I argue for a new delimitation of what Kant means by ‘cognition [Erkenntnis]’, on the basis of the intermediate, transitional place that Kant gives to cognition in the ‘progression [Stufenleiter]’ of our representations and our consciousness of them. I show how cognition differs from mental acts lying earlier on this progression—such as sensing, intuiting, and perceiving—and also how cognition differs from acts lying later on this progression—such as explaining, having insight, and comprehending. I also argue that cognition should not be (...) confused with ‘knowledge [Wissen]’, insofar as knowledge represents the culmination of a separate orthogonal progression of acts of ‘holding-true’. Along the way, I show how having in focus the specific progression from representation, to consciousness, to cognition allows us to better appreciate the architectonic significance of the progression of Kant’s analysis in the first Critique, and also helps to illuminate the unity of Kant’s account of cognition itself across its variety of forms. (shrink)

In the section of the Antinomy of pure Reason Kant presents three notions of infinity. By investigating these concepts of infinity, this paper highlights important ‘building blocks’ of the structure of the mathematical antinomies, such as the ability of reason of producing ascending and descending series, as well as the notions of given and givable series. These structural features are discussed in order to clarify Ernst Zermelo’s reading of Kant’s antinomy, according to which the latter is deeply rooted in the (...) tendency of the mind of producing “creative progress” and “inclusive closure”. The aim of this paper is to explain in which sense and why Kant’s treatment of the antinomies attracts the attention of Zermelo in the early 1900s and which aspects of his second axiomatic system have been inspired by Kant’s philosophy. Thus, by reading Kant’s antinomy ‘through Zermelo’s eyes’—with emphasis on the concept of regressive series in indefinitum and on that of regressive series ad infinitum – this paper identifies the echoes of Kant’s work in the making of the ZFC set theory. (shrink)

This article discusses the quiddity of the empty set from its epistemological and linguistic aspects. It consists of four parts. The first part compares the concept of _nihil privativum_ and the empty set in terms of representability, arguing the empty set can be treated as a negative and formal concept. It is argued that, unlike Frege’s definition of zero, the quantitative negation with a full scope is what enables us to represent the empty set conceptually without committing to an antinomy. (...) The second part examines the type and scope of the negation in the concept of _nihil privativum_ and the empty set. In the third part the empty set is interpreted as a rigid abstract general term. The uniqueness of the empty set is explained via a widened version of Kripke’s notion of rigidity. The fourth part proposes a construction for the pure singleton, comparing it with Zermelo’s conception of singletons with the Ur-elements. It is argued that the proposed construction does not face the criterion and ontological inflation problems. The first conclusion of the article is that the empty set can be construed as a negative, formal and unique abstract general term, with quantitative negation full in scope. The second conclusion is that the pure singleton constructed out of the empty set construed in this way overcomes the criterion and ontological inflation problems. (shrink)

I argue for a new delimitation of what Kant means by ‘cognition [Erkenntnis]’, on the basis of the intermediate, transitional place that Kant gives to cognition in the ‘progression [Stufenleiter]’ of our representations and our consciousness of them. I show how cognition differs from mental acts lying earlier on this progression—such as sensing, intuiting, and perceiving—and also how cognition differs from acts lying later on this progression—such as explaining, having insight, and comprehending. I also argue that cognition should not be (...) confused with ‘knowledge [Wissen]’, insofar as knowledge represents the culmination of a separate orthogonal progression of acts of ‘holding-true’. Along the way, I show how having in focus the specific progression from representation, to consciousness, to cognition (and beyond) allows us to better appreciate the architectonic significance of the progression of Kant’s analysis in the first Critique (and beyond), and also helps to illuminate the unity of Kant’s account of cognition itself across its variety of (empirical, mathematical, philosophical) forms. (shrink)

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 (...) can be characterized as cardinal or ordinal. This essay argues that Kant's conception of number includes both cardinal and ordinal elements; it suggests that the cardinal elements provide the basis of a conception of number in general, while the ordinal elements contribute to specifying the exact size of particular collections. In considering these elements, roles for intuition begin to emerge, setting the stage for a reevaluation of the role of intuition in Kant's arithmetic. (shrink)