1. Recent Debates and Metaphysical Versus Mathematical Interpretations

I begin with a few rather broad remarks about my approach to working on Kant occasioned by Anderson’s and Heis’ comments. My introduction to Kant’s philosophy of mathematics began with the work of Michael Friedman, Charles Parsons, and Jaakko Hintikka. Parsons had explored Kant’s understanding of magnitudes and Friedman had drawn a connection between Kant’s treatment of magnitudes and the Euclidean tradition, but I realized that Kant’s theory of magnitudes not only expressed a deeply different way of thinking about mathematics from our views today and that it had far greater implications for his philosophy of mathematics and his account of all our experience than had been appreciated. I then reconstructed Kant’s views informed by the long tradition stretching back to Euclid, quite intentionally setting aside some of the primary issues in recent debates concerning Kant’s philosophy of mathematics, in particular, the role of the singularity and immediacy of intuition, Kant’s arguments for the syntheticity of mathematics, and the role of intuition in mathematical inferences. I let Kant’s understanding of magnitudes rather than those debates guide my investigation with the aim of gaining a fresh perspective. The point was to allow Kant’s theory of magnitudes to reframe Kant’s account of mathematical cognition and the nature of human experience to provide a new orientation and more accurate interpretation. I believe this approach reveals a previously overlooked role for intuition in representing magnitudes and a much tighter connection between mathematics and the objects of experience than had been appreciated. It also leads to a fundamental rethinking of the singularity of intuition and lays the groundwork for an enriched understanding of the various roles of intuition in mathematics and our cognition of experience.

My original aim in writing KMW was to first reconstruct Kant’s theory of magnitudes and to then consider Kant’s accounts of geometry, arithmetic, algebra, and the foundations of mathematical physics, but the first task took more space than expected and I had to postpone the latter for a further book. Since a full account of the syntheticity of mathematical judgments and the role of intuition in mathematical inferences requires taking into account the differences between these four subfields of mathematics, I also postponed a systematic treatment of these issues until the second book. While KMW identifies multiple roles for intuition in mathematical cognition that are illuminated by his theory of magnitudes and provides material to address these issues, it does not focus on them or provide all the details. For someone well versed in recent debates, such as Heis, KMW might appear not only simultaneously original and refreshing but also less than fully satisfying. Heis takes me to task for not focusing on the syntheticity of mathematical judgments and the justification of mathematical inferences. In a closely related criticism, he finds my account of the role of intuition “blurry.” I will attempt to provide a clearer picture in what follows without delving too deeply into details that will be addressed in the second book.

Anderson makes a claim about the broad character of my approach and asks me to comment on it. He sees a debate in the coming decades between, on the one hand, a resurgent metaphysical turn in Kant interpretation, and on the other, a mathematical strand of Kant interpretation that eschews first-order metaphysical theorizing and instead focuses on transcendental arguments concerning the conditions of the possibility of the exact sciences and cultural practices. Anderson sees my work as in the latter camp.

There is an important sense in which this is correct. My interest in Kant’s philosophy has first and foremost concerned the presuppositions of mathematics and natural science and I was inspired to tackle these issues in no small part by Michael Friedman, whose work continues the mathematical strand of interpretation that began in the late-19th and early-20th centuries and to which Anderson alludes. Moreover, although I do not think, as some philosophers such as Strawson have, that one can simply jettison Kant’s transcendental idealism and “transcendental psychology” in order to salvage his most valuable insights, I also think there is much of value in Kant even for those who are not committed to his transcendental idealism or transcendental psychology. It is also the case that I was not particularly drawn to resolving the thorny textual and interpretive challenges presented by his transcendental idealism. So, I attempted to focus on Kant’s philosophy of mathematics and science while leaving open what the best interpretation of his transcendental idealism might be.

While these aspects of my relation to Kant’s philosophy naturally align me with what Anderson calls the mathematical strand of Kant interpretation, my initial lack of engagement with Kant’s transcendental idealism was not an expression of the anti-metaphysical reading that is central to that strand. In fact, I think the development of two-aspect metaphysical views and the revival of (broadly) phenomenalist interpretations in recent years is a valuable and important area of research into Kant’s metaphysics. I see no tension taking Kant’s metaphysics seriously while appreciating his work on the exact sciences and I think that they can be mutually informative.

There is another important way in which my approach differs from the mathematical strand Anderson identifies. The latter emphasizes Kant’s transcendental method employed to uncover the presuppositions of mathematics, natural sciences, and cultural practices, applying that method to later developments in our understanding of those fields. It focuses on the presuppositions themselves, rather than an account of how they are grounded in our most basic cognitive capacities and how they make knowledge possible. As I have noted, I see Kant as providing a foundation of mathematical cognition that aims to do just that, which means that, despite radical differences in method and aims, his project partially overlaps with contemporary, empirically based cognitive science. I do not think that the strand of interpretation Anderson endorses has been sympathetic to this aspect of Kant’s project. Nevertheless, I believe that it is in fact Kant’s central project, that one must take it seriously if one wishes to understand Kant’s philosophy of mathematics and its connection to experience, and finally, that Kant’s careful theorizing about mathematical cognition still has lessons for us today.

2. Heis’ Master Argument

Heis takes on board the centrality of Kant’s theory of magnitudes for which I have argued but then reframes its relation to Kant’s philosophy of mathematics. Heis’ presentation provides a different way of thinking about Kant’s theory of magnitudes from my interpretation, and engaging with it is instructive, despite (and indeed because) I do not recognize my own understanding of Kant’s project in it. An account of why I do not helps situate my own interpretation.

Heis identifies a “Master Argument” consisting of two halves, which establish that mathematics is the science of quantity and that the science of quantity is mathematics. Heis believes that I focus on the second half of the Master Argument, so he directs his attention there. Heis provides an eight-step reconstruction of that half of Kant’s Master Argument in accord with my interpretation of Kant’s theory of magnitudes, which invites challenges to both Kant and my interpretation.

Unfortunately, Heis thinks that my appeal to Kant’s theory of magnitudes in support of the second half of the Master Argument is deficient in two closely related ways. He thinks it falls flat because I fail to make use of two additional features of Kant’s philosophy of mathematics mentioned in the previous section: the syntheticity of mathematical judgments and the logical role of intuition in mathematical proofs. Heis finds my not focusing on these topics surprising, since they have been two major points of debate in Kant scholarship, while I think not focusing on them, at least not at first, has benefits. But as Heis frames the issues, he thinks that each of these features are potentially viable candidates for supplementing the otherwise deficient Master Argument. Heis also believes that my account of the role of intuition in the theory of magnitudes is “blurry.” Clarifying its role would explain whether and how it can support the claims that mathematical judgments are synthetic and that mathematical inference depends on intuition. These are the two underlying criticisms of my interpretation, which I am eager to address. Before doing so, however, I would like to make several points about Heis’ Master Argument, because I think his reconstruction misrepresents Kant and my interpretation in several ways and bringing them out is instructive.

Argument reconstruction is a useful tool for reconstructing Kant’s views; indeed, as Chapter 3 of KMW illustrates, I rely on it heavily. And of course, it is useful for then evaluating those views—whether in relation to Kant’s other philosophical commitments (which is my emphasis in Chapter 3 and elsewhere), or from the perspective of his rationalist predecessors and contemporaries, or from the perspective of Russell or Frege, or from our modern standpoint. It puts pressure on specific claims that are essential to Kant’s account, and responding to that pressure is fruitful. But I did not recognize Kant or my interpretation in Heis’ reconstruction, and I think it paints a misleading picture of both, for several reasons. First, I did not focus on what Heis calls the Master Argument because Kant does not. Of course, Heis does not claim that Kant presents his position as a Master Argument; it is a reconstruction of an argument on his behalf in accordance with my interpretation. But I do not see Kant as starting from some independent characterizations of mathematics on the one hand and a science of quantity on the other and then arguing for a biconditional connecting them. Rather, I think Kant takes as his starting point the claim that mathematics is the science of magnitudes and their measure and attempts to fill out a positive theory of what this amounts to in a way that both shapes and takes into consideration his account of human cognition more generally. That positive theory emerged in the critical period to include, for example, the claim that mathematics is rational cognition by means of construction. I think that in working out his account, Kant was not primarily focused on arguing against his rationalist predecessors and contemporaries or on providing an argument that they would not find, by their own lights, question-begging, even if by the time he wrote the Critique of Pure Reason he did mount arguments against their views. If his theory as a whole provides a more satisfactory and plausible account that would itself count in its favor.Footnote ¹ As a result, I concentrated on what I took to be an already sufficiently challenging task whose completion has until now proved elusive, which was simply to reconstruct Kant’s positive views as faithfully and nonanachronistically as possible to arrive at his actual views.Footnote ² I also think that Kant’s positive views are more philosophically interesting than whether he has an argument his rationalist adversaries would accept. So, I focused on explicating Kant’s theory of magnitudes and its role in mathematical and human cognition more generally, while postponing a full evaluation of his philosophy of mathematics and account of cognition until we have a clearer picture of Kant’s views in their entirety. And the starting point of the mathematical part of his project is the claim that mathematics is the science of magnitudes and their measure. In order to understand why, it is helpful to back up to the development of Kant’s critical philosophy.

As I understand it, the essays Kant wrote in the Prize Essay period from roughly 1762–3, that is, The Only Possible Argument in Support of the Demonstration of the Existence of God, An Attempt to introduce the Concept of Negative Magnitudes in Philosophy, and Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality (a.k.a. the Prize Essay), left him convinced that the conceptual rationalist program of Leibniz and Wolff could not succeed. More particularly, Negative Magnitudes reflects his conviction that real opposition of magnitudes, such as the opposition of forces or motions, could not be accounted for by means of the logical opposition of concepts. Nearly simultaneously, writing the Inquiry convinced him that mathematical method differed deeply from philosophical method and that close attention to basic cases of mathematical cognition, especially geometrical cognition, reveal a dependence on the representation of concrete particulars in intuition.Footnote ³ Kant was, in this respect, sensitive to mathematical practice; he reflected carefully on what it is like to engage in quite basic mathematical acts of cognition and on the grounds of their certainty, especially in geometry. This further persuaded him that mathematics was not adequately explained by the rationalist accounts of concepts and truth. I believe Kant’s focus in the Inquiry on the certainty of mathematical demonstrations suggested to him that intuition was playing a substantive role in grounding and justifying mathematical judgments, not just a heuristic role that acquaints us in a confused way with truths that reduced to relations among concepts.

Moreover, Kant would have been acquainted with Wolff’s attempt to radically reform Euclidean geometry in his Elementa Matheseos Universae by introducing a philosophical definition of similarity and attempting to derive the truths of geometry from definitions. Wolff gives Leibniz credit for suggesting the use of this definition of similarity to overhaul Euclid. Nevertheless, a close reading of his attempt reveals a non sequitur in the proof of the very first proposition, revealing the inherent inadequacy of Wolff’s grounding of geometry in philosophical definitions.Footnote ⁴ While this failure was not a decisive proof that the Leibnizian and Wolffian (and in particular the Leibnizian) approach could not be made to work, I believe it gave Kant further reason to think that rationalists could not remain consistent with their principles and explain mathematical cognition.

Kant’s understanding of intuition was at this point still nascent; he had not yet taken the step of regarding space and time as ideal or intuition as a priori, had not yet worked out his mature formulation of the analytic/synthetic distinction, nor had he fleshed out his understanding of our cognition of magnitudes, but he had broken with Leibnizian and Wolffian rationalism and was now on the path toward his critical philosophy. I believe that Kant’s reflections on magnitudes and mathematics in the Prize Essay period deeply influenced the development of his critical philosophy and spurred him to work out a new and revolutionary theory of human cognition informed by his understanding of mathematical cognition and his conviction that it relied on intuition.Footnote ⁵

Finally, in rethinking the nature of mathematical cognition, Kant begins with the view that at its foundations mathematics is a science of magnitudes and their measure because it was a commonly shared view among mathematicians. As I have argued and Heis notes, it was a familiar and uncontroversial position before 1850; this understanding of mathematics was expressed in particular by Abraham Kästner, Leonhard Euler, and by Christian Wolff. This was not surprising, since it was a venerable view with a long history tracing back to in Euclid’s Elements. Nevertheless, it had been neither seriously examined nor adequately developed with the aim of explaining mathematical cognition, so it became part of Kant’s task to take on that work. Kant was starting with a widely shared though underdeveloped understanding of mathematics with a two-millennia pedigree.

In summary, I think Kant was convinced that the rationalist program was deeply flawed, but rather than focusing on finding nonquestion-begging arguments that his rationalist predecessors and contemporaries would accept, he aimed to develop a new, positive account to replace it. An overall account of mathematical and human cognition that is more coherent and plausible than the theories the rationalists offered, that more closely reflected mathematical practice, and that was grounded in a widely held view of the relation between mathematics and magnitudes with its roots in Euclid’s Elements all would speak in its favor. Kant builds a case with his overall account, not in linear step-by-step arguments tracing two directions of a biconditional. And in this way, I think Heis’ Master Argument is misleading. By the critical period, of course, Kant also argues against his rationalist predecessors, in particular, by attempting to leverage his mature understanding of the analytic/synthetic distinction and his claim that we have synthetic a priori cognition. Whether his arguments against them are successful or whether they would have found them question-begging is a further matter.

A few comments on Lanier Anderson’s valuable book The Poverty of Conceptual Truth are called for at this point, since it directly bears on whether Kant attempts to provide nonquestion-begging arguments against his rationalist predecessors and contemporaries. Anderson rightly points out the striking fact that in 1781 the Leibnizian–Wolffian philosophy was quite firmly entrenched in Germany, and yet by 1790 it had been almost entirely abandoned in favor of a Kantian framework. He believes that explaining this relatively sudden change requires a nonquestion-begging argument against the Leibnizian–Wolffian philosophy, and he structures his book about around this demand.

Anderson deftly outlines the metaphysical program of Leibniz’s and Wolff’s philosophies and shows how, in reaction to Leibnizian and Wolffian rationalism, Kant’s analytic/synthetic distinction developed through both methodological and epistemological versions before settling on a logical–semantic version. In particular, Anderson underscores the way in which the Leibnizian and Wolffian philosophy was thoroughly conceptualist and grounded truth in concept containment, which renders all truths analytic in Kant’s mature understanding of the analytic/synthetic distinction. He also shows how Wolff aspired to arrange all concepts into a hierarchy and to assimilate all truths to containment relations between concepts in that hierarchy. Anderson describes how Kant came to appreciate the inadequacy of their approach to account for truths Kant would classify as synthetic, especially relations of equality expressed in judgments such as 7 + 5 = 12. I think Anderson’s work on these themes is excellent, though I differ on one point of emphasis. One can get the impression from his work that the development of Kant’s critical philosophy was first made possible by his mature formulation of the analytic/synthetic distinction, which according to Anderson emerged in about 1772, while I see that development beginning much earlier, spurred by an appreciation of the inadequacy of Wolff’s reform of geometry, the essays of the Prize Essay period of 1762–3, and reflections on Hume, while his settling on the mature analytic/synthetic distinction in 1772 allowed him to express his own view more clearly and to sharply contrast it to the Leibnizian–Wolffian philosophy.

Anderson notes that to Kant’s rationalist opponents, Kant’s pronouncements in the Critique of Pure Reason that mathematical judgments are synthetic despite being a priori and hence necessary will sound like mere table-thumping. Moreover, Kant’s negative argument faces a problem because they have a ready response to his claim that mathematical judgments are synthetic, which Anderson calls the “Recuperation Strategy”: evidence that a predicate belongs with necessity to a subject in a judgment is simply dispositive evidence that the predicate is contained in the subject, so that any purportedly a priori synthetic proposition is analytic after all, even if we did not at first recognize it to be so. From a Kantian perspective, the Recuperation Strategy appears ad hoc. Nevertheless, Eberhard makes use of it in his sustained attacks on Kant’s philosophy in the Philosophisches Magazin beginning in 1788, to which Kant attempts to respond in On a Discovery According to which Any New Critique of Pure Reason Has Been Made Superfluous by an Earlier One, published in 1790. Anderson does not think that the rationalist will be moved by Kant’s insistence in the Critique and elsewhere that, for example, search as we may, the concept of 12 cannot be found in the concept of 7 + 5, so that Kant requires a further argument. Anderson finds two such arguments on Kant’s behalf. One is specific to Wolff’s insistence that mathematical propositions such as 7 + 5 = 12 have a place in a hierarchy of conceptual truths; as Anderson shows, it appears impossible to find a place for such a truth in the Wolffian hierarchy. The other argument is broader and deeper in targeting not just the hierarchy of conceptual truths on which Wolff insisted, but the Leibnizian account of truth in terms of concept containment. Here, Anderson draws on my argument that intuition is required for the representation of the mathematical properties of magnitudes because it depends on the representation of strictly homogeneous manifolds and their properties and relations. Anderson focuses on equality and in particular the equality expressed in equations such as 7 + 5 = 12, providing his own illuminating argument that it cannot be reduced to the relation between concepts, an argument that Anderson thinks the rationalist would have to accept.Footnote ⁶

I believe that looking for arguments that various philosophers at different times would find non-question-begging is valuable insofar as it allows us to formulate challenging questions and hold Kant’s feet to the fire. It is also valuable because it helps us relate Kant not just to his rationalist predecessors, but to later mathematicians and philosophers—such as Russell and Frege—who resisted Kant’s claim that arithmetic requires intuition and believed that conceptual representation and logic, properly understood, were sufficient for arithmetic. These later criticisms rejected Kant’s claim that mathematical judgments are synthetic in light of developments in logic and a shifting understanding of analyticity that is not confined to concept containment; it also led Russell, Hintikka, and Friedman to more closely consider the role of the singularity of intuition in mathematical proofs. Anderson and Heis have this larger context in mind, which provides a broader motivation to search in Kant for a nonquestion-begging argument that mathematics is not merely conceptual. Nevertheless, I think insisting on a nonquestion-begging argument in reconstructing Kant’s views can occlude Kant’s positive views, which have been my focus.

Returning to Heis’ Master Argument, I think there are a few other features that are potentially misleading. Heis refers to the science of quantity. The vocabulary in Kant’s day was not settled; moreover, the terms “magnitude” and "quantity" were used ambiguously to refer to something that can be measured and to the result of its measurement—referring, for example, to a piece of lumber itself as a magnitude and to its length, for example, 6 feet, as its magnitude—where the former is paradigmatically some concrete particular. Kant attempted to resolve the ambiguity by distinguishing between two senses of magnitude, quanta and quantitas, where “quanta” refers to what can be measured and “quantitas” refers to its measurement. Thus, “quanta” gives a name to the concrete sense of magnitudes already in play. In Kant’s terminology, the science of magnitudes and their meaure become the the science of quanta and their quantitas which is the starting point of his investigation. Kant then asked himself what it is about such quanta that allows of measurement and distinguishes them from those things that do not. As Heis makes clear in footnote 3 [XX], by “science of quantity,” he means the science of quantitas alone, which he identifies with the categories of quantity. Heis drops out reference to quanta, though he goes on to say that an alternative that seems Kantian and consistent with KWM is that mathematics is the science of quantitas “inasmuch as it is” the quantity of quanta. This is almost there, but it still singles out quantitas. I believe Heis focuses on quantitas because he wants a characterization of the science of quantity that does not already implicitly invoke representations of concrete particulars; he wants Kant to argue that they are required.Footnote ⁷ I will address this challenge next, but it would have been closer to Kant’s views to use “magnitudes” rather than quantity, and to take his starting point to be the view that mathematics is the science of quanta and their quantitas. Henceforth I will refer to the science of magnitudes and their measurement simply as the science of magnitude, and substitute “magnitude” where Heis uses “quantity.”

A second potentially misleading feature concerns the conclusion of the Master Argument and hence the scope of what Heis states is the main claim of my book. I see Kant as developing a general theory of magnitudes, which includes the synthetic a priori metaphysical principles that objects of experience are extensive magnitudes and that all of their real properties have an intensive magnitude. The general theory and these principles in particular support the view that the world is mathematical. In formulating the conclusion of the Master Argument as the claim that mathematics is the science of magnitude, Heis drops the second half of what he quotes as the main claim of my book, that is, that the world of experience is a world of magnitudes; shortening it to “mathematics is the science of magnitude” threatens to obscure the broader theory and its centrality to Kant’s views. It all depends in how we understand “mathematics” and “science” in this claim. I would just like to dispel any mistaken impression one might have that my main claim and Heis’ formulation of the conclusion of the Master Argument only concern pure mathematics and a proper science of magnitudes, where a proper science might be erroneously construed as restricted to magnitudes constructed a priori in pure intuition, such as a triangle in pure geometry. That is not my intent. Nor is it Heis’.Footnote ⁸

A final point concerns the structure of the Master Argument. Heis thinks that the second direction of the biconditional—that if a science concerns magnitude, then it is mathematics—is more difficult to argue, and that this is the direction I am following. But in light of what I said earlier, I think that Kant’s unfolding of the claim that mathematics is the science of magnitudes supports a broad theory of magnitudes that, if anything, supports both directions of the biconditional, with neither direction being more difficult.Footnote ⁹ Moreover, I would say that my interpretation, if anything, addresses both directions. The fact that I focus on the role of intuition in making it possible to represent magnitudes may suggest that I am moving from the science of magnitudes to mathematics (as Heis states on p. 4 [XX]), but in focusing on representations that intuition makes possible, I am thereby in no small part articulating necessary conditions of those representations, and a chain of such necessary conditions backing up from intuition to mathematical cognition would provide an argument that mathematics is the science of magnitude—the first of Heis’ biconditionals.

Despite these misgivings with Heis’ Master Argument, however, his analysis leads to some deep and insightful criticisms. Broadly speaking, the first of his two main criticisms is his discomfort with the clarity of the role of intuition in mathematical cognition, including in mathematical judgments and inferences. As Heis points out, I issued promissory notes in KMW with the intention of cashing in those notes in a second book, because a full account needs to consider the differences between geometry, arithmetic, and algebra.Footnote ¹⁰ Sections 3–6 will provide a brief synopsis of the role of intuition in mathematical cognition in general with discussions of the particular elementary branches of mathematics. Heis’ other main criticism concerns the relation between empirical cognition and pure mathematical cognition, and Nunez as well as Heis object to my account of construction. I will address these two issues in Section 7.

3. The Role of Intuition in Representing Strict Homogeneity

Heis is interested in the role of intuition in mathematical cognition quite generally, but his concern with my interpretation centers on the role of intuition that arises from Kant’s theory of magnitudes. Since on my interpretation intuition is what makes it possible to represent a strictly homogeneous manifold, his discomfort is more specifically with this role. And hovering in the background are the two questions that have been the focus of recent debates: what is the role of the representation of strict homogeneity in grounding synthetic a priori judgments in mathematics and its role in mathematical inferences? Does the theory of magnitudes support these features of mathematical cognition, or does the theory need to appeal to them for extra support?

I will begin in this section by reprising why Kant thinks that the representation of strict homogeneity is required for mathematical cognition, but before doing so I would like to remind that though KMW focuses on a previously neglected role for intuition in representing a strictly homogeneous manifold, I allow that in Kant’s view there are other roles for intuition in mathematical cognition as well; once Kant came to think that intuition played a central role in mathematical cognition, he was open to various roles it might play both in mathematics and in human cognition more generally.

The notion of strict homogeneity derives from the Eudoxian theory of proportions found in Book V of Euclid’s geometry. According to that theory, magnitudes included only the continuous spatial magnitudes: lines, areas, volumes, and angles. Lines are homogeneous with lines, areas with areas, and so on, but there is no homogeneity across these types of spatial magnitude. As I discuss in KMW Section 6.4, Euclid does not define magnitude, but the first five definitions of Book V of the Elements implicitly define homogeneous magnitudes as the sorts of things that can measure each other. I argue in Chapter 9 of KMW that the notion of measure is part of a general theory of pure concrete measurement that governs magnitudes. It also governs numbers conceived of as collections of units, which later in the Greek mathematical tradition are classified as discrete magnitudes. This theory of measurement presupposes that homogeneous magnitudes stand in part/whole relations and in a total ordering of comparative size relations. It also presupposes that homogenous magnitudes can be composed into larger magnitudes that are homogeneous with them. Moreover, homogeneous magnitudes can stand in ratios to one another. Finally, homogeneous magnitudes are capable of standing in the relation of equality. These presuppositions guarantee that homogeneous magnitudes are governed by the propositions concerning ratios and proportions developed in Books V and VII, which describe the mathematical relations among magnitudes. This is why it is reasonable for Kant to take as his starting point the claim that mathematics is the science of magnitude.

Since only magnitudes that are homogeneous with each other stand in these mathematical relations to each other, there is something about their homogeneity that is fundamentally tied to their mathematical character. I believe that sometime during or after the Prize Essay period, Kant reflected on what that something is, and first considering geometry, he concluded that underlying the homogeneity of each of the kinds of determinate space (i.e., lines, areas, etc.) is the strict homogeneity of space itself. Each of the kinds of determinate space are composed of parts that are qualitatively identical and can be composed into wholes that are qualitatively identical with those parts and hence homogeneous with them, yet at the same time are quantitatively more than those parts.Footnote ¹¹ The synthesis of a homogeneous manifold results in more than what you started with but is nevertheless homogeneous with it—if you extend a straight line segment, the result is still a line; it only differs in being more line relative to that which you started, and this is a purely quantitative difference. The synthesis of composition is at the heart of Kant’s account of determinate magnitudes, a form of synthesis unique to homogeneous magnitudes (B201n). Taking his cue from the Greek mathematical tradition, Kant extends this theory to cover discrete magnitudes as well, and hence number conceived of as collections of units, where we abstract from all qualitative features of that which we call a one. Finally, Kant holds that concepts on their own cannot represent a strictly homogeneous manifold, while intuition makes that first possible. (See KMW Chapter 7 for the full argument and textual support.) This is the most fundamental role for intuition in the representation of magnitudes and hence in mathematics—it makes it possible for us to represent quanta and their quantitative determinations, that is, their quantitas.

4. Why Do Arithmetic and Algebra Require the Representation of Quanta?

As I noted earlier, Heis would like Kant to argue that the representation of quanta is required for mathematical cognition and not get it for free. Heis raises a two-level challenge against the claim that quantitas requires the representation of quanta. He first allows that geometry concerns quanta, such as lines, circles, areas, etc. That seems like a reasonable concession, at least for the sake of argument: synthetic geometry proceeds through construction of figures in space, which are quanta, and yields propositions about those quanta. But Heis thinks that Kant is question-begging in assuming that arithmetic and algebra do. I stated earlier that Kant is sensitive to mathematical practice, especially to geometrical practice, but it might seem that he is overgeneralizing from geometry to all of mathematics and is insensitive to arithmetical and algebraic practice. Although I had postponed a full treatment until the second book, I will begin to address these issues here, starting with arithmetic.

5. Why Does Any Mathematical Cognition Require Intuition?

6. The Role of Intuition: Concepts, Judgments, and Inferences

In the previous section, I defended the claim that the representation of quanta in intuition is required to supply sensible content to the concepts of quantitas and quanta in general. In Section 6.a I briefly consider the same role for the representation of quanta in the subfields of geometry, arithmetic, and algebra. The discussion is brief because the content and objective validity of geometrical concepts constitute Kant’s paradigm case for the need for the representation of quanta in intuition and is the most plausible, and I already discussed the role of intuition in providing sensible content and objective validity to the concepts of arithmetic and algebra in Section 4. In Section 6.2, I turn to the role of intuition in grounding synthetic mathematical judgments, first for quanta and quantitas in general and then for geometry, arithmetic, and algebra. I then point out a few important additional roles for intuition in mathematical cognition in Section 6.3, before considering some difficulties in distinguishing between the role of intuition in providing sensible content to concepts and grounding synthetic judgments in Section 6.4. Section 6.5 addresses the role of intuition in justifying nonanalytic inferences in mathematics in general and then for geometry, arithmetic, and algebra. We will see that determining Kant’s views and making the case for a role for intuition become increasingly difficult; Section 6.6 argues that Kant’s views are opaque because he continually returns to the notion of the construction of a concept whenever he argues for the role of intuition in mathematics.

7. The Relation Between Empirical Cognition and Pure Mathematical Cognition

I now turn to a general issue raised by both Heis and Nunez, though in different ways: the relation between empirical cognition, such as cognition of a house, and pure mathematical cognition. Both think that I am, to use Nunez’s formulation, “taking the conditions of possibility of experience and those of the exact sciences to be more intertwined than they are in Kant” (Nunez, p. 3 [XX]). There are two points at which they press this complaint; the role of strict homogeneity in empirical cognition and my account of construction. Heis objects to the claim that our representation of spatiotemporal magnitudes requires the representation of pure spatiotemporal regions. Both Heis and Nunez object to what I say about construction. I will address each point in turn. The upshot of my responses is that the conditions of the possibility of experience and those of the exact sciences are in fact very deeply intertwined.

8. Strict Homogeneity: Further Clarifications

I responded earlier to Heis’s concern that qualitatively distinguishable magnitudes such as red and blue blocks do not require the representation of strict homogeneity as a condition of their composition. Nunez raises a few additional worries one might have about the claim that mathematical cognition requires strict homogeneity, which gives me a chance to further clarify its role. The first concerns how the account can accommodate the qualitative features of mathematical objects, for example, that the diagonal of a cube is the cube root of the sides, or that 12 has the distinctive property of being the sole sum of 5 + 7 (Nunez, p. 2 [XX]). As Nunez puts it, these objects have specific differences and hence qualities and hence are also conceptualizable. As a result, mathematical objects can be distinguished conceptually by their qualitative differences—instances of cubes can be distinguished from other kinds of solids and 12 can be distinguished from other numbers, for example. But if concepts can be used to distinguish mathematical objects, we have no need for intuition to make it possible to distinguish them.

The claim is not, however, that the only way to distinguish any already conceptualized mathematical object is by means of intuition, that is, by distinguishing distinct parts or locations of a qualitatively indistinguishable manifold. Nor is the claim that you cannot conceptualize mathematical properties that make essential reference to strict homogeneity. With the aid of intuition, we can certainly form concepts, such as the concept of the diagonal of a cube, that depend on intuition for their content; the role of intuition in representing a strictly homogeneous magnitude is at a deeper level than this concept formation. The claim is that concepts on their own, without the aid of intuition in representing strict homogeneity, cannot represent quanta such as a cube at all, and hence cannot represent the synthesis of composition which is characteristic of them and which allows both their representation and their mathematical treatment. Nunez finishes this line of argument with the claim that “it looks like each in-itself identical part of space and time will have properties that are specific to it.” This is missing the heart of the view. Each of the parts of space are self-identical, but the point is that they are all qualitatively identical, and that concepts cannot represent qualitatively identical manifoldness without the aid of intuition. With the aid of intuition, however, we can represent the manifold by means of concepts. And the possibility of cognizing the measure of the diagonal of a cube (the quantitas of this line segment relative to one of the sides) already depends on the fact that the line is composed of a strictly homogeneous quantum. Mathematical concepts are ultimately concepts of magnitude, which are sensible concepts, and depend on intuition for their content.

Consider the most general case I considered in KMW Section 5.2. One might think that you could form a concept of a strict homogeneity—a concept of a manifold (a manyness or a muchness thought through the category of plurality) that is numerically different without being qualitatively different—by unifying concepts that do not essentially refer to intuition into one concept, for example, by employing concepts of plurality, difference, quality, and negation. Let us suppose for the sake of argument that were possible. Note first, however, that the concept is defined with a negative condition, namely, “a manifold without qualitative difference” and that without appeal to intuition, there is nothing that answers to this concept. More importantly, however, it would at best have a logical content; unless it is given content by its reference to intuition, we cannot grasp what it would be to be a strictly homogeneous manifold, because concepts on their own are incapable of representing such a manifold. Yet intuiting a strictly homogeneous manifold allows you to reflect a concept from the content of your intuiting to acquire content for that concept. All mathematical concepts are ultimately concepts of magnitude, and all concepts of magnitude are sensible concepts whose content makes essential reference to intuition, and in particular, to their property of being a homogeneous manifold.

Similarly, the concept of the ratio of a diagonal of a cube to its side is a pure sensible concept that makes essential reference to a strictly homogeneous manifold. The ratio of a diagonal of a cube to its side is conceptualizable, but only because we can represent the cube and the diagonal in intuition by means of representing a strictly homogeneous manifold, which is required to represent a line at all. And the same applies to the concept of 12 and its relation to 7 and 5 . On Kant’s understanding, the concept of 12 ultimately makes essential reference to a manifold of units as such, that is, units that are not qualitatively distinguished. Moreover, the concept of addition makes essential reference to a kind of composition that concepts can acquire through reflection on the synthesis of composition of a homogeneous manifold. Finally, the concept of equality makes essential reference to identity of quantity without qualitative difference. Again, these are pure sensible concepts whose content includes essential reference to strictly homogeneous manifolds.

Nunez reformulates his question in another way, but I take him to be raising a related but distinct question. He asks why the fact about cubes that the diagonal is root three of one of its sides is a synthetic truth. Why is it not a part of the concept cube? Similarly, why is the unique fact that 12 = 5 + 7 contained in the concept of 12 synthetic? Why is not this contained in the concept 12? (Nunez, p. 3 [XX]).

The answer is that these are propositions about the diagonal of cubes and the number 12 that make claims about quantitas, which concern how big something is. As I discussed earlier in Section 6, in the arithmetical case, the bare concept of unifying 7 + 5 is not enough to ground the proposition concerning its relation to 12; you need to compose 5 to 7 and that requires a synthesis of composition of a strict homogeneity. Concepts on their own cannot represent the special sort of composition grounding this proposition; all we could think through concepts alone is the unification of concepts with each other, which is the result of a quite different sort of synthesis. Of course, once we compose magnitudes, we can form a concept of the synthesis of composition of a homogeneous manifold, but this concept is subsequent to the synthesis; its sensible content depends on synthesizing such a manifold.

This is not to say that Kant’s understanding of the quality/quantity distinction and the qualities of quanta and quantitas are perspicuous. In fact, I think that Kant uses the notion of quality in two ways, one that contrasts with the notion of quantum and the other that contrasts with quantitas, the latter of which is closely related to the Leibnizian–Wolffian contrast between quality and quantity. Kant’s views are hard to untangle, which might be an artifact of his attempting to reform the Leibnizian–Wolffian theory of quantity while also reworking the Euclidean theory of magnitudes. Sections 8.3 and 8.4 of KMW were my best attempt to address these issues, while Section 8.5 directly considered the qualities of magnitudes, such as why Kant considers the straightness or curvature of a line to be a quality, which bear directly on Nunez’s worries.

9. Equality

Anderson asks me to say more about the relation of equality and the role that intuition plays in allowing its representation. To begin, there are two sorts of equality in the broad sense: the equality of continuous magnitudes, such as volumes, and the equality of collections of things regarded as units, that is, equimultiplicity. Kant treats such collections as discrete magnitudes, so equality in the broad sense covers both continuous and discrete magnitudes.

As I discuss in Chapter 9, the relation of equality is crucial to bridging the gap between our cognition of the mereology of magnitudes grounded in the categories of quantity (unity, plurality, and allness), and a full account of the mathematics of magnitudes. The part-whole relation allows a strict partial ordering of magnitudes of a particular kind, such as volumes, with respect to size: one magnitude is greater than another if the latter is a fully contained proper part of the former; it is partial because it does not order disjoint or merely overlapping magnitudes of the same kind. Kant, however, states that one magnitude is greater than another if the latter is equal to a proper part of the former, which can be used to induce a total ordering of magnitudes of a particular kind with respect to size. Moreover, as I mentioned in the Précis, the relation of equality is the linchpin of what I call the “general theory of concrete measurement,” which I argue in Chapter 9 is a tacitly presupposed theory supporting both Euclidean geometry and Kant’s philosophy of mathematics. The relation of equality plays three roles in that theory: first, in the composition of equal units in measuring; second, in the equality between the totality of composed equal units and what is measured; and third, in adding equal units of one magnitude an equimultiple number of times that one adds equal units of another magnitude. This last role makes not just simple measurement but the Eudoxian theory of proportions possible, which in turn makes the mathematical treatment of magnitudes possible.

In Chapter 8, I argue that Kant accepts the traditional philosophical definitions, found in his rationalist predecessors of equality as identity of quantity and similarity as identity of quality, but that he radically revises it by introducing his distinction between quanta and quantitas and arguing that the former is a strictly homogeneous manifold that requires intuition for its representation. Given the importance of the relation of equality, I consider, in Section 9.8, where the relation of equality might best fit into Kant’s architectonic scheme, speculating that equality defined as the identity of quantity might belong to the concepts of reflection, in particular to the concepts of identity and difference, as described in the Amphiboly of the CPR.

But there is also something more substantive to be gleaned from the Amphiboly with respect to equality. I have placed great weight on the Amphiboly discussion of identity and difference, which I discuss in KMW Section 7.5; Kant argues in the Amphiboly against Leibniz’s principle of the identity of indiscernibles, and on the basis of what Kant says there, I have argued that because intuition is a qualitatively identical manifold, intuition allows us to represent bare numerical diversity. As a consequence, intuition makes it possible to represent two quanta, insofar as they are homogeneous manifolds represented intuition, as numerically diverse despite having the same quantitas, that is, being quantitatively identical, and hence equal. As mentioned earlier, all appearances as regards their intuition are extensive magnitudes; they may be discernible with regard to some qualities or other, but intuition is what allows us to represent them, merely insofar as they are quanta, as numerically diverse while also being equal. The representation of equality among magnitudes is, of course, fundamental to their mathematical treatment, and as I indicated earlier, plays three roles in the Euxodian theory of proportions.

Whether arithmetical or algebraic equations such as 7 + 5 = 12 or a = √b, which also concern quantitas, are statements of identity or equality is a further issue; it depends on the relation between symbolic and ostensive construction and the level of abstraction expressed in these equations, an issue I plan to address in my next book. Regardless, the representation of equality is fundamental to mathematical cognition.

10. Kant’s Theory of Magnitudes and Kantian Nonconceptualism

In Chapter 4 of KMW, I give an interpretation of extensive magnitudes that avoids a threat of an infinite regress in our representation of them. I point out that my account undermines an argument nonconceptualists have used to argue for their position, but I studiously avoided saying more about the conceptualist/nonconceptualist debate. Despite my efforts, Anderson wants to draw me in and presses me to say more about the implications of my interpretation for the debate.

There has been a remarkable amount of work on the specifically Kantian version of the conceptualist/nonconceptualist debate in the last decades, which has included disagreements about what conceptualism and nonconceptualism means or should mean in thinking about Kant’s philosophy. It has connections to and was influenced by, while differing from, the version found in the contemporary philosophy of mind (see Allais, Reference Allais and Schulting2016). Even the specifically Kantian version of the debate has also transformed from its origins in Wilfred Sellars’ articulation of the Myth of the Given. While all agree that concepts are required for cognition, there were unclarities or differences of opinion about whether the locus of the debate concerns perception or intuition; the consensus appears now to be focused on intuition, though Kant’s claims concerning apprehension and perception still play a role in claims about intuition. There have been different notions of object in play, as well as different notions of content. The debate has also included several ways of drawing a distinction between strong and moderate conceptualism (Land, Reference Land2015b, Reference Land and Schulting2016) or between conceptualism and intellectualism (McClear, 2016). One could defend a moderate conceptualism according to which intuitions require a sensible synthesis which in turn requires spontaneity and the activity of the understanding, whether or not that involves concepts. And a nonconceptualist could argue for the claim that intuition does not constitutively require concepts, even if in fact all human intuitions are transformed by concepts (see Allais, Reference Allais and Schulting2016). Both the conceptualists and the nonconceptualists have refined and moderated their claims and the positions are not as far apart as they once were, though room for substantial disagreement remains. One might shun an attempt to enter the fray, and Nunez praises me for avoiding it in my book. Yet, I agree with Allais that though the debate has not been definitively settled, it has occasioned a more nuanced understanding of Kant’s views, and I believe that connecting the controversy with Kant’s theory of magnitudes would be mutually beneficial. Unfortunately, there was no way for me to do justice to the lively debates in my book, much less for me to do so here, but I will venture to mention four ways in which Kant’s theory of magnitude bears on it.

The first concerns the resolution to the threat of infinite regress in Kant’s account of extensive magnitudes, which I addressed in Chapter 4 of my book. In a nutshell, Kant states that an extensive magnitude is one in which the representation of the parts makes possible the representation of a whole and therefore necessarily precedes the latter, and those parts are synthesized through a successive synthesis of productive imagination. But if each of these parts is in turn an extensive magnitude, then they require a representation of their parts, and so on. The resolution draws on the distinction between determinately and indeterminately represented intuitions. An extensive magnitude is a determinate intuition, which is the result of the successive synthesis, while the synthesized parts are only indeterminately represented.

There are, however, different understandings of these indeterminate intuitions. According to Lucy Allais and Sasha Golob, the need for indeterminately represented parts provides an argument for a kind of intuitive atomism and hence for nonconceptualism, because in their view the parts must be represented particulars prior to being synthesized in order to serve their function.Footnote ⁵⁸ I argue, however, that this misconstrues the nature of these indeterminately represented parts and their synthesis. The latter is, in the first instance, a continuous synthesis of a continuous homogeneous manifold starting from a point, in which no particular parts are represented or presented.Footnote ⁵⁹ I argue that Kant’s discussion of the continuity of magnitudes strongly supports this reading and in fact explains the applicability of the calculus to appearances, which is crucial for the foundations of natural science. A proper understanding of Kant’s theory of magnitudes and their mathematical character therefore undercuts one argument for nonconceptualism, but I was careful to state in the book that doing so did not settle the conceptualist/nonconceptualist debate, and I refrained from saying more; what follows is the more for which Anderson asks.

The second consequence of Kant’s theory of magnitudes is a broader point: it introduces a complication that alters the framework of the debate. Kant thinks that a fundamental role of intuition is to allow us to represent a continuous homogeneous manifold whose parts are only indeterminately represented, but the role of intuition in the representation of such a manifold appears to be at odds with the singularity of intuition, which is widely construed as requiring a representation of a singular object, that is, a particular individual. Taking the case of an outer representation, it requires a representation that differentiates a spatial object from the surrounding space and from other objects. This understanding of the singularity of intuition is quite natural given Kant’s focus in the Critique of Pure Reason on the conditions for the cognition of objects of experience, a robust notion of object the cognition of which requires a great deal in Kant’s account. I argue in Chapter 5, however, that the singularity of intuition is not incompatible with the representation of a homogenous manifold because the singularity of intuition is best understood as a way of representing singularly, which allows for the representation of a continuous homogeneous manifold—similar to the way in which a map can represent a continuous expanse of space without a determinate representation of the parts of the expanse. This representation is more basic than, and in fact underlies and makes possible, the singular representation of a singular object; in fact, it is the ultimate source of the singularity of the representation of the singular object.

This understanding of the place of the representation of a homogenous manifold obviously has direct relevance for understanding Kant’s account of the conditions for cognizing singular objects. It is also directly relevant to the conceptualist/nonconceptualist debate, which has almost entirely focused on the conditions for the singular representation of singular objects, where singularity is usually construed as the representation of an object distinguished from its surround and other objects. Since there is a role for intuition in representing a homogeneous manifold that is at a more basic level than the representation of singular objects, the conceptualist and nonconceptualist alike must address whether and how concepts might be required at each level. It thus adds further complexity to the debate. One could be a nonconceptualist about intuition’s role in representing a continuous indeterminate homogeneous manifold, but a conceptualist about the representation of singular objects in it, or one could either be a nonconceptualist or a conceptualist about both. A few participants in the conceptualist/nonconceptualist debate have taken this further complexity into account, in particular, Emily Carson and Thomas Land; I will return to their work below.

The third way in which Kant’s theory of magnitudes bears on the debate concerns its origins in Wilfred Sellars. While I think the recent versions and refinements of Kantian conceptualism and nonconceptualism are of value in their own right, the history and origins of the debate, tracing back through many recent writers and futher back through the work of Robert Hanna, John McDowell, and Wilfred Sellars, is also of philosophical interest. In Chapter 5, I claim that the singularity of intuition is best understood as a way of representing singularly; in a recent article, I have further defended that claim and criticized Sellars’ understanding of Kant.Footnote ⁶⁰ My analysis relies on the distinction between intuiting and intuited in a manner similar to Sellars, but the position I argue for conflicts with Sellar’s criticism of Kant’s account of intuition and the form of intuition. Sellars believes that Kant uses “intuition” ambiguously for a this-such intuition, on the one hand, and an impression–intuition, on the other.Footnote ⁶¹ Sellars construes the former as representing singular objects, such as a “this-cube,” which requires intuition for reference to a “this” and a sortal concept for the “such.” Impression–intuition, in contrast, is a more basic notion of intuition that corresponds to a manifold of sense impressions; it belongs to sheer receptivity and is nonconceptual and supplies the brute fact or constraining element of perceptual experience. Those familiar with the Myth of the Given will see where this is going—Sellars thinks Kant errs in countenancing the notion of an impression-intuition.

I argue that Sellars is simply mistaken in identifying a more basic notion of intuition with a manifold of sense impressions. But what is important for present purposes is Sellars’ further argument. Sellars thinks that Kant’s ambiguity in the notion of intuition is reflected in two corresponding notions of form of intuition. Sellars argues that construing space as the form of intuition in the basic sense is incoherent and nonsense. His argument ultimately rests on the view that the more basic notion of intuition is non-representational, so that “intuition” in the basic sense can only refer to mental states as such (as opposed to representational contents); since mental states are not spatial, space cannot be their form. Sellars’ argument depends on the idea that having objective purport is only achieved with this-such intuitions of singular objects, which involve concepts. There is no room in Sellars for the idea that intuition allows us to represent a homogeneous manifold singularly, which is what in turn allows us to represent singular objects.

More is required to fill out this critique of Sellar’s way of reading Kant, but the upshot is that Sellars thinks that this-such intuition is the only coherent notion of intuition in Kant, and because he assumes that the singularity of intuition consists in the representation of a singular object, and hence a this-such, it already involves concepts. Sellars overlooks the more basic role of intuition in singularly representing a homogeneous manifold, whose form is to represent singularly. Sellars may wish to argue that the possibility of singularly representing a homogeneous manifold, insofar as this is representational and has objective purport, requires concepts. Kant’s manner of speaking about intuitions at least strongly suggests that they have objective purport, that is, that intuitions as such have intentional content. If one thought that having objective purport itself requires concepts, then conceptualism would entail that the role of intuition in representing a homogeneous manifold already in some way involves concepts. But that argument depends on the conditions for having representational content at all, not on the conditions for the representation of a singular object as a this-such.

The fourth way in which Kant’s theory of magnitudes bears on the debate concerns whether the role of the categories in mathematical cognition provides an argument for conceptualism. In pure mathematics, the representation of a mathematical object such as a triangle requires construction of a concept of magnitude, that is, exhibiting a priori an intuition corresponding to the concept. So, concepts are required for any intuition of a mathematical object, and conceptualism holds for the objects of pure mathematics.

At an earlier point in my work, I thought that the thorough intertwining of the conditions of possibility of experience and those of the exact sciences provided a possible argument for conceptualism about the intuition of empirical objects as well. I was further influenced in this direction by the thoughtful work of Emily Carson and Thomas Land, both of whom were arguing for a form of conceptualism based in part in reaction to my work on Kant’s theory of magnitudes.Footnote ⁶²

While writing Chapter 5 and reflecting on Allais’ nonconceptualism, however, I became less sanguine about this line of thought; I was also influenced at that time by exchanges with Tyler Burge and Matt Boyle, who convinced me that I was reading too much into empirical cognition under the influence of my focus on Kant’s philosophy of mathematics. I did not leave room for the possibility that in the case of empirical cognition, my representation of a determinate intuition is responsive to an indeterminate intuition that presents not merely a continuous indeterminate homogeneous manifold, as in the case of pure a priori mathematical cognition, but presents a singular object in that manifold in virtue of the affection of sensibility by the thing in itself. The continuous synthesis of the manifold in generating a determinate empirical intuition would, in the first instance, still be a continuous synthesis of the homogeneous manifold, but it would be responsive to the singular object presented in it.Footnote ⁶³ The presentation of this singular object would be indeterminate insofar as it has not yet been brought under the synthesis of composition under the categories of quantity that generates a determinate intuition of the object. This opens the door for a nonconceptualist interpretation of empirical intuition, and makes room for different notions of objects here, whether “perceptual particulars,” or particular property instances located in particular regions of space.Footnote ⁶⁴ The synthesis of the continuous indeterminate homogeneous manifold would in either case be carried out in response to the indeterminate intuition of a singular object presented in it. I discussed this possible nonconceptualist position in KMW Sections 4.5.2 and 5.5.2; I prescinded from endorsing it myself in an effort to stay out of the fray, but it is a position that I think is reasonable to defend and in fact one to which I am sympathetic.

Conceptualists of all sorts will balk, of course. They might argue that intuition of any sort of singular object or perceptual particular is a representation of a this-such and hence requires concepts, in line with Sellar’s view. Or they might argue that any objective purport whatsoever requires concepts—or, adopting a weaker version of conceptualism, at least a spontaneous synthesis of the understanding. The real upshot of all this is that both the conceptualist and nonconceptualist need to take into account that the most basic role of intuition is to represent a homogeneous manifold rather than a singular object, but at least as far as I can see right now, I do not think that Kant’s theory of magnitudes settles these questions one way or another.Footnote ⁶⁵

11. Determinate and Indeterminate Magnitudes

Nunez calls into question my interpretation of indeterminate magnitudes and the distinction between them and determinate magnitudes (Nunez, p. 5 [XX]). I briefly review my account before addressing Nunez’s specific complaint. Although in a few instances Kant refers to intuitions as determining, in virtually all contexts determination is an act of the spontaneity of the understanding, and moreover it almost always employs concepts.Footnote ⁶⁶ In Chapter 2 of KMW, I make much of the distinction between indeterminate and determinate intuitions and magnitudes, using it to explain the different treatments of both space and time and of mathematics in the Transcendental Aesthetic and the Transcendental Analytic. I argue that the discussion of space, time, and mathematics in the Transcendental Aesthetic abstracts as far as possible from the contributions of the understanding, and hence from the determining activity of the understanding under concepts, while it is precisely the task of the Transcendental Analytic to take those contributions of the understanding into account. Thus, when Kant states in the Transcendental Aesthetic that space and time are magnitudes, he has in mind indeterminate magnitudes, while the Axioms of Intuition concerns determinate magnitudes. I argue in Chapter 2 of KMW that in the B-edition of the formulation of the Axioms principle, it would have been more precise for Kant to have said that all determinate intuitions are extensive magnitudes.

The determination of intuitions in the representation of a determinate magnitude (quantum) is effected through a synthesis of composition of the homogeneous manifold of intuition in accordance with the categories of quantity. In the first instance, the composition is a continuous synthesis of a continuous and homogeneous manifold, paradigmatically exemplified in the continuous generation of the representation of a line carried out in the continuous homogeneous manifold of space.Footnote ⁶⁷ The line is a determinate magnitude because it is determined through this synthesis and because the line is contained within boundaries—the starting point and endpoint of the synthesis—and these boundaries are themselves determined through the synthesis. But, and this is crucial, it is not having boundaries which alone constitute its being a determinate magnitude; it is the fact that the understanding runs through the manifold in an act of continuous synthesis. At the same time, no particular potentially representable parts of the line are determinately represented; that is, none of its potentially representable parts are themselves determinate magnitudes. This is what avoids the infinite regress in the representation of extensive magnitudes, as I discussed earlier.

In contrast to the representation of determinate continuous quanta, the representations of space and time as wholes discussed in the Transcendental Aesthetic are not represented through a continuous synthesis of their manifolds, nor are they represented as bounded. They are indeterminate magnitudes.

I have just described the determination of continuous quanta, but there is another level of determination of quantitas, which is the “how big” of a quantum. The representation of a determinate quantum is more basic and does not require the determination of its quantitas. The determination of quantitas only occurs when we measure something in accordance with a unit of measure, where the unit, the plurality of units, and the totality of that plurality fall under the categories of unity, plurality, and allness. Significantly it also presupposes the notion of equality, which is required for measurement. It is one thing to simply represent a determinate continuous quantum, for example, a line, and quite another to determine its quantitas, for example, how many shorter line segments are equal to it, or, for an empirical example, how many feet long a piece of lumber is. As I argue in my book, one of Kant’s important innovations is to draw this distinction. It certainly seems to be the case that we can represent something continuous and extended in space without measuring how many equal units of some specified unit it contains. Moreover, as I argue in Chapter 9 of KMW, Kant is quite sensitive to the further requirements of measurement, in particular, the need for equality. Kant’s account of our representation of quanta—its grounding in the categories of quantity and the nature of the synthesis of composition—do not require the relation of equality.

In addition to continuous quanta and their quantitas, there are also discrete quanta, that is collections of things insofar as they are units, and their quantitas, that is, their measure in answer to the question “how big?” Here, too it is one thing to represent a determinate collection, for example, a collection of shoes in my closet, and another to determine its quantitas according to some unit, for example, how many shoes or how many pairs of shoes there are. So, in addition to the distinction between indeterminate and determinate quanta, we also have to keep in mind the distinction between quantum and quantitas and the distinction between continuous and discrete quanta and the quantitas of each.

Distinguishing between the different senses of magnitude is further complicated by the fact that, in Kant’s view, the representation of the units of discrete quanta ultimately depends on the representation of continuous quanta, and because the determination of the quantitas of a continuous quanta requires regarding the continuous quanta as composed of equal units, and hence presupposes the representation of discrete quanta and their measurement. These entanglements reflect a long history of grappling with the relationship between continuous magnitudes and number. There are also further complications with Kant’s account, which make his views difficult to decipher; two salient examples include his conception of number, which includes both cardinal and ordinal elements, and his doctrine of schemata.

What is most important for the issue at hand are the distinctions between the representation of indeterminate continuous quanta, determinate continuous quanta, and the quantitas of these quanta. Nunez objects to my interpretation of indeterminate quanta in the proof of the thesis of the First Antinomy (A462n/B454n) and offers his own interpretation. Kant argues there against the possibility of representing the world as an infinite given whole of simultaneously existing things. Kant interprets this possibility as requiring the representation of the magnitude [Größe] of the world as a quantum, by which he means the quantitas of this quantum, that is, with the representation of the measure of its size. I will not discuss the proof as a whole and will focus on what Kant says about representing an indeterminate quantum.

Kant claims that unless a magnitude of a quantum is given within certain boundaries of any intuition, then it can only be thought through the synthesis of its parts, and the totality of the quantum only through the completed synthesis, or through the repeated addition of units to each other. What is of interest is the exception to his claim expressed in the unless clause—the case in which a magnitude of a quantum is given within certain boundaries of any intuition. A footnote clarifies that he means the case in which we intuit an indeterminate quantum as whole when it is enclosed within boundaries; in that case, we do not need to construct its totality through measurement, because the boundaries determine its completeness. How does Kant understand our thinking an indeterminate quantum as enclosed within boundaries?

As I just noted earlier, the representation of a determinate quantum requires a synthesis of composition that runs through a homogeneous manifold and also determines its limits through its starting and ending, as exemplified in the drawing of a line. Since space has multiple dimensions, it is possible to use the representation of determinate lines to delimit an area, such a square. The area is enclosed within boundaries and is thereby intuited as a whole, but the homogeneous manifold of space contained within those boundaries has not been determined through a successive synthesis of its manifold and hence is not itself a determinate quantum. Similarly, we can represent six square planes to delimit a volume, namely, a cube. The volume is enclosed within boundaries and intuited as whole, but the space contained within those boundaries has not been determined through a continuous successive synthesis of its manifold. It follows that representing indeterminate quanta as a whole in this way precludes directly measuring it by specifying a unit and synthesizing those units in a totality and thereby determining its quantitas. Its quantitas can nevertheless be indirectly determined if the quantitas of the delimiting sides are determined; if I know the length of the sides of a square the quantitas of the enclosed area is indirectly determined, and similarly for the quantitas of the sides of a cube and the enclosed volume. What is important is that Kant is allowing for two sorts of indeterminate quanta; space and time as infinite given wholes discussed in the Transcendental Aesthetic, and indeterminate quanta whose manifolds are not synthesized and yet are represented as a whole within boundaries.

Nunez construes the distinction between indeterminate and determinate magnitudes in a quite different way. In his view, indeterminate magnitudes are represented in empirical apprehension and determinate magnitudes are represented through a priori construction, where he understands a priori construction as requiring measurement (Nunez, p. 5ff [XX]). In short, a determinate magnitude is one that has been measured. Note first that this understanding of a priori construction ignores the distinction between the representation of a quantum and the representation of its quantitas in pure a priori cognition; on Nunez’s view, one cannot represent a quantum such as a line without measuring it. As I noted earlier, I do not think it is plausible, nor do I think it is Kant’s view, that whenever we represent a line by drawing it, we must measure it, determining how many equal units of some chosen unit of measure are contained in it.

According to Nunez’s reading of the First Antinomy, we can empirically apprehend an indeterminate magnitude within boundaries, where the apprehension involves the deployment of the categories of quantity and allows us to empirically apprehend it as a quantum prior to determining its quantitas through an a priori construction. This gives Nunez an account of the conditions of the possibility of experience that are less intertwined with the conditions for mathematical cognition and the exact sciences than the account I give. Empirical cognition does not require measurement.

Unfortunately, I do not think this can work. Note that Nunez preserves the distinction between quanta and quantitas in the case of empirical apprehension while rejecting it for the construction of geometrical figures in imagination, which makes his rejection of it in the latter case even less plausible. Moreover, and I think this is decisive, the argument of the Axioms of Intuition turns on Kant’s claim that appearances can only be apprehended and taken up into empirical consciousness through the same synthesis as that through which the representations of a determinate space or time are generated, and Kant states explicitly that this is the synthesis required for the representation of a quantum, so it is at the deeper level of quanta that empirical apprehension and mathematical cognition are intertwined, not just when the quantitas of an empirical intuition is determined. Moreover, everything Kant says in the Axioms about the determination of spaces and times, the synthesis of composition of a homogeneous manifold, and the concept of magnitude (Quanti) applies to mathematical objects as well, while nothing he says or argues presupposes the relation of equality or measurement; there is nothing beyond part/whole relations. It is the fact that apprehension of appearances and mathematical cognition are already fused at the level of quanta that makes appearances susceptible to measurement. It is true that mathematical cognition really gets going once we introduce measurement and hence the determination of quantitas. But our apprehension of appearances and mathematical cognition are already fused at the level of quanta, even though only the synthesis of pure a priori mathematical cognition is elective and counts as construction in the proper sense.

Once again, I am grateful for the thoughtfulness and depth of the comments provided by Anderson, Heis, and Nunez, which range from broadly programmatic to detailed issues of textual interpretation. I hope my responses have clarified my interpretation and hence Kant’s philosophy of mathematics and its relation to human experience.