The publication of Daniel Sutherland’s Kant’s Mathematical World (Sutherland Reference Sutherland2022; hereafter: KMW) is a major event in Kant scholarship and a significant event for our understanding of Kant’s philosophy of mathematics and its place in the critical philosophy. The book is deep, detailed, and original. For those of us who have read everything Sutherland has written in the last 20 years, many ideas will be familiar, but it is entirely different seeing the ideas all together and spelled out in a complete narrative. Having read the book, I now understand his view much more deeply than before and have found the resulting picture more persuasive and compelling. This is an excellent book—a book, like all of the best books, that requires and rewards careful reading and re-reading.

The main claim of Sutherland’s book is that “in Kant’s account, mathematics is a science of magnitudes, and the world of experience is a world of magnitudes” (1). A closely related secondary claim is that

In this paper, I will be reflecting on this main claim and the secondary claim. But before I jump in, I’d like to point out an initially surprising feature of Sutherland’s book. Despite the importance of the topic from the 1780s up to today, KMW is the first book-length treatment of Kant’s philosophy of mathematics (at least in English). But there has been a lively, high-quality debate over Kant’s philosophy of mathematics in the last few decades. There have been two principal subjects of this debate: the nature of Kant’s argument that mathematics is synthetic, and the Russell–Friedman thesis that intuition plays a logical role in mathematical proofs. Surprisingly, neither topic is mentioned explicitly in Sutherland’s book. In some ways, this is refreshing and highlights the originality of Sutherland’s approach. And, of course, Sutherland is planning a second volume, where presumably these topics will be discussed explicitly. But I worry that the omission of these two issues in KMW ends up doing mischief, in the sense that it leaves the argument for Sutherland’s main claim and secondary claim incomplete. Or so I will argue.

I will begin with an opinionated recapitulation of the main argument of the book. Sutherland’s main claim, that “mathematics is a science of magnitude,” is on its surface familiar and uncontroversial, both as an interpretation of Kant and as a position in the philosophy of mathematics before roughly 1850. But, as with so many other aspects of Kantian philosophy—the second analogy, the categorical imperative, and the parallelogram rule for adding velocities—the true novelty comes not in the principle itself but in the distinctive ground that Kant provides for the claim. Indeed, Kant says this explicitly:

Kant insists, then, that the claim that mathematics is the science of quantity is not true by definition but is the result of a deeper characterization of it as a constructive science. He then sketches an argument from the premise that mathematics is constructive to the conclusion that mathematics concerns quantity via some intermediary steps deploying the concepts of a quantum and a priori intuition. This argument is clearly present in Kant’s text, and Kant himself took it to be deeply important. And yet it really has not been explored adequately before Sutherland’s book, and it is, I think, one of the great merits of the book to have given it the serious consideration it deserves. In a pretentious mood, and without Sutherland’s endorsement, I will call Kant’s argument for the conclusion that mathematics is the science of magnitude the “Master Argument” of Kant’s philosophy of mathematics.

A few observations about the Master Argument. The argument brings together, as Sutherland’s reading makes clear and I will momentarily summarize, all the chief features of Kant’s philosophy of mathematics. Note that the conclusion of the argument, that mathematics is the science of quantity, is really a biconditional—if something is a science of quantity, then it is mathematics in Kant’s sense; and if something is mathematics in Kant’s sense, then it’s a science of quantity. The argument thus runs in two directions. The passage I just quoted from Kant goes in one direction (if a science is mathematical in the sense of constructive, then it concerns quantity); the argument in the other direction (if a science concerns quantity, then it must be mathematical in Kant’s sense) is the direction that Sutherland is following when he attributes to Kant a hitherto underappreciated argument for the essential role of pure intuition in the representation of magnitudes:

Sutherland’s idea is that the cognition of magnitudes requires pure intuition, which (now filling in the rest of the argument) makes possible cognition through construction, that is, mathematics.Footnote ¹ Thus, the direction in the Master Argument that proceeds from quantity to mathematics in Kant’s sense is the direction Sutherland focuses onFootnote ² (and the more difficult direction!).

I present the argument in a sequence of steps that link together a series of concepts in a chain whose broad outline can be sketched using the detailed discussions in KMW. The broad outline is given in Figure 1.

Figure 1.

Outline of the proof: the science of quantity is mathematics.

Figure 1. Long description

The flowchart consists of eight sequential steps, each marked by a blue box on the left and connected by downward-pointing arrows.

* Step 1: Quantity (Quantitas). A quantum is a homogeneous manifold in intuition (B 20 3). It is the concrete thing that has a quantity. A quantity is more abstract; it is the quantitative property of the concrete quantum. Since a quantitas is a quantity of a quantum.

* Step 2: Quantum. An extensive quantum is a quantum in which the representation of the part makes possible the representation of the whole. Since only extensive quanta make the part/whole structure of quanta cognitively accessible (K M W, 9 2).

* Step 3: Extensive Quantum. Since the only extensive quanta are spatial and temporal quanta.

* Step 4: Spatially (and/or temporally) extended things. Since all spatio-temporally extended things have a pure form: pure space and time. And so representing a spatio-temporally extended thing requires representing a pure spatio-temporal region.

* Step 5: Pure spatial (and/or temporal) regions. Since pure spatio-temporal regions are composed of strictly logically homogeneous parts. Thus: a quantum is a homogeneous manifold in intuition.

* Step 6: Strictly logically Homogenous Parts. Since concepts cannot represent strictly logically homogenous parts. Thus: a quantum is a homogeneous manifold in intuition.

* Step 7: Intuition of a quantum. To construct is to exhibit a priori an intuition corresponding to a concept (A 71 4 / B 7 42).

* Step 8: Construction. Mathematics is rational cognition from the construction of concepts (A 71 3 / B 7 41).

The final arrow points to the concluding term: Mathematics.

Now this sketch of the Master Argument is still quite incomplete. For one thing, the links in the chain are concepts; if it is to be an argument, we need to be linking premises and conclusions, and the arrows need to be implication relations. As I fill the outline in, I will try to be precise about the particular claims being made, and I will try also to be faithful to Sutherland’s ideas (though I may be going beyond what he feels comfortable with). Here is a proposal for the Master Argument, with more filled in:

Looking over the argument, there are at least four steps whose argumentative details I find difficult to fill in. In the first step, we are arguing that mathematics, as a system of cognitions concerning quantity,Footnote ³ requires the representation of quanta, i.e., not just the concept <quantum>, but the representation of particular quanta—what Sutherland calls “concrete quanta.”Footnote ⁴ Now, geometry uncontroversially concerns quanta—things like line segments, areas, and angles; but arithmetic and algebra, not so obviously. After all, number is the schematization of the concept quantitas (A142), and Kant says explicitly that “mathematics does not merely construct magnitudes (quanta), as in geometry, but also mere magnitude (quantitatem), as in algebra” (A717/B745). Sutherland sometimes talks of “mathematical objects,”Footnote ⁵ which I take to be pure extensive magnitudes, i.e., pure spatio-temporal regions. If we knew ahead of time that mathematics has as its objects these “mathematical objects,” this first step would be complete. But this is not the kind of thing that can be assumed at the get-go: why should we think that arithmetic and algebra require the representation of quanta? And why should we assume that even geometry should require the representation of concrete, singular quanta (as opposed to the concept <quantum> or concepts of species of quanta, such as the concepts <line> or <circle>)? Granted, of course, that every quantitas is the quantity of some quantum. But why would it follow that a science of quantitas must represent particular, concrete quanta?Footnote ⁶

In the second step, we argue from the claim that mathematics requires the representation of quanta to the claim that it requires the representation of specifically extensive quanta. In an extremely illuminating discussion, Sutherland argues that

Extensive magnitudes, i.e., spatial and temporal magnitudes, “manifest” the part–whole structure because they are precisely those magnitudes where the “representation of the part makes possible the representation of the whole” (A162/B203). Kant uses many examples to illustrate this: e.g., we measure the relative intensities of two light sources (an intensive magnitude) by measuring the distance (an extensive magnitude) at which the two lights appear equally intense. Now, if we knew that the mathematical representation of intensive magnitudes had to proceed as, for example, Newton does it in Principia, where velocities and forces are represented as line segments and areas, then this step would be secure. But why must it be this way? Put another way, “cognitively accessible” is a relative notion: accessible to whom or what? And for what kind of cognition? Now, if we knew ahead of time that mathematical proof is constructive and thus requires drawing general conclusions a priori from singular representations of individuals, then again this step would proceed smoothly. But that’s supposed to be the conclusion, not a premise.

In the fourth step, we move from the claim that the science of quantity requires the representation of spatio-temporal magnitudesFootnote ⁷ to the claim that it requires pure spatio-temporal regions. An important difference between spatio-temporal magnitudes (for instance, physical bodies such as blue and red wooden blocks) and pure spatio-temporal regions (for instance, cubical and pyramidal regions of space) is that the latter, but not the former, are what Sutherland calls “strictly logically homogeneous.” Indeed, according to Sutherland, Kant holds that “quanta, that is, concrete magnitudes, exhibit strict homogeneity; that is, the parts are specifically identical yet numerically diverse” (203).Footnote ⁸ This is a puzzling claim, because Kant asserts in the “Axioms of Intuition” that “all appearances are, as regards their intuition, extensive magnitudes,” and “all intuitions are extensive magnitudes.” But my intuitions of blue and red blocks are not strictly homogeneous—one is red and one is blue—and the blocks are indeed appearances, and so still magnitudes, despite all that. So it cannot be quite right that

1. 1. appearances are magnitudes,

2. 2. magnitudes are homogeneous, and

3. 3. the homogeneity of a magnitude is strict logical homogeneity.

Perhaps, on Sutherland’s view, Kant should have written something more qualified, such as “all appearances are, with respect to their form, magnitudes,” or “all appearances are, inasmuch as they are regarded a priori, magnitudes” or “all appearances contain magnitudes.”Footnote ⁹

Sutherland seems to me to qualify his claim that quanta are strictly logically homogeneous. When he considers the case of composing blue and red blocks into a larger three-dimensional “grouping,” he denies that the qualitative differences among the blue and red blocks prevent them from being mathematically combined. He clarifies: “Kant’s account requires that only those features that are directly represented as mathematically composed be represented as strictly homogeneous” (KMW, 215, emphasis mine). But if we allow that two extensive magnitudes can be mathematically combined as long as they are homogeneous with respect to a composable feature, then this undermines the necessity of Sutherland’s “mathematical objects.” The science of quantity would need to deploy only representations that are homogeneous with respect to certain features. And plain old spatio-temporal objects, such as red and blue blocks, would suffice just fine after all.

In the sixth step, we arrive at the secondary claim of KMW, that “there is a role for intuition in representing magnitudes, and consequently in … mathematical cognition” (23). The difficulty is understanding what precisely is being claimed here—what precisely the role of intuition in mathematical cognition is. Sutherland summarizes the argument of this step in the following way:

So the necessity of intuition in mathematics, as the science of magnitude, follows ultimately from the necessity of representing strict homogeneity. But what precisely is the role of the representation of strict homogeneity in the science of magnitude, and so what precisely is the role for intuition? For instance, one very minimal role for intuition would be to demonstrate the possibility of quanta and thus the objective reality of the concept <magnitude>.Footnote ¹⁰ This role, though important, is still minimal, since, on Kant’s view, intuition is required to prove the objective reality of any concept whatsoever, and thus mathematics would be no more an intuitive science than any other. Compare this to a view according to which the science of quantity requires, for its demonstrations, a singular representation of a concrete individual quantum, some of whose properties can be immediately and a priori known. This view would require a maximal role for pure intuition, a role that would be distinctive of the science of mathematics. Is this maximal role the kind of “important role” that step six is supposed to secure?

I’ve been expressing a discomfort with the clarity of Sutherland’s conclusion that mathematics, as the science of quantity, requires an important role for pure intuition. As the conclusion to step 6 of the Master Argument, a possible way to sharpen the claim would be to look at step 7 and fill in from there: were the science of quantity to be constructive, what would the role of intuition in the science of quantity have to be? Given the purposes of the book, Sutherland understandably does not devote a considerable amount of time to the notion of construction, beyond a portion of Chapter 2. However, I have reservations about the account of construction provided there. Let me explain.

In Chapter 2, Sutherland examines the relations between Kant’s views of determination, construction, and the concept of a quantum. Two of the goals in this chapter are to show that pure mathematics requires the concept of a quantum, and that the same activity—the determination of a space and time through figurative synthesis—is a precondition of both pure mathematical cognition and the experience of an appearance.Footnote ¹¹ In the process of making this claim, Sutherland argues convincingly that mathematical construction requires the determination of space and time through a figurative synthesis and that a representation so synthesized is a representation of a quantum. But Sutherland also argues for the converse claim: “the cognition of a determinate space or time requires construction in intuition” (KMW 45n). And this converse claim needs a separate defense: from the fact that all constructions are determinations, it does not follow that all determinations are constructions. Now the converse claim is not needed to achieve the two goals I listed above for Chapter 2. But nevertheless, I believe that the converse claim is false, and it will be instructive to see why.

Construction for Kant has a special meaning—it is the kind of cognitive procedure that a mathematician carries out in a demonstration that makes it possible for her to draw a general conclusion from the properties of a concrete but representative individual (A734-5/B762-3). The conditions for construction, according to Kant, are strict: among other things, a concept can be constructed only if it possesses a (real) mathematical definition (A727-31/B755-9). These conditions are strict enough, according to Kant, that all kinds of sciences fail to meet them, most famously philosophy, but also some of the natural sciences (other than physics). Thus, representing a quantum through determining a space, time, or quality is different from constructing the concept of that quantum. When I visually perceive a dog, I determine the space it occupies: I intuit it as dog-shaped. But that is very different from a mathematical construction. It is simply impossible for me to construct the concept of <dog> or even <dog-shaped>, because, as empirical concepts, they are incapable of (real, mathematical) definitions and so also not susceptible to construction. Hence, geometers have no theorems about dogs or dog shapes.Footnote ¹² Determining a space and constructing are therefore not co-extensive. One can represent a quantum in intuition without constructing the concept of that quantum.

Though this point, as I mentioned, does not affect Sutherland’s ability to achieve the argumentative goals in Chapter 2,Footnote ¹³ I do think that it does affect the book’s ability to clarify the “role for intuition in representing magnitudes … in both mathematical cognition and the cognition of appearances” (KMW, 23). Unless we are clear on the different roles that the representation of magnitude plays in the apprehension of an appearance and in mathematical construction, the precise role of intuition in mathematics will also be blurry. And unless we are clear on the precise role of intuition in mathematical cognition (and how it differs from its related role in experience), the clear outlines of the Master Argument of Kant’s philosophy of mathematics will remain elusive.

In Lanier Anderson’s book The Poverty of Conceptual Truth, he cites Sutherland’s earlier work (which was expanded and systematically presented in KMW) to present an interpretation of Kant’s argument that elementary mathematics is synthetic and must make substantive and essential use of pure intuition. In the book, Anderson sometimes appeals to Sutherland’s work on the homogeneity of magnitudes in a way that suggests what one might call a “quick and dirty” argument for the syntheticity of elementary mathematical truths:

Anderson’s claim is that the representation of a magnitude requires the representation of numerical difference despite conceptual identity, and so truths that “rest on” the “comparison” of logically homogeneous non-identicals must be synthetic. The second passage identifies a specific kind of truth that rests on the comparison of homogeneous non-identicals: truths concerning the equality of non-identicals.

Read in the quick and dirty way, this argument would show that any judgment concerning magnitudes would be synthetic, or at least any judgment that asserts that non-identicals are equal would be synthetic. But read this way, the argument is too quick and, indeed, dirty. There are many analytic judgments concerning magnitudes. There are analytic judgments that deploy the concept <magnitude>: “All magnitudes are homogeneous”; “A triangle is a magnitude.” There are analytic judgments that deploy concepts of specific species of magnitudes: “All triangles are three-sided” (A716/B744). There are even judgments that Kant himself identifies as analytic that assert the “equality of non-identicals”: the Euclidean common notion “Equals added to equals are equal” (A164/B204-5). But the quick and dirty argument would imply that all of these analytic judgments are in fact synthetic.

Of course, Anderson’s argument need not be read in the quick and dirty way, and, in fact, there are other places in Anderson’s book where he suggests an argument that depends not just on the fact that elementary mathematical truths concern homogeneous non-identicals but on very specific facts about how mathematical reasoning works. Indeed, in the first passage quoted above, Anderson claims that a truth must be synthetic if it rests on “this sort of comparison” (229). But what is the kind of comparing that mathematicians do? In Chapter 9 of Anderson Reference Anderson2015, Anderson’s goal is to investigate “how … claims were actually sustained in elementary mathematics as Kant knew it” and the “basic argumentative patterns of even elementary mathematics” (233). Based on an analysis of four propositions in Euclid’s Elements I, he argues that those basic argumentative patterns include, among others, (i) proving a general theorem by drawing an individual concrete figure and (ii) intuitively ascertaining when one figure in a diagram is a proper part of, or spatially overlaps with, another figure in a diagram.

This then is the kind of “comparison” that leads to syntheticity: the comparison of the spatial overlap of concrete individuals. Here, of course, Anderson is knee-deep in discussion of those old hobby horses, the nature of Kantian analyticity and the Russell–Friedman thesis, which KMW avoids.Footnote ¹⁴ But in avoiding these hobby horses, KMW does not clearly identify what specific “important role” intuition plays “in mathematical cognition by allowing us to represent … properties” of magnitudes (KMW, 58–9). For instance, is the role of intuition in mathematics exhausted by the minimal kind of role it plays in grounding analytic judgments such as “Equals added to equals are equal,” or does it play the more maximal role of representing immediately the spatial overlap of concrete individuals?

What matters is the particular way in which particular magnitudes, or particular concepts that pertain to magnitudes, or the concept <magnitude> itself, are deployed in mathematics. For if we want to know the essential role of intuition in mathematics, and it is through the representation of magnitudes as (strictly logically) homogeneous that intuition enters into mathematical reasoning, then we need to know specifically how magnitudes (or concepts of/the concept of magnitude) are represented in mathematics. Kant’s own argument from A714/B742 (quoted above) starts with a characterization of how mathematical cognition works—by construction, as the a priori exhibition of an intuition corresponding to a concept. There’s a lot of meat in that characterization of construction and so enough to allow Kant to derive his conclusion that mathematics can only concern magnitudes. What I’m hoping for is a similarly meaty characterization of the “science of magnitude” that will enable us to derive the conclusion that the science of magnitudes requires intuition in a particularly robust way—a way that will entail that the science of magnitude is mathematics.

A comparison with what Kant says about physics in the introduction to the Metaphysical Foundations of Natural Science (Ak 4:470) is instructive. He begins with a meaty characterization of physics—it is the proper science of the nature of matter. From this characterization, he infers that physics must provide an a priori demonstration of the actuality of matter, and so must be constructive and thus mathematical.

Earlier I mentioned Kant’s view that Euclid’s common notions, such as The whole is equal to its parts and Equals added to equals are equal, are analytic. For that reason, a subject can form the judgment and come to know this truth with a comparatively minimal contribution from intuition: there need not be any intuitive “third thing” (A155/B194) that grounds the relation between the component concepts in the judgment. Suppose all the judgments of mathematics were analytic in this way and all of the inferences of mathematical proofs were just formal-logical. Intuition would still have a role here, in securing the objective reality of the concepts. The concept <equals>, for instance, is a determination of the concept <magnitude>, and so to know that the concept <equals> is not empty, we’d need to know that a magnitude, as a homogeneous multiplicity, is possible—a task that KMW argues requires Kantian intuition. But this strikes me as small potatoes, and surely Kant (and Sutherland’s Kant) wants more than this. But the question is how to show it. The Master Argument, which I’ve reconstructed from KMW, proceeds in six steps from the initial claim about the science of quantity to the conclusion that the science of quantity requires an intuitive representation, and I found difficulties above on how to close up the first, second, and fourth steps. Surely one reason for these difficulties is that the conclusion of the sixth step—that a science of magnitude requires intuition—is blurry, and so it’s hard to know how to fill in the steps until the goal of the argument comes into clearer focus.

In order to make the argument clearer and to fill in the gaps, there are three proposals that I suggest may be worth trying out, each of which supplements the outline of the Master Argument by appealing to some extra feature of the science of quantity to complete the argument. On the first proposal, what is missing is a further explication of the notion of “(proper) science,” on analogy with the introduction to the Metaphysical Foundations of Natural Science (Kant Reference Kant, Allison, Heath, Hatfield, Friedman, Allison and Heath2002), where Kant argues from the claim that the science of nature is a proper science, despite concerning determinate natural things, to the claim that it must be mathematical in form and must deploy pure intuition. So the first suggestion would be to exploit analogous facts about mathematics as a proper science of quantity in order to argue that it requires the a priori representation not just of quantity but of determinate concrete quanta. And perhaps this could close the gaps I identified.Footnote ¹⁵

A second proposal would draw on additional features of the science of quantity, namely, that it is a priori and that it proceeds by considering representative individuals in order to justify general claims. The fact that mathematics proceeds by considering representative individuals would secure step one—to reason about quantities, we would need to represent individuals that have those quantities, namely, individual quanta—and step four would be secured by mathematics’ a priority—the representation of pure spatial regions would be a priori, but not the representation of blue and red blocks. (Similarly for step 2.)

A third suggestion adds in the consideration that the science of quantity is a system of synthetic a priori judgments about quantity. This seems to me most promising. Being synthetic, the judgments about the science of quantity would require some third thing besides the concepts involved, and this third thing would be an individual that has a quantity, namely an individual quantum (thus, step 1). Being a priori, again, the blue and red blocks could not be that third thing, and so the science of quantity would require pure extensive magnitudes, that is, the pure spatial regions (step 4). And step 2 might go through, since only an extensive magnitude could exhibit its quantitative properties in a way accessible to synthetic a priori cognition.Footnote ¹⁶

I asserted at the beginning that Sutherland’s relative silence on two large issues in Kant interpretation—the nature of Kant’s argument for syntheticity and the Russell–Friedman thesis concerning mathematical proof—may end up doing mischief. Now we see why. In order to understand Kant’s argument for the main claim of Sutherland’s book, that is, the Master Argument for the conclusion that mathematics is the science of magnitude, and in order to understand Sutherland’s secondary claim, that the representation of magnitudes in mathematics requires pure intuition, we need ultimately to bring in considerations about the character of mathematical proof and the synthetic character of mathematics—the two topics Sutherland largely avoids.

I intend all of this as a friendly suggestion and invitation for further discussion. After all, Sutherland has promised a second book where these issues may be resolved and the mischief eliminated. But let me close now by thanking Daniel for the honor of commenting on a book so rich and so important.