists precisely in the depth and acuity of his insight into the state of the mathematical exact sciences as he found them, and, although these sciences have radically changed in ways, this circumstance in no way diminishes Kant's achievements. Friedman M 2 further indicates that the highly motivation to uncover Kant's philosophy of mathematics comes from the fact that Kant was deeply immersed in the textbook mathematics of the eighteenth century. Since Kant's philosophy of mathematics 3 was developed relative to a specific body of mathematical practice quite distinct from that which currently obtains, our reading of Kant must not ignore the dissonance between the ontology and methodology of eighteenth-and twentieth-century mathematics. The description of Kant's philosophy 150 of mathematics involves the discussion of Kant's perception on the basis validity of mathematical knowledge which consists of arithmetical knowledge and geometrical
LOCKE AND KANT ON MATHEMATICAL KNOWLEDGE
Both Locke and Kant sought, in different ways, to limit our claims to knowledge in general by comparing it to our knowledge of mathematics. On the one hand, Locke thought it a mistake to think that mathematics alone is capable of demonstrative certainty. He therefore tried to isolate what it is about mathematics that makes it thus capable, in the hope of showing that other areas of inquiry -morality, for example -admit of the same degree of certainty. Kant, on the other hand, attributed much of the metaphysical excess of philosophy to the attempt by metaphysicians to imitate the method of mathematicians. He therefore sought to limit that excess by examining the mathematical method, like Locke, in order to isolate what is special about mathematics that accounts for its certainty.
Kant's Philosophy of Mathematics VOLUME I the critical philosophy and its roots, 2020
Volume I of a TWO VOLUME ANTHOLOGY _Kant's Philosophy of Mathematics VOLUME II: reception and influence_ forthcoming The late 1960s saw the emergence of new philosophical interest in Kant's philosophy of mathematics, and since then this interest has developed into a major and dynamic field of study. In this state-of-the-art survey of contemporary scholarship on Kant's mathematical thinking, Carl Posy and Ofra Rechter gather leading authors who approach it from multiple perspectives, engaging with topics including geometry, arithmetic, logic, and metaphysics. Their essays offer fine-grained analysis of Kant's philosophy of mathematics in the context of his Critical philosophy, and also show sensitivity to its historical background. The volume will be important for readers seeking a comprehensive picture of the current scholarship about the development of Kant's philosophy of mathematics, its place in his overall philosophy, and the Kantian themes that influenced mathematics and its philosophy after Kant. Engages with a lively and emerging field which will connect Kantian studies with mathematical philosophy in innovative ways Brings together authors from different schools of thought to provide readers with a full spectrum of contemporary approaches to Kant's philosophy of mathematics Explores how Kant's mathematical thought developed over time, with chapters organised thematically to aid readers' navigation of the issues.
The fact of modern mathematics: Geometry, logic, and concept formation in Kant and Cassirer
2007
This new "Kantian" theory of modern mathematics, Cassirer argues, is inconsistent with the traditional theory of concept formation by abstraction. Drawing on earlier Neo-Kantian interpretations, Cassirer argues that Kant's theory of concepts as rules undermines the traditional theory of concept formation, and he gives a "transcendental" defense of the new logic of Frege and Russell. (In an appendix, I discuss the contemporaneous accounts of concept formation in Gottlob Frege and Hermann Lotze.) vi TABLE OF CONTENTS
La question épistémologique de l'applicabilité des mathématiques
2018
International audienceThe question of the applicability of mathematics is an epistemological issue that was explicitly raised by Kant, and which has played different roles in the works of neo-Kantian philosophers, before becoming an essential issue in early analytic philosophy. This paper will first distinguish three main issues that are related to the application of mathematics: (1) indispensability arguments that are aimed at justifying mathematics itself; (2) philosophical justifications of the successful application of mathematics to scientific theories; and (3) discussions on the application of real numbers to the measurement of physical magnitudes. A refinement of this tripartition is suggested and supported by a historical investigation of the differences between Kant's position on the problem, several neo-Kantian perspectives (Helmholtz and Cassirer in particular, but also Otto Hölder), early analytic philosophy (Frege), and late 19th century mathematicians (Grassmann, D...