Kant and the Neo-Kantians on Mathematics (Oliva) (original) (raw)
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The sensible foundation for mathematics: A defense of Kant's view
Studies in History and Philosophy of Science Part A, 1990
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
Existence or Possibility? Kantian Conundrum of Mathematical Objects
In this paper I argue that there is no plausible candidate for a Kantian picture of mathematical objects. I, first, present a rudimentary picture of Kant's epistemology and cognition and extend this picture to mathematical cognition. The usual justification for Kantian mathematical cognition relies on the concept of pure intuition or a priori intuition in Kant. However, just citing this source does not suffice for the concepts of mathematical objects or so I argue. It turns out that filling in the details of how pure intuitions figure into mathematical cognition leads to insurmountable problems in specifying any candidate for mathematical objects. As a bonus, I try and show that Parsons's roughly-Kantian picture also suffers from similar problems.
Redrawing Kant's Philosophy of Mathematics
South African Journal of Philosophy, 2013
This essay offers a strategic reinterpretation of Kant’s philosophy of mathematics in his Critique of Pure Reason via a broad, empirically based reconception of Kant’s conception of drawing. It begins with a general overview of Kant’s philosophy of mathematics, observing how he differentiates mathematics in the Critique from both the dynamical and the philosophical. Second, it examines how a recent wave of critical analyses of Kant’s constructivism takes up these issues, largely inspired by Hintikka’s unorthodox conception of Kantian intuition. Third, it offers further analyses of three Kantian concepts vitally linked to that of drawing. It concludes with an etymologically based exploration of the seven clusters of meanings of the word "drawing" to gesture toward new possibilities for interpreting a Kantian philosophy of mathematics.
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.
Kant's Schematism and the Foundations of Mathematics
2005
The theory of schematism was initiated by I. Kant, who, however, was never precise with respect to what he understood under this theory. I give-based on the theoretical works of Kant-an interpretation of the most important aspects of Kant's theory of schematism. In doing this I show how schematism can form a point of departure for a reinterpretation of Kant's theory of knowledge. This can be done by letting the concept of schema be the central concept. I show how strange passages in, say, the first Critique are in fact understandable, when one takes schematism serious. Likewise, I show how we-on the background of schematism-get a characterization of Kant's concept of 'object'. This takes me to an analysis of the ontology and epistemology of mathematics. Kant understood himself as a philosopher in contact with science. It was science which he wanted to provide a foundation for. I show that, contrary to Kant's own intentions, he was not up-to-date on mathematics. And in fact, it was because of this that it was possible for him to formulate his rather rigid theory concerning the unique characterizations of intuition and understanding. I show how phenomena in the mathematics of the time of Kant should have had an effect on him. He should have remained more critical towards his formulation and demarcation of intuition, understanding and reason.
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.