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)

Der Kommentar zur Kritik der reinen Vernunft bietet eine textnahe Erschließung der zentralen Begriffe, Thesen und Argumentationsgänge von Kants Hauptwerk auf aktuellem Forschungsstand. Es ist der erste Kommentar zur KrV, der den gesamten Text in der Fassung der ersten und zweiten Auflage gleichmäßig und lückenlos berücksichtigt. Davon profitieren vor allem die „Transzendentale Dialektik“ und die „Methodenlehre“, die in früheren Gesamtkommentaren meist nicht hinreichend berücksichtigt worden sind. Die Beiträge wurden nach einheitlichen Richtlinien verfasst, wobei unterschiedliche Herangehensweisen und Interpretationsansätze zur Geltung kommen. (...) Um dem Leser dieses Kommentars die Orientierung zu erleichtern, ist jeder Beitrag in drei bzw. vier Teile untergliedert: Der erste Teil behandelt die Stellung und Funktion des kommentierten Textabschnitts in der KrV, der zweite Teil gibt einen Überblick über Inhalt und Aufbau des Abschnitts, der dritte Teil enthält den eigentlichen Textkommentar. Wo dies sinnvoll erschien, wurden in einem vierten Teil wichtige Interpretationsfragen angesprochen. Hinweise auf weiterführende Spezialliteratur am Ende eines jeden Beitrags, eine Auswahlbibliographie zu Kant und zur Kritik der reinen Vernunft sowie ein Namen- und ein Sachregister machen den Band zu einem komfortablen Arbeitsmittel für den Benutzer. Die vorliegende zweite Auflage trägt der Tatsache Rechnung, dass der Band mittlerweile als Standardwerk der Kant-Forschung gilt. Mit Beiträgen von Henry Allison, Karl Ameriksm Reinhard Brandt, Wolfgang Carl, Konrad Cramer, Jean Ferrari Eckart Förster, Volker Gerhardt, Paul Guyer, Otfried Höffe, Hansgeorg Hoppe, Rolf-Peter Horstmann, Heiner F. Klemme, Lothar Kreimendahl, Béatrice Longuenesse, Georg Mohr, Birgit Recki, Alain Renaut, Peter Rohs, Gerhard Seel, Dieter Sturma, Bernhard Thöle, Eric Watkins, Marcus Willaschek. (shrink)

What is the relationship between the world and logic, between intuition and language, between objects and their quantitative determinations? Rationalists, on the one hand, hold that the world is structured in a rational way. Representationalists, on the other hand, assume that language, logic, and mathematics are only the means to order and describe the intuitively given world. In World and Logic, Jens Lemanski takes up three surprising arguments from Arthur Schopenhauer’s hitherto undiscovered Berlin Lectures, which concern the philosophy of language, (...) logic, and mathematics. Based on these arguments, Lemanski develops a new position entitled ‘rational representationalism’: the world is always structured by human beings according to linguistic, logical, and mathematical principles, but the basic vocabulary of these structural descriptions already contains metaphors taken from the world around us. (shrink)

Logic diagrams have seen a resurgence in their application in a range of fields, including logic, biology, media science, computer science and philosophy. Consequently, understanding the history and philosophy of these diagrams has become crucial. As many current diagrammatic systems in logic are based on ideas that originated in the 18th and 19th centuries, it is important to consider what motivated the use of logic diagrams in the past and whether these reasons are still valid today. This paper proposes that (...) transcendental philosophy was a key inspiration for the development of logic diagrams and that such diagrams can be employed in transcendental arguments, even after the linguistic turn. (shrink)

In 1764, Kant published his Observations on the Feeling of the Beautiful and the Sublime and in 1790 his influential third Critique, the Critique of the Power of Judgment. The latter contains two parts, the ‘Critique of the Aesthetic Power of Judgment’ and the ‘Critique of the Teleological Power of Judgment’. They reveal a new principle, namely the a priori principle of purposiveness (Zweckmäßigkeit) of our power of judgment, and thereby offer new a priori grounds for beauty and biology within (...) the framework of Kant's transcendental philosophy. They also unite the previous two Critiques, the Critique of Pure Reason and the Critique of Practical Reason. Besides contributing to general and systematic aspects within his transcendental philosophy, Kant's aesthetics also offers new insights into old problems. It deals with feeling versus experience, subjectivity versus objectivity, disinterested pleasure, aesthetic universality, free and adherent beauty, the sensus communis, genius, aesthetic ideas, beauty as the symbol of morality, beauty of nature versus beauty of art, the sublime, and the supersensible. In this article I will limit myself to this critical aesthetics of Kant. But I will also discuss the ugly and the possibility of beauty in mathematics and see whether Kant's theory can successfully explain or deal with them. I will also compare his theory with philosophical ideas from a very different tradition, namely from Confucius, not only as a challenge to Kant's theory, but also because there is a growing interest from the Chinese side in combining ideas from Confucius and Kant, an interest that might well become influential in both East and West during the 21st century. (shrink)

According to what Jonathan Bennett calls the Kant–Frege view of existence, Frege gave solid logical foundations to Kant’s claim that existence is not a real predicate. In this article I will challenge Bennett’s claim by arguing that although Kant and Frege agree on what existence is not, they agree neither on what it is nor on the importance and justification of existential propositions. I identify three main differences: first, whereas for Frege existence is a property of a concept, for Kant (...) it is a relational property pertaining between the concept and intuition of an object. Second, whereas for Frege truth about individuals presupposes their existence, for Kant truth is in many cases independent of the existence of objects. Third, whereas Frege binds logic to existence and removes modalities from logic, for Kant existence is a modal category that is emphatically removed from the domain of logic and set in the core of metaphysics. Due to these differences in Kant’s and Frege’s theories of existence, Frege cannot be seen as giving logical clarity to Kant’s view. (shrink)

Abstract: Kant follows a substantial tradition by defining judgment so that it must involve a relation of concepts, which raises the question of why he thinks that single-term existential judgments should still qualify as judgments. There is a ready explanation if Kant is somehow anticipating a Fregean second-order account of existence, an interpretation that is already widely held for separate reasons. This paper examines Kant's early (1763) critique of Wolffian accounts of existence, finding that it provides the key idea in (...) his mature model of existential judgment, which is in fact sharply opposed to the Fregean strategy. By relating this to Kant's theory of judgment in general—in particular, to his claim for an isomorphism between the assertoric function of judgment and the category of existence—a preliminary case is made that absolute positing, far from being a marginal special case, accomplishes the primary function of judgment. This argument shows the importance of distinguishing between contexts in which Kant is treating judgment as a vehicle for inference (e.g. pure general logic) and contexts in which he is treating it, more robustly, as the cognition of an object. (shrink)

In his lectures on general logic Kant maintains that the generality of a representation (the form of a concept) arises from the logical acts of comparison, reflection and abstraction. These acts are commonly understood to be identical with the acts that generate reflected schemata. I argue that this is mistaken, and that the generality of concepts, as products of the understanding, should be distinguished from the classificatory generality of schemata, which are products of the imagination. A Kantian concept does not (...) provide mere criteria for noting sameness and difference in things, but instead reflects the inner nature of things. Its form consists in the self-consciousness of a capacity to judge (i.e. the Concept is the ‘I think’). (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)

In this paper, I shall discuss the assumption that Kant’s method in the Critique of Pure Reason is a transcendental analysis of experience. In order to do this, I will consider the conception of transcendental analysis that Marburg Neo-Kantianism powerfully introduced into Kant interpretation. I shall first develop a general model of transcendental analysis in the Marburg sense. I shall then ask whether this is a suitable model for the interpretation of the first Critique. Kant’s distinction between the analytic and (...) synthetic method in the Prolegomena seems to contradict this, given that it is a distinction between two different mathematical methods of demonstration. However, I shall argue that any such reading is incompatible with the Critique, and that transcendental analysis is indeed a legitimate interpretative model for understanding Kant. I shall finally turn to Strawson’s conception of descriptive metaphysics, which has strong affinities to Kant and the Marburg school. Most importantly, Strawson helps us to understand three significant aims at which transcendental analysis is directed. Finally, comparing transcendental analysis to dominant conceptions of analysis in philosophy, I shall argue that it constitutes a variety of philosophical analysis sui generis. (shrink)

Dennis Schulting offers a thoroughgoing, analytic account of the first half of the Transcendental Deduction of the Categories in the B-edition of Kant's Critique of Pure Reason that is different from existing interpretations in at least one important aspect: its central claim is that each of the 12 categories is wholly derivable from the principle of apperception, which goes against the current view that the Deduction is not a proof in a strict philosophical sense and the standard reading that in (...) the Deduction Kant only gives an account of the global applicability of the categories to experience. This novel approach enables a reappraisal of Kant's controversial claim that transcendental self-consciousness is not only a necessary condition of objective experience but also sufficient for it. The book provides an extensive analysis of Kant's theory of transcendental apperception and also explains why the argument of the Transcendental Deduction is both a regressive and a progressive argument. (shrink)

In questo articolo si analizzano congiuntamente l'origine delle idee e il loro uso regolativo. A questo scopo si comincia considerando la connessione tra uso logico della ragione e definizione della ragione in generale come facoltà dei principi. Si considera poi come la rappresentazione di un incondizionato derivi necessariamente dall'uso logico, e come quindi questo implichi necessariamente un uso reale della ragione. Si considera poi come le varie rappresentazioni dell'incondizionato diano origine alle idee trascendentali. Infine, si considera l'uso regolativo delle idee (...) alla luce della pratica di inferire dal quale derivano. (shrink)