This perfectly titled book delivers exactly what it promises. Even a casual reader of the first Critique can see that Kant is concerned to show that mathematics applies to the world, but Sutherland shows how deeply intertwined mathematics and the world of experience are: mathematics is not ‘an abstract and independently grounded body of knowledge that can be applied to the world: it is immanent in the world as we experience it’ (p. 283). What underlies the fundamentally mathematical character of the world is the idea that ‘mathematics is a science of magnitudes and the world of experience is a world of magnitudes’ (p. 1). Because the theory of magnitudes that Sutherland develops is ‘at the foundation of the theory of mathematical cognition’ but also ‘of Kant’s understanding of human cognition generally’ (p. 82), the book has implications well beyond the philosophy of mathematics. It sheds special light on Kant’s account of intuition and its role in cognition generally. The book is therefore essential reading for anyone interested in Kant’s theoretical philosophy.

Broadly speaking, this book examines the interplay between Kant’s theory of magnitudes, the theory of mathematical cognition and the theory of human cognition. Chapters 1–4 show that Kant’s philosophy of mathematics and his account of the mathematical character of the world rest on a theory of magnitude, and that this theory is fundamental to the account of experience in the first Critique. The primary focus here is on the Axioms of Intuition. Chapter 5 provides a novel reading of the singularity of intuition compatible with Sutherland’s reading of its role in representing magnitudes. Chapters 6–9 develop the theory of magnitudes in more detail, providing historical context and showing how Kant reshapes the theory of magnitude and the rationalist metaphysics of quantity in order to ground his account of mathematical cognition.

So what is the theory of magnitude? In what sense does it ground Kant’s philosophy of mathematics and his account of the mathematical character of the world and our cognition of it? On the one hand, there is the notion of magnitude handed down in the Euclidean tradition. On the other hand, Sutherland explains how Kant tweaks and deepens that traditional theory, introducing a definition of magnitude suited to explain our cognition of magnitudes in terms of our more basic cognitive capacities, in particular, the categories of quantity and the pure forms of space and time (p. 8); but ‘he does so in a way that lays the foundation for and retains what he sees as valuable in the traditional theory of magnitudes’ (p. 19).

For Euclid, magnitudes are characterized by their role in the Eudoxian theory of proportions: they are the sorts of things that can be combined and composed and stand in ratios and proportions. Magnitudes are homogeneous when they can stand in ratios with one another: lines with lines, areas with areas, volumes with volumes, angles with angles. Kant defines magnitudes as homogeneous manifolds, but they are what Sutherland calls ‘strictly homogeneous’: the manifold parts of any magnitude are all homogeneous with each other (p. 198). This is crucial for Sutherland’s account of the role of intuition in Kant’s philosophy of mathematics. Since concepts on their own cannot represent strict homogeneity, whereas intuition makes possible the representation of strict homogeneity, the strict homogeneity of magnitudes explains the essential role for intuition in cognition of magnitudes.

Why this modification of the notions of homogeneity and magnitude? Sutherland offers a mathematical explanation for the change to strict homogeneity: ‘only the representation of a strictly homogeneous manifold allows the representation of the sort of composition required for the Eudoxian theory of magnitudes and characteristic of mathematical cognition’ (p. 214), where the composition of strictly homogeneous parts ‘results in a whole that is greater than those parts’ (p. 213). Sutherland cites a passage from the Amphiboly which asserts this important feature of space: ‘for a part of space, even if it may be completely similar and equal to another [part] is nevertheless outside of it, and for this very reason is a different part from that which is added to it in order to constitute a larger space’ (p. 213; cf. A264/B320). This important property of the mereology of space is intimately tied to the representation of a strictly homogeneous manifold in intuition: the composition of a strictly homogenous manifold ‘yields more of the same’ (p. 213), thereby making possible our cognition of the part/whole structure of space in a way that concepts cannot. But Kant makes it clear here that ‘multiplicity and numerical difference are already given by space itself as the condition of outer appearances’: this feature therefore ‘holds of everything that exists simultaneously in the various positions in space’ (A264/B320). Here we begin to see how Sutherland’s analysis of the theory of magnitude in terms of strict homogeneity sheds light on Kant’s general theory of cognition.

The crucial point in the Critique where the theory of magnitude, the theory of mathematical cognition and the theory of cognition in general meet is in the Axioms of Intuition. The Axioms are mathematical principles that explain the applicability of pure mathematics to objects of experience, but the most important aim of the Axioms is, as Sutherland puts it, ‘to explain why and how the categories of quantity, under conditions of sensibility, are employed in the cognition of appearances’ (p. 272). The argument of the Axioms (pp. 61–2; B202–3) begins with the claim that the apprehension of appearances requires a synthesis of the manifold that generates determinate spaces or times. That synthesis in turn requires a composition of the homogeneous and the consciousness of the synthetic unity of this homogeneous manifold. But consciousness of the homogeneous manifold in intuition in general, insofar as through it the representation of an object becomes possible, just is the concept of a magnitude (quanti). It follows that all appearances are magnitudes (quanta).

The key claim, of course, is that the apprehension of appearances requires a composition of the homogeneous and consciousness of the synthetic unity of this homogeneous manifold. It’s clear why the synthesis generating representations of determinate spaces or times requires composition of a strictly homogeneous manifold: space and time are just such manifolds. In this way, Kant’s notion of magnitude answers to the demands of a theory of cognition in general and a theory of mathematical cognition specifically. The category of magnitude thus defined is essential to the account of our cognition of appearances in space and time.

In addition, the conditions making possible our cognition of space and appearances also make possible the representation of the sort of composition required for the Eudoxian theory of magnitudes. In this way, the Axioms reveal a necessary condition of the applicability of mathematics to the world: the cognition of objects of appearance must be subject to the same synthesis of composition as the synthesis underlying mathematical cognition. The concept of magnitude is that in virtue of which Kant ‘fuses our cognition of mathematics and objects of experience in the required way’ (p. 283). This is a wonderful statement of the upshot of the book. The theory of magnitudes is informed by and tweaked to fit both mathematical cognition and our fundamental cognitive capacities. By tweaking the notion of magnitude in both directions, Kant makes it possible to understand the world of experience as a world of mathematical magnitudes, thereby explaining the mathematical character of the world.

The added value of the emphasis on the theory of magnitudes is that it allows Sutherland to shed new light on the nature of our fundamental cognitive capacities – space and time as forms of intuition, the synthesis of composition, the role of intuition and of the categories – by diving deeper into the mathematical notion of magnitude. A further example of the fruitfulness of this approach is Sutherland’s solution to what he calls ‘the extensive magnitude regress problem’ (p. 98).

So far, I have only discussed magnitudes in general. But the notion of extensive magnitude is at the heart of Sutherland’s project. The Axioms establish the claim that all appearances are not just magnitudes but extensive magnitudes. That is, they are such that the representation of the parts makes possible the representation of the whole and therefore necessarily precedes the whole. Their centrality for Kant’s philosophy of mathematics results from the fact that only extensive magnitudes manifest their part–whole structure in a way that makes that structure cognitively accessible. The crucial claim in the extensive magnitude argument is that the representation of a determinate space or time (which is part of the content of an appearance) is possible only through a successive synthesis from part to part. As a result, all determinate spaces and times, therefore all appearances, are represented as aggregates of previously given parts out of which they are composed (p. 95). This accounts for the representation of the composition required for the Eudoxian theory of magnitudes.

Why is it that determinate spaces and times require a successive synthesis? Since only space and time require successive synthesis, it must be that the nature of space and time themselves imposes this constraint (p. 95). In the A Deduction, Kant seems to take this as evident: ‘it is obvious that if I draw a line in thought, or think of the time from one noon to the next … I must necessarily first grasp one of these manifold representations after another in my thoughts. But if I were always to lose the preceding representations (the first parts of the line, the preceding parts of time…) from my thoughts, and not reproduce them when I proceed to the following ones, then no whole representation … not even the purest and most fundamental representations of space and time, could ever arise’ (A99). Kant repeats these examples in the B edition: ‘I cannot represent to myself any line … without drawing it in thought, i.e., successively generating all its parts from one point … It is exactly the same with even the smallest time’ (B204).

This raises a problem confronting the account of extensive magnitudes and successive synthesis. Kant argues that all spaces and times and therefore all appearances are continuous in the sense that ‘no part of them is the smallest (no part is simple)’ (A169/B211). But we’ve seen that they are also such that the representation of the parts makes possible the representation of the whole. What then are the parts whose representation makes possible the representation of the whole? If they are themselves extensive magnitudes, then since there is no smallest part, it seems we have an infinite regress. But how are the parts represented, if not as extensive magnitudes?

If we don’t want to go the way of intuitive atomism (and Sutherland gives us compelling reasons not to), then to cut off the regress, we need a distinction between the representation of a whole extensive magnitude and the representation of its parts that allows us to deny that the parts are themselves extensive magnitudes. Since the relevant feature of extensive magnitudes is that they are determinate spaces and times, one solution would be to say that the parts are only indeterminately represented (and therefore aren’t themselves extensive magnitudes). The challenge then is to explain what the indeterminate representation of parts amounts to.

Sutherland’s solution, taking a cue from the Second Antinomy, is to take the indeterminate representation of parts to be the representation of parts that are ‘as yet unarticulated’: that is, the merely potential and as-yet-undivided parts that are nevertheless given in the whole intuition (p. 112). This doesn’t yet solve the regress though. The synthesis of composition is a synthesis of part to part. How are we to understand this if the parts aren’t represented as parts? Kant’s main example of the successive synthesis from part to part underlying our cognition of extensive magnitudes is the drawing in thought of a line ‘by successively generating all its parts from one point’. Similarly, a determinate magnitude of time, Kant says, depends on the successive progress from one moment to another, where through all the parts of time and their addition ‘a determinate magnitude is finally generated’ (A162/B203). Sutherland appeals to this passage to reinforce the claim that the parts synthesized are not themselves determinate magnitudes, but they are nonetheless represented: they are ‘successively generated’ from a point. At B155, Kant identifies the description of a space – the ‘pure act of the successive synthesis of the manifold in outer intuition’ – with motion. He thereby appeals to our understanding of motion as both continuous and successive in order to make explicit the special nature of the synthesis of composition that generates representations of determinate magnitudes. The continuity of the motion or synthesis ensures that each of the potentially determinately represented parts of the line is represented, but not explicitly represented as parts.

This has been pointed out by others, for example, by Thomas Land (Reference Land and Schulting2016: 168–9), but Sutherland takes it a step further. Underlying this solution is a claim that an undifferentiated continuous homogeneous manifold can be intuitively represented. But this gives rise to another question: how is this compatible with the singularity of intuition? If one understands the singularity of intuition to consist in representing individuals, then it seems that what should be intuitively represented are individual parts of space, rather than an undifferentiated continuous manifold. What is needed is an account of the singularity of intuition that allows for the intuitive representation of a continuous homogeneous manifold. Sutherland’s solution is to take the intuition as a mode of representing singularly that includes both the representation of singular objects like tables and chairs and houses but also, more fundamentally, the direct representation of an indeterminate homogeneous manifold. This is where things get a little more challenging.

Sutherland acknowledges that this understanding of intuition is ‘not easy to characterize’ (p. 142). His appeal to an admittedly imperfect map analogy is not especially helpful. Despite this, the considerations that bring Sutherland’s argument to this point provide independent reason to accept the underlying idea. He has identified as the most fundamental feature of Kantian intuition that it makes it possible to represent and cognize ‘an extended undifferentiated stuff’ (p. 125), a ‘muchness’ – as opposed to a manyness – that doesn’t include a representation of its parts. The representation of a singular object like a house depends on an underlying representation of such ‘an extended undifferentiated stuff’ (p. 125). The primary role for intuition is thus to represent ‘a substratum for conceptual determination’ (p. 143) which in turn makes possible the representation of a singular object. The most fundamental role of intuition is therefore not primarily to provide objects for our concepts, but to provide the substratum for the representation of those objects. Failure to recognize this is what leads to the mistaken view that it follows from the nature of intuition alone that it can represent only singular objects, including spatial parts. This is an excellent example of the fruitfulness of Sutherland’s emphasis on Kant’s theory of magnitude. That there remain details to be worked out concerning this new characterization of intuition does not take away from the value of the insight that makes the characterization necessary.

Kant’s Mathematical World is an important contribution to the Kant literature which will force a rethinking of fundamental aspects of Kant’s theoretical philosophy and spark debate among scholars of Kant’s theory of cognition and philosophy of mathematics.