Kant’s Mathematical World: Mathematics, Cognition, and Experience (henceforth KMW) argues that Kant’s philosophy of mathematics is grounded in a theory of magnitudes. The 17th and 18th centuries saw remarkable and rapid developments in mathematics. Those advances were accompanied by shifts in the understanding of mathematics as a field, which included the steadily increasing centrality of the notion of number, changing views of the proper methods of mathematics (including the required level of certainty and the means of attaining it), and shifting attitudes toward the foundations of mathematics. Despite the advances and the change in views, however, attitudes toward the foundations of mathematics still reflected ancient ways of thinking. Kant and his contemporaries still described mathematics as the science of magnitudes and their measure, a view whose heritage stretched back to Greek mathematics. Some eminent mathematicians characterized mathematics in this way in mathematical textbooks, although they sometimes seem to be perfunctory introductory comments before passing on to the primary topic of the text and are not serious attempts at foundations.Footnote 1 Regardless, the prevailing attitude in Kant’s time was that, when it came to the general nature of mathematics and its foundations, mathematics should be understood as the science of magnitudes and their measure. Unlike Descartes or Leibniz, Kant was not a mathematician in addition to a philosopher. As a philosopher, Kant is above all interested in foundational issues in mathematics, but more particularly in the foundations of mathematical cognition. He took the theory of magnitudes to be at the foundation of both mathematics and mathematical cognition. KMW argues that Kant deepened and transformed the theory of magnitudes inherited from the Greek mathematical tradition while reworking the 18th-century metaphysics of quantity. KMW focuses on providing a general account of the framework of Kant’s theory of magnitudes and his account of human experience, and it includes its implications for his understanding of intensive and extensive magnitudes, the role of intuition in mathematical cognition and experience, the nature of intuitive and conceptual representation, the representation of continuity, the singularity of intuition, the metaphysics of quantity, and more. Most importantly, it argues that Kant’s theory of magnitudes forges a tight bond between his philosophy of mathematics and his account of human experience. That is, the conditions of the possibility of experience are intertwined with the conditions of the possibility of mathematical cognition at the deepest level and accounts for the mathematical character of the world. As a consequence, one cannot properly understand Kant’s account of experience without understanding his theory of magnitudes and its relation to mathematical cognition.
The goal of KMW is first and foremost to accurately represent Kant’s views on the foundations of mathematical cognition and its relation to human experience. There is, of course, much more that can be said about Kant’s philosophy of mathematics than this book covers; it lays out the general framework while leaving further elaboration of its consequences for his understanding of geometry, arithmetic algebra, and the foundations of mathematical physics to a second book. But even those elaborations do not cover all the richness of Kant’s views on various topics thoughtfully and fruitfully addressed by scholars of Kant’s mathematics in the last 60 years.Footnote 2 Moreover, there is certainly a great deal more to be learned about Kant’s views from a closer examination of mathematics as it was practiced in Kant’s time. KMW focuses on recovering a very different way of thinking about mathematics from our contemporary way of thinking after mathematics was arithmetized, logicized, and axiomatized in the 19th century.
1. Foundations of Mathematics and Mathematical Cognition
The title of KMW refers to Kant’s foundations of mathematical cognition rather than to Kant’s foundations of mathematics. This is intended to reflect Kant’s own approach to the philosophy of mathematics, which contrasts with our modern orientation. Philosophy of mathematics today encompasses a broad range of issues, from general philosophical questions about the ontology and epistemology of numbers to more particular issues, such as whether category theory or set theory provides the best foundation for mathematics, or the implications of the existence of Woodin cardinals for our understanding of real numbers. These more particular issues have bearing on the more general, but it is at the more general philosophical level, especially the epistemological, that contemporary philosophy of mathematics overlaps with Kant’s concerns.
At the same time, Kant’s focus on epistemology includes a more specific demand that is not found in contemporary philosophy of mathematics. Kant aims to provide an account of the nature of our cognitive capacities that make human knowledge, including mathematical knowledge, possible. That is, Kant aims to explain the possibility of mathematical knowledge by tracing the conditions of that knowledge back to our most basic mental capacities, and hence ultimately to the categories and to our pure forms of intuition, space, and time.Footnote 3 Kant’s program thus looks quite different from the issues that concern most contemporary philosophers of mathematics. In fact, it more closely resembles current work in cognitive science that focuses on mathematical cognition, despite the quite radical difference between Kant’s philosophical method and their empirical scientific methods; Kant believes that careful philosophical reflection on the nature of mathematical cognition can inform us about the conditions that make it possible and that those reflections both inform and are informed by reflection on the conditions of human cognition more generally. Moreover, and equally important, Kant’s analysis of the conditions of cognition are situated within the broader context of explaining how and why successful cognitions justify our claims to knowledge.Footnote 4 Kant’s program and contemporary cognitive science also differ in another way. Contemporary work on mathematical cognition by and large presupposes our contemporary understanding of mathematics and its foundations when probing for answers. For example, their investigations of when and how we acquire cognition of natural numbers looks to a contemporary analysis of the natural numbers provided by the philosophy of mathematics, and on that basis, they ask how we represent the fundamental properties of cardinal or ordinal numbers. While answering these questions is undoubtedly informative and important to understanding mathematical cognition, it should at least be an open question whether the most basic elements of human mathematical cognition happen to coincide with the fundamental properties identified by our contemporary foundations of mathematics; whether, for example, the sameness of cardinality of collections is represented by means of one-to-one correspondence or, say, by some more fundamental recognition of the equimultiplicity of two collections.Footnote 5 Although Kant’s project overlaps with contemporary work in cognitive science, it differs not only in method and broader orientation, but also in offering a different perspective by treating mathematics as a science of magnitudes and their measurement.
2. Overview of Kant’s Mathematical World
The book is divided into two parts with an interlude on Euclid and the Euclidean tradition. After an introductory chapter, Part I offers a close reading of the texts to establish that Kant’s theory of magnitudes is fundamental to his thinking about all mathematical cognition and his account of experience. The first two sections of the System of Principles, the Axioms of Intuition and the Anticipations of Anticipation, establish the applicability of the categories of quantity and quality to appearances, and thereby, the applicability of mathematics to them and their properties. They do so by establishing that appearances are, with respect to their intuition, extensive magnitudes, and that sensation and the real properties of objects corresponding to them have an intensive magnitude. The Axioms of Intuition play the most fundamental role in Kant’s theory of magnitudes. It is widely recognized that the Axioms and Anticipations concern the applicability of mathematics to experience, but many have thought that these sections and their principles only concern the applicability of mathematics to experience. This is understandable, since the System of Principles as a whole focuses on the conditions for the possibility of experience. Nevertheless, the Axioms of Intuition in particular reveals much more about the nature of mathematical cognition and pure mathematics than has been appreciated. Unfortunately, the structure of the Critique encourages misunderstandings. Both the Transcendental Aesthetic and the Axioms of Intuition make important claims about space, time, and mathematics that appear to overlap in some ways and contradict in others. Each discusses the possibility of mathematics and lays claim to establishing the applicability of mathematics to experience. In addition, each discusses space and time in quite different ways, particularly with respect to the relative priority of parts to whole. This has led to some confusions among commentators, some of whom hold that the Axioms account of mathematics duplicates the Transcendental Aesthetic, or that the Transcendental Aesthetic concerns topological features of space while the Axioms concerns the introduction of a metric, or that the two sections of the Critique are simply inconsistent. In Chapter 2, I endorse an interpretation of the two sections according to which the Transcendental Aesthetic abstracts from the conditions under which the understanding determines our cognition of spaces and times and our mathematical cognitions, while the Axioms takes those conditions into account. Thus, the theory of magnitudes which emerges in the Axioms is at the basis of his account of all determinate spaces and times of both empirical and mathematical cognition, and of all mathematical cognition, pure as well as applied, topological as well as metrical.
Chapter 2 clears the way for a closer consideration of the Axioms in Chapter 3, which explains Kant’s concept of magnitude and reconstructs his two-stage argument for the principle of the Axioms, neither of which have not been properly understood. Crucially, Kant distinguishes between to senses of magnitude [Größe]: quanta, and quantitas. A Quantum is something insofar as it can be measured, while quantitas is the determination of a quantum that answers the question how big it is(i.e., how much or how many it is) according to some unit of measure. Kant defines a magnitude as a homogeneous manifold in intuition in general.Footnote 6 I argue against a textual emendation suggested by Hans Vaihinger and widely adopted that adds the concept of unity to Kant’s definition of magnitude. The emendation suggests, and many have thought, that this secures the correspondence between the concept of magnitude and all three categories of quantity. I argue instead that the concept of magnitude Kant defines, namely quanta, corresponds to the category of plurality, while his more abstract notion of magnitude, quantitas, corresponds to all three categories. This clarification shows that Kant’s most basic concept of magnitude concerns a manifold that is homogeneous, and opens the door to a new interpretation of the arguments for the principle of the Axioms. I argue that the argument proceeds in two steps, the first securing that appearances as well as all intuitions are magnitudes, while the second establishes that they are all extensive magnitudes. I also argue that the first step reveals that in Kant’s view, the synthesis that generates a determinate space or time underlies all pure mathematical cognition as well as all apprehension and perception of objects.
Chapter 4 turns to Kant’s distinction between extensive and intensive magnitudes. Kant argues that determinate spaces and times are extensive magnitudes, while the real properties of objects of experience (and our sensations of them) have an intensive magnitude. After explaining how Kant draws the distinction, the chapter addresses a threat of an infinite regress in the representation of extensive magnitudes. Kant defines extensive magnitudes as those in which the representation of the parts makes possible and necessarily precedes the representation of the whole. If the parts are themselves extensive magnitudes, an infinite regress ensues. The solution to the regress draws a distinction between the representation of an extensive magnitude, which is determinate, and the representation of its parts, which are indeterminate and not themselves represented as extensive magnitudes. I argue against two possible accounts of how to understand the indeterminate representation of parts, one of which has played a role in recent Kantian conceptualist/nonconceptualist debates. I then argue for a solution that appeals to a continuous synthesis underlying our representation of continuous magnitudes. On the account I defend, intuition makes it possible to directly represent a continuous manifold in space and time and in that sense only indeterminately represents its parts.
My claim that intuition is what allows the direct representation of a continuous manifold appears to be in tension with the singularity of intuition, which is usually construed as the representation of a singular object. Chapter 5 addresses this tension. I consider the nature of intuitive representation more closely, and develop a new interpretation of singular representation. I argue that it is better to understand the singularity of intuition as a mode of representing singularly, that this mode of representing is compatible with the representation of a continuous manifold, and that the latter in fact underlies and makes possible the singular representation of singular objects. I then consider Kant’s understanding of the distinction between concrete and abstract in order to explain the relation between the singularity of intuition and in concreto representation, and explain the sense in which quanta and quantitas are relatively abstract concepts whose objects may nevertheless be concrete.
These topics constitute Part I of the book, which proceeds as far as possible in reconstructing Kant’s views through a close reading of the texts. A more in-depth examination of Kant’s views requires an understanding of the theory of magnitudes that descended through the Euclidean tradition. Chapter 6 is an interlude between the two main parts of the book that provides a focused and compressed account of this history. It begins with an examination of the Elements without presupposing prior acquaintance with it. In particular, it describes the theory of magnitudes and theory of numbers presented in Books V and VII of the Elements. It then briefly discusses the long Euclidean tradition that followed and touches on a few of Kant’s immediate predecessors. While necessarily incomplete and selective, it brings out some of the most relevant features of that tradition for understanding Kant’s views.
Part II of the book then delves more deeply into Kant’s views against this backdrop. Chapter 7 shows how Kant reworks the Euclidean theory of magnitudes described in Chapter 6. It is at the heart of my interpretation of Kant. Euclid does not explicitly define “magnitude” or “homogeneity.” Instead, the meaning of these terms and the mathematical properties of magnitudes are revealed by their role in the theory of proportions. Kant attempts to do better by giving an account of what it is to be a magnitude and what makes our cognition of them possible, refashioning the theory of magnitudes at its foundations while preserving its consequences. As mentioned, Kant defines magnitude (quantum) as a homogeneous manifold in intuition in general; I argue that Kant here means homogeneous in the sense of what I call “strict” homogeneity, that is, a manifold that exhibits numerical difference with no qualitative or specific difference. Thus, the manifold of space is strictly homogeneous because the manifold exhibits the "numerical" difference of its indeterminately represented parts without any qualitative difference. Concepts on their own cannot represent such a manifold because concepts on their own can only represent specific differences. On the other hand, intuition makes the representation of a strictly homogeneous manifold possible. I also argue for two further claims; according to Kant, the categories of quantity make the cognition of the part-whole relations of magnitudes possible, and that the synthesis of composition underlying the representation of magnitudes is distinctive of mathematical cognition and the cognition of the mathematical properties of appearances. I then show how Kant’s definition of homogeneity corresponds to the Euclidean.
I next pivot to Kant’s reaction to the German rationalist metaphysics of quantity. Chapter 8 reviews how Leibniz, Wolff, and Baumgarten attempt to employ metaphysical definitions of quantity, quality, equality, and similarity in a way that incorporates mathematics into a larger metaphysical framework. It then shows how Kant fundamentally reformulates their metaphysics by introducing his distinction between quanta and quantitas and assigning a role for intuition in representing quanta, which radically departs from their understanding of the metaphysics of quantity. Just as Kant transformed the Euclidean theory of magnitudes at its foundations while preserving the theory of proportions, Kant transformed the metaphysics of quantity found in Leibniz, Wolff, and Baumgarten at its foundations while retaining its basic architecture. The result is Kant’s innovative understanding of mathematical cognition and its relation to the mathematical character of experience.
The work of the previous chapters reveals that Kant’s account of magnitudes, the role of the categories of quantity, and his understanding of composition are all fundamentally mereological. This leaves a gap between Kant’s mereological foundation of magnitudes and the rich mathematical notion of magnitudes found in Euclid and the Euclidean tradition, which raises questions about the adequacy of Kant’s account. Does Kant’s mereological understanding of magnitude fall short of accounting for the mathematical nature of magnitudes found in Euclid’s theory of proportions and required for an adequate explanation of mathematical cognition? Chapter 9 addresses this gap. I argue that analysis and reconstruction of the assumptions of Euclid’s and Kant’s theories of magnitude reveal that both tacitly presuppose what I characterize as a general theory of pure concrete measurement. It is concrete because it is grounded in the relations among concrete magnitudes rather than in abstract notions of quantity. It is pure in the sense that Euclidean geometry is pure and abstracts from empirical considerations. And it is general because it does not presuppose particular numbers. The chapter argues that the relation of equality plays the pivotal role in bridging the gap between the mereology and mathematics of magnitudes. It also shows that Kant was aware that equality plays this pivotal role.
Kant almost certainly had more to say about the foundations of the general theory of magnitude and its relation to mathematics and the world, but it is at this point that we reach the limits of Kant’s explicit theorizing in either published or unpublished texts about the nature and cognition of magnitudes. The contents of this book outline the basic framework of Kant’s theory of magnitudes and its role in both Kant’s account of mathematical cognition and in the mathematical character of experience. It also sets the stage for a more focused examination of Kant’s views of geometry, arithmetic, algebra, and the foundations of mathematical physics, which will appear in future work.
Daniel Sutherland is a Professor at the University of Illinois Chicago. His primary focus of research is Kant’s theoretical philosophy, particularly his philosophy of mathematics and science. He also has interests in Kant’s Third Critique, contemporary theories of mathematical cognition, and philosophy of mind.
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