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Arithmetic from Kant to Frege: Numbers, Pure Units, and the Limits of Conceptual Representation

Published online by Cambridge University Press:  16 October 2008

Daniel Sutherland
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There is evidence in Kant of the idea that concepts of particular numbers, such as the number 5, are derived from the representation of units, and in particular pure units, that is, units that are qualitatively indistinguishable. Frege, in contrast, rejects any attempt to derive concepts of number from the representation of units. In the Foundations of Arithmetic, he softens up his reader for his groundbreaking and unintuitive analysis of number by attacking alternative views, and he devotes the majority of this attack to the units view, with particular attention to pure units. Since Frege, the units view has been all but abandoned. Nevertheless, the idea that concepts of number are derived from the representation of units has a long history, beginning with the ancient Greeks, and was prevalent among Frege's contemporaries. I am not interested in resurrecting the units view or in righting wrongs in Frege's criticisms of his contemporaries. Rather, I am interested in the program of deriving concepts of number from pure units and its history from Kant to Frege. An examination of that history helps us understand the units view in a way that Frege's criticisms do not, and in the process uncovers important features of both Kant's and Frege's views. I will argue that, although they had deep differences, Kant and Frege share assumptions about what such a view would require and about the limits of conceptual representation. I will also argue that they would have rejected the accounts given by some of Frege's contemporaries for the same reasons. Despite these agreements, however, there is evidence that Kant thinks that space and time play a role in overcoming the limitations of conceptual representation, while Frege argues that they do not.

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Royal Institute of Philosophy Supplements , Volume 63 , October 2008 , pp. 135 - 164
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Copyright © The Royal Institute of Philosophy and the contributors 2008

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References

1 Frege (1884), English translation (1950). Frege commits a total of 27 sections at the beginning of the Foundations to attacking his contemporaries on the concept of number. He begins with his crucial arguments that number is neither a property of external things nor something subjective, a theme that reappears throughout his polemic. He then devotes the next 16 sections to attacking the idea that numbers could be derived from units (10 sections attack pure units in particular, and two are specifically directed against an appeal to space and time).

2 Friedman (1992a), pp. 110–3, points out the connection between Kant's theory of magnitudes and the Eudoxian theory. I develop a broader interpretation of Kant's theory of magnitudes and its relation to the Eudoxian theory in Sutherland (2004).

3 Although the theory treats both continuous and discrete magnitudes, the treatment of the continuous and the discrete are split between Books V and VII respectively. Algebra traces its origin through an arithmetical tradition leading back to Diophantus, but the development of algebra and its application to geometry in the early modern period led to its being viewed as a universal mathematics of all magnitudes, corresponding to the Eudoxian theory that bridges both continuous and discrete magnitudes. For a fuller discussion of this point, see Sutherland (2006).

4 For a defense of the claim that Kant's analysis of homogeneity is an extension of the Greek notion of homogeneity, see Sutherland (2004).

5 Quotations from the Critique of Pure Reason closely follow, with occasional modifications, the Paul Guyer and Allen Wood (1997) translation. All other references to Kant's work will be to volume and page number, separated by a colon, of the Akademie edition of Kants gesammelte Schriften.

6 Kant states:

According to mere concepts of the understanding, it is a contradiction to think of two things outside of each other that are nevertheless fully identical in respect of all their inner determinations (of quality and quantity); it is always one and the same thing thought twice (numerically one) (Ak 20:280).

A few pages later Kant adds that conceptual representation alone would ‘bring the whole of infinite space into a cubic inch and less …’ (20:282). This work is translated as ‘What Real Progress Has Metaphysics Made in Germany since the Time of Leibniz and Wolff?’, in H. Allison and P. Heath (eds.), (2002) Theoretical Philosophy after 1781, The Cambridge Edition of the Works of Immanuel Kant (Cambridge: Cambridge University Press), 337–424.

7 Kant makes this clear in the Amphiboly, which emphasizes this feature of space and at the same time its composability:

(…) multiplicity and numerical difference are already given us by space itself as the condition of outer appearances. For a part of space, although completely similar and equal to another part, is nevertheless outside of it, and is for that very reason a different part from that which abuts it to constitute a larger space (A264/B320).

He reaffirms this later in the Amphiboly, stating:

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) (A282/B338).

8 See Sutherland (2004). Geometry provides the paradigm for the role of intuition in the representation of a homogeneous manifold. To quickly summarize the broader view: the categories of quantity—unity, plurality, and allness—are used to cognize the part-whole relations of continuous regions in space, which provides a mereological basis for our cognition of the composition relations among the parts of space. (It is worth noting that the role of intuition in allowing us to represent a homogeneous manifold has been overlooked, but it is not the only role for intuition in mathematical cognition).

9 I will come back to this relation below.

10 My comments rely primarily on Jakob Klein (1968), especially Chapter 6; §1 of Stein (1990); and William Tait (2005), especially §9.

11 Interpretation is difficult, but arithmos in this sense might conceivably belong only to the particular collection counted, so that the eight of these sheep would be distinct from the eight of those sheep. Rogers Albritton first suggested to me that considering numbers as abstract particulars is at the very least an interpretative possibility that requires consideration. Klein seems at least to make room for this reading; see Klein (1968), 46–7. However, arithmos as counting number might be thought of as a species under which collections of a particular size fall, so that the eight of these sheep is the same eight as the eight of those sheep. On this view, particular number words, such as ‘eight’ can be viewed as a common name of collections of a particular size. See Stein (1990), 163–4, and Tait (2005), 238–9.

12 The generality of such a representation might be thought a necessary presupposition of the enumeration that results in counting-numbers, and hence it might be thought to be more fundamental. See Klein (1968), 49.

13 Plato's Republic, trans. by G. Grube, (Indianapolis: Hackett, 1974, revised by C.D.C. Reeve in 1992), 526 A; quoted in Klein (1968), 24.

14 I say more below about only attending to the action of construction in my discussion of schemata below.

15 Interpreting Kant's notion of homogeneity along these lines is prominent in various interpretations, in particular Longuenesse (1998).

16 For a more detailed defense of this view, see Sutherland (2004).

17 For more on this additional support, see Sutherland (2006).

18 There are two cases to consider. First, if we call to mind a general representation of any seven samurai, then we do not want to represent the individual samurai as possessing particular distinguishing properties, and the deeper homogeneity of intuition allows us to accomplish this. Second, if we have seven particular samurai before us and we wish to represent them as seven, the homogeneity of intuition may allow us to abstract from their distinguishing properties, in the way that we can appeal to our fingers in the representation of 5 + 7 = 12. In both cases, however, the account may be more complicated; it may be that once we have attained pure a priori concepts of numbers and pure a priori cognition of arithmetical truths such as 5 + 7 = 12, we can directly apply those concepts (and truths) to particular collections. Either way, however, there is a further issue that I will address below: Kant may think that the role of the homogeneity of intuition is to allow the concepts of number to arise or to establish their objective validity in a way that does not undermine their generality.

19 Friedman (1992a), pp. 112–22, develops this line of thought with some care. For important modifications to this view, see Friedman (2000).

20 Friedman (1992a), pp. 122–29, identifies the schemata for geometrical constructions with Euclidean constructions, and points out that in Kant's view, the problem of deriving general conclusions from particular geometrical figures is solved by an appeal to schemata rather than images (p. 90).

21 Schultz had been Kant's student and later wrote a review of Kant's Inaugural Dissertation that influenced the development of Kant's thought. Encouraged by Kant, he published a summary and explanation of Kant's views (Schultz, 1784). Kant thought well enough of it that at one point he planned to use it as a textbook for his metaphysics course. Schultz (1789) is less of a summary and includes more philosophical analysis. (Here below, all references to Schultz are to this latter work with page numbers in brackets). In addition, Schultz wrote at least seven reviews supporting Kant's philosophy.

Schultz lived in Königsberg and became its court chaplain; he also received a professorship in mathematics at the university, most likely with Kant's support. The two would have had opportunity to converse, but they nevertheless communicated at least in part through letters; we have at least a dozen from Kant to Schultz. See Kuehn (2001), and James C. Morrison introduction to the English translation of Schultz (1784).

22 Schultz (1789), 221–3, also identifies two postulates of arithmetic, both of which concern magnitudes and emphasize homogeneity. In keeping with my interpretation Schultz is attempting to articulate the way in which Kant's philosophy of arithmetic rests on Kant's more general theory of magnitudes.

23 Schultz also echoes Kant when he describes the consequences of attempting to represent pure units without appealing to intuition:

The help of intuition is required, without which the concept of their plurality would be wholly impossible; rather, the understanding would have to think them all together as absolutely [schlechterdings] eadem numero, that is, as only one (226).

24 I am setting aside debates about various connotations that have been attributed to Anzahl and Zahl by philologists; the distinction I describe is paramount. For more details on these disputes, see Stosch (1772); Adelung (1793); Eberhard (1819), among others.

25 I would like to thank William Tait for prompting me to make this last point explicit.

26 Lipschitz (1877).

27 It is possible that Lipschitz would simply reject the pure plurality problem; his views might approach those of Mill in this respect. Unfortunately, a discussion of Mill is beyond the scope of this paper.

28 Schröder (1873). All references to Schröder will be to this work (page numbers in brackets).

29 I will use ‘oners’ because Schröder prefers ‘einer’ to ‘eins’ and because it will help avoid confusion between the numeral one and the (or a) number one.

30 In a collection, the oners are separated from each other by plus signs (p. 5), a representation of number he calls their ‘Urform’ (p. 22). Note that Schröder says that the sum of oners is a natural number, not that it has a natural number. Schröder holds that the numbers consisting of distinct collections of oners are themselves distinct even when they are equipollent. In this respect, his Zahl resembles the first interpretation of Greek arithmos mentioned in footnote 11 above.

31 One might think that in Schröder's view, we could take the oners themselves as pure units, that is, that we could represent them while abstracting from all their distinguishing properties other than their spatial relations, in the way that I have suggested Kant may think of using dots on a page. It is not entirely clear what the illustration relation consists in, but I think that on the most sympathetic reading of Schröder's position consistent with his claims of maximal abstraction, the oners represent pure units. As noted in §1 above, there are two sorts of purity at stake in the representation of pure units: a lack of distinguishing characteristics among a collection of units, and a lack of any distinguishing characteristics at all. Oners may have (or at least approximate) the first sort of purity, while clearly failing to achieve the second. See footnote 32 for more on oners and the first sort of purity. I would like to thank Charles Parsons, whose comments led me to distinguish the oners and the units they illustrate.

32 It is possible that the distinctness of the oners is intended to represent the distinctness of units illustrated by means of them (see footnote 31). In practice, we distinguish between strokes or between tokens of the numeral ‘1’ by their location in space and time, not by any qualitative differences, such as imperfections or the darkness of their ink. It is possible that Schröder may make at least implicit appeal to space to distinguish the numeral tokens, thereby implicitly adopting the Kantian position. Nevertheless, he nowhere states that a necessary condition of reaching a general representation of number is distinguishing between oners by spatial or temporal location alone. He does say that the fact that number consists of spatially separated components that can be generated at any time brings advantages (p. 9), but that is as far as he goes.

33 Jevons (1874). All references to Jevons here below will be to this text (page numbers in brackets).

34 Thomae (1880/1898). All references to Thomae will be to this work (with page numbers in brackets).

35 Thomae significantly revised his account of number concepts in the 1898 edition, greatly expanding it and attempting to take into account the work of Dedekind and Frege, and in particular Frege's criticisms. He seems not to have really grasped the importance of Frege's advances; he still holds to a units view based on the purported empirical abstraction found in childhood concept acquisition, and he even states that his new account does not differ substantially from Frege's. Nevertheless, he makes interesting further points relevant to the units view, which I will unfortunately not be able to explore here.

36 Frege has various arguments undermining an appeal to units, including arguments that 0 and 1 cannot be accounted for and that arithmetical operations cannot be made to correspond to the union and dissolution of units, but most of his attention is directed against the possibility of representing the units and it is on this argument that I will focus.

37 Frege does think of a cardinal number n as the object that is the collection of concepts under which exactly n things fall, but that is another matter.

38 See, for example, the headings preceding §55 and the heading to §62. I noted above that Frege does not acknowledge the distinct accounts Schröder gives of Anzahl and Zahl; this may reflect Frege's conviction that there is no distinct account to be given. I am indebted to William Tait (2005), 38–9 and 242–3, for a helpful treatment of Frege's view of number, identity, and equality in contrast to the views of those Frege discusses. Tait believes that Frege's rejection of the conception of number as a collection leads him to misread Euclid, Hume, and Schröder; see footnote 40 below for a discussion of Hume and footnote 42 below for a discussion of Schröder.

39 See Austin's (1950) English translation of Frege (1884), p. II, note.

40 For example, when Hume articulates what has become known as ‘Hume's Principle,’ he states that those numbers (i.e. collections of things) that can be put into a one-to-one correspondence are called equal. When Frege discusses Hume's Principle in §63, he uses the term ‘gleich,’ and goes on to discuss ‘Gleichheit’. As Tait (2005), p. 239, has argued, Frege seems to misread Hume here. Hume is using ‘number’ to refer to collections, and the relevant notion of gleich is of equality between collections, but Frege seems to take Hume to use ‘number’ to refer to singular objects and to interpret ‘gleich’ as identity. In that case, translating ‘gleich’ as ‘identity’ would capture Frege's misunderstanding of Hume, but it would not be true to what Hume said or meant. In this case, Austin translates ‘gleich’ as ‘equal,’ since that was Hume's original term, and then translates Frege's use of ‘Gleichheit’ as ‘equality or identity’—the former term is a nod to what Hume in fact said, while the latter is how Austin thinks Frege interpreted him.

41 It may be that Austin only has the strict sense of identity in mind, for he seems to have some compunction about rendering Frege's description of Thomae's use of ‘gleich’ as ‘identical.’ In §34, Frege states: ‘Thomae calls the individual member of his set a unit, and says that “units are identical to each other” …’. Austin usually stays very close to the original, but in this case he drops the quotation and renders it: ‘Thomae … says in so many words that units are identical with each other …’ (my italics). If Austin also had the looser sense of identity in mind, he could have translated Frege more directly. This, of course, assumes that Thomae had this looser sense of identity in mind, as I argued above, but perhaps Austin was unsure of Thomae's meaning.

It is worth noting that in §34, Frege quotes three passages from Thomae and Lipschitz in a way that suggests the first and second passage are from Thomae and the third from Lipschitz. Austin breaks up the sentence and clearly attributes the second passage to Thomae, although it in fact belongs to Lipschitz.

42 Tait (2005), p. 242, maintains that Frege understands ‘gleich’ to mean strict identity, and that Frege imposes this reading on those he discusses. In Frege's §36 discussion of Hume's principle, where ‘gleich’ is applied to equipollent collections, I think he is most likely correct; see footnote 40 above. Tait does not, however, explicitly distinguish between this use of ‘gleich’ and its application to elements among a collection, and he does not explicitly consider the possibility that Frege uses ‘identity’ in the looser sense when describing the purported identity of units among a collection that his opponents, such as Schröder, espouse.

43 This would be so if qualitative identity implied substitutability. It is on the basis of §65 that Tait (2005), p. 235, ascribes the principle of identity of indiscernibles to Frege.

44 Frege's argument is somewhat more complicated than this summary suggests; he presents a dilemma, and I am drawing out what he says concerning the first horn.

45 It is perhaps surprising to attribute to Frege a limitation on conceptual representation; his views on what is conceptually representable would appear to be, if anything, quite liberal. Nevertheless, it seems to me that Frege's argument points in this direction.

46 Asserting a version of PII applied to points of space would be a departure from the traditional understanding of the principle. Leibniz only applies PII to substances or real beings and their phenomenal manifestations: monads, ‘pieces of matter,’ atoms, and sensible things such as leaves and drops of water. In his view, PII does not apply to abstractions or ideal beings, and because he holds that spaces and times are merely ideal, he does not think that PII applies to them (as Tait (2005), fn. 29, points out). In fact, Leibniz states that points of space and time are distinct despite being indistinguishable, and he even compares them to pure units in §27 of Leibniz's Fifth Paper of the Leibniz–Clarke Correspondence; see H. G. Alexander (1956). Kant would also reject the application of PII to points of space, but for a different reason. In his view, PII does not apply to any objects of sensibility, whether or not they are substances, and that includes qualitatively identical parts of space and points of space. Kant focuses on parts of space. In his view points are limits of spaces and hence presuppose parts of space, and the same diversity without specific difference holds for them as for qualitatively identical parts of space.

47 Frege's statement concerning points of space reads:

It is only in themselves, and neglecting their spatial relations, that points of space are identical to one another [einander gleich]; if I am to think of them together, I am bound then to consider them in their collocation in space, or else they fuse irretrievably into one (§41).

The phrases ‘neglecting their spatial relations’ and ‘if I am to think of them together [zusammenfassen]’ and ‘I am bound to consider them’ leave open the possibility that Frege intends to be discussing only our representation of points of space rather than points of space themselves. This would be consistent with his argument strategy; that is, with his attempt to show that Schröder, Jevons, and Thomae cannot represent units as both pure and distinct, and hence cannot account for our representation of number by appealing to our representation of units. If Frege is making a point about our representation of points of space, then he may be implicitly invoking an intensionalized version of PII—that is, if someone represents x and y as indistinguishable, then that person represents x and y as identical. Nevertheless, as noted in the previous section, Frege asserts an un-intensionalized version of PII in §34, and §65 seems to support it as well. On balance, I think that it is likely that Frege implicitly applies un-intensionalised PII to points of space in §41. I would like to thank Walter Edelberg for suggesting intensionalized versions of PII to me and pressing me for clarification. Un-intensionalized and intensionalized versions of PII are closely related to each other in both Leibniz and Kant; there is much more to say on this point than I can include in this paper.

48 One can question whether Frege's position makes sense. William Tait has argued that relations between points of space will not distinguish those points unless the relations themselves can be distinguished, which presupposes the distinctness of points of space. Yet Frege has just denied that points of space in themselves are distinct, see Tait (2005), p. 235. If an adequate response to this criticism can be given, it will depend upon difficult issues concerning the nature of relations, distinctness, and identity, issues that I cannot adequately address in this paper.

49 In fact, he holds that even a single point of space has something sui generis that distinguishes it from, say, a moment in time, and ‘of which there is no trace in the concept of number’ (§41).

50 Much more can and needs to be said on this point, but I will have to postpone an explication and defense for another occasion.

51 Parsons (1984) explored Kant's conceptions of magnitude and their relation to mathematical cognition and to the categories of quantity. Friedman (1992a), 110–113, drew a connection between Kant's account of algebra and the Eudoxian theory of proportions and he subsequently encouraged me to explore the connection between the Eudoxian theory and Kant's theory of magnitudes. My work can thus be viewed as an extension of Parsons' and Friedman's lines of investigation and I have benefited greatly from their work. Discussions with William Tait revealed that he had addressed some of the same issues I was exploring. I have benefited immensely from Tait's (2005) paper and from regular discussions with him in the spring of 2007 and 2008. I would also like to thank those who responded to versions of this paper delivered at the Central Meeting of the American Philosophical Association, April 2007; the Kant and Philosophy of Science Today conference at the Royal Institute of Philosophy, University College London, July 2007; and the UCSB Department of Philosophy, February 2008. I am particularly grateful to Michael Friedman, Robert Howell, and Charles Parsons for detailed comments, as well as Brandon Look for comments and for sharing a draft of his paper on Leibniz's principle of identity of indiscernibles. Special thanks are due the members of the UIC Philosophy of Mathematics Reading Group, especially Bill Hart and Walter Edelberg, for their many insightful remarks, which substantially improved both my understanding of the philosophical issues and this paper. I am also grateful to Marcus Giaquinto for discussion and for comments on the penultimate draft. Finally, I would like to acknowledge the generous support of the National Science Foundation; this paper is based upon work supported by them under Grant No. 0452527.