Kant and the determinacy of intuition

A central issue in debates about Kant and nonconceptualism concerns the nature of intuition. There is sharp disagreement among Kant scholars about both whether, prior to conceptualization, mere intuition can be considered conscious and, if so, how determinate this consciousness is. In this article, I argue that Kant regards pre-synthesized intuition as conscious but indeterminate. To make this case, I contextualize Kant's position through the work of H.S. Reimarus, a predecessor of Kant who influenced his views on animals, infants, and the role of attention. I use Reimarus to clarify Kant's otherwise ambiguous commitments on the determinacy of intuition in animals and newborns, and the role attention, concepts, and judgment play in making intuited contents determinate. This contextualization helps to shed light on Kant's discussion of pre-synthesized intuition in the threefold synthesis of the A-Deduction by demonstrating that Kant's theory of mind in the deduction offers transcendental grounding for empirical accounts of infant development like Kant and Reimarus's. The upshot is a Kant at odds with many recent interpretations of his theory of mind: pre-synthesized intuition is conscious but indeterminate.     

Urbild und Abbild. Leibniz, Kant und Hausdorff über das Raumproblem

The article attempts to reconsider the relationship between Leibniz’s and Kant’s philosophy of geometry on the one hand and the nineteenth century debate on the foundation of geometry on the other. The author argues that the examples used by Leibniz and Kant to explain the peculiarity of the geometrical way of thinking are actually special cases of what the Jewish-German mathematician Felix Hausdorff called “transformation principle”, the very same principle that thinkers such as Helmholtz or Poincaré applied in a more general form in their celebrated philosophical writings about geometry. The first two parts of the article try to show that Leibniz’s and Kant’s philosophies of geometry, despite their differences, appear to be preoccupied with the common problem of the impossibility to grasp conceptually the intuitive difference between two figures (such as a figure and its scaled, displaced or mirrored copy). In the third part, it is argued that from the perspective of Hausdorff’s philosophical-geometrical reflections, this very same problem seems to find a more radical application in Helmholtz’s or Poincaré’s thought experiments on the impossibility of distinguishing distorted copies of our universe from the original one. I draw the conclusion that in Hausdorff’s philosophical work, which has received scholarly attention only recently, one can find not only an original attempt to frame these classical arguments from a set-theoretical point of view, but also the possibility of considering the history of philosophy of geometry from an uncommon perspective, where especially the significance of Kant’s infamous appeal to “intuition” can be judged by more appropriate standards.     

The mathematical form of measurement and the argument for Proposition I in Newton’s Principia

Newton characterizes the reasoning of Principia Mathematica as geometrical. He emulates classical geometry by displaying, in diagrams, the objects of his reasoning and comparisons between them. Examination of Newton??s unpublished texts (and the views of his mentor, Isaac Barrow) shows that Newton conceives geometry as the science of measurement. On this view, all measurement ultimately involves the literal juxtaposition??the putting-together in space??of the item to be measured with a measure, whose dimensions serve as the standard of reference, so that all quantity (which is what measurement makes known) is ultimately related to spatial extension. I use this conception of Newton??s project to explain the organization and proofs of the first theorems of mechanics to appear in the Principia (beginning in Sect. 2 of Book I). The placementof Kepler??s rule of areas as the first proposition, and the manner in which Newton proves it, appear natural on the supposition that Newton seeks a measure, in the sense of a moveable spatial quantity, of time. I argue that Newton proceeds in this way so that his reasoning can have the ostensive certainty of geometry.     

What Does It Mean That “Space Can Be Transcendental Without the Axioms Being So”?

In 1870, Hermann von Helmholtz criticized the Kantian conception of geometrical axioms as a priori synthetic judgments grounded in spatial intuition. However, during his dispute with Albrecht Krause (Kant und Helmholtz über den Ursprung und die Bedeutung der Raumanschauung und der geometrischen Axiome. Lahr, Schauenburg, 1878), Helmholtz maintained that space can be transcendental without the axioms being so. In this paper, I will analyze Helmholtz’s claim in connection with his theory of measurement. Helmholtz uses a Kantian argument that can be summarized as follows: mathematical structures that can be defined independently of the objects we experience are necessary for judgments about magnitudes to be generally valid. I suggest that space is conceived by Helmholtz as one such structure. I will analyze his argument in its most detailed version, which is found in Helmholtz (Zählen und Messen, erkenntnistheoretisch betrachtet 1887. In: Schriften zur Erkenntnistheorie. Springer, Berlin, 1921, 70–97). In support of my view, I will consider alternative formulations of the same argument by Ernst Cassirer and Otto Hölder.     

Looking for laws in all the wrong spaces: Kant on laws,the understanding,and space

Prolegomena §38 is intended to elucidate the claim that the understanding legislates a priori laws to nature (the ‘Legislation Thesis’). Kant cites various laws of geometry as examples and discusses a derivation of the inverse‐square law from such laws. I address 4 key interpretive questions about this cryptic text that have not yet received satisfying answers: (a) How exactly are Kant's examples of laws supposed to elucidate the Legislation Thesis? (b) What is Kant's view of the epistemic status of the inverse‐square law and, relatedly, of the legitimacy of the geometric derivation of that law? (c) Whose account of laws, the understanding, and space is Kant critiquing in the passage? (d) What positive account of the relationship between laws, the understanding, and space is Kant offering in the passage? My answer to (d) depends crucially on my answers to (a)–(c). As I interpret Kant, he holds that a wide range of a priori laws—including geometric laws, the inverse‐square law, and the universal laws discussed in the Analytic of Principles—are ‘grounded’ (a technical term defined in the paper) in categorial syntheses rather than the intrinsic nature of the space given to us in pure intuition.     

Local axioms in disguise: Hilbert on Minkowski diagrams

While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski??s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating ??rigorously?? with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as ??drawn formulas??, and formulas as ??written diagrams??, thus suggesting that the former encapsulate propositional information which can be extracted and translated into formulas. In the case of Minkowski diagrams, local geometrical axioms were actually being produced, starting with the diagrams, by a process that was both constrained and fostered by the requirement, brought about by the axiomatic method itself, that geometry ought to be made independent of analysis. This paper aims at making a twofold point. On the one hand, it shows that Minkowski??s diagrammatic methods in number theory prompted Hilbert??s axiomatic investigations into the notion of a straight line as the shortest distance between two points, which start from his earlier work focused on the role of the triangle inequality property in the foundations of geometry, and lead up to his formulation of the 1900 Fourth Problem. On the other hand, it purports to make clear how Hilbert??s assessment of Minkowski??s diagram-based reasoning in number theory both raises and illuminates conceptual compatibility concerns that were crucial to his philosophy of mathematics.     

Reading Kant’s doctrine of schematism algebraically

Kant’s investigations into so-called a priori judgments of pure mathematics in the Critique of Pure Reason (KrV) are mainly confined to geometry and arithmetic both of which are grounded on our pure forms of intuition, space, and time. Nevertheless, as regards notions such as irrational numbers and continuous magnitudes, such a restricted account is crucially problematic. I argue that algebra can play a transcendental role with respect to the two pure intuitive sciences, arithmetic and geometry, as the condition of their possibility. It follows that Kant’s schematism of the concept of magnitude ought to be quantitatively represented in algebraic formulas in general, and also undergo several modifications in order to suit continuities.     

The relationship between space and mutual interaction: Kant contra Newton and Leibniz

Kant claims that we cannot cognize the mutual interaction of substances without their being in space; he also claims that we cannot cognize a ‘spatial community’ among substances without their being in mutual interaction. I situate these theses in their historical context and consider Kant’s reasons for accepting them. I argue that they rest on commitments regarding the metaphysical grounding of, first, the possibility of mutual interaction among substances-as-appearances and, second, the actuality of specific distance-relations among such substances. By illuminating these commitments, I shed light on Kant’s metaphysics of space and its relation to Newton and Leibniz’s views.     

Sensations as Representations in Kant

This paper defends an interpretation of the representational function of sensation in Kant's theory of empirical cognition. Against those who argue that sensations are ‘subjective representations’ and hence can only represent the sensory state of the subject, I argue that Kant appeals to different notions of subjectivity, and that the subjectivity of sensations is consistent with sensations representing external, spatial objects. Against those who claim that sensations cannot be representational at all, because sensations are not cognitively sophisticated enough to possess intentionality, I argue that Kant does not use the term ‘Vorstellung’ to refer to intentional mental states exclusively. Sensations do not possess their own intentionality, but they nevertheless perform a representational function in virtue of their role as the matter of empirical intuition. In empirical intuition, the sensory qualities given in sensation are combined with the representation of space to constitute the intuited appearance. The representational function of sensation consists in sensation being the medium out of which intuited appearances are constituted: the qualities of sensations stand in for what the understanding will judge (conceptualize) as material substance.     

Is “space” a concept? Kant,Durkheim, and French Neo-Kantianism

According to Kant, all humans share a basic form of spatial representation—space is an “a priori intuition.” Durkheim felt that Kant's a priori stance blocked the kind of empirical inquiry that would show human spatial representation to be, on the contrary, quite diverse. Durkheim's claim raises the issues in intellectual history and philosophy addressed in this paper. First, the paper traces Durkheim's reading of Kant through the nineteenth-century French neo-Kantians Renouvier and Hamelin. Second, it argues that Kant's and Durkheim's projects are not, after all, genuine competitors. The result is to reassert the sharp distinction between epistemological and sociological approaches to spatial representation that Durkheim and others tried to collapse. © 1996 John Wiley & Sons, Inc.     

Diagrams as sketches

This article puts forward the notion of ??evolving diagram?? as an important case of mathematical diagram. An evolving diagram combines, through a dynamic graphical enrichment, the representation of an object and the representation of a piece of reasoning based on the representation of that object. Evolving diagrams can be illustrated in particular with category-theoretic diagrams (hereafter ??diagrams*??) in the context of ??sketch theory,?? a branch of modern category theory. It is argued that sketch theory provides a diagrammatic* theory of diagrams*, that it helps to overcome the rivalry between set theory and category theory as a general semantical framework, and that it suggests a more flexible understanding of the opposition between formal proofs and diagrammatic reasoning. Thus, the aim of the paper is twofold. First, it claims that diagrams* provide a clear example of evolving diagrams, and shed light on them as a general phenomenon. Second, in return, it uses sketches, understood as evolving diagrams, to show how diagrams* in general should be re-evaluated positively.     

Kant on Duty in the Groundwork

Barbara Herman offers an interpretation of Kant??s Groundwork on which an action has moral worth if the primary motive for the action is the motive of duty. She offers this approach in place of Richard Henson??s sufficiency-based interpretation, according to which an action has moral worth when the motive of duty is sufficient by itself to generate the action. Noa Latham criticizes Herman??s account and argues that we cannot make sense of the position that an agent can hold multiple motives for action and yet be motivated by only one of them, concluding that we must accept a face-value interpretation of the Groundwork where morally worthy actions obtain only when the agent??s sole motive is the motive of duty. This paper has two goals, one broad and one more constrained. The broader objective is to argue that interpretations of moral worth, as it is presented in the Groundwork, depend on interpretations of Kant??s theory of freedom. I show that whether we can make sense of the inclusion of nonmoral motives in morally worthy actions depends on whether the ??always causal framework?? is consistent with Kant??s theory of freedom. The narrow goal is to show that if we adopt an ??always causal?? framework for moral motivation, then Herman??s position and her critique of the sufficiency-based approach fail. Furthermore, within this framework I will specify a criterion for judging whether an action is determined by the motive of duty, even in the presence of nonmoral motives. Thus, I argue Latham??s conclusion that we must accept a face-value interpretation is incorrect.     

Constructive geometrical reasoning and diagrams

Modern formal accounts of the constructive nature of elementary geometry do not aim to capture the intuitive or concrete character of geometrical construction. In line with the general abstract approach of modern axiomatics, nothing is presumed of the objects that a geometric construction produces. This study explores the possibility of a formal account of geometric construction where the basic geometric objects are understood from the outset to possess certain spatial properties. The discussion is centered around Eu, a recently developed formal system of proof (presented in Mumma (Synthese 175:255?C287, 2010)) within which Euclid??s diagrammatic proofs can be represented.     

Idealization and external symbolic storage: the epistemic and technical dimensions of theoretic cognition

This paper explores some of the constructive dimensions and specifics of human theoretic cognition, combining perspectives from (Husserlian) genetic phenomenology and distributed cognition approaches. I further consult recent psychological research concerning spatial and numerical cognition. The focus is on the nexus between the theoretic development of abstract, idealized geometrical and mathematical notions of space and the development and effective use of environmental cognitive support systems. In my discussion, I show that the evolution of the theoretic cognition of space apparently follows two opposing, but in truth, intrinsically aligned trajectories. On the epistemic plane, which is the main focus of Husserl??s genetic phenomenological investigations, theoretic conceptions of space are progressively constituted by way of an idealizing emancipation of spatial cognition from the concrete, embodied intentionality underlying the human organism??s perception of space. As a result of this emancipation, it ultimately becomes possible for the human mind to theoretically conceive of and posit space as an ideal entity that is universally geometrical and mathematical. At the same time, by synthesizing a range of literature on spatial and mathematical cognition, I illustrate that for the theoretic mind to undertake precisely this emancipating process successfully, and further, for an ideal and objective notion of geometrical and mathematical space to first of all become fully scientifically operative, the cognitive support provided by a range of specific symbolic technologies is central. These include lettered diagrams, notation systems, and more generally, the technique of formalization and require for their functioning various cognitively efficacious types of embodiment. Ultimately, this paper endeavors to understand the specific symbolic-technological dimensions that have been instrumental to major shifts in the development of idealized, scientific conceptions of space. The epistemic characteristics of these shifts have been previously discussed in genetic phenomenology, but without devoting sufficient attention to the constructive role of symbolic technologies. At the same time, this paper identifies some of the irreducible phenomenological and epistemic dimensions that characterize the functioning of the historically situated, embodied and distributed theoretic mind.     

Is Intuition Necessary for Defending Platonism?

G?del asserts that his philosophy falls under the category of conceptual realism. This paper gives a general picture of G?del’s conceptual realism’s basic doctrines, and gives a way to understand conceptual realism in the background of Leibniz’s and Kant’s philosophies. Among philosophers of mathematics, there is a widespread view that Platonism encounters an epistemological difficulty because we do not have sensations of abstract objects. In his writings, G?del asserts that we have mathematical intuitions of mathematical objects. Some philosophers do not think it is necessary to resort to intuition to defend Platonism, and other philosophers think that the arguments resorting to intuition are too na?ve to be convincing. I argue that the epistemic difficulty is not particular to Platonism; when faced with skepticism, physicalists also need to give an answer concerning the relationship between our experience and reality. G?del and Kant both think that sensations or combinations of sensations are not ideas of physical objects, but that, to form ideas of physical objects, concepts must be added. However, unlike Kant, G?del thinks that concepts are not subjective but independent of our minds. Based on my analysis of G?del’s conceptual realism, I give an answer to the question in the title and show that arguments resorting to intuition are far from na?ve, despite what some philosophers have claimed.     

Hume on space, geometry, and diagrammatic reasoning

Hume??s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume??s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume??s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume??s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume??s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume??s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume??s most original contributions. First, Hume??s epistemological model invokes the ??confounding?? of indivisible minima to explain the appearance of spatial continuity. Second, Hume??s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the ??loose judgements?? of the vulgar.     

Peirce’s ‘Prescision’ as a Transcendental Method

Abstract

In this Paper I interpret Charles S. Peirce’s method of prescision as a transcendental method. In order to do so, I argue that Peirce’s pragmatism can be interpreted in a transcendental light only if we use a non‐justificatory understanding of transcendental philosophy. I show how Peirce’s prescision is similar to some abstracting procedure that Immanuel Kant used in his Critique of Pure Reason. Prescision abstracts from experience and thought in general those elements without which such experience and thought would be unaccountable. Similarly, in the Aesthetics, Kant isolated the a priori forms of intuition by showing how they could be abstracted from experience in general, while experience in general cannot be thought without them. However, if Peirce’s and Kant’s methods are similar in this respect, they reached very different conclusions.     

GARRETT ON THE THEOLOGICAL OBJECTION TO HUME'S COMPATIBILISM

Between 1927 and 1936, Martin Heidegger devoted almost one thousand pages of close textual commentary to the philosophy of Immanuel Kant. This article aims to shed new light on the relationship between Kant and Heidegger by providing a fresh analysis of two central texts: Heidegger's 1927/8 lecture course Phenomenological Interpretation of Kant's Critique of Pure Reason and his 1929 monograph Kant and the Problem of Metaphysics. I argue that to make sense of Heidegger's reading of Kant, one must resolve two questions. First, how does Heidegger's Kant understand the concept of the transcendental? Second, what role does the concept of a horizon play in Heidegger's reconstruction of the Critique? I answer the first question by drawing on Cassam's model of a self-directed transcendental argument (‘The role of the transcendental within Heidegger's Kant’), and the second by examining the relationship between Kant's doctrine that ‘pure, general logic’ abstracts from all semantic content and Hume's attack on metaphysics (‘The role of the horizon within Heidegger's Kant’). I close by sketching the implications of my results for Heidegger's own thought (‘From Heidegger's Kant to Sein und Zeit’). Ultimately, I conclude that Heidegger's commentary on the Critical system is defined, above all, by a single issue: the nature of the ‘form’ of intentionality.     

The Importance of Cartesian Triangles: A New Look at Descartes's Ontological Argument

In this paper, I argue that commentators have missed a significant clue given by Descartes in coming to understand his 'ontological' proof for the existence of God. In both the analytic and synthetic presentations of the proof throughout his writings, Descartes notes that the proof works 'in the same way' as a particular geometrical proof. I explore the significance of such a parallel, and conclude that Descartes could not have intended readers to think that the argument consists of some kind of intuition. I argue that for Descartes the attribute of existence is a 'second-order' attribute that is demonstrated to belong to the idea of God on the basis of 'first-order' attributes. The proof, properly understood, is in fact a demonstration. Having brought to light the geometrical parallels between the ontological and geometrical proofs, we have new evidence to resolve the 'intuition versus demonstration' controversy that has characterized much of the discussion of Descartes's ontological argument.     

On A- and B-theoretic elements of branching spacetimes

This paper assesses branching spacetime theories in light of metaphysical considerations concerning time. I present the A, B, and C series in terms of the temporal structure they impose on sets of events, and raise problems for two elements of extant branching spacetime theories??McCall??s ??branch attrition??, and the ??no backward branching?? feature of Belnap??s ??branching space?Ctime????in terms of their respective A- and B-theoretic nature. I argue that McCall??s presentation of branch attrition can only be coherently formulated on a model with at least two temporal dimensions, and that this results in severing the link between branch attrition and the flow of time. I argue that ??no backward branching?? prohibits Belnap??s theory from capturing the modal content of indeterministic physical theories, and results in it ascribing to the world a time-asymmetric modal structure that lacks physical justification.