Kant says patently conflicting things about infinity and our grasp of it. Infinite space is a good case in point. In his solution to the First Antinomy, he denies that we can grasp the spatial universe as infinite, and therefore that this universe can be infinite; while in the Aesthetic he says just the opposite: ‘Space is represented as a given infinite magnitude’. And he rests these upon consistently opposite grounds. In the Antinomy we are told that we can have (...) no intuitive grasp of an infinite space, and in the Aesthetic he says that our grasp of infinite space is precisely intuitive. (shrink)

This is a philosophical investigation of the nature of the limits of thought. Drawing on recent developments in the field of logic, Graham Priest shows that the description of such limits leads to contradiction, and argues that these contradictions are in fact veridical. Beginning with an analysis of the way in which these limits arise in pre-Kantian philosophy, Priest goes on to illustrate how the nature of these limits was theorised by Kant and Hegel. He offers new interpretations of Berkeley's (...) master argument for idealism and Kant on the antimonies. He explores the paradoxes of self reference, and provides a unified account of the structure of such paradoxes. The book concludes by tracing the theme of the limits of thought in modern philosophy of language, including discussions of the ideas of Wittgenstein and Derrida. (shrink)

"Kant and the Capacity to Judge" will prove to be an important and influential event in Kant studies and in philosophy.

This paper discusses an apparent contrast between Kant’s accounts of the mathematical antinomies in the first Critique and in the Prolegomena. The Critique claims that the antitheses are infinite judgements. The Prolegomena seem to claim that they are negative judgements. For the Critique, theses and antitheses are false because they presuppose that the world has a determinate magnitude, and this is not the case. For the Prolegomena, theses and antitheses are false because they presuppose an inconsistent notion of world. The (...) paper argues that the contrast between the two works is only apparent, and provides an interpretation which removes it. (shrink)

This series publishes outstanding monographs and edited volumes that investigate all aspects of Kant's philosophy, including its systematic relationship to other philosophical approaches, both past and present. Studies that appear in the series are distinguished by their innovative nature and ability to close lacunae in the research. In this way, the series is a venue for the latest findings in scholarship on Kant.

In the First Antinomy of The Critique of Pure Reason, Kant drew two conclusions from the argument he gives. First, Kant took his argument to show that the referent of the concept of ‘world’ does not exist as a thing in itself. For at B532 he says.

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 ...

Does matter consist of the simple or is it divisible into infinity? This is the question posed by the second antinomy of the Critique of Pure Reason. In this first comprehensive systematic study of the antinomy of division, its derivation, the proofs for thesis and antithesis as well as the resolution are analysed. The developmental and historical dimensions of the topic are also discussed. The study shows that although the antinomy of division is on the one hand a critique of (...) metaphysics, it nevertheless achieves a positive result for Kant's transcendental philosophy: On the one hand, the resolution of the antinomy represents a conceptual sharpening of realism and idealism as well as of the transcendental concept of appearance. On the other hand, it shows that the structure of matter is conditioned by a priori determinations of reason and understanding. These insights are highly relevant not only for Kant's enterprise of an a priori foundation of the natural sciences, but also for the problem of the soul. (shrink)

In der Reihe werden herausragende monographische Untersuchungen und Sammelbände zu allen Aspekten der Philosophie Kants veröffentlicht, ebenso zum systematischen Verhältnis seiner Philosophie zu anderen philosophischen Ansätzen in Geschichte und Gegenwart. Veröffentlicht werden Studien, die einen innovativen Charakter haben und ausdrückliche Desiderate der Forschung erfüllen. Die Publikationen repräsentieren damit den aktuellsten Stand der Forschung.

It's well-known that Kant believed that intuition was central to an account of mathematical knowledge. What that role is and how Kant argues for it are, however, still open to debate. There are, broadly speaking, two tendencies in interpreting Kant's account of intuition in mathematics, each emphasizing different aspects of Kant's general doctrine of intuition. On one view, most recently put forward by Michael Friedman, this central role for intuition is a direct result of the limitations of the syllogistic logic (...) available to Kant. On this view, Kant's reasons for introducing intuition are taken to be logical or mathematical, rather than philosophical. The other tendency, which I shall try to develop here, emphasizes an epistemological or phenomenological role for intuition in mathematics arising out of what may loosely be called Kant's ‘antiformalism.’This paper, which focuses specifically on the case of geometry, falls into two parts. First, I consider Kant's discussion of intuition in the Metaphysical Exposition of the concept of space. (shrink)

Kant says patently conflicting things about infinity and our grasp of it. Infinite space is a good case in point. In his solution to the First Antinomy, he denies that we can grasp the spatial universe as infinite, and therefore that this universe can be infinite; while in the Aesthetic he says just the opposite: ‘Space is represented as a given infinite magnitude’ (A25/B39). And he rests these upon consistently opposite grounds. In the Antinomy we are told that we can (...) have no intuitive grasp of an infinite space, and in the Aesthetic he says that our grasp of infinite space is precisely intuitive. (shrink)

In papers published between 1930 and 1935. Zermelo outlines a foundational program, with infinitary logic at its heart, that is intended to (1) secure axiomatic set theory as a foundation for arithmetic and analysis and (2) show that all mathematical propositions are decidable. Zermelo's theory of systems of infinitely long propositions may be termed "Cantorian" in that a logical distinction between open and closed domains plays a signal role. Well-foundedness and strong inaccessibility are used to systematically integrate highly transfinite concepts (...) of demonstrability and existence. Zermelo incompleteness is then the analogue of the Problem of Proper Classes, and the resolution of these two anomalies is similarly analogous. (shrink)

There is evidence in Kant of the idea that concepts of particular numbers, such as the number 5, are derived from the representation of units, and in particular pure units, that is, units that are qualitatively indistinguishable. Frege, in contrast, rejects any attempt to derive concepts of number from the representation of units. In the Foundations of Arithmetic, he softens up his reader for his groundbreaking and unintuitive analysis of number by attacking alternative views, and he devotes the majority of (...) this attack to the units view, with particular attention to pure units. Since Frege, the units view has been all but abandoned. Nevertheless, the idea that concepts of number are derived from the representation of units has a long history, beginning with the ancient Greeks, and was prevalent among Frege's contemporaries. I am not interested in resurrecting the units view or in righting wrongs in Frege's criticisms of his contemporaries. Rather, I am interested in the program of deriving concepts of number from pure units and its history from Kant to Frege. An examination of that history helps us understand the units view in a way that Frege's criticisms do not, and in the process uncovers important features of both Kant's and Frege's views. I will argue that, although they had deep differences, Kant and Frege share assumptions about what such a view would require and about the limits of conceptual representation. I will also argue that they would have rejected the accounts given by some of Frege's contemporaries for the same reasons. Despite these agreements, however, there is evidence that Kant thinks that space and time play a role in overcoming the limitations of conceptual representation, while Frege argues that they do not. (shrink)

Les recherches de Zermelo sur la théorie des ensembles et les fon­dements des mathématiques se divisent en deux périodes : de 1901 à 1910 et de 1927 à 1935. Elles s’effectuent en même temps que les deux projets de recherche sur les fondements des mathématiques de David Hilbert et de ses collaborateurs à Göttingen ; durant la première période, Hilbert élaborait son premier programme d’axiomatisation, auquel Zermelo souscrivait totalement. La seconde période correspond au développement du programme formaliste de Hilbert que (...) Zermelo rejette en raison de son caractère finitiste. son travail est à cette époque dirigé contre Hilbert. (shrink)

Les recherches de Zermelo sur la théorie des ensembles et les fon­dements des mathématiques se divisent en deux périodes : de 1901 à 1910 et de 1927 à 1935. Elles s’effectuent en même temps que les deux projets de recherche sur les fondements des mathématiques de David Hilbert et de ses collaborateurs à Göttingen ; durant la première période, Hilbert élaborait son premier programme d’axiomatisation, auquel Zermelo souscrivait totalement. La seconde période correspond au développement du programme formaliste de Hilbert que (...) Zermelo rejette en raison de son caractère finitiste. son travail est à cette époque dirigé contre Hilbert. (shrink)

In the First Antinomy of The Critique of Pure Reason, Kant drew two conclusions from the argument he gives. First, Kant took his argument to show that the referent of the concept of ‘world’ does not exist as a thing in itself. For at B532 he says.