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Kant on Arithmetic, Algebra, and the Theory of Proportions D a n i e l S ut h e r l a n d *

kant’s philosophy of mathematics has both enthralled and exercised philosophers since the appearance of the Critique of Pure Reason. Neither the Critique nor any other work provides a sustained and focused account of his mature views on mathematical cognition, forcing readers to glean what they can from disparate contexts. Despite these hurdles, Kant’s views have been of great interest to philosophers of mathematics. They have also been of interest to philosophers wishing to understand Kant’s philosophy more generally, since Kant maintains that mathematical judgments are synthetic a priori and that the type of synthesis underlying mathematics is the same as that underlying the perception of objects (CPR, B 202–03, A 163/B 204).1 Our understanding of Kant’s views has been greatly improved during the last four decades by work on Kant’s philosophy of mathematics.2 1  This paper will focus on giving an account of Kant’s philosophy of mathematics. It will not aim to draw out its implications for his theory of cognition more generally. In the standard way, I will refer to the Critique of Pure Reason using ‘A’ for the first edition pagination and ‘B’ for the second. Other references will be to Kant’s Gesammelte Schriften, (Berlin: G. Reimer, subsequently Walter de Gruyter & Co., 1902–) by volume and page number. A reference to a lecture will be followed by the best estimate of the date the lecture was given. All translations from the German are my own, although I have closely consulted Immanuel Kant, Critique of Pure Reason, ed. and trans. Paul Guyer and Allen W. Wood (Cambridge: Cambridge University Press, 1998); Immanuel Kant, Prolegomena to Any Future Metaphysics, ed. and trans. Gary Hatfield (Cambridge: Cambridge University Press, 1997); and Immanuel Kant, Lectures on Metaphysics, ed. and trans. Karl Ameriks and Steve Naragon (Cambridge: Cambridge University Press, 1997). 2  The contributors are too numerous for an exhaustive listing. The relatively recent interest was spurred by the work of Jaakko Hintikka and Charles Parsons; subsequent authors include Philip Kitcher, Gordon Brittan, Manley Thompson, J. Michael Young, Arthur Melnick, and Michael Friedman, among others. A helpful selection of these papers and references to others published before 1992 is found in Kant’s Philosophy of Mathematics: Modern Essays [KPM], ed. Carl Posy (Dordrecht: Kluwer, 1992). A few works not in that collection, as well as more recent work, include: J. Michael Young, “Kant on the Construction of Arithmetical Concepts” [“Kant on the Construction”], Kant-Studien 73 (1982): 17–46; Michael Friedman, Kant and the Exact Sciences [KES], (Cambridge, Mass: Harvard University Press, 1992), as well as “Geometry, Construction and Intuition in Kant and His Successors,” in Between Logic and Intuition: Essays in Honor of Charles Parsons, ed. Gila Sher and Richard Tieszen (Cambridge:

* Daniel Sutherland is Associate Professor of Philosophy at the University of Illinois at Chicago. Journal of the History of Philosophy, vol. 44, no. 4 (2006) 533–558

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Despite these advances, however, I think the recent work has largely neglected the importance of Kant’s theory of magnitudes and the extent to which that theory is influenced by the Eudoxian theory of proportions presented in Euclid’s Elements. As a consequence, an important feature of Kant’s philosophy of mathematics has been overlooked: the role of intuition in the representation of magnitudes.3 Eudoxus developed the theory of proportions to provide a mathematical treatment of continuous magnitudes, in particular spatial magnitudes, and there is strong evidence for this interpretation with respect to Kant’s geometry. It is unclear, however, what bearing this account has on discrete magnitudes, and hence arithmetic. In general, Kant’s claim that intuition is required for mathematical cognition has seemed less comprehensible and plausible in the case of arithmetic than in the case of geometry. This paper will attempt to make it more comprehensible and plausible by focusing on the place of arithmetic in Kant’s theory of magnitudes. Kant’s views on arithmetic and discrete magnitudes can be best clarified by considering two closely related issues. First, does Kant think that there is a universal mathematics common to all mathematical disciplines, and if so, how does it relate to arithmetic? Second, what is the role of intuition in our cognition of arithmetical propositions? I will argue that Kant thought of algebra as an overarching universal mathematics that includes arithmetic, that it takes magnitudes as its object of study, and that algebra expresses the Eudoxian theory of proportions.4 Furthermore, on Kant’s view, intuition plays a crucial role in arithmetic through representing discrete magnitudes. My investigation will require discussion of the Greek mathematical tradition and an examination of the development of algebra in the early modern period, since early modern conceptions of arithmetic were strongly influenced by both. In the first part of the paper, I examine Kant’s arithmetic in relation to the theory of proportions. Section 1 sets the stage with a brief introduction to the Greek conCambridge University Press, 2000), 62–100; Emily Carson, “Kant on Intuition in Geometry,” Canadian Journal of Philosophy 27 (1997): 489–512, as well as “Kant on the Method of Mathematics,” Journal of the History of Philosophy 37 (1999): 629–52; and Lisa Shabel, “Kant on the ‘Symbolic Construction’ of Mathematical Concepts” [“Symbolic Construction”], Studies in History and Philosophy of Science 29A (1998): 589–621. 3  I have argued for this interpretation in “Kant’s Philosophy of Mathematics and the Greek Mathematical Tradition” [“Kant’s Philosophy of Mathematics”], Philosophical Review, 113 (2004): 157–201. 4  Michael Friedman draws the connection between Kant’s mathematics, in particular his algebra, and the Eudoxian theory of proportions, in Friedman, KES, 107–14. It was Michael Friedman who first encouraged me to look at Kant’s views on the Eudoxian theory of proportions in relation to his views on magnitude, and his work is a departure point for my interpretation. Other work touching on issues related to Kant’s algebra does not focus on the theory of proportions, but is illuminating in other ways. It includes C. D. Broad, “Kant’s Theory of Mathematical and Philosophical Reasoning,” Proceedings of the Aristotelian Society 42 (1941): 1–24; Jaakko Hintikka, “Kant’s Mathematical Method,” Monist 51 (1967): 352–75; Charles Parsons, “Kant’s Philosophy of Arithmetic” [“Kant’s Arithmetic”], in Mathematics in Philosophy (Ithaca: Cornell University Press, 1983); Charles Parsons, “Arithmetic and the Categories” [“Arithmetic and Categories”], Topoi 3 (1984): 109–22; Manley Thompson, “Singular Terms and Intuitions in Kant’s Epistemology,” Review of Metaphysics 26 (1972): 168–89; Philip Kitcher, “Kant and the Foundation of Mathematics,” Philosophical Review 84 (1975): 23–50; and Gordon Brittan, Jr., “Algebra and Intuition,” in KPM, 315–39. See also J. Michael Young, “Kant on the Construction,” and Gottlieb Martin, Arithmetic and Combinatorics: Kant and his Contemporaries [AC], trans. Judy Wubnig (Carbondale and Evansville: Southern Illinois University Press, 1985). Lisa Shabel, “Symbolic Construction” provides a particularly helpful and careful investigation of Kant’s views on algebra and their relation to the early modern tradition of geometrical problem-solving.

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ception of number and the Eudoxian theory of proportions. Section 2 describes Kant’s project of explaining mathematical cognition by explaining the cognitions presupposed by the theory of proportions. It suggests a natural way of assimilating arithmetic and number into his account of geometrical cognition, but raises three problems with this assimilation. These problems motivate a closer examination of the relationship between arithmetic and algebra, which I investigate in the second part of the paper. Sections 3 and 4 provide needed background on the relation between arithmetic and algebra in the early modern period and in Kant. They argue that Kant thought of algebra as a universal mathematics and that algebra expresses the Eudoxian theory of proportions. With this important background in place, Section 5 argues for a different understanding of Kant’s conception of arithmetic and number from that suggested in Section 2.

1. the greek conception of number and the theory of proportions The way in which numbers were understood by Greek mathematicians and philosophers (in particular, by Plato and Aristotle) is a matter of considerable debate. Nevertheless, one can identify several important features of the Greek conception of numbers that had far reaching consequences. Plato distinguished between numbers of things, on the one hand, and “pure” numbers, on the other. The former are identified with the things numbered and are paradigmatically collections of a kind of sensible object, such as cattle or army camps. The latter are accessible only to thought and their study belongs to the philosopher.5 Our ability to count collections of things depends upon our thinking of the pure numbers, although someone engaged in practical counting may not be aware of this fact.6 Both sorts of number presuppose a unit as its basis. Plato holds that the numbers of things (i.e., the numbered) consist of units that can be distinguished from each other, such as the cows that make up a cattle-number. In contrast, the pure units presupposed by pure numbers “are each equal to every other and are not in the least different,” that is, they are indistinguishable from each other.7 As is well known, Aristotle rejects Plato’s claim that pure numbers have an independent existence, arguing that they are mere abstractions whose being depends upon the objects from which they are abstracted. Nevertheless, Aristotle’s abstractions are also pure numbers, and the units presupposed by them are also indistinguishable, for we abstract from all differences and think of undifferentiated units.8 The Greek no5  I set aside whether Plato thinks of these numbers as Forms or as “intermediates.” For a detailed account of the Greek conception of number and its role in Greek mathematics, see Jacob Klein, Greek Mathematical Thought and the Origin of Algebra [GMT], trans. Eva Brann (New York: Dover Publications, 1968). The following summary of the Greek conception of number relies on Klein’s work. See Klein, GMT, 22 and 46–47 on pure numbers. 6  Klein GMT, 71, 60. 7  Plato, The Republic, ed. G. R. F. Ferrari, trans. Tom Griffith (Cambridge: Cambridge University Press, 2000), 526A. Cf. Plato, Philebus, trans. Dorothea Frede (Indianapolis: Hackett Pub. Co., 1993), 56D–E. See Klein GMT, 22–24, for a discussion of these texts. 8  “… in mathematical number no unit is in any way different from any other” (Aristotle, The Complete Works of Aristotle [CW], ed. and trans. Jonathan Barnes [Princeton: Princeton University Press, 1984], Met. M.6, 1080a21). “… mathematical number consists of undifferentiated units …” (Aristotle, CW, Met. M.7, 1081a20). “In general, to differentiate the units in any way is an absurd fiction; and by

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tion of pure units persisted in the Greek mathematical tradition and, as we shall see below, influenced Kant. Plato also maintains that pure units are indivisible;9 hence, only whole numbers are, properly speaking, numbers. Pythagoreans before Plato shared this view, which is also found in Euclid’s Book 7, where Euclid defines “number” as a collection of composed units, and then avoids fractions of the unit.10 This conception of number leads to difficulties, however. The Pythagoreans developed the first theory of ratios, and in their approach, ratios between magnitudes (such as between two lines or between two areas) were expressed as ratios between numbers. Because numbers were limited to whole numbers, however, the theory could not be applied to magnitudes more generally, as they had originally thought. If a square has sides of length 1, then a diagonal of the square has a length of √2. The latter is today called an irrational number precisely because it cannot be expressed as the ratio of two whole numbers; these two magnitudes are also said to be incommensurable because there is no unit that will measure both. In other words, if some multiple of a unit measures one magnitude, no multiple of that unit will measure the other magnitude—no matter how small one takes the unit to be. Thus, there were ratios between magnitudes which could not be expressed by the ratio between two whole numbers; Greek mathematicians eventually recognized this fundamental limitation of the Pythagorean theory of ratios. Eudoxus responded to this difficulty by developing an ingenious treatment of ratios. His approach turns on defining a proportion (i.e., the sameness of two ratios) in a way that does not appeal to numbers. In Book 5, Euclid defines a ratio as “a sort of relation with respect to size between two homogeneous magnitudes” and follows Eudoxus in his definition of sameness of ratios:11 Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and the third, and any equimultiples whatever of the second and the fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. (Euclid, Elements, Book 5, Def. 5)

In modern algebraic notation, the definition would read: For any four magnitudes a, b, c and d and any two positive natural numbers m and n, a :b = c :d iff for all m, n: ma > nb → mc > nd ma = nb → mc = nd ma < nb → mc < nd. a fiction I mean that which is brought in forcibly to suit a hypothesis. For neither in quantity nor in quality do we see unit differing from unit …” (Aristotle, CW, Met. M.7, 1082b2–5). See Klein, GMT, 103–05. Aristotle suggests a similar notion of purity for geometrical objects in CW, Met. K—a point to which I shall return. 9  Plato, The Republic, 525E. 10  See Euclid, Elements, ed. and trans. T. L. Heath (New York: Dover Publications, 1956), Book 7, Def. 2. It is thought that Theatetus, a Platonist, was the source for Euclid’s “arithmetical” Books 7–9. See Klein, GMT, 43. 11  For a lucid account of the Eudoxian theory of proportions, see Howard Stein, “Eudoxus and Dedekind: On the Ancient Greek Theory of Ratios and Its Relation to Modern Mathematics,” Synthese 84 (1990): 163–211. My exposition draws from his account. See also Friedman KES, 110–13, which draws this connection between the Eudoxian theory of proportions and Kant’s views in particular.

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In other words, one pair of magnitudes stands in the same ratio as another if and only if the comparative size relation of the first pair (greater, equal, or less) is the same as the comparative size relation of the second pair under all equimultiple transformations.12 Comparative size and multiplication are relations and operations on magnitudes themselves, not on numbers corresponding to the sizes of magnitudes. As a consequence, the Eudoxian theory of proportions does not use numbers to express ratios and can be applied to incommensurable magnitudes. The Greek concept of number, the problem of incommensurables, and the resulting Eudoxian theory of proportions had far-reaching consequences for the relation between arithmetic and geometry in the Greek mathematical tradition handed down through Euclid. Numbers counted as magnitudes, and in particular, discrete magnitudes.13 Because numbers cannot stand in many of the ratios that continuous magnitudes can, however, numbers appear to be just a special, limited case of magnitudes more generally. As a result, the Euclidean tradition focused on the continuous magnitudes of geometry rather than on arithmetic and developed mathematics as far as possible without appeal to numbers. The Greek mathematical tradition passed down through Euclid was shaped by this priority of geometry over arithmetic and influenced thinkers into the eighteenth-century, including Kant.14 There were further important features of the theory of proportions that influenced Kant’s philosophy of mathematics. Euclid uses the phrase ‘homogeneous magnitude’ to designate the sorts of things that are capable of standing in ratios.15 Euclid primarily has in mind things like lines, areas, and numbers; two lines can stand in a ratio, for example, and the numbers 4 and 7 can stand in the ratio 4:7.16 12  The definition of sameness of ratios allows that two pairs of magnitudes can stand in the same ratio even if the pairs are inhomogeneous with each other; that is, the magnitudes a and b must be homogeneous with each other, as must c and d, but a and b need not be homogeneous with magnitudes c and d. For example, two lines can stand in the same ratio as the numbers 1 and 5 and in the same ratio as the areas of two triangles. Proportions are useful and powerful in part because they allow us to make claims that relate ratios of different kinds of magnitudes. 13  Euclid states that a ratio is a relation in respect of size between two magnitudes of the same kind (Elements, Book 5, Def. 3). Since Euclid clearly thinks that numbers can stand in ratios, they too are magnitudes. 14  The story is much more complex than this quick summary implies, but this characterization will be adequate for the aims of my paper. 15  Homogeneous magnitudes are not quite coextensive with those things that are capable of standing in a ratio. Euclid leaves open the possibility that two homogeneous magnitudes might not stand in a ratio to one another, for he states that in order to have a ratio, each magnitude must be capable, when multiplied, of exceeding each other (Euclid, Elements, Book 5, Def. 4). This is known as the Archimedean property, and it rules out ratios between homogeneous magnitudes if one is infinitely small or infinitely large with respect to the other. Although not all homogeneous magnitudes stand in ratios, what is important about homogeneous magnitudes is that they are the sorts of things that can stand in ratios with each other and that do stand in ratios if we set aside infinitesimal and infinite magnitudes. 16  It is not entirely clear what kinds of things the Greeks counted as magnitudes. Although the kinds mentioned were clearly paradigmatic, angles, weights, and other things may have been counted as magnitudes as well. Aristotle, who followed Eudoxus and preceded Euclid, does not use the term magnitude when discussing ratios and proportions, and he may have wished to reserve the term for geometrical magnitudes. See Ian Mueller, “Homogeneity in Eudoxus’s Theory of Proportion” [“Homogeneity”], Archive for the History of the Exact Sciences 7 (1970–71): 1–6. Euclid, on the other hand, thought of numbers as magnitudes, since magnitudes are characterized as what can stand in ratios, and numbers can stand in ratios.

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Thus, lines are homogeneous with lines, planes with planes, and numbers with numbers. A line cannot stand in a ratio to a number, just as an area cannot stand in a ratio to a volume, because there is no sense that can be given to the comparative size relations between them. Homogeneous magnitudes, in contrast, can stand in comparative size relations. Furthermore, homogeneous magnitudes can compose a magnitude that is homogeneous with them and greater than each.17 Lines can be combined to form longer lines, spaces to form larger spaces, and so on. This is a sense of composition particular to homogeneous magnitudes and is essential to their mathematical character. Books 5 and 7 of the Elements establish sixty-four propositions concerning the ratios and proportions of magnitudes, which include laws governing the ratios between magnitudes under the operations of addition and subtraction, the alteration of terms of ratios, and the addition and subtraction of proportions. These propositions govern the addition and subtraction of numbers as well as the composition and decomposition of other magnitudes; they thereby express the basic mathematical properties of magnitudes. We have seen that the Eudoxian theory of proportions rests on the composition of magnitudes and the comparative size relations of equal, greater than, and less than. This foundation for the theory of proportions can be narrowed, however, since one magnitude being ‘greater than’ another can be defined as the second being ‘equal to a part of’ the first. ‘Less than’ can be defined similarly. Thus, composition, part-whole relations, and equality and are sufficient to account for the basic mathematical properties of magnitudes. The fact that the theory of proportions expresses the basic mathematical properties of magnitudes and can be accounted for by the composition, part-whole relations, and equality of homogeneous magnitudes shaped Kant’s philosophy of geometry.18 It also shaped Kant’s philosophy of arithmetic and algebra, or so I will argue.

2. kant’s explanation of the possibility of mathematical cognition 2.1. Kant on the Mathematical Cognition of Continuous Magnitudes What follows is a condensed summary of Kant’s theory of magnitudes.19 Kant’s philosophy of mathematics rests on his theory of magnitudes, which in turn rests on the Eudoxian theory of proportions. One aim of the Critique is to explain the possibility of synthetic a priori cognition, which includes mathematical cognition. Although Kant argues that mathematical cognition is synthetic a priori, he is less clear on exactly how mathematical cognition works; he does not, for example, devote a section of the Critique to explain it, leaving one to sift through many relatively brief passages. However, Kant states more than is usually recognized

17  The requirement of homogeneity for standing in ratios may not have been a part of Eudoxus’ original theory, but it was a requirement by the time of Euclid. See Ian Mueller, “Homogeneity,” 1–6. 18  I defend this interpretation of Kant’s philosophy of mathematics in Sutherland, “Kant’s Philosophy of Mathematics.” 19  For a fuller account, see Sutherland, “Kant’s Philosophy of Mathematics.”

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about the nature of mathematical cognition in the Axioms of Intuition.20 Kant thinks of mathematical cognitions as cognitions of magnitudes.21 Since his view of magnitudes derives from the Euclidean tradition, his account of mathematical cognition turns on the cognitions that make the theory of proportions possible. Those cognitions include the cognition of comparative size relations by means of the cognition of equality and part-whole relations. They also include the cognition of the part-whole composition relations of magnitudes. In Kant’s theory, the part-whole composition relations are cognized by applying the categories of quantity—unity, plurality, and totality—to intuition. Thus, the categories of quantity provide a mereological foundation for mathematical cognition. Kant defines a magnitude as a homogeneous manifold in intuition, which reflects the Eudoxian-Euclidian conception of homogeneous magnitudes in Book 5 of the Elements.22 The Greek and Kantian notions of homogeneity attempt to capture the special sort of mathematical composition of which magnitudes are capable—a composition in which parts of the same kind can be composed to make more of exactly the same kind. Elements of these kinds (such as lines, or areas, or volumes) are said to be homogeneous with each other, and I will refer to them as ‘mathematically homogeneous.’ In his attempt to give an account of mathematical cognition, Kant focuses on the composition of mathematically homogeneous magnitudes. He thought that a condition of such cognition is the representation of numerical difference without qualitative difference. He states that a manifold of numerical difference without qualitative difference distinguishes quantity from quality. Because this notion of homogeneity expresses a condition of cognition that does not allow the representation of any qualitative difference at all, I have called it ‘strict homogeneity.’ On Kant’s view, concepts on their own can represent only qualitative differences, and hence on their own they cannot represent a homogeneous manifold. By contrast, intuition can represent numerical difference without qualitative difference, and as a consequence, intuition allows us to represent magnitudes—a previously neglected role for the intuitions of space and time in Kant’s philosophy of mathematics.23 At this point, we might rightly ask: Why does Kant think strict homogeneity is required for mathematics? The beginning of an answer is found in Plato and Aristotle. The parts of strictly homogeneous magnitudes are qualitatively indis20  I discuss the Critique’s lack of a section devoted explicitly to mathematical cognition and argue for the role of the Axioms in Kant’s philosophy of mathematics in “The Point of Kant’s Axioms of Intuition,” Pacific Philosophical Quarterly 86 (2005): 135–59. 21  For an explication of Kant’s concept of magnitude and the arguments that rest on it in the Axioms of Intuition, see Sutherland, “The Role of Magnitude in Kant’s Critical Philosophy” [“Role of Magnitude”], Canadian Journal of Philosophy 34 (2004): 411–42. 22  Euclid’s definition is “A ratio is a sort of relation in respect of size between two homogeneous magnitudes” (Euclid, Elements, Book 5, Def. 3, original emphasis). 23  This is not to say that the only role intuition plays in mathematical cognition is to represent a strictly homogeneous manifold. In fact, I think Kant envisions multiple roles for intuition, such as the representation of successiveness, as I make clear below. Kant also thinks that the representation of the part-whole relations of a strictly homogeneous manifold makes possible the representation of the mathematical homogeneity presupposed by the Eudoxian theory of magnitudes. For more details, see Sutherland, “Kant’s Philosophy of Mathematics.”

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tinguishable and therefore correspond to the Platonic and Aristotelian purity of mathematical units described in Section 1. Aristotle expresses a similar conception of the purity of geometrical objects: the mathematician investigates abstractions (for in his investigation he eliminates all the sensible qualities, e.g. weight and lightness, hardness and its contrary, and also heat and cold and other sensible contraries, and leaves only the quantitative and continuous, sometimes in one, sometimes in two, sometimes in three dimensions, and the attributes of things qua quantitative and continuous, and does not consider them in any other respect, and examines the relative positions of some and the consequences of these, and the commensurability and incommensurability of others, and the ratios of others; but yet we say there is one and the same science of all these things—geometry … (Met.K.3, 1061a28)

Kant is therefore not alone in thinking that mathematics considers quantity apart from qualitative properties. More specifically, we saw above that the mathematical homogeneity presupposed by the Eudoxian theory of proportions is characterized by the representation of composition, part-whole, and equality relations. Kant holds that representing these relations of a strictly homogeneous manifold allows us to cognize mathematical homogeneity, that is, the mathematical relations among magnitudes. The fact that space consists of pure numerical difference without qualitative difference explains the special composition of which spatial magnitudes are capable. Furthermore, determinate spaces are extensive magnitudes, which means that the representation of a determinate space presupposes the representation of its parts. As a consequence, we can represent the part-whole relations of a spatial magnitude and between spatial magnitudes, allowing us to cognize their comparative size relations. This forges a connection between strict homogeneity and the mathematical character of magnitudes.24 The importance of Kant’s theory of magnitudes and the role of the Eudoxian theory of proportions in it have not been fully appreciated, but the evidence for it is, I think, quite strong. 25 Kant primarily has in mind continuous magnitudes and, in particular, the continuous spatial magnitudes of geometry. On the other hand, arithmetic concerns numbers and discrete magnitudes—collections of discrete and even disconnected things. The question is: How does arithmetic fit into Kant’s account?

2.2. Kant on the Mathematical Cognition of Arithmetical Propositions Discussions of Kant’s arithmetic often focus on Kant’s claim that intuition is required to represent particular numbers. In the Introduction to the Critique, for example, Kant states that in cognizing that 5 + 7 = 12, we must seek assistance in intuition to represent the ‘5’ and ‘7’ by use of fingers or dots on a page (CPR, B 15–16). If we reread these passages in light of the cognitive presuppositions of the theory of proportions, the roles of composition, part-whole relations, and equality come to the fore. These features of Kant’s discussions of arithmetic strongly 24  There is much more to be said about the representation of part-whole relations and the relations between space, time, and intensive magnitudes. For details and a full defense of this interpretation, see Sutherland, “Kant’s Philosophy of Mathematics.” 25  I present the evidence in Sutherland, “Kant’s Philosophy of Mathematics.”

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suggest that his understanding of it is embedded within the more general theory of magnitudes. For example, in the Introduction of the Critique, Kant argues for the syntheticity of arithmetic propositions: To be sure, one might initially think that the proposition “7 + 5 = 12” is merely an analytic proposition that follows from the concept of a sum [Summe] of seven and five in accordance with the principle of contradiction. Yet if one considers it more closely, one finds that the concept of the sum of 7 and 5 contains nothing more than the unification [Vereinigung] of both numbers in a single one, through which it is not at all thought what this single number is which comprehends the two of them. The concept of twelve is by no means already thought merely by my thinking of that unification of seven and five, and no matter how much I analyze my concept of such a possible sum I will still not find twelve in it. (CPR, B 15–16, my emphasis)

Kant focuses on what is lacking in the mere concept of a sum or unification of the numbers. Kant next invokes the use of fingers or dots to represent the numbers in intuition. He then returns to the concept of a sum: I now put together [zutun] the units that I have previously taken together in order to constitute the number 5 one after the other to the number seven, and thus see the number 12 arise. That 7 should be put together [hinzugetan] with 5 I have, to be sure, thought in the concept of a sum = 7 + 5, but not that this sum is equal to the number 12. (CPR, B 15–16, my emphasis in bold)

Kant’s point is that the mere concept of a sum as the unification of two numbers includes the idea that they should be put together, but that the concept falls short of what is needed, namely, actually putting them together. Without intuition, we cannot cognize that actually putting them together is equal to 12. He concludes that the actual composing of units cannot be represented without intuition. Kant makes the same point in the Axioms of Intuition (CPR, A 164/B 205).26 We can think of unifying of two numbers, but until we carry out the composition in intuition, we cannot determine what the sum is to which they are equal. The connection between numbers and the intuitive synthesis underlying composition is made even more forcefully in the Schematism, where Kant says that number “is a representation that sums up [zusammenfaßt] the successive addition of one to one (homogeneous) [successive Addition von Einem zu Einem (gleichartigen)] (CPR, B 182, my italics).”27 Composition is of central importance to Kant. He also calls it ‘figurative synthesis,’ and it operates in the drawing of a line. In Kant and the Exact Sciences, Michael Friedman stresses the importance of this fundamental synthesis, and in particular, the importance of successive synthesis for the representation of successive addition and the role of time in allowing us to represent that succession. Developing a suggestion of Charles Parsons, Friedman argues that Kant appeals to time in order to represent the successive iteration of operations upon which mathematics rests.28 I 26  “I do not think the number 12 either in the representation of 7 nor in that of 5 nor in the representation of the composition [Zusammensetzung] of the two (that I ought to think this in the addition [Addition] of the two is not here at issue; in the case of the analytic proposition, the question is only whether I actually think the predicate in the representation of the subject)” (CPR, A 164/B 205, my emphasis in bold). 27  He adds that number is “nothing but the unity of the synthesis of the manifold of a homogeneous intuition in general …” (CPR, B 182). 28  Friedman, KES, 121; Parsons, “Arithmetic and Categories,” 116.

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think this is right, but I think there are other important features of this synthesis. Kant reserves the term ‘composition’ for the special synthesis that generates, and only generates, representations of magnitudes.29 Kant describes it as the “synthesis of the homogeneous in everything that can be considered mathematically.” The synthesis of composition underlies the mathematical synthetic principles (CPR, B 201 n.). Kant’s synthesis of composition corresponds to the special composition of which homogeneous magnitudes and only homogeneous magnitudes are capable, and it is the representation of this composition in intuition that is required in arithmetic as well as the generation of continuous magnitudes. Some of these same passages also suggest that arithmetical composition requires representing that particular numbers added are equal to their particular total, and this in turn appeals to part-whole relations. Kant states that the bare concept of a sum equal to 7 + 5 does not give us the thought that this sum is equal to the number 12. In a letter to Schultz in 1788, Kant states that if I view 3 + 4 as the expression of a problem—namely, to find for the numbers 3 and 4 a third = 7 that can be considered as the complementum ad totum of the others—then the solution is effected through the simplest action, which requires no particular formula of resolution, namely through the successive addition that brings forth the number 4, only set into operation as a continuation of the enumeration of the number 3. (10:556)

Kant mentions the successiveness of addition in this passage, but he also states that a complementum ad totum is required, so that 3 and 4 compose the complementary parts of the whole number 7. In the A-Deduction as well, Kant implies that a number requires being seen as a whole (CPR, A 99). Moreover, following the table of categories in §11, he states that number belongs to the category of totality, by which he means that number requires not merely a multitude of parts, but a cognition of the whole, which is made possible by the category of totality (CPR, B 111). Apart from Kant’s theory of magnitudes, these comments are suggestive. In the larger context of this theory of magnitudes, they take on added significance. To cite just one example from this larger context, Kant states in a lecture on metaphysics that the homogeneous manifolds of space and time allow us to grasp the part-whole relations of magnitude: the category of magnitude [Größe], as a homogeneous many that together constitutes [ausmacht] one; this cannot be grasped without space and time. (29:979, 1794–95)

By “category of magnitude” Kant means the categories of quantity. Space and time allow us to grasp that a homogeneous many together constitute a one—that is, that homogeneous parts constitute a whole—and this is what is required for grasping the categories of quantity. I have been arguing that in Kant’s view, arithmetic, just as much as geometry, is grounded in the theory of proportions, and our cognition of arithmetical truths rests on the same cognitions that underlie it. I have done so by citing passages 29  Kant sometimes uses the term ‘composition’ (Zusammensetzung) in a broader sense that appears to be synonymous with ‘combination’ [Verbindung], but he is quite clear on the scope of the narrower sense in the Critique.

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concerning arithmetic which emphasize the role of composition, part-whole relations, the mereological function of the categories of quantity, and the cognition of equality. Rather then citing more evidence, I will turn to the question of how the discrete magnitudes of arithmetic are related to the continuous magnitudes of the theory of proportion.

2.3. The Assimilation of Cognition of Discrete Magnitudes with Continuous Magnitudes If, as I have argued, Kant’s theory of arithmetic also rests on the theory of proportions, then his account of numbers and discrete magnitudes might be assimilated with the account of continuous magnitudes. Kant’s focus on geometry and geometrical examples throughout his work, and his account of the continuous synthesis underlying composition, suggest that he gives pride of place to continuous magnitudes. Moreover, in the Axioms of Intuition, Kant denies that a discrete magnitude is really a ‘magnitude’—a term that properly belongs only to continuous magnitude (CPR, A 170–71/B 212). He is not entirely clear on this point, for he allows elsewhere, even in the Critique, that there are discrete magnitudes (cf. CPR, B 554).30 Nevertheless, the Axioms of Intuition passage gives priority to continuous magnitudes. Furthermore, assimilating discrete magnitudes into the account of continuous magnitudes parallels the Euclidean tradition and its emphasis on geometry over arithmetic. The structure of Euclid’s Elements suggests that numbers are simply a special case of magnitudes. As noted earlier, in Book 7, Euclid defines ‘number’ as a collection of units; the propositions of Book 7 then develop a theory of ratios and proportions between numbers that parallels—in many cases simply duplicates—the Eudoxian theory of proportions in Book 5. This suggests that numbers are just a special case of magnitudes.31 Thus, it would be quite natural for Kant to treat numbers as a special case of magnitudes more generally. How might numbers be assimilated into Kant’s account of continuous magnitudes? Aristotle suggests a possibility in the Metaphysics: each number is said to be many because it consists of ones and because each number is measurable by one … And one and many in numbers are in a sense opposed, not as contrary, but as we have said some relative terms are opposed; for inasmuch as one is measure and the other measurable, they are opposed … plurality is number and the one is measure. (Met. I, 1056b– 1057a) For measure is that by which quantity is known; and quantity qua quantity is known either by a “one” or a number, and all number is known by a “one.” (Met. I, 1052b, 20–23) being one in the strictest sense, if we define it according to the meaning of the word, is a measure … (Met. I, 1053b, 4)

I will return to Kant’s apparent ambivalence on this point in Section 5 below. The Greek conception of number gave Euclid reason not to consolidate the accounts of magnitudes in Books 5 and 7. Book 5 assumes the existence of a fourth proportional for any three; expressed algebraically, it assumes that for every homogeneous magnitude pair a, b and another magnitude c, there is a magnitude x homogeneous with c such that a :b = c :x. This condition is not fulfilled on the Greek conception of numbers; for example, there is no whole number corresponding to 2:5 = 1:x. (I would like to thank Bill Tait for pointing this out to me). 30  31 

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Aristotle’s view seems to be that one is the measure of number. On this view, the counting of discrete quantities could be thought of as a kind of measuring. As in measuring, we specify a unit with which to count and then identify the units one by one, thinking of them in a progression of totals. For example, we might specify that our units are matching pairs of shoes and then count off the pairs of shoes in a room. This approach might give us a unified account of discrete and continuous magnitudes. As Julia Annas puts it in an elaboration of Aristotle’s suggestion: “Whereas we regard it as natural to think of measurement as applied arithmetic, Aristotle regards counting as a kind of pure measuring.”32 Book 7 of Euclid’s Elements also suggests an interpretation of counting as measuring. Ten of the twenty-two definitions refer to measurement. For example, Euclid states, “A number is part of a number, the less of the greater, when it measures the greater” (Elements, Book 7, Def. 3).33 In the early modern period, it was not uncommon for all of mathematics to be described as the science of measurement. A mathematician turned prolific philosopher, Christian Wolff successfully promoted a scholastic version of Leibnizianism in Germany a generation before Kant. In his Mathematisches Lexicon, Wolff states that mathematics “is a science to measure everything that allows of measurement.”34 Leonhard Euler’s Elements of Algebra appeared in German in 1770 and was enormously influential and popular; it was printed over thirty times in three editions and six languages. He begins: Whatever is capable of increase or diminution is called magnitude, or quantity … Mathematics, in general, is the science of quantity; or, the science that investigates the means of measuring quantity.35

Euler’s first example of a unit of measure is “a louis, a crown, a ducat, or some other coin,” which is used to measure how many there are in a collection of coins.36 Kant expresses a similar view in lectures on mathematics delivered between 1762 and 1764, which Herder attended and recorded. Kant states: Mathematics is a science of measuring the magnitude of things, or how many times something is posited in a thing … the measure of magnitude (plurality) is the unit, or one. For example: every number, measure, weight = herd, mile, etc.. Line, surface [sic.]. (29:49; see also 29:59)

In addition to describing mathematics as the science of measurement, Kant also treats the number one and a herd as examples of units of measure for counting. It was also not uncommon to hold that there was a cognitive requirement for counting: the things counted must be thought of as of the same kind, and hence as falling under the same counting-concept. I cannot count oranges and apples, Julia Annas, “Aristotle, Number and Time,” The Philosophical Quarterly 25 (1975), 99. Euclid means an “aliquot part,” that is, a part that is a submultiple. See Heath, Elements, Vol.

32  33 

2, 280. 34  Mathematisches Lexicon [ML] (Leibzig: Johann Friedrich Gleditschens seel. Sohn, 1716); reproduced in Christian Wolff: Gesammelte Werke, I. Abteilung, Band 11 (Hildesheim: Georg Olms Verlag, 1965), 863. 35  Leonhard Euler, Elements of Algebra [EA] , trans. Rev. John Hewlett (New York: Springer Verlag, 1972), xxxiii. 36  Euler, EA, 1–2.

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for example, using the concept “oranges;” I must ascend at least to the common concept “pieces of fruit.” Wolff states in his Elements of All Mathematical Sciences that, “[w]hen one takes together many individual things of one kind, a number arises there from.”37 Wolff’s further discussion makes it clear that in order to count a collection, one must think of a property or set of properties of the things that allows one to represent them as the same kind.38 Euler also states that one must “fix at pleasure upon any one known magnitude of the same species with that which is to be determined, and consider it as the measure or unit.”39 Kant suggests a similar view in the Herder lectures: A whole out of many things of one and the same kind, clearly expressed, is number. The unity that is used as a measure in mathematics is here a known concept. (29:52)

The cognitive condition of finding a counting-concept allows us to think of the counted items of the same kind. This sameness of kind is a sort of homogeneity, for the objects fall under the same species concept. This homogeneity for counting provides a parallel to the mathematical homogeneity required for continuous magnitudes. As it stands, the suggestion that counting be thought of as a kind of measuring is mere metaphor. A natural way to flesh out this metaphor in order to assimilate counting with measuring would be to think of numbers as represented by lengths. (One might even think that number concepts could arise from the synthesis that generates a line when the lengths of lines are marked off by a unit.) This would provide a natural way for Kant to assimilate the cognition of arithmetic into his account of the cognition of continuous magnitudes, and hence to the theory of proportions. There are obstacles to this interpretation, however. The first is that Kant’s descriptions of the synthesis of composition underlying arithmetic differ from the synthesis underlying continuous magnitudes or descriptions of the synthesis of composition in general. Kant’s references to the latter usually refer either simply to a synthesis of the homogeneous or to a synthesis of part to part. In contrast, his descriptions of the synthesis of composition underlying arithmetic usually refer to the synthesis of units. For example, Kant states in the Prolegomena that “arithmetic brings forth [bringt zu Stande] even its number-concepts through successive putting together of units [Einheiten] in time” (4:283). In the Introduction passage of the Critique, Kant refers to the units that are put together to constitute the number 5. In the Schematism, Kant states that number is a representation that comprises the “successive addition of one to one (homogeneous)” (CPR, A 142/B 182; see also A 105). In the Axioms, he again refers to the synthesis of homogeneous units underlying arithmetic; and in the Anticipations he states that every number is

37  Christian Wolff, Anfangsgründe aller Mathematischen Wissenschaften [AMW] (1st ed., 1710; 7th ed., Franckfurt und Leipzig: Rengerschen Buchhandlung, 1750–77); reproduced in Christian Wolff: Gesammelte Werke [GW] I. Abteilung, Band 12 (Hildesheim: Georg Olms, 1973), 38. 38  Wolff, AMW, 39. 39  Euler, EA, 2.

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grounded by a unit (CPR, A 164/B 205, A 170–71/B 212).40 Kant thinks that the synthesis of composition underlying arithmetic differs in an important way from the synthesis of composition underlying continuous magnitudes. This difference would not undermine assimilation with numbers to lengths, if the lengths were represented as consisting of units. Unfortunately, as mentioned above, Kant does not appeal to lines to represent numbers, but to fingers; he also refers to Johann Andreas Segner’s work, Elementa Arithmeticae, Geometricae et Calculi, which uses dots to represent numbers (CPR, B 15). Segner may also use lines to represent numbers, but if so, it is all the more significant that Kant does not refer to them. In the System of Principles, Kant refers to fingers, beads of an abacus, strokes, or points placed before the eyes (CPR, A 240/B 299). On the other hand, in a draft of a letter to Rehberg in 1790, Kant mentions our ability to construct equations in geometry using lines to stand for numbers and the wondrous impression this makes on beginning students of algebra (14:58). In the context of explaining the nature of arithmetical cognition, however, Kant always refers to discrete and disconnected units. I think we must take his examples at face value. We should understand Kant’s paradigm of arithmetical cognition to be of units that are both discrete and disconnected. There is a second problem concerning Kant’s understanding of the relation between arithmetic and the Eudoxian theory of proportions. Kant gave lectures on mathematics between 1762 and 1764, and in his lectures on arithmetic, Kant singles out Archimedes, Nichomachus, Diophantus, and Psellus (29:51).41 The last three figures are part of an arithmetic tradition quite distinct from the Euclidean geometrical tradition and the theory of proportions it contained. That arithmetic tradition developed techniques for calculating the solutions to numerical problems, and it strongly influenced the development of algebra in the early modern period.42 If Kant assimilated arithmetic into the Eudoxian theory of proportions found in Euclid, why does Kant refer to this distinct arithmetical tradition? Finally, there is a third problem concerning homogeneity. The homogeneity required for counting is importantly different from the strict homogeneity required for mathematical composition. The homogeneity of merely falling under a common counting-concept is not anywhere as strict as numerical difference without any qualitative difference whatsoever. The homogeneity of counting hardly requires intuition. So why would intuition be required for the combination underlying arithmetic? Addressing the issues I have raised requires a closer look at the connection between arithmetic and algebra in the early modern period and in Kant.

40  Kant’s term is ‘Einheit’ in these passages, which can mean either “unity” or “unit.” Unity and the conditions for its cognition are, of course, of central importance in Kant’s philosophy. Nevertheless, the ambiguity between unity and unit has, I think, obscured passages concerning arithmetic in which Kant means unit. A unit presupposes a unity, but also implies that the unity is employed as a measure or is something counted. Translating ‘Einheit’ as unity loses this connotation and can lead to ambiguities concerning precisely which unity Kant has in mind. 41  Kant’s mathematical lectures were based on Wolff’s AMW. Wolff mentions these figures, among others, in Kurtzter Unterricht von den Vornehmsten Mathematischen Schriften [KU] (Franckfurt und Leipzig: Rengerschen Buchhandlung, 1750); reproduced in GW, I. Abteilung, Band 12. 42  Klein, GMT, 26–36.

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3. arithmetic, algebra, and magnitudes The arithmetic tradition to which Kant alludes maintained the Greek conception of number as a collection of units. The practical demands of calculation, however, led to a liberalization of the concept of number to include rational numbers. By conceiving of the unit as subject to stipulation rather than fixed, the Greek arithmeticians were able to divide a unit as needed and treat the parts of the unit as new units.43 Rational numbers were accepted because they were reducible to ratios between whole numbers; irrational numbers were still excluded. Diophantus of Alexandria (fl. 250 A.D.) was the most important and influential of the three figures of the arithmetic tradition Kant mentions.44 His Arithmetic greatly improved the available techniques of numerical calculation and increased the level of abstraction. His methods allowed him to solve first and second degree determinate and indeterminate equations, and he can be regarded as the first Greek algebraist.45 Developments in algebra through the sixteenth-century primarily concerned its application to numerical problems.46 At the end of the sixteenth- and the beginning of the seventeenth- century, François Viète was inspired by Diophantus to make important breakthroughs in algebra and his application of it to geometry.47 Descartes’s development of analytic geometry further improved the application of algebra to geometry and began a robust era of geometrical problem solving in the early modern period.48 Even in the sixteenth- and the seventeenth- centuries, however, algebra was still closely tied to its roots in arithmetic. In fact, the distinction between arithmetic and algebra was not easily drawn. In German, arithmetic was called ‘Rechenkunst,’ or the art of calculating, whereas 43  Klein states: “Nothing now stands in the way of changing the unit of measurement in the course of the calculation and of transforming all the fractional parts of the original unit into ‘whole’ numbers consisting of the new units of measurement” (Klein, GMT, 112). According to Klein, they were encouraged in this development by Aristotle’s view that what counts as a unit depends upon one’s specification. 44  Kant singles him out as “the most subtle arithmetician” (29:52). Nichomachus of Gerasa (fl. 100 A.D.) authored an arithmetic that was significantly less developed than that of Diophantus. Michael Psellus of Constantinople (1018–1078) wrote a great number of compendia, including works on arithmetic. He used what may have been the only remaining Greek copy of Diophantus’s Arithmetic. At the fall of Constantinople, that copy was taken to Italy, where it was rediscovered during the Renaissance. See entries for Psellus and Diophantus in Dictionary for Scientific Biography, ed. Charles Coulston (New York: Charles Scribner’s and Sons, 1970–80). 45  T. L. Heath, Diophantus of Alexandria: A Study in the History of Greek Algebra (New York: Dover Publications, 1964); Klein, GMT, 126. 46  See Henk J. M. Bos, Redefining Geometrical Exactness: Descartes’s Transformation of the Early Modern Concept of Construction [RGE] (New York: Springer Verlag, 2001), 129. There was nevertheless an early appreciation of the parallels between geometrical problems and algebraic equations, as I will discuss in what follows. 47  Klein, GMT, 150–85. Viète was also influenced by the Arabic algebraic tradition. He, like some others in the early modern period, thought that Diophantus and the Greek algebraic tradition influenced the Arabic authors. In fact, he thought of himself as reviving algebra from the “barbaric” influence of the Arab tradition; see Klein, GMT, 153. It is now thought that the most important and influential Arab authors, such as Al Khwarizmi, were not influenced by Greek sources, but by Persian and Indian sources. See B. L. van der Waerden, A History of Algebra: From Al-Khwarizmi to Emmy Noether [HA] (New York: Springer Verlag, 1985), 13. Nevertheless, in the Renaissance and early modern period it was thought that Diophantus was the source of Arabic algebra, making Diophantus’ presumed influence twofold: indirectly through Arab sources and directly on Viète. 48  See Bos, RGE, for an extended and thorough treatment.

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algebra was called ‘Buchstabenrechenkunst,’ or the art of calculating with letters—that is, with letters that stand in for or represent as yet unknown numbers and indeterminate numbers. Kant distinguishes between “arithmetic” and “algebra” in this way in his lectures on mathematics (29:49). But the deep difference between arithmetic and algebra is not simply marked by whether letters are used; this is made clear by reflecting on the (lack of) difference between calculating what the sum of 5 + 7 equals and calculating what x is in “x = 5 + 7.” In fact, in his Mathematical Lexicon, Wolff distinguishes algebra speciosa from algebra numerosa; the latter counts as algebra although it only employs numbers.49 Algebra was described as a method for finding solutions using equations, but a method of finding solutions to purely numerical equations would have belonged to arithmetic. Algebra was also characterized by the particular method in which it solves equations—that is, by adding, subtracting, multiplying, or dividing both sides of an equation by identical terms.50 These operations, however, can also be carried out on numerical equations.51 Despite similarity and overlap, however, paradigm cases of solving a set of equations using letters and paradigm cases of carrying out a numerical calculation were distinguishable. Roughly speaking, algebra was the method of finding unknown values through reducing sets of equations by substituting or combining the equations to eliminate unknowns. As it developed, algebra was increasingly viewed as a general method for finding solutions, rather than as an arithmetical technique of calculation. The important point is that the connection between arithmetic and algebra was close in both development and activity. They were so close that it was not uncommon to characterize algebra as generalized arithmetic.52 Kant’s discussions of algebra reflect this close relation to arithmetic. In his outline of arithmetic, he states that algebra uses general signs for numbers (29:49). Moreover, in both sections 2 and 4 of the Inquiry, Kant refers to algebra as general arithmetic (2:278, 2:282). Although these are early pre-critical works, Kant also identifies algebra with general arithmetic in a letter to Schulz in 1788 (10:555). In the Critique, Kant distinguishes two kinds of magnitude, quantitas and quanta; arithmetical formulae and algebra are grouped together under the rubric of quantitas, in explicit contrast to the quanta of geometry (CPR, B 204–05, A 717/B 745).53 The connection between arithmetic and algebra in Kant is, therefore, quite strong. Algebra was not merely applied to the calculation of numbers, however. It was also applied to geometry. In the early modern period, Viète applied algebra to geometrical problems, as did Descartes, famously, in developing analytic geometry. Algebra emerged as a general method applied to solve problems concerning all kinds of magnitudes, and this promoted thinking of it as a general method of problem-solving. This generality led to the reconciliation of the Euclidean and Wolff, ML, 35–37. van der Waerden, HA, 4–5. 51  As Kant does in the Herder lectures on mathematics (29:53–55). 52  In 1657, Wallis wrote that “Universal algebra is in truth arithmetic, not geometry, and must therefore be explained rather on arithmetic than geometric principles” (quoted in Klein, GMT, 215). Wolff states that ‘arithmetica literalis’ is a synonym for ‘algebra’ (ML, 178). 53  I discuss the distinction between quanta and quantitas in more detail in Sutherland, “Role of Magnitude,” and in Sutherland, “Kant on Fundamental Geometrical Relations,” Archiv für Geschichte der Philosophie 87 (2005): 117–58. 49  50 

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arithmetical traditions, and raised hopes of a universal mathematics. For example, Wallis (1616–1703), who strongly identified arithmetic and algebra, thought of them as constituting a mathesis universalis—a fact that Wolff highlights in his Mathematisches Lexicon.54 Kant too thinks of algebra as a universal mathematics; in particular, he thinks of it as expressing a general doctrine of magnitudes. In §4 of the Inquiry, Kant states that the object of mathematics is magnitude and that algebra is the general doctrine of magnitudes (2:282). This is not a view limited to the pre-critical period. In a 1789 letter to Carl Rheinhold, Kant states that algebra concerns mere magnitudes without qualities and that algebra allows us to represent the ratios between magnitudes (11:42). In the previously mentioned Schultz letter of 1788, Kant states that general arithmetic (i.e., algebra) is a general doctrine of magnitudes from which the rest of mathematics benefits. Finally, Kant calls algebra a Mathesis Universalis in his mathematics lectures (29:49); and in the Metaphysical Foundations of Natural Science, Kant glosses ‘mathesis’ as a general doctrine of magnitude (4:489). There are a few passages that apparently contradict the claim that the objects of algebra are magnitudes. These passages suggest that algebra either has no object, or at best provides us with the concept of a thing in general. In the Discipline of Pure Reason, for example, Kant states that “algebra wholly [gänzlich] abstracts from the constitution [Beschaffenheit] of the object that is to be thought in accordance with such a concept of magnitude” (CPR, A 163–64/B 204). Kant’s emphasis on the completeness of abstraction might lead one to think that algebra has no object at all, or has as its object a thing in general.55 Such a view would be easily reinforced by the fact that algebra was conceived as a technique or method of solving problems rather than as a discipline with its own domain of objects. The passages we have been examining, however, reveal that algebra does not abstract completely from the constitution of the object, if that means abstracting from the character of the object as a magnitude. As the letter to Rheinhold of 1789 shows, algebra concerns mere magnitudes without qualities. It therefore abstracts from qualities, but not their purely quantitative properties, that is, their character as magnitudes. This is even apparent in the Discipline of Pure Reason passage just quoted, which states that “algebra wholly abstracts from the constitution of the object that is to be thought in accordance with the concept of magnitude” (CPR, A 717/B 745, my emphasis). Thus, algebra does have an object, namely, magnitude. At the same time, it does not have its own special domain of objects within mathematics, since magnitude is the object of all mathematics. This point is important for the following reason. If Kant holds that algebra is the general doctrine of magnitudes, and if, as I have argued, Kant’s theory of magnitudes rests on the Eudoxian theory of proportions, then Kant’s understanding of algebra will also rest on this theory of proportions. The next section of this paper will argue that algebra can be viewed as expressing the Eudoxian theory of proportions and that Kant, in fact, viewed it that way.

Wolff, ML, 869. See, for example, Parsons “Kant’s Arithmetic,” 134–35, and Friedman, KES, 112–22, esp. 113, 114, and 122; but see also 114 n. 34. 54  55 

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4. algebra and the theory of proportions The connection between algebra and the Eudoxian theory of proportions is not difficult to make. The theory of proportions presented in Euclid’s Books 5 and 6 proceeds entirely without reference to numbers and does not rely on the manipulation of symbolic notation for its proofs. Nevertheless, the propositions in Books 5 and 6 are often called ‘geometrical algebra’ because they establish relationships between magnitudes that correspond to algebraic statements. They are, for that reason, most easily and most commonly explained to modern readers with the aid of algebraic notation, as we find in Heath’s commentary on the Elements and in my explanation of the Eudoxian-Euclidean definition of proportions above. The nature of geometrical algebra is already clear in Book 5, Proposition 4: “If a straight line be cut at random, the square on the whole is equal to the squares on the segments, and twice the rectangle contained by the segments.” In other words, the square made up of sides a + b is equal to the square with sides a, the square with sides b, and two rectangles with sides a and b. This corresponds to the algebraic formula “(a + b)2 = a 2 + b 2 + 2ab,” an equation established in algebra by multiplying out and combining like terms. The Greeks gave such problems an entirely geometrical interpretation.56 The theory of proportions found in Euclid’s Books 5 and 6 is called ‘geometrical algebra,’ in particular, because it is applied by Euclid to geometrical relations, but the theory itself concerns all magnitudes. The reconciliation of arithmetical algebra and geometrical algebra in the early modern period made it possible to view algebra as a mathesis universalis, a universal mathematical method that expressed the Eudoxian theory of ratios. Some of the most important early-modern algebraists emphasized the connection between the Diophantine algebra of numbers and the Greek theory of proportions. Algebra, as we have noted, employs equations in the resolution of problems. Viète directly connects this conception of algebra with the Eudoxian theory when he states that “a proportion can be called the construction of an equation, and an equation the solution of a proportion.” In fact, Viète draws from the Eudoxian theory of proportions in Euclid (in particular, Books 5 and 7) in formulating his algebra. He posited an algebraic method common to both the calculation of numbers and the Eudoxian theory of proportions for magnitudes.57 Descartes shared similar views. He developed analytic geometry by applying algebra to geometrical problems; at the same time, he thought of proportions as the key to a mathematical understanding of the world. He thought of algebra as the mathesis universalis and also held that the mathesis universalis is a general theory of proportions and equations.58 He also states in the Regulae that the subject of proportions and equations is magnitudes.59 Wallis thought that because arithmetic was 56  During the early modern development of algebra, before Viète was inspired by Diophantus to allow reference to indeterminate quantities, algebraists sometimes turned to geometrical proofs of such equations because the geometrical magnitudes are specified indeterminately. See van der Waerden, HA, 8, 38. 57  Klein, GMT, 160–61. 58  Klein, GMT, 197, 209. 59  Descartes, Rules for the Direction of the Natural Intelligence, ed. and trans. George Heffernan (Atlanta: Rodopi, 1998), Rule XIV.

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more abstract than geometry, the universal algebra was arithmetical rather than geometrical.60 At the same time, Wallis thought of algebra as explicitly tied to the theory of proportions; for example, he attempted to prove the Eudoxian theory of proportions algebraically. Furthermore, he thought that a ratio underlies every number and stated that the “whole of arithmetic seems, on closer inspection, to be nothing other than a theory of ratios.”61 Wolff also drew the connection between algebra and the theory of proportions, and he emphasized the importance of the latter for mathematics. He does not directly discuss the theory of proportions in his German Elements of All Mathematical Sciences, but in his expanded Latin version, the Elements of Universal Mathematics, he added a full independent section on ratios and proportions immediately following the sections on arithmetic.62 Wolff explains this addition in his Mathematical Lexicon under the heading of ‘Ratio.’ He used algebraic notation to present the theory of numerical proportions in his German Elements because it is easier for beginning students; but he says that this algebraic presentation would not satisfy anyone who valued rigor. He states that what is needed is the Eudoxian theory of ratios and proportions, which motivated him to add the section on proportions to his Latin Elements. Thus, he holds that algebraically expressed relations are the ratios and proportions of the Eudoxian theory. Significantly, he also states that the theory of proportions is the soul of mathematics.63 This was the early-modern context for Kant’s own thinking, and it helps explain why Kant equates algebra with a general doctrine of magnitudes and at the same time grounds his theory of magnitudes in the Eudoxian theory of proportions. It also explains why Kant makes special mention of ratios in his discussions of algebra. In the Discipline of Pure Reason, for example, he states that algebra “designates the general concept of magnitudes in accordance with their different ratios …” (CPR, A 717/B 745).64 Later in the Discipline, he states that the procedure of the algebraists with their equations is one which “represents signs of the concepts, especially those of the ratios of magnitudes” (CPR, A 734/B 762). Kant lays the same emphasis on ratios in the letter to Rheinhold (cited earlier): The mathematician cannot make the smallest assertation about any object whatsoever without exhibiting it in intuition (or, if we are considering only magnitudes without qualities, as in algebra, exhibiting the magnitude ratios thought under symbols). (11:42)65

In summary, Kant thinks that algebra, that is, general arithmetic, expresses the Eudoxian theory of proportions. With this background in place, we are now in 60  In his view, geometry is subordinate to arithmetic. Algebra is universal arithmetic, and geometry applies the universal assertions of arithmetic to its objects. See Klein, GMT, 215–16. 61  Klein, GMT, 220. 62  Wolff, Elementa Matheseos Universae [EMU], in Christian Wolff: Gesammelte Werke II. Abteilung, Band 29 (Hildesheim: Georg Olms, 1968), 92–98. Wolff’s German Elements (Anfangsgründe aller Mathematischen Wissenschaften) [AMW] first appeared in 1710; and the Latin Elements [EMU] in 1730. 63  Wolff, ML, 1169–1170. 64  Buchstabenrechnung … den allgemeinen Begriff der Größen nach den verschiedenen Verhältnissen derselben auch bezeichnet hat … The German term ‘Verhältniss’ can be translated as ‘relation,’ but in mathematical contexts, it was used for ratios. 65  The term Kant uses is ‘Grössenverhältnisse.’ See previous footnote.

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a position to return to the conception of arithmetic and number in the early modern period.

5. arithmetic and number 5.1. Number in the Early Modern Period As I noted earlier, the Greek conception of number as a collection of indivisible units led to the problem of incommensurables. This caused a divergence between the Diophantine arithmetic tradition and the Euclidean geometrical tradition. These two traditions nevertheless merged in the early modern period. How did the conception of number change to make this reconciliation possible? The earlymodern transformation of the conception of number is quite complex; it includes the development of views about the nature of indeterminate quantities as well as a shift away from a concrete conception of number.66 I will abstract from these complexities to make a few points relevant to our present inquiry. In 1585, the Flemish mathematician Simon Stevin took a bold step when he allowed his conception of number to be guided by operations on numbers and the very general similarity between arithmetic operations and geometrical constructions.67 He rejected the Greek definition of ‘number’ as a collection of units. He thereby rejected the view that numbers are essentially discrete and claimed that numbers are just as continuous as continuous magnitudes; he explicitly endorsed the view that there are irrational numbers. His reasoning here is instructive. He argued that the part is of the same material as the whole. Because √8 is a part of 8, and 8 is a number, √8 is also a number.68 Stevin’s conception of number was based on numerical operations, not on notions about their natures inherited from the Greek mathematical tradition. Stevin’s conception of a continuum of numbers came to be accepted at least implicitly by later mathematicians. Descartes, Newton, and Leibniz, for example, held that there was a number corresponding to every ratio between two lines.69 Descartes’s Geometry, in particular, opened the way to viewing numbers as lines and vice versa. The new conception of number did not come easily, however. Wallis adopted the new view of number under the influence of Stevin, Descartes’s Geometry, and the parallels between arithmetic operations and geometrical constructions. As noted above, he thought that geometry is subordinate to arithmetic and that geometry applies the universal assertions of arithmetic to its objects. Wallis’s view requires that a number can express the ratio between any two lines, including incommensurable lines. At the same time, Wallis went to great lengths to make his statements conform to those of the Greeks. In fact, he went as far as inconsistency; he states

66  See Klein, GMT for an extended discussion of the representation of indeterminate and unknown quantities. 67  Despite similarities between arithmetic operations and geometrical constructions, there were important differences between them, which helps to explain why Stevin’s approach was not embraced sooner. See Bos, RGE, 130–32. 68  Klein, GMT, 191–95. 69  Johannes Tropfke, Geschichte der Elementarmathematik, Band 1, 4. auflage (New York: Walter de Gruyter, 1980), 137; referred to by van der Waerden, HA, 69.

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at one point that all numbers are constituted out of units, which does not allow one to express the ratio between incommensurable lines.70 The commingling of the new and Greek conceptions of number was not uncommon and persisted into the eighteenth-century. Wolff used both conceptions, for example. He states in his German Elements that when one takes together several single things of the same kind, a number arises, and that for this reason Euclid defines number as a multitude of units.71 In his Latin Elements, however, Wolff defines a number as that which relates to unity in the way that one straight line relates to another straight line. He explains in the Mathematical Lexicon that he only used Euclid’s definition in the German Elements because it is very clear to beginners, and that he changed the definition in the Latin Elements so that it would include irrationals. He then points out the parallel between geometrical constructions and arithmetic operations, saying that the utility of this parallel becomes apparent when one applies algebra to geometry, and refers explicitly to Descartes’s Geometry.72 Thus, Wolff employs the Greek conception of number, but only in his earlier and less scholarly work. Euler’s views on the two conceptions of number are also noteworthy. In 1738 and 1740, Euler published a two-part Introduction to Arithmetic designed for use in the schools. Like Wolff, he used the Greek conception of number as a collection of units.73 His Elements of Algebra of 1770 provides quite a different conception of number, however. In the first pages, Euler states that number is nothing but the proportion of one magnitude to another arbitrarily assumed as a unit. This would include irrational numbers, since it would include the ratio between the diagonal of a square and its side. Thus, he can go on to say that all magnitudes may be expressed by numbers.74 He addresses the nature of irrational numbers in the following discussion of square roots: These irrational quantities, though they cannot be expressed by fractions, are nevertheless magnitudes of which we may form an accurate idea. That is because, however concealed the square root of 12, for example, may appear, we are not ignorant that it must be a number, which, when multiplied by itself, would exactly produce 12; and this property is sufficient to give us an idea of the number, since it is in our power to continually approach its value.75

Euler accepts that the square root of 12 is a number because its square is a number. Euler resembles Stevin in allowing his conception of number to be guided by Wallis, Mathesis Universalis, 56; referred to by Klein, GMT, 217. Wolff, EMU, 35. 72  Wolff, ML, 944–45. 73  Euler, Einleitung zur Rechen-Kunst zum Gebrauch des Gymnasii bei der Kaiserlichen Academie der Wissenschaft in St. Petersburg [ERK]; reprinted in Euler Opera Omnia, ser. 3 v. 2 (Geneva: Tuebner Lipsiae et Berolini, 1942), 1–303. Like many of the mathematical texts of the time, it took a rather practical approach, but Euler nevertheless insisted on acquainting his readers with the grounds of the rules of arithmetic. He holds that above all, we need a clear concept of number: “When many pieces [Stücke] of one kind [Art] are at hand, this plurality [Vielheit] is indicated through a number. And for this reason one understands through a number, how many pieces are spoken of.” He adds that the best way to attain a clear concept of number is through consideration of that which is called one, for number indicates how many of the same sort is intended. Euler gives rubles as an example of the kind of “pieces” (Stücke) that are numbered. His arithmetic includes fractions, but does not discuss irrational quantities. 74  Euler, ERK, 2. 75  Euler, ERK, 40. 70  71 

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operations on numbers. Thus, like Wolff, Euler employs the Greek conception of number, but only in his earlier and less scholarly work.

5.2. Arithmetic and Number in Kant In Section 2, I argued that Kant bases his philosophy of mathematics on a theory of magnitudes that reflects the Eudoxian theory of proportions. We saw that this approach suggests a rather straightforward account of arithmetic that assimilates discrete magnitudes into Kant’s account of continuous magnitudes and interprets numbers as lengths of lines. There were three problems with attributing this approach to Kant, however. We are now in a position to address them and to provide a better account. The first problem was that Kant does not explain our cognition of numbers by appeal to the cognition of lengths. Instead, his discussions of arithmetical cognition appeal to discrete magnitudes such as fingers or dots. Furthermore, as explained in Section 2.3, Kant thinks that the synthesis of composition underlying arithmetic is a synthesis of discrete and discontinuous units (CPR, B 15, A 164/B 205, A 170–71/B 212, A 240/B 299; 4:283). Finally, he states explicitly in the Critique that there is a unity grounding every number (CPR, A 170–71/B 212). These passages show that Kant’s views reflect the Greek conception of number rather than the modern. Kant allows rational numbers, which can be expressed as a ratio of two numbers consisting of units. He does not, however, admit irrational numbers, as he states in the 1790 letter to Rehberg concerning the representation of irrational quantities.76 Like Segner and like Wolff and Euler in their more popular works, Kant adhered to the more traditional Greek conception of number.77 Kant holds that there is a kind of synthesis through which we can represent the composition of discrete and disconnected unities into collections. He thinks either that this arithmetical synthesis differs from the synthesis that generates the representation of a continuous magnitude in the drawing of a line, or that it is the same underlying synthesis carried out in a distinctly different manner. In addition, the categories of quantity provide a mereological foundation for mathematics in two different ways, for they can be employed in the cognition of the part-whole relations of discrete collections as well as continuous magnitudes. Therefore, Kant does not attempt to directly assimilate arithmetic into the theory of proportions by thinking of numbers as represented by lines or thinking of counting as measuring units of length. Kant refers to the arithmetic tradition 76  Kant does not quite say that there are no irrational numbers. He states that no number can be found which clearly and completely represents an irrational magnitude in concepts. He then states that “[t]he understanding … cannot bring forth the complete number-concept, but must accede to entering into an infinite approximation to the number …” (11:208). Note that Kant calls that which is approached a ‘number.’ The implied contrast, however, is between a number-concept with a determinate ratio to unity, and the concept afforded by the algebraic representation of 1:x::x:2. (See Friedman, KES, 109–12, for a helpful discussion of this issue.) Moreover, in a draft of his letter, Kant does deny that an irrational number is a number (14:57). Since this is what is strictly required by his definition of number, it is most likely what he meant. 77  As mentioned in Section 2.3, Kant may have shown some ambivalence about the status of discrete magnitudes. If so, the emerging new conception of number, which does not distinguish between discrete and continuous magnitudes, may be the cause. Nevertheless, Kant fairly clearly endorses the Greek conception of number.

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in his lectures on mathematics because he, like others in the early modern period, thought of arithmetic as about collections of discrete units, which in the paradigm case are disconnected in the way that dots on a page are disconnected. This view of number is compatible with representing numbers as discretely cognized lengths of lines so long as one keeps in mind that there are ratios between lines for which there is not a corresponding ratio between numbers. The second problem concerned Kant’s reference to Nichomachus, Diophantus, and Psellus. Why would Kant refer to this arithmetic tradition, which was distinct from the Euclidean tradition, if he thought that mathematical cognition was grounded in the Eudoxian theory of proportions? As noted, Kant endorses the Greek conception of number as a collection of units; this is a conception that both the Euclidean and the arithmetic tradition shared. The two approaches did differ in important ways: the aims of Euclid were very different from those of the Greek Arithmeticians, and Euclid took only whole numbers to be numbers, while the latter admitted rationals. Nevertheless, the propositions of Euclid’s arithmetical books (which apply the theory of proportions to the special case of numbers) were consistent with the claims of the arithmetic tradition. That is why Kant can refer to the arithmetical books of Euclid in the same lectures in which he mentions Nichomachus, Diophantus, and Psellus. Kant states that “[a]lready 2000 years ago Euclid quite demonstratively explained the properties of numbers in Books 7–9, which in newer editions have been grievously omitted. Taquet [sic.] translated them” (29:52).78 The fact that he regards their omission as a serious mistake is further evidence that Kant thought that arithmetic was best understood as a part of the Eudoxian theory of proportions. Finally, we have seen that the arithmetical tradition developed into algebra, which Kant thinks expresses the theory of proportions. Thus, Kant holds that, despite their early divergence, the arithmetical tradition and the Euclidean theory of proportions are two routes to the same view of mathematics—a view based on the nature of magnitudes. The third problem concerned homogeneity. I argued that concepts alone cannot represent strict numerical difference without qualitative difference and that intuition can, so that intuition allows us to represent the strict homogeneity required for mathematical cognition. If counting only required the homogeneity of falling under a common concept, however, then intuition would not be needed to represent the units added. Does this mean that intuition is not required for the representation of the discrete magnitudes of arithmetic? Although Kant does not think of numbers as represented by lengths, he still holds that arithmetic is grounded in the representation of a strict homogeneity. The homogeneous units to which Kant refers differ from each other numerically but not qualitatively. We saw in Section 1 that the Greeks thought of arithmetic as grounded in “pure” units, whose purity consisted in qualitative indistinguishability. 78  Kant is either referring to a late edition of Tacquet’s Elementis planae et solidae (1701), which by then may have included the arithmetical books (as suggested by 29:683 n. 5211), or to his Arithmeticae theoria et praxis (1656). See Wolff, KU, 8. The arithmetical books were omitted in many if not most editions of Euclid; they were nevertheless available in Kant’s day.

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Kant does not make the strict homogeneity requirement explicit for discrete magnitudes, but he does suggest it in the Amphiboly attack on the identity of indiscernibles, where the numerical difference between qualitatively identical objects (such as two drops of water) rests upon the numerical difference between regions of space. Kant states: Multiplicity and numerical difference are already given by space itself as the condition of outer appearances. For a part of space, even though it might be completely similar and equal to another, is nevertheless outside of it, and is for that very reason a different part from that which abuts it to constitute a larger space. (CPR, A 264/B 320)

Kant thinks that space itself gives us multiplicity and numerical difference; the strict homogeneity of space allows us to differentiate these discrete and disconnected, qualitatively identical objects. Moreover, Kant mentions in this context that strictly homogeneous parts of space can be composed to constitute larger spaces. This is precisely the sort of composition that is at the basis of the mathematical homogeneity presupposed by the Euclidean conception of magnitudes. Without strict homogeneity, we would not be able to cognize the mathematical character of arithmetic composition; we could not even acquire arithmetical concepts, which arise through a successive putting together of strictly homogeneous units. In fact, the qualitatively identical units to which Kant refers in the Amphiboly could themselves be put to mathematical use. Qualitatively identical water drops are discrete and disconnected and are only distinguished by the spaces they occupy. Hence they can serve as pure units for arithmetic. The spaces mentioned in the passage above are adjoining, but later in the Amphiboly, Kant refers to regions of space more generally: The concept of a cubic foot of space, wherever and however often I think it, is in itself always completely the same. Yet two [distinct] cubic feet of space are nevertheless distinguished in space merely through their locations (numero diversa) … (CPR, A 282/B 338)

These underlying spaces are what Kant has in mind when he states that we can use dots on a page to aid us in our arithmetical cognition; the dots are intuitively and not qualitatively distinguished, and in that sense they serve as pure units. This also allows us to understand Kant’s reference to the role of fingers in counting: although the fingers are qualitatively different from each other, in mathematical cognition, they aid us in thinking of units that are not distinguished in any other way but intuitively. Kant is not explicit about the strict homogeneity requirement for arithmetic in the Critique. Nevertheless, as I argued in Section 2.1, he holds that the synthesis of composition of continuous magnitudes requires a strict homogeneity. Since the synthesis of composition underlying arithmetic only differs in requiring units, it will also require strict homogeneity. In his lectures on metaphysics, Kant states that numerical difference is a difference of multiple instances of a lowest species, which includes a difference among discrete, qualitatively identical objects. Kant then states that the homogeneity of magnitudes is numerical difference without qualitative difference (28:504, late 1780s). Kant’s account is general, so the

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homogeneity of discrete magnitudes would also consist in numerical difference without qualitative difference.79 Furthermore, Kant’s friend and defender Johann Schultz makes the strict homogeneity requirement explicit in his work Examination of the Kantian Critique of Pure Reason: When I am to add 5 to 7, for example, I must first individually represent to myself the unities of the number 5, and hence represent 1 + 1 + 1 + 1 + 1. Now every one of these units, with regard to quality and quantity, thus in itself, is completely identical, and thus the understanding cannot differentiate them by the smallest inner characteristic, consequently it cannot differentiate them through concepts. It is therefore only possible through outer or sensible characteristics, that is, through those that do not consist in a concept but an intuition … Things that in respect of their inner characteristics are completely identical, and hence completely indistinguishable for the mere understanding, we cannot distinguish in any other way than by representing them at different places of space or in different points in time … 80

Schultz disagrees with Kant on the existence of axioms of arithmetic—a serious difference.81 On other matters, however, he hews fairly close to Kant, and I think the view that Schultz represents here is Kant’s. Thus, arithmetic, like the theory of proportions for continuous magnitudes, requires strict homogeneity, and hence intuition. This is not to say that Kant believed one can only count pure spatio-temporal units. As mentioned above, empirically perceived objects such as fingers will do, so long as one uses them as an aid to think about pure indistinguishable spaces. It is pure spatio-temporal units, however, that provide an a priori foundation for arithmetic.82 These features of Kant’s philosophy of arithmetic explain why Kant is reluctant to allow irrational numbers and thinks that geometry has a role to play in showing 79  For a detailed discussion of this issue and relevant passages, see Sutherland “Kant’s Philosophy of Mathematics.” 80  Johann Schultz [Schulze], Prüfung der Kantischen Critik der reinen Vernunft, erster Theil (Königsberg: G. L. Hartung, 1789) zweiter Theil (Königsberg: Friedrich Nicolovius, 1792); reproduced in Aetas Kantiana (Bruxelles: Impression Anastaltique: Culture et Civilation, 1968), 225–26. 81  Gottfried Martin (AC, ch. 1) has argued that Schultz’s view must have been inspired by Kant. This seems extremely unlikely, however. As Charles Parsons has pointed out, Kant states in the Critique of Pure Reason that there are no axioms concerning quantitas, i.e. concerning the answer to the question “How big is something?” Kant then denies that numerical propositions such at “7 + 5 = 12” are axioms (CPR, A 164/B 205). If he had thought arithmetic had axioms, he would have made that clear here. Furthermore, Kant explicitly says in a letter to Schultz in 1788 that arithmetic has no axioms (10:555). See Parsons, “Kant’s Arithmetic,” 122–23. 82  The view that arithmetical cognition rests on pure units was endorsed by nineteenth-century philosophers such as Johannes Thomae, Elementare Theorie der analytischen Functionen (Halle. a.S.: L Nebert, 1880); Rudolf Lipschitz, Lehrbuch der Analysis (Bonn: M. Cohen & Sohn, 1877); and Ernst Schröder, Lehrbuch der Arithmetik und Algebra (Leipzig: B.G. Teubner, 1873). I think that this is no coincidence and that they were likely influenced by Kant’s views and Schultz’s defense of Kant. Nevertheless, the views of these philosophers are primarily known today as targets of Frege’s trenchant (although often dismissive) criticisms; see his Foundations of Arithmetic, ed. and trans. J. L. Austin (Evanston: Northwestern University Press, 1980) §§34–44. Hasn’t Frege shown us that any attempt to ground arithmetic in pure units is implausible and hence uninteresting? I hope to have made Kant’s philosophy of arithmetic not only more comprehensible but also more plausible by placing it in the context of his theory of magnitudes, the theory of proportions, and his understanding of algebra. If Thomae, Lipschitz, and Schröder were influenced by Kant, then a better understanding of Kant’s views will make their views more comprehensible and Kant’s historical influence more interesting, which will in turn, I think, make Frege’s criticisms more interesting.

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us that the concept of √2 is not an empty concept.83 The conception of number based on a unit entails that we cannot represent √2 as a complete concept; the best we can do will always be an approximation. We can nevertheless apply arithmetic to lengths and demonstrate that the ratio between the side of a square and its diagonal corresponds to √2, and hence that the diagonal of a square is a determinate representation of an object corresponding to the concept √2. A more complex account of Kant’s philosophy of arithmetic emerges than one might have expected. Kant does not simply assimilate arithmetic into the theory of magnitudes by thinking of numbers as represented by lengths and thinking of arithmetic as a kind of measuring. He retains the traditional conception of numbers as requiring a unit. Kant’s account draws on the development of arithmetic into algebra and an interpretation of algebra as expressing the Eudoxian theory of proportions. His appeal to the Eudoxian theory of proportions allows him to give a unified account of our cognition of discrete magnitudes and continuous magnitudes, thus allowing him to give a unified account of geometric, arithmetic, and algebraic cognition. My aim has been to clarify the relation between Kant’s views of arithmetic, algebra and the theory of proportions. I have also argued that in Kant’s view, arithmetical cognition requires the representation of a strictly homogeneous manifold, a previously neglected yet important role for intuition in Kant’s philosophy of arithmetic. I think these results will be of interest to anyone looking for either a historical understanding of the philosophy of mathematics or simply a perspective quite different from the modern one. I also think that they will be of interest to anyone wishing to understand Kant’s views of human cognition more generally.84

Kant’s letter to Rehberg, September 25, 1790 (11:208, 14:57). See footnote 76 above. The work on which this paper is based owes much to conversations with and comments by Robert Adams, Tyler Burge, John Carriero, Michael Friedman, and Charles Parsons. Earlier versions of this paper was presented at the Pacific Meeting of the American Philosophical Association in March of 2003 and at the Midwest Meeting of the North American Kant Society in October of 2003; I would especially like to thank Michael Friedman for his incisive commentary on the first occasion and the participants at both. I also thank Lisa Downing and Bill Hart for comments on an earlier version of this paper and two anonymous reviewers for their helpful criticisms. I am also grateful to the participants in my seminar on Kant’s philosophy of mathematics taught at University of Illinois at Chicago in Fall 2005 and at Harvard University in Spring 2006. 83  84