I argue that Kant’s distinction between the cognitive roles of sensibility and understanding raises a question concerning the conditions necessary for objective representation. I distinguish two opposing interpretive positions—viz. Intellectualism and Sensibilism. According to Intellectualism all objective representation depends, at least in part, on the unifying synthetic activity of the mind. In contrast, Sensibilism argues that at least some forms of objective representation, specifically intuitions, do not require synthesis. I argue that there are deep reasons for thinking that Intellectualism is (...) incompatible with Kant's view as expressed in the Transcendental Aesthetic. We can better see how Kant’s arguments in the first Critique may be integrated, I suggest, by examining his notion of the 'unity' [Einheit] of a representation. I articulate two distinct ways in which a representation may possess unity and claim that we can use these notions to integrate Kant’s arguments in the Aesthetic and the Transcendental Deduction without compromising the core claims of either Sensibilism or Intellectualism—that intuition is a form of objective representation independent of synthesis, and that the kind of objective representations that ground scientific knowledge of the world require synthesis by the categories. (shrink)

The article presents Leibniz's preoccupation (in 1675?6) with the difference between the notion of infinite number, which he regards as impossible, and that of the infinite being, which he regards as possible. I call this issue ?Leibniz's Problem? and examine Spinoza's solution to a similar problem that arises in the context of his philosophy. ?Spinoza's solution? is expounded in his letter on the infinite (Ep.12), which Leibniz read and annotated in April 1676. The gist of Spinoza's solution is to distinguish (...) between three kinds of infinity and, in particular, between one that applies to substance, and one that applies to numbers, seen as auxiliaries of the imagination. The rest of the paper examines the extent to which Spinoza's solution solves Leibniz's problem. The main thesis I advance is that, when Spinoza and Leibniz say that the divine substance is infinite, in most contexts it is to be understood in non-numerical and non-quantitative terms. Instead, for Spinoza and Leibniz, a substance is said to be infinite in a qualitative sense stressing that it is complete, perfect and indivisible. I argue that this approach solves one strand of Leibniz's problem and leaves another unsolved. (shrink)

Kant's account of space as an infinite given magnitude in the Critique of Pure Reason is paradoxical, since infinite magnitudes go beyond the limits of possible experience. Michael Friedman's and Charles Parsons's accounts make sense of geometrical construction, but I argue that they do not resolve the paradox. I argue that metaphysical space is based on the ability of the subject to generate distinctly oriented spatial magnitudes of invariant scalar quantity through translation or rotation. The set of determinately oriented, constructed (...) geometrical spaces is a proper subset of metaphysical space, thus, metaphysical space is infinite. Kant's paradoxical doctrine of metaphysical space is necessary to reconcile his empiricism with his transcendental idealism. (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)

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)

In this paper, I propose a novel interpretation of the role of the understanding in generating the unity of space and time. On the account I propose, we must distinguish between the unity that belongs to determinate spaces and times – which is a result of category-guided synthesis and which is Kant’s primary focus in §26 of the B-Deduction, including the famous B160–1n – and the unity that belongs to space and time themselves as all-encompassing structures. Non-conceptualist readers of Kant (...) have argued that this latter unity cannot be the product of categorial synthesis. While they are correct that this unity is not the product of any particular act of category-guided synthesis, I argue that conceptualists are right to nevertheless attribute this unity to the understanding. I argue that it is a result of what we can think of as the ‘original’ synthesis of understanding and sensibility themselves – it is a synthesis, moreover, in which the whole is logically prior to the parts. (shrink)

In theCritique of Pure Reason,Kant argues for two principles that concern magnitudes. The first is the principle that ‘All intuitions are extensive magnitudes,’ which appears in the Axioms of Intuition (B202); the second is the principle that ‘In all appearances the real, which is an object of sensation, has an intensive magnitude, that is, a degree,’ which appears in the Anticipations of Perception (B207). A circle drawn in geometry and the space occupied by an object such as a book are (...) paradigm examples of extensive magnitudes, while the intensity of a light is a paradigm example of an intensive magnitude. These principles justify and explain the possibility of applying mathematics to objects of experience. The Axioms principle also explains the possibility of any mathematical cognition at all. (shrink)

Jonathan Lear; XII*—Aristotelian Infinity, Proceedings of the Aristotelian Society, Volume 80, Issue 1, 1 June 1980, Pages 187–210, https://doi.org/10.1093/aris.

This paper aims to show that a proper understanding of what Leibniz meant by “hypercategorematic infinite” sheds light on some fundamental aspects of his conceptions of God and of the relationship between God and created simple substances or monads. After revisiting Leibniz’s distinction between (i) syncategorematic infinite, (ii) categorematic infinite, and (iii) actual infinite, I examine his claim that the hypercategorematic infinite is “God himself” in conjunction with other key statements about God. I then discuss the issue of whether the (...) hypercategorematic infinite is a “whole”, comparing the four kinds of infinite outlined by Leibniz in 1706 with the three degrees of infinity outlined in 1676. In the last section, I discuss the relationship between the hypercategorematic infinite and created simple substances. I conclude that, for Leibniz, only a being beyond all determinations but eminently embracing all determinations can enjoy the pure positivity of what is truly infinite while constituting the ontological grounding of all things. (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)

Descartes’s metaphysics posits a sharp distinction between two types of non-finitude, or unlimitedness: whereas God alone is infinite, numbers, space, and time are indefinite. The distinction has proven difficult to interpret in a way that abides by the textual evidence and conserves the theoretical roles that the distinction plays in Descartes’s philosophy—in particular, the important role it plays in the causal proof for God’s existence in the Meditations. After formulating the interpretive task, I criticize extant interpretations of the distinction. I (...) then propose an alternative at whose core is the idea that whereas the indefinite is a structural, iterative notion, designating the absence of an upper bound, the infinite is an ontic notion, signifying being in general, or what is, without qualification. (shrink)

In the Resolution of the Second Antinomy of the first Critique and the Dynamics chapter of the Metaphysical Foundations of Natural Sciences, Kant presents his critical views on mereology, the study of parts and wholes. He endorses an unusual position: Matter is said to be infinitely divisible without being infinitely divided. It would be mistaken to think that matter consists of infinitely many parts—rather, parts “exist only in the representation of them, hence in the dividing”. This view, according to which (...) parts are created through division somehow, was criticized as obscure early on, and has not received much attention since. Against this trend, I show how a coherent position, which I call Mereological Conceptualism, can be extracted from the sparse textual basis. (shrink)

Many historical and philosophical studies treat infinity as an exclusively quantitative notion, whose proper domain of application is mathematics and physics. The main aim of this paper is to disentangle, by critically examining, three notions of infinity in the early modern period, and to argue that one—but only one—of them is quantitative. One of these non-quantitative notions concerns being or reality, while the other concerns a particular iterative property of an aggregate. These three notions will emerge through examination of three (...) central figures in the period: Locke, Descartes, and Leibniz. (shrink)

This important collection of more than twenty original essays by prominent Kant scholars covers the multiple aspects of Kant’s teaching in relation to his published works. With the Academy edition’s continuing publication of Kant’s lectures, the role of his lecturing activity has been drawing more and more deserved attention. Several of Kant’s lectures on metaphysics, logic, ethics, anthropology, theology, and pedagogy have been translated into English, and important studies have appeared in many languages. But why study the lectures? When they (...) are read in light of Kant’s published writings, the lectures offer a new perspective of Kant’s philosophical development, clarify points in the published texts, consider topics there unexamined, and depict the intellectual background in richer detail. And the lectures are often more accessible to readers than the published works. This book discusses all areas of Kant's lecturing activity. Some essays even analyze in detail the content of Kant's courses and the role of textbooks written by key authors such as Baumgarten, helping us understand Kant’s thought in its intellectual and historical contexts. Contributors: Huaping Lu-Adler; Henny Blomme ; Robert Clewis; Alix Cohen; Corey Dyck; Faustino Fabbianelli; Norbert Fischer; Courtney Fugate; Paul Guyer; Robert Louden; Antonio Moretto; Steve Naragon; Christian Onof; Stephen Palmquist; Riccardo Pozzo; Frederick Rauscher; Dennis Schulting; Oliver Sensen; Susan Shell; Werner Stark; John Zammito; Günter Zöller. (shrink)

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)

This volume contains essays that examine infinity in early modern philosophy. The essays not only consider the ways that key figures viewed the concept. They also detail how these different beliefs about infinity influenced major philosophical systems throughout the era. These domains include mathematics, metaphysics, epistemology, ethics, science, and theology. Coverage begins with an introduction that outlines the overall importance of infinity to early modern philosophy. It then moves from a general background of infinity up through Kant. Readers will learn (...) about the place of infinity in the writings of key early modern thinkers. The contributors profile the work of Descartes, Spinoza, Leibniz, and Kant. Debates over infinity significantly influenced philosophical discussion regarding the human condition and the extent and limits of human knowledge. Questions about the infinity of space, for instance, helped lead to the introduction of a heliocentric solar system as well as the discovery of calculus. This volume offers readers an insightful look into all this and more. It provides a broad perspective that will help advance the present state of knowledge on this important but often overlooked topic. (shrink)

I use recent work on Kant and diagrammatic reasoning to develop a reconsideration of central aspects of Kant’s philosophy of geometry and its relation to spatial intuition. In particular, I reconsider in this light the relations between geometrical concepts and their schemata, and the relationship between pure and empirical intuition. I argue that diagrammatic interpretations of Kant’s theory of geometrical intuition can, at best, capture only part of what Kant’s conception involves and that, for example, they cannot explain why Kant (...) takes geometrical constructions in the style of Euclid to provide us with an a priori framework for physical space. I attempt, along the way, to shed new light on the relationship between Kant’s theory of space and the debate between Newton and Leibniz to which he was reacting, and also on the role of geometry and spatial intuition in the transcendental deduction of the categories. (shrink)

An observation and a thesis: The observation is that, whatever the connection between Kant’s philosophy and Hilbert’s conception of finitism, Kant’s account of geometric reasoning shares an essential idea with the account of finitist number theory in “Finitism”, namely the idea of constructions f from ‘arbitrary’ or ‘generic’ objects of various types. The thesis is that, contrary to a substantial part of contemporary literature on the subject, when Kant referred to number and arithmetic, he was not referring to the natural (...) or whole numbers and their arithmetic, but rather to the real numbers and their arithmetic. (shrink)

This volume contains essays that examine infinity in early modern philosophy. The essays not only consider the ways that key figures viewed the concept. They also detail how these different beliefs about infinity influenced major philosophical systems throughout the era. These domains include mathematics, metaphysics, epistemology, ethics, science, and theology. Coverage begins with an introduction that outlines the overall importance of infinity to early modern philosophy. It then moves from a general background of infinity up through Kant. Readers will learn (...) about the place of infinity in the writings of key early modern thinkers. The contributors profile the work of Descartes, Spinoza, Leibniz, and Kant. Debates over infinity significantly influenced philosophical discussion regarding the human condition and the extent and limits of human knowledge. Questions about the infinity of space, for instance, helped lead to the introduction of a heliocentric solar system as well as the discovery of calculus. This volume offers readers an insightful look into all this and more. It provides a broad perspective that will help advance the present state of knowledge on this important but often overlooked topic. (shrink)

I argue that Kant's claim in the “Transcendental Aesthetic” of the Critique of Pure Reason that space and time are immediately given in intuition as infinite magnitudes is undercut by his general theory of mathematical knowledge. On this general theory, pure intuition does not give objects of any determinate magnitude at all, but only forms of possible objects. Specifically, what pure intuition itself yields is the recognition that any determinate space or time is part of a larger one, but it (...) requires an inference of reason to go from that to the claim that space and time are infinite. I further argue that this result is consistent with Kant's claim in the second-edition “Transcendental Deduction” that the unity of space and time are the products of synthesis, but also means that the unity of space and time as objects cannot be used a premise in the Deduction but can only be regarded as a conclusion of the deduction and the following “System of Principles.”. (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 den aktuellsten Stand der Forschung.