It Adds Up After All: Kant's Philosophy of Arithmetic in Light of the Traditional Logic

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  • Philosophy and Phenomenological Research Vol. LXIX, No. 3, November 2004

    It Adds Up After All: Kants Philosophy of Arithmetic in Light of the Traditional Logic


    Stanford University

    Officially, for Kant, judgments are analytic iff the predicate is contained in the sub- ject. I defend the containment definition against the common charge of obscurity, and argue that arithmetic cannot be analytic, in the resulting sense. My account deploys two traditional logical notions: logical division and concept hierarchies. Division separates a genus concept into exclusive, exhaustive species. Repeated divisions generate a hierar- chy, in which lower species are derived from their genus, by adding differentia(e). Hierarchies afford a straightforward sense of containment: genera are contained in the species formed from them. Kants thesis then amounts to the claim that no concept hier- archy conforming to division rules can express truths like 7+5=12. Kant is correct. Operation concepts () bear two relations to number concepts: and are inputs, is output. To capture both relations, hierarchies must posit overlaps between concepts that violate the exclusion rule. Thus, such truths are synthetic.

    One core thesis of Kants philosophy of mathematics is that mathematical knowledge is synthetic. (Synthetic judgments are defined in opposition to analytic ones, whose predicate concept is contained in (A 6/B the sub-

    Work on this paper was supported by a fellowship at the Stanford Humanities Center, which I gratefully acknowledge. The ideas benefitted from exchanges with Solomon Feferman, Michael Friedman, Gary Hatfield, David Hills, Nadeem Hussain, Beatrice Longuenesse, John MacFarlane, Elijah Millgram, John Perry, Lisa Shabel, Alison Sim- mons, Daniel Sutherland, Pat Suppes, Ken Taylor, Jennie Uleman, Tom Wasow, Allen Wood, and Richard Zach, as well as from audience suggestions after talks at Berkeleys HPLM working group, the William James discussion group at Stanford, the Stanford Humanities Center, HOPOS 2002, and philosophy department colloquia at New York University, Villanova University, the University of Wisconsin, Milwaukee, and the Uni- versity of Utah. Finally, I am indebted to very helpful comments from two anonymous reviewers for this journal. Kants works are cited according to the pagination of the Akademie edition (Ak.), with the exception of the Critique ofpure Reason, where 1 follow the standard A/B format referring to the pages of the first (=A) and second (=B) editions. I have made use of (and largely follow) the translations listed among the references. Works of Kant, Leib- niz, and Aristotle are cited according to the abbreviations listed there. Other works are identified by date of publication (with original publication date of the relevant edition added in brackets).


  • ject.) Kants thesis has met with two kinds of objection. First, critics have complained that the view is unsustainable, especially in the case of arithme- tic, where the role of construction in intuition is less obvious than it is in g e ~ m e t r y . ~ Second, even prior to questions of its correctness, the Kantian doctrine has been rejected as unclear, based on a general skepticism that there is any real distinction resting on containment-or indeed any intelligible distinction at all- between analytic and synthetic judgments.

    The two forms of criticism intersect in complaints about the frustratingly thin character of Kants reasoning about the non-analyticity of arithmetic. In a typical passage, Kant writes,

    To be sure, one might initially think that ... 7+5=12 is a merely analytic proposition ... . 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 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 that unification of seven and five, and no matter how long I analyze my concept of such a possible sum, I will still not find twelve in it. [B 151

    Apparently, Kant does not so much argue here, as pound the table. Instead of explaining what is revealed when one considers it more closely, he simply restates his point in the more emphatic form of a challenge: Analyze all you want; you will never find the predicate in the subject. Other treatments merely repeat the same invitation to consider what is thought in the sum concept (see e.g., A 164B 205). Meanwhile, the Critique is silent on what must be considered the pressing questions for the view: How are claims about what is contained in concepts supposed to be defended? How can we know that a purported analysis of a concept is complete, or correct? Without answers, Kant remains open to the rejoinder that mathematics only seems synthetic because of failures in his own analyses. Deeper analysis might reveal the appropriate containment relations. The perceived deficiencies of Kants discussion have contributed to general skepticism about his definition of analyticity as concept containment. Many philosophers have therefore come to accept Quines influential dismissal of the Kantian definition as merely metaphorical, and so hopelessly inadequate to sustain a principled distinction between logically different types of proposition (Quine 1961 [1953], 20-l).4

    In Euclidean proof, diagrams convey information essential to the success of the demon- strations. The construction of such intuitive representations is thus indispensable for geometry in the form in which Kant knew the science, as shown by a number of illumi- nating recent studies (Friedman 1985, 1992, ch. 1; Shabel 1998, 2003; Carson 1997; Sutherland, forthcoming, and unpublished). It is less obvious, though, how diagrammatic reasoning is necessary in arithmetic. The point has been explored in penetrating work by Parsons 1983, 110-49; Friedman 1992, ch. 2; and Shabel 1998. Even many Kant scholars have adopted this broadly Quinean line: see, e.g., Beth 195617, 374; Beck 1965.77-80; Bennett 1966,7; Brittan 1978, 13-20; Allison 1983, 73-5; Kitcher


  • This paper begins from a reconsideration of Kants analytidsynthetic dis- tinction. I will show, contrary to currently widespread opinion, that Kant deploys a clear and defensible notion of concept containment, which emerges in light of traditional, early modern logical ideas, and their appropriation in the metaphysics of Christian Wolff. Once we understand it, that notion of containment provides resources for a compelling argument that arithmetic must be synthetic, sensu Kant. The result does not yet provide a full account of arithmetic knowledge: it immediately raises, but does nothing to address, the famous Kantian question about how such synthetic a priori knowledge is possible. Still, it does successfully undermine what I take to be Kants initial target- the Wolffian position that all mathematical truth is fundamentally conceptual, and can be reconstructed in strictly syllogistic form.

    This negative conclusion is important because in light of the Wolffian stance, Kant needs independent reason to reject the analyticity of mathematics before he can convincingly motivate his well known positive theory of such knowledge, based on the role of pure intuition in mathematical argument and concept formation. Consider, in Kants estimation Wolffians treat intuitions as confused concepts, thereby denying his exclusive distinction between con- cepts and intuitions (see A 44/B 61-2, A 270-1B 326-7). Therefore, direct inference from the intuitive contribution to mathematical practice to the non- conceptual, or non-analytic, character of mathematical judgment would beg the question against Wolffians. (Asserting the need for intuition would not establish a non-conceptual component of mathematical cognition, if intui- tions were themselves conceptual, as Wolffians hold by Kants own lights.) For this reason, Kants claims about the role of intuition in mathematical knowledge are not happily understood as mere restatements, or explications, of his thesis that mathematics is synthetic. Rather, they should be seen as offering a solution to a difficulty that arises once mathematics is already seen to be synthetic-viz., explaining how mathematics achieves what mere analysis of concepts cannot. If this is right, though, then Kant needs direct considerations, independent of the full theory of pure intuition, to support the initial negative claim that mathematical judgments are non-conceptual. It is those I will in~est igate .~

    1990, 13, 27; and to some extent, Parsons 1992.75. (Longuenesse 1998 (at 275-6, et pas- sim) is a notable exception.) Kitcher (1990, 27) offers an especially clear (and clearly Quine-influenced) expression of the idea. An early version of this criticism was leveled already by MaaD already in 1789. See Allison 1973, 42-5, for discussion. In this sense, I propose that our understanding of Kants overall account of the synthetic- ity of mathematics should begin a step earlier than the arguments that have so far received most critical attention. Most recent interpretations have concentrated on Kants positive claims about the role of intuition in mathematics, whether due to skepticism about containment analyticity (Hintikka 1967; Kitcher 1975; Parsons 1983, 110-49), or simply out of interest in the richer questions within philosophy of mathematics that can be addressed via the full theory of pure intuition (Friedman 1985, 1992; Shabel 2003), or


  • To anticipate the main argument, Kants denial that the concept is contained in the sum concept amounts to a claim that we cannot con- struct a concept hierarchy, conforming to the rules of traditional logical division, which establishes a containment relation between and ~ 7 + 5 > . ~ In section 1, I explain these logical notions; section 3 shows that Kants claim is correct. It follows not only that arithmetic is synthetic, as Kant understood the term, but also that there are deep, principled limitations on the expressive power of logical systems of the sort appropriate to Wolf- fian metaphysics. Kants result thereby deals a fatal blow to the Wolffian program to reconstruct all genuine scientific knowledge in privileged logical form. The same expressive limitations illuminate the motivations of Kants broader philosophy, because they raise a problem about how synthetic judg- ment is possible at all- a problem Kant aimed to solve through a general theory of cognitive synthesis. The theory of mathematical construction in pure intuition should be understood as one prominent part of that broader theory of synthesis.

    On this account, the logical system of analytic conceptual relations will turn out to be quite weak, but if I am right, that was just Kants point. Mere logic (sensu the traditional logic of concepts) cannot underwrite synthetic judgments. Kant does not think that this is af law of the traditional logic. It is a limit of that logic, to be sure. But the limitation does not arise because logic is inadequate (see B viii); Kants point is rather just that mathematical knowledge (as, indeed, synthetic knowledge generally) is non-logical in char- a ~ t e r . ~ The first step in articulating Kants insight is to outline the character- istics of traditional logic on which it depends.

    both. These accounts have shed a great deal of light on Kants substantive story about how synthetic and a priori mathematical knowledge is possible, and I take such an account of mathematical intuition to be a necessary complement to the results explored here. But if I am right, Kants initial claim that mathematics k synthetic needs to be established independently-on the basis of considerations proper to the logic of con- cepts-in such a way as to help underwrite the conceprlintuition distinction itself. Of course, I need not (and do not) deny that once such arguments have disabled the Wolf- fian stance, Kant can also offer additional reasons supporting the non-analyticity of mathematics from within the critical system, where those arguments would assume his full positive theory of the role of pure intuition, and thus also a sbict conceprlintuition dis- tinction. Thanks to Lisa Shabel for discussion. Angle brackets (< >) indicate the mention of a concept. In this respect, the basic thought behind my reading is indebted to seminal ideas proposed by Michael Friedman (1985, 1992, chs. I-2), who systematically develops the thesis that Kants philosophy of mathematics responds to deep expressive limitations of the tradi- tional logic. Friedman focuses attention on the work done (for Kant) in mathematical argumentation by intuition, showing that its role could only be captured via powerful tools of modern logic unavailable within the traditional logic. I aim to identify a separate set of considerations that can be framed within the t e r n of the traditional logic itself.


  • 1. The Analytic/Synthetic Distinction and the Traditional Logic of Concepts

    A . On the Logical Basis of Kants Distinction Kant introduces the analytic/synthetic distinction this way:

    In all judgmen ts... the relation of a subject to a predicate [can be] thought ... in two different ways. Either the predicate B... is (covertly) contained in this concept A; or B lies entirely out- side the concept A, though to be sure it stands in connection with it. In the first case 1 call the judgment analytic, in the second synthetic. Analytic judgments are thus those in which the connection of the predicate is thought through identity, but those in which this connection is thought without identity are to be called synthetic judgments. One could also call the former judgments of clarification, and the latter judgments of amplification, since ... the latter ... add to ... the subject a predicate that was not thought in it at all, and could not be extracted from it through any analysis. [A 6-7/B 101

    To present-day ears, the passage has the ring of a stipulative definition, tell- ing us how Kant will use the terms analytic and synthetic (as applied to judgments*). For Kants original audience, by contrast, it must have carried the force of a substantive thesis, not mere stipulation. Talk of the subjects containing the predicate would have been familiar, but not as the characteriza- tion of some special subclass of judgments. Rather, it pretended to be a per- fectly general account of true judgment as such. Leibniz connects the idea to the logic of the proposition: The predicate or consequent is always in the subject or antecedent, and the nature of truth in general or the connection between the terms of a statement consists in this very thing (AG 31). One underlying thought seems to be this: A judgment is a relation between con- cepts; But the logical nature of a concept is to have conte...