Officially, for Kant, judgments are analytic iff the predicate is "contained in" the subject. I defend the containment definition against the common charge of obscurity, and argue that arithmetic cannot be analytic, in the resulting sense. My account deploys two traditional logical notions: logical division and concept hierarchies. Division separates a genus concept into exclusive, exhaustive species. Repeated divisions generate a hierarchy, in which lower species are derived from their genus, by adding differentia(e). Hierarchies afford a straightforward sense of containment: (...) genera are contained in the species formed from them. Kant's thesis then amounts to the claim that no concept hierarchy conforming to division rules can express truths like '7+5=12.' Kant is correct. Operation concepts ( ) bear two relations to number concepts: and are inputs, is output. To capture both relations, hierarchies must posit overlaps between concepts that violate the exclusion rule. Thus, such truths are synthetic. (shrink)

This essay reassesses the relation between Kant and Kripke on the relation between necessity and the a priori. Kripke famously argues against what he takes to be the traditional view that a statement is necessary only if it is a priori, where, very roughly, what it means for a statement to be necessary is that it is true and could not have been false and what it means for a statement to be a priori is that it is knowable independently (...) of experience. Call such a view the Entailment Thesis. Along with many Kant scholars, Kripke thinks that Kant endorses the Entailment Thesis. Thus Kripke and many others take his arguments against the Entailment Thesis to tell against Kant and to mark an important point of disagreement with him. I will argue that this is a mistake. Kant does not endorse the Entailment Thesis that Kripke and many others attribute to him. He does endorse two quite different theses concerning the relation between necessity and the a priori, as he conceives them. One is a matter of definition and the other is a very substantial philosophical thesis indeed—to establish it is the aim of the entire Critique of Pure Reason. But Kripke’s arguments against the Entailment Thesis tell against neither of Kant’s theses, as they involve crucially different conceptions of necessity and the a priori. This superficial lack of disagreement masks deep disagreements, but these result from divergent views regarding matters such as realism, modal epistemology, and philosophical methodology; views which Kant does a lot, and Kripke very little, to argue for. (shrink)

This paper reconstructs the Peircean interpretation of Kant's doctrine on the syntheticity of mathematics. Peirce correctly locates Kant's distinction in two different sources: Kant's lack of access to polyadic logic and, more interestingly, Kant's insight into the role of ingenious experiments required in theorem-proving. In this second respect, Kant's analytic/synthetic distinction is identical with the distinction Peirce discovered among types of mathematical reasoning. I contrast this Peircean theory with two other prominent views on Kant's syntheticity, i.e. the Russellian and the (...) Beckian views, and show how Peirce's interpretation of Kant solves the dilemma that each of these two views faces. I also show that Hintikka's criterion for Kant's synthetic judgments, i.e. a new individual introduced by the -instantiation rule, does not capture the most important characteristic of Peirce's theorematic reasoning, i.e. the process of choosing a correct individual. (shrink)

What is central to the progression of a sequence is the idea of succession, which is fundamentally a temporal notion. In Kant's ontology numbers are not objects but rules (schemata) for representing the magnitude of a quantum. The magnitude of a discrete quantum 11...11 is determined by a counting procedure, an operation which can be understood as a mapping from the ordinals to the cardinals. All empirical models for numbers isomorphic to 11...11 must conform to the transcendental determination of time-order. (...) Kant's transcendental model for number entails a procedural semantics in which the semantic value of the number-concept is defined in terms of temporal procedures. A number is constructible if and only if it can be schematized in a procedural form. This representability condition explains how an arbitrarily large number is representable and why Kant thinks that arithmetical statements are synthetic and not analytic. (shrink)

The aggregate EIRP of an N-element antenna array is proportional to N 2. This observation illustrates an effective approach for providing deep space networks with very powerful uplinks. The increased aggregate EIRP can be employed in a number of ways, including improved emergency communications, reaching farther into deep space, increased uplink data rates, and the flexibility of simultaneously providing more than one uplink beam with the array. Furthermore, potential for cost savings also exists since the array can be formed using (...) small apertures. (shrink)

I examine how Kant argues for the transcendental ideality of space. I defend a reading on which Kant accepts the ideality of space because it explains our (actual) knowledge that mathematical judgments are necessarily true. I argue that this reading is preferable over the alternative suggestion that Kant can infer the ideality of space directly from the fact that we have an a priori intuition of space. Moreover, I argue that the reading I propose does not commit Kant to incoherent (...) modal views. If we carefully distinguish between different senses of modality, the fact that our spatial form of intuition is (in some sense) contingent does not undermine the claim that this form can explain how our mathematical judgments are (in some sense) necessary. (shrink)

Robert Hanna argues for the importance of Kant's theories of the epistemological, metaphysical, and practical foundations of the "exact sciences"--relegated to the dustbin of the history of philosophy for most of the 20th century. In doing so he makes a valuable contribution to one of the most active and fruitful areas in contemporary scholarship on Kant.

The article examines how Salomon Maimon’s concept of number as ratio can be used to demonstrate that arithmetical judgments are analytical. Based on his critique of Kant’s synthetic a priori judgments, I show how this notion of number fulfills Maimon’s requirements for apodictic knowledge. Moreover, I suggest that Maimon was influenced by mathematicians who previously defined number as a ratio, such as Wallis and Newton. Following an analysis of the real definition of this concept, I conclude that within the framework (...) of Maimon’s philosophy, arithmetical judgments cannot be analytical, nor is arithmetic an objectively necessary science, but rather only subjectively necessary. We should also cast doubt on his claim that we can create real objects from pure concepts of the understanding. (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)

The ArgumentThe main subject examined in this paper is Immanuel Kant's controversy withPhilosophisches Magazinregarding Kant's new theory of judgments. J. A. Eberhard, editor ofPhilosophisches Magazin, and his colleagues wanted to vindicate the Wollfian traditional concept of judgments by undermining Kant's claims. As will be demonstrated, their arguments were effective mainly in exposing the ambiguity that was inherent in Kant's concept of the synthetic a priori; an ambiguity that resulted from Kant's desire—central to his critique of metaphysics—to present judgments pertaining to (...) mathematics, (dogmatic) metaphysics, and pure natural science as judgments which shared a common form. Exposing this ambiguity was not the intended result, and it was insufficient for the purpose of vindicating the Wollfian tradition. The contributors toPhilosophisches Magazinignored the important properties shared by the class of judgments falling under Kant's concept of synthetic a priori judgments. They also ignored the fact that their position was unable to account for the logical phenomena that motivated Kant to present a new theory of judgments. On the other hand, Kant's theory of judgments was insensitive to the important differences that exist among the distinct types of judgments falling under his concept of a synthetic a priori judgment. This latter point is clearly shown in the controversy regarding the novelty of Kant's concept of a synthetic a priori judgment, and in the controversy regarding the function of intuitions within synthetic judgments.A result of the controversy was that Kant's concept of the synthetic a priori, which he believed to be an exact concept, was revealed to be a metaphor: no more than an invitation to view certain intellectual fields in the light of others. On the other hand, Eberhard and his colleagues failed to come up with satisfactory answers to Kant's questions within their traditional concept of judgment. Both parties refused to acknowledge this result. Consequently, the search for a new logic, a new architectonic order, and a new unity within reason became a general problem for the new generation of philosophers. (shrink)

The direct proof of transcendental idealism, in the Transcendental Aesthetic of Kant's First Critique, has borne the brunt of enormous criticism. Much of this criticism has arisen from a confusion regarding the epistemological nature of the arguments Kant proposes with the alleged ontological conclusions he draws. In this paper I attempt to deflect this species of criticism. I concentrate my analysis on the Metaphysical Expositions of Space and Time. I argue that the argument form of the Metaphysical Expositions is that (...) of disjunctive syllogism and that Kant's primary target of attack is that of Leibnizian relativism. I provide a detailed analysis and reconstruction of the arguments of the Metaphysical Expositions, defending Kant against various claims of argumentative invalidity and incoherence. I conclude by identifying what can properly be inferred regarding the ontological nature of space and time, given my reconstructions of the arguments, and by suggesting the manner in which Kant can deflect objections from the other major proponent of transcendental realism, namely, Newtonian absolutism. (shrink)