Redrawing Kant's philosophy of mathematics

This essay offers a strategic reinterpretation of Kant's philosophy of mathemat- ics in Critique of Pure Reason via a broad, empirically based reconception of Kant's conception of drawing. It begins with a general overview of Kant's philosophy of mathematics, observing how he differentiates mathematics in the Critique from both the dynamical and the philosophical. Second, it examines how a recent wave of critical analyses of Kant's constructivism takes up these issues, largely inspired by Hintikka's unorthodox conception of Kantian intuition. Third, it offers further analyses of three Kantian concepts vitally linked to that of drawing. It concludes with an etymologically based exploration of the seven clusters of meanings of the word drawing to gesture toward new possibilities for interpreting a Kantian philosophy of mathematics.     

The Pleasures of Contra‐purposiveness: Kant,the Sublime,and Being Human

Serious doubts have been raised about the coherence of theories of the sublime and the usefulness of the concept. By contrast, the sublime is increasingly studied as a key function in Kant's moral psychology and in his ethics. This article combines methodological conservatism, approaching the topic from within Kant's discussion of aesthetic judgment, with reconstruction of a conception of human agency that is tenable on Kantian grounds. I argue that a coherent theory of the sublime is possible and useful, and the experience of the sublime is significant for our self‐conception as agents. However, the chief interest in the sublime is not moral.     

Deleuze's Use of Kant's Argument from Incongruent Counterparts

The aim of this paper is to explore Deleuze's use of Kant's argument from incongruent counterparts, which Kant uses to show the existence of what he calls an “internal difference” within things. I want to explore how Deleuze draws out an important distinction between the concept and the Idea, and provides an incisive account of his relationship to both the Kantian and Leibnizian projects. First, I look at Kant's use of the argument to provide a refutation of the Leibnizian account of space, before showing how this criticism in fact rests on the question of the conceptual determination of object. Second, I show how Deleuze develops a taxonomy of difference on the basis of his reading of Kant's argument. Finally, I look at what Deleuze sees as the limitations in Kant's understanding of this concept and Deleuze's attempt to overcome these limitations through the introduction of the notion of the Idea, which will provide a genetic and nonconceptual account of the object. In doing so, I show why Deleuze takes the formulation of an adequate account of difference to be one of the central aims of his own metaphysics.     

Kant on the normativity of taste: The role of aesthetic ideas

For Kant, the form of a subject's experience of an object provides the normative basis for an aesthetic judgement about it. In other words, if the subject's experience of an object has certain structural properties, then Kant thinks she can legitimately judge that the object is beautiful—and that it is beautiful for everyone. My goal in this paper is to provide a new account of how this ‘subjective universalism’ is supposed to work. In doing so, I appeal to Kant's notions of an aesthetic idea and an aesthetic attribute, and the connection that Kant makes between an object's expression of rational and the normativity of aesthetic judgements about it.     

The German Reformation and the Mathematization of the Created World

This article explores the emergence of mathematics and mathematical method as a means of defining authoritative truth in the thought of some scholars in the German Reformation. Against the background of Martin Luther's critique of Aristotelian philosophy, Philip Melanchthon presented mathematics as an ideal discipline for preparing the mind to understand God. His approach drew on the work of humanist mathematicians such as Regiomontanus. It finds resonances in the work of the Basel humanist Simon Grynaeus, and (in a less mathematically informed way) in the thought of Peter Ramus. These discussions about the divine nature and certainty of mathematical truth formed the context within which Johannes Kepler's Platonist astronomy emerged.     

The influence of Platonism on mathematics research and theological beliefs

An age-old debate in the philosophy of mathematics is whether mathematics is discovered or invented. There are four popular viewpoints in this debate, namely Platonism, formalism, intuitionism, and logicism. A natural question that arises is whether belief in one of these viewpoints affects the mathematician’s research? In particular, does subscribing to a Platonist or a formalist viewpoint influence how a mathematician conducts research? Does the area of research influence a mathematician’s beliefs on the nature of mathematics? How are the beliefs regarding the nature of mathematics connected to theological beliefs? In order to investigate these questions, five professional research mathematicians were interviewed. The mathematicians worked in diverse areas within analysis, algebra, and within applied mathematics, and had a combined 160 years of research experience. Although none of the mathematicians wanted to be pigeonholed into any one category of beliefs, the study revealed that four of the mathematicians leaned towards Platonism, which runs contrary to the popular notion that Platonism is an exception today. This study revealed that beliefs regarding the nature of mathematics influenced how mathematicians’ conducted research and were deeply connected to their theological beliefs. The findings are presented in the form of vignettes that give an insight into the mathematical and theological belief structures of the mathematicians.     

The Sound of Bedrock: Lines of Grammar between Kant,Wittgenstein, and Cavell

In ‘Aesthetics Problems of Modern Philosophy’ Stanley Cavell proposes, first, that Kant's characterization of judgments of beauty may be read as a Wittgensteinian grammatical characterization, and, second, that the philosophical appeal to ‘what we say and mean’ partakes of the grammar of judgment of beauty. I argue first that the expression of the dawning of an aspect partakes of the grammar of judgments of beauty as characterized by Kant, and may also be seen—on a prevailing way of thinking about concepts and how they relate to their instances—to have the same kind of significance that judgments of beauty have according to Kant. And then I argue that there are good (Wittgensteinian) reasons for being suspicious of the prevailing conception of concepts, and therefore good reasons for being suspicious of the proposed understanding of the significance of aspect perception—an understanding that has attracted many readers of Wittgenstein's remarks on aspects. This leads me to suggest that it is actually the philosophical appeal to ordinary language that has the kind of significance that the Kantian picture attributes to judgments of beauty and to the seeing of aspects. In this way, I offer a way to vindicate Cavell's second proposal.     

Aesthetic Judging and The Intentionality of Pleasure

I point out some unclarities in Allison's interpretation of Kant's aesthetic theory, specifically in his account of the free play of the faculties. I argue that there is a tension between Allison's commitment to the intentionality of the pleasure involved in a judgment of beauty, and his view that the pleasure is distinct from the judgment, and I claim that the tension should be resolved by rejecting the latter view. I conclude by addressing Allison's objection that my own view fails to accommodate judgments of non-beauty or ugliness.     

Kant on Informed Pure Judgments of Taste

Two dominant interpretations of Kant's notion of adherent beauty, the conjunctive view and the incorporation view, provide an account of how to form informed aesthetic assessments concerning artworks. According to both accounts, judgments of perfection play a crucial role in making informed, although impure, judgments of taste. These accounts only examine aesthetic responses to objects that meet or fail to meet the expectations we have regarding what they ought to be. I demonstrate that Kant's works of genius do not fall within either of these categories. The distinguishing features of these works, namely, originality and exemplarity, become unrecognizable on these interpretations because originality and exemplarity lie in the work's ability to exceed one's expectations concerning its form and content. They contribute to artistic beauty through alternative transformation methods distinct from that of abstraction, namely, concept expansion and repudiation. These additional accounts of transformation lead to a rather surprising outcome: works of genius turn out to be paradigm cases where one can and indeed ought to form informed pure judgments of taste.     

Aesthetic Judgment and Perceptual Normativity

I draw a connection between the question, raised by Hume and Kant, of how aesthetic judgments can claim universal agreement, and the question, raised in recent discussions of nonconceptual content, of how concepts can be acquired on the basis of experience. Developing an idea suggested by Kant's linkage of aesthetic judgment with the capacity for empirical conceptualization, I propose that both questions can be resolved by appealing to the idea of “perceptual normativity”. Perceptual experience, on this proposal, involves the awareness of its own appropriateness with respect to the object perceived, where this appropriateness is more primitive than truth or veridicality. This means that a subject can take herself to be perceiving an object as she (and anyone else) ought to perceive it, without first recognizing the object as falling under a corresponding concept. I motivate the proposal through a criticism of Peacocke's account of concept‐acquisition, which, I argue, rests on a confusion between the notion of a way something is perceived, and that of a way it is perceived as being. Whereas Peacocke's account of concept‐acquisition depends on an illicit slide between these two notions, the notion of perceptual normativity allows a legitimate transition between them: if someone's perceiving something a certain way involves her taking it that she ought to perceive it that way, then she perceives the thing as being a certain way, so that the corresponding concept is available to her in perceptual experience.     

Dependence on Wrongdoing in the Consumption of Meat: A Kantian Analysis

Kant's ethics is used by some as a defence of the exploitation of animals and is criticised by others for not recognising any moral relevance of the plight of animals. These appeals overlook the broad applicability of Kant's principles. In this article, I argue that Kant's ethics implies a duty to abstain from most meat and some other animal products derived from farming. I argue that there is a Kantian principle not to choose goods that have been derived from wrongdoing, with certain qualifications. This principle isolates the wrong of using others to commit wrongdoing on one's behalf. As has been argued by others, Kant's ethics implies that animal farming as we know it in our society almost universally involves wrongdoing and the slaughter of animals is especially tied to wrongdoing. I argue for a broad sense in which these ideas together imply that choosing farmed meat, and probably other animal products, is treating animal industry workers as mere means. Thus, we have a Kantian duty to abstain from these products.     

The Principle of Sufficient Reason,the Ontological Argument and the Is/Ought Distinction

Kant's attack on metaphysics consists in large part in his attack on a principle that he names the Supreme Principle of Pure Reason. This principle, it is not often noticed, is the Principle of Sufficient Reason [PSR]. In interpreting this principle as such, I argue that Kant's attack on the PSR (and thereby his attack on dogmatic metaphysics as a whole) depends on Kant's claim that existence is not a first‐order predicate. If existence isn't what Kant calls a real predicate, the PSR is false. While this constitutes a powerful Kantian argument against dogmatic rationalism, it also poses a problem for Kant. For, as I argue, if the PSR is true, Kant's argument that existence isn't a first‐order predicate is false. In this sense, Kant's attack on the PSR is begging the question vis‐á‐vis radical metaphysicians (e.g. Spinoza). Concluding the paper I suggest relying on Kant's 'is'/'ought' distinction in avoiding this circularity, thereby reinforcing the Kantian critique.     

Realizing the Good: Hegel's Critique of Kantian Morality

Although the best‐known Hegelian objection against Kant's moral philosophy is the charge that the categorical imperative is an ‘empty formalism’, Hegel's criticisms also include what we might call the realizability objection. Tentatively stated, the realizability objection says that within the sphere of Kantian morality, the good remains an unrealizable ‘ought’ – in other words, the Kantian moral ‘ought’ can never become an ‘is’. In this paper, I attempt to come to grips with this objection in two steps. In the first section of the paper, I provide an initial reading of the objection, according to which Hegel agrees with Kant's formulation of the realizability problem but disagrees with the specific Kantian solution, namely, with the Kantian idea of the highest good and the doctrine of the postulates. In the second section, I go on to argue that this reading is potentially too superficial and offer a more far‐reaching interpretation whereby Hegel is ultimately targeting fundamental distinctions (between, for instance, reason and sensibility) of Kant's moral theory. I end by employing these more far‐reaching results of Hegel's objection to sketch some features of Hegel's alternative ethical view.     

Reply to the Comments of Longuenesse and Ginsborg

In this discussion I respond to some of the criticisms raised by Béatrice Longuenesse and Hannah Ginsborg to my account of Kant's aesthetic theory presents in Kant's Theory of Taste.     

Form and Matter in Kantian Political Philosophy: A Reply

This paper responds briefly to four reviews of Force and Freedom. Valentini and Sangiovanni criticize what they see as the excessive formalism of the Kantian enterprise, contending that the Kantian project is circular, because it defines rights and freedom together, and that this circularity renders it unable to say anything determinate about appropriate restrictions and permissions. I show that the appearance of circularity arises from a misconstrual of the Kantian idea of a right. Properly understood, Kantian rights are partially indeterminate, but not in a way that causes problems for the account. Ronzoni and Williams seek to broaden the reach of public right, arguing that Kant's abstract approach overlooks pressing questions of social and political life, (Ronzoni) and that public right should allow for democratic deliberation about purposes that go beyond the requirement that a state provide a rightful condition for its members (Williams). I argue that the Kantian view makes room for these factors, but that each must be understood in relation to the formal constraints of right.     

Aesthetic Licence: Foucault's Modernism and Kant's Post-modernism

Recently criticism and theory have maintained that Kant's aesthetic theory is central to modernism, and have used Foucault's archaeology to interrogate that modernism. This paper suggests that archaeology ultimately cannot escape Kant's hold because it depends on Kantian theses. The first section will consider how a recent exponent of an 'archaeological' viewpoint characterizes Kant's theory and will set out the critical role Kant ascribes to art. The second section compares Kant and Foucault to argue that despite appearances their projects turn out to be substantially coterminal. My interest in comparing these critics is not only to be provocative but also to show that post-modernist thinking, at least in the guise of Foucault, needs and uses standards that Kant proposes.     

Comments on Guyer

Paul Guyer's paper “Naturalistic and Transcendental Moments in Kant's Moral Philosophy” raises a set of issues about how Kantian ethics should be understood in relation to present day “philosophical naturalism” that are very much in need of discussion. The paper itself is challenging, even in some respects iconoclastic, and provides a highly welcome provocation to raise in new ways some basic questions about what Kantian ethics is and what it ought to be. Guyer offers us an admirably informed and complex argument, both historical and philosophical, that tangles with some of the most difficult problems in Kant's moral philosophy. It begins with some ambitious and controversial claims about Kant's moral philosophy prior to the Groundwork of 1785. It then offers an interpretation, and also a fundamental criticism, of the Groundwork's attempt to establish the moral law based on the idea of freedom of the will. And finally, it raises – and expresses some opinions on – the large and vexed questions of the relationship between transcendental philosophy and philosophical naturalism, and whether Kantian ethics can be made consistent with a naturalistic philosophical outlook. In these comments I will have something to say on each of these three topics, without pretending (any more than Guyer does) to have exhausted what might be said about them.     

Quantifying Inner Experience?—Kant's Mathematical Principles in the Context of Empirical Psychology

This paper shows why Kant's critique of empirical psychology should not be read as a scathing criticism of quantitative scientific psychology, but has valuable lessons to teach in support of it. By analysing Kant's alleged objections in the light of his critical theory of cognition, it provides a fresh look at the problem of quantifying first‐person experiences, such as emotions and sense‐perceptions. An in‐depth discussion of applying the mathematical principles, which are defined in the Critique of Pure Reason as the constitutive conditions for mathematical‐numerical experience in general, to inner sense will demonstrate why it is in principle possible to justify a quantitative structure of psychological judgments on the grounds of Kant's critical thinking. In conclusion, it will propose how Kant's critique could be used in a constructive way to develop first steps towards a transcendental foundation of psychological knowledge.     

THE POINT OF KANT'S AXIOMS OF INTUITION

Abstract: Kant's Critique of Pure Reason makes important claims about space, time and mathematics in both the Transcendental Aesthetic and the Axioms of Intuition, claims that appear to overlap in some ways and contradict in others. Various interpretations have been offered to resolve these tensions; I argue for an interpretation that accords the Axioms of Intuition a more important role in explaining mathematical cognition than it is usually given. Appreciation for this larger role reveals that magnitudes are central to Kant's philosophy of mathematics and to the part that intuition plays in it.     

Between the Bounds of Experience and Divine Intuition: Kant's Epistemic Limits and Hegel's Ambitions

Hegel seeks to overturn Kant's conclusion that our knowledge is restricted, or that we cannot have knowledge of things as they are in themselves. Understanding this Hegelian ambition requires distinguishing two Kantian characterizations of our epistemic limits: First, we can have knowledge only within the “bounds of experience”. Second, we cannot have knowledge of objects that would be accessible only to a divine intellectual intuition, even though the faculty of reason requires us to conceive of such objects. Hegel aims to drive a wedge between these two characterizations, showing that we can have knowledge beyond Kant's bounds of experience, yet without need of divine intuition. And attention to such knowledge is supposed to show that we have no legitimate need to even conceive of divine intuition and its objects—and no need to conclude that our own knowledge is restricted by comparison, or that we cannot know things as they are in themselves. I focus here on the initial case Hegel uses to introduce this extended argument strategy: we can have more knowledge of natural kinds and laws than would be allowed by Kant's bounds of experience.