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

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

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

R. LANIER ANDERSON

Stanford University

Officially, for Kant, judgments are analytic iff the predicate is “contained in” the subject. 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 hierarchy, 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. Kant’s thesis then amounts to the claim that no concept hierarchy conforming to division rules can express truths like ‘7+5=12.’ Kant is correct. Operation concepts (<7+5>) bear two relations to number concepts: <7> and <5> are inputs, <12> 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 Kant’s 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 Berkeley’s 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. Kant’s 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).

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ject.) Kant’s thesis has met with two kinds of objection. First, critics have complained that the view is unsustainable, especially in the case of arithmetic, 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 Kant’s 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 Kant’s discussion have contributed to general skepticism about his definition of analyticity as “concept containment.” Many philosophers have therefore come to accept Quine’s 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 illuminating 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

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This paper begins from a reconsideration of Kant’s analytidsynthetic distinction. 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 Kant’s 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 Kant’s estimation Wolffians treat intuitions as confused concepts, thereby denying his exclusive distinction between concepts 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 intuitions were themselves conceptual, as Wolffians hold by Kant’s own lights.) For this reason, Kant’s 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 passim) 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 Kant’s overall account of the syntheticity of mathematics should begin a step earlier than the arguments that have so far received most critical attention. Most recent interpretations have concentrated on Kant’s 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, Kant’s denial that the concept <12> is contained in the sum concept <7+5> amounts to a claim that we cannot construct a concept hierarchy, conforming to the rules of traditional logical division, which establishes a containment relation between <12> and ~ 7 + 5 > . ~ In section 1, I explain these logical notions; section 3 shows that Kant’s 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. Kant’s 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 Kant’s broader philosophy, because they raise a problem about how synthetic judgment 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 Kant’s 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); Kant’s 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 Kant’s 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 Kant’s 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, Kant’s initial claim that mathematics k synthetic needs to be established independently-on the basis of considerations proper to the logic of concepts-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 distinction. 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 Kant’s philosophy of mathematics responds to deep expressive limitations of the traditional 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 Kant’s 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, telling us how Kant will use the terms ‘analytic’ and ‘synthetic’ (as applied to judgments*). For Kant’s original audience, by contrast, it must have carried the force of a substantive thesis, not mere stipulation. Talk of the subject’s containing the predicate would have been familiar, but not as the characterization of some special subclass of judgments. Rather, it pretended to be a perfectly 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 concepts; But the logical nature of a concept is to have content, i.e., to contain other concepts; So, the obvious relation to posit as the logical basis of the proposition is containment - one concept contains the other. Thus, “the connection between the terms of a statement consists in this very thing,” viz., containment (my ital.). Many eighteenth century German philosophers, including the pre-critical Kant, accepted such a predicate-in-subject account of judgment, which renders all judgments analytic, in the terms of the contain-

’ Kant uses ‘analytic’ and ‘synthetic’ to mark two different distinctions, one between arguments or methods of proof, and the other between judgments. I focus on the distinction ofjudgrnents, which is new with Kant (see B 19, and more generally, Allison 1973). The two distinctions bear some interesting analogies, but I postpone full discussion to another occasion. Leibniz articulates this core doctrine in numerous places, some of which were known to Kant. See, e.g., “On the General Characteristic” (L 226), “Universal Synthesis and Analysis” (L 231-2). “Discourse on Metaphysics” (8 8; AG 40-I), and New Essays (NE 486). For discussion, see Adams 1994, 70-1.


ment definition.” Seen within the formative context of the Critique, then, Kant’s opening claim that the relation between subject and predicate “is possible in two different ways” (A 6/B 10; my ital.) already, and controversially, introduces his critical rejection of German rationalism.‘’

The salience of containment for Kant’s main rationalist targets gave him good reason to privilege it in the official definition of analyticity I quoted. Kant then goes on to explain other central features of analyticities in terms of containment: they are true by identity or contradiction because their component concepts contain the same content at least in part (so that denying the predicate of the subject amounts to a contradiction); likewise, they are merely explicative because the predicate does not go beyond what is contained in the subject.” These additional features still deserve separate mention, though, because they tie Kant’s new notion to core rationalist ideas. Since analyticities are “thought through identity,” treating philosophy as a body of analytic truths would entail Wolff‘s thesis that the principles of identity and contradiction are the basic principles of all kn0w1edge.I~ Moreover, analytic “judgments of clarification” are discovered by the method of analysis, which decomposes a concept’s content into component concepts (marks, differentiae); e.g., gold is a yellow metal, so <gold> contains <yellow> and <metal> as marks (see Prol., 267). Analysis renders a concept distinct by teasing

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See FS (Ak. 2: 60-I), but the view is also present in various versions in OP, NM, I, and ID. In an important sense, I would argue, it is the acceptance of the predicate-in-subject principle that makes these works pre-critical. The anti-rationalist force of Kant’s point is made apparent as the discussion proceeds. After sketching the two kinds of judgment, he goes on to suggest that essentially all important cognition falls on the synthetic side, including empirical judgments (A 7-8B 11- 12), mathematics (B 14-17); a priori (pure) natural science (B 17-18); and metaphysics ( B 18; cf. A 10). That is, a rationalist account of knowledge based on analysis of concepts is doomed to failure for virtually all significant bodies of knowledge. (The topic sentences introducing each of these claims are emphasized in B, so as to highlight this aspect of the Introduction’s argumentative organization.) Recall from the quoted passage, Kant first defines analyticity as containment, and continues, “Analytic judgments are thus those in which the connection of the predicate is thought through identity ... . One could also call [them] judgments of clarification ... since through the predicate [they] do not add anything to the concept of the subject, but only break it up ... into its component concepts” (A 6-7/B 1 0 my ital.). That is, analyticities count as identities or clarifications because the predicate is already contained in the subject, and so does not depart from or add to it. Other key Kantian discussions of analyticity tend similarly to rest on ultimate explications via the containment idea (see A 15UB 190-1; OD Ak. 8: 228, 232). For this reason, I am skeptical of efforts to replace the containment definition with some other mark of analyticity (e.g., truth by the principle of contradiction, or non-ampliativeness), despite powerful and interesting developments of such a stance by van Cleve (1999, 17-21) and Allison (1973, 53-6; 1983, 73-8). I must postpone fuller treatment of the complex textual issues about the relation between containment and other marks of analyticity in Kant’s usage (though some discussion is available in Anderson, forthcoming). Thanks to Daniel Sutherland for discussion. Kant prominently criticizes Wolff s thesis in the Prolegomena introduction (Prol. 4: 270).

R . LANIER ANDERSON

apart its marks to yield an articulated and explicit grasp of the ~ontent . ’~ It thereby settles the pattern of relations among affiliated concepts, by identifying which marks are contained in a concept, what further marks are contained in the initial ones, etc. Kant’s containment definition evokes this standard contemporary picture of explication by analysis.

Despite the scorn which such containment has often received post-Quine, it can be given clear sense in the traditional logic. In particular, as de Jong (1995) also notes, it can be understood by appeal to the logical division of concepts, which supports Aristotelian definition.” Division separates a concept’s extension into sub-classes, each corresponding to a specific way of having that predicate; e.g., the genus cnumben may be divided into <even> and <odd>. Two rules of division were standard: 1) a division must be complete, so that the species taken together exhaust the entire genus, and 2) the members of the division must be exclusive, so that no species can be pmh- cated of any other. That is, divisions are exclusive and exhaustive disjunctions.I6 Since the species concepts cover proper parts of the generic extension, we can see a species (e.g., <even>> as composed out of, and therefore defined by, the genus itself (here, <number>), plus some differentia that marks off its particular way of having the genus concept (e.g., <divisible by two>). It is then natural to think that, at least for suitably regimented conceptual systems, the composition of concepts could be reconstructed through division, and every concept could receive an Aristotelian definition.

Given the rules of division, a notion of analytic containment falls out di- rectly. Conceptual relations generated by division can be represented in a genuslspecies hierarchy, where a genus is contained in the species concepts formed from it, and they are contained under it. On this picture, a concept’s content includes the concepts contained in it (i.e., above it in the tree), and its extension comprises the concepts contained under it. The two forms of containment (content and extension) are strongly reciprocal. This means, first, that everything in the extension of a concept, A , contains A as part of its content, and conversely, everything included in the content of A covers A as part of its extension. Then, second, Kant holds that

l4 For a typical discussion of how analysis achieves distinct concepts, see Wolff 1965 [I7541 (“Deutsche Logik”), 126-34, followed by Baumgarten 1973 [1761], 8-12, and Meier (AV 851 15-39; in Ak., 16: 296-341). These discussions anticipate a conception of logic as essentially involved with “explicitation,” recently defended by Brandom 1994. An Aristotelian definition gives the genus to which a term belongs, and identifies the differentia that separates it from others also falling under that genus; for example, humans are rational animals-i.e., animals (genus) of a rational sort (differentia). Port Royal formulates these rules in the chapter on classification (i.e., logical division); see Arnauld and Nicole 1996 [1683], 124-5. Kant treats logical division and its rules in his logic lectures at Ak. 24: 273, 760-2, and 925-28, and in the Jasche Logic at Ak. 9: 146-8; compare Meier 1914 [1752], $5 285-91; Ak. 16: 612-19.

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In regard to the logical extension of concepts, the following universal rules hold 1. What belongs to or contradicts higher concepts also belongs to or contradicts all lower concepts that are contained under those higher ones; and 2. conversely: What belongs to or contradicts all lower concepts also belongs to or contradicts their higher concept. [Logic, Ak. 9: 981

In effect, these rules define equivalence conditions for conceptual contents and extensions. They entail that concepts with the same extension also have the same content, and vice-versa. Not only must any two such concepts must include the same marks “belonging to” their contents or extensions, but they must also each exclude the very same marks, which “contradict” the content or extension. (Concepts sharing the same content and extension in this way are “convertible” or “reciprocal” [Wechselbegriffe]; Logic, Ak. 9: 98, also Ak. 24: 261, 755, 912). In this sense, conceptual extension and content cannot come apart: any difference in content entails a difference in logical extension, and conversely. (Note that the characterization of both contents and “logical extensions” is thoroughly “intensional” in our modem sense- a point to which I shall return.”)

To see the picture, consider the following partial hierarchy of concepts, based on ideas familiar in the eighteenth century from Linnaeus’s (1964 [ 17351) work on biological classification:

<Animate Substance>

/\

Here, more specific concepts are obtained by adding differentiae to a higher genus; e.g., a quadruped is an animal with four limbs and hair, a glirine” is a quadruped with two front incisors, and so on. It follows immediately that, for example, <glirine> already contains <quadruped>, <animal>, etc. The analysis of a concept into its more abstract components amounts to subtraction of marks as one moves up the tree, and this underwrites the containment idea. What remains after subtraction was contained in the original concept all along. Kant makes this conception of logical abstraction explicit in his mar-

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The intensional character of Kant’s account of relations among concepts was not unusual in early modern logic, and for him, extensions in the logical sense are always sets of concepts, not objects (see Logic, Ak. 9: 95-10), But for a crucial qualification, cf. note 28 below. In Linnaeus’s classification, the order Glires included animals now classified as rodents, distinguished by having two prominent front incisors in each jaw.

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ginal note to Meier’s treatment of higher and lower concepts: “Abstraction [Absonderung] is subtraction” (R 2885; Ak. 16: 559). The division-based account of containment relations between lower and higher concepts is also fairly explicit in Kant’s logic lectures and the Jasche Logic. For example, the “Vienna Logic” defines containment (conceptual content and logical extension) via genushpecies relations along the lines just sketched (Ak. 24: 91 l), and then continues, “From the highest concept, consequently, the lowest ones are determined, and this ... comprehends the correct determination of all things” (Ak. 24: 912). In the logical context, such “determination” of concepts is governed by division: ‘The determination of a concept in regard to everything possible that is contained under it, insofar as things are opposed to one another, i.e., distinct from one another, is called the logical division of the concept” (Logic, Ak. 9: 146).*’ Reading the two passages together, we see that for Kant, “the correct determination of all things” is to be reached via logical division of concepts, from highest to lowest.

Thus, a look at his logical writings shows that Kant’s persistent talk of “containment” in the account of analyticity should be understood by reference to logical division and concept hierarchies. If the rules of division are observed, then 1) every species concept includes its genus, so judgments connecting them are true by virtue of what the concepts contain; and 2) since the members of a division exclude one another, they (and their subspecies) are never predicable of one another, and all (affirmative) predications represented by the hierarchy are of the genus/species type.*’ The rules of division are thus

l9 Thus, logical abstraction via analysis and logical determination through division a re inverse operations, which articulate the two reciprocal forms of containment (conceptual content versus conceptual extension): so, “To take apart a concept and to divide it are thus quite different things. In taking a concept apart I see what is contained in it (through analysis), in dividing it I consider what is contained under it” (Logic, Ak. 9: 146; see also J 15, Ak. 9: 99). Kant views the extension to negative categorical judgments as straightforward (A 6B lo), but it is worth noting because it highlights the importance of the exclusion rule. Negative judgments of the form ‘No S is P‘ are analytic iff the subject and predicate concepts exclude one another (i.e., fall under opposed members of a division). Thus, for example, ‘No unlearned person is learned’ is analytic, “since the mark (of unlearned- ness) is now comprised in the concept of the subject” so that the subject and predicate are exclusive (A 153/B 192). In general, the principle of contradiction serves as the principle for analyticities via such considerations of concept exclusion: “For the contrary of that which as a concept already lies and is thought in the cognition of the object is always correctly denied, while the concept itself must necessarily be affirmed of it, since its opposite would contradict the object” (A 151/B 190-1). Note, in this passage Kant applies the reasoning both to cases of explicitly contradictory (“opposite”) concepts (e.g., <unlearned>/<learned>), and to merely “contrary” concepts whose mutual exclusion would be established by division. Thus, at bottom, ‘No youth is old’ (cf. A 152-3/B 192) or ‘No fish are quadrupeds’ should be counted as analytic for the same reasons as ‘No unlearned person is learned’: i.e., the division of <animal> into <fish>, <quadruped>, etc., is exclusive, just as <learned>/<unleamed> is an exclusive division of <person>. Thanks to Daniel Sutherland for discussion.

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sufficient to guarantee that a hierarchy encodes only analyticities. Analysis can then be understood as an objective matter of locating concepts in the true hierarchy, which “comprehends the correct determination of all things” (Ak. 24: 912).

By contrast, the necessary condition for a hierarchy to represent pairwise containment relations between concepts is weaker than the rules of division. As long as every lower concept, S, is wholly contained under its higher concept, P (so that Ss must be P ) , we can think of P as included in s, even if (contrary to the rules) the division of P was not exhaustive or exclusive. Leibniz envisioned such a system, treating containment as a purely syntactic property based on his hypothesized universal characteristic, or “alphabet of human thought.”2’ For example, a concept represented by a concatenation of three primitive symbols in the characteristic, ABC, would contain not only A, B, and C , but also the complex components AB, and BC. Under such assumptions, a hierarchy violating the

B C

rules of division, like the one at right?2 could nonetheless represent containment relations- e.g., B is contained in AB, in BC, and in ABC. In these circumstances, the strong reciprocity of content and extension assumed by Kant, which is guaranteed by the exclusion rule in particular, may also be violated (e.g., AB and BC have the same extension, but different contents.)

But typical analytic judgments (like ‘Quadrupeds are animals’) are not expressed in the privileged terms of Leibnizian characteristic, and when we return to the task of developing a general notion of analyticity, it turns out that relaxing the strict division rules undermines the clarity of the representation of containment. A division hierarchy can precisely capture containment relations only because it shows explicitly which concepts are contained in a given concept and which ones are not, thereby providing a fully general crite- rion of a n a l y t i ~ i t y . ~ ~ If the rules can be violated, by contrast, we cannot infer that two concepts are incompatible from their positions in the hierarchy. (The standard ground for the inference would be that the concepts fall under different member species of a division. But absent the exclusion rule, that fact no

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22 See, e.g., “General Inquiries about the Analysis of Concepts and of Truths” (P 47-87). The hierarchy represents divisions which are not exclusive, because the two species of B (AB and B C ) have a common subspecies, ABC. The division may not be exhaustive either: there may be infinitely many elementary concepts, and so there could always be another concept (say, BD) alongside AB and BC in the division. Kant certainly saw the point that an adequately clear theory of containment required the specification of both conceptual inclusions, and conceptual exclusions. His “universal rules” about “the logical extension of concepts” (quoted above) assert not only that what “belongs to” higher concepts also belongs to their lower ones, but also that what contradicts the former contradicts the latter (Logic, Ak. 9: 98).

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longer prevents their overlap in some common subspecies.) That is, exclusion is no longer conveyed by the shape of the hierarchy at all, and we can infer it only where we discover actually contradictory marks. Without the division rules, then, exclusion would have to be represented in the very syntax of the concepts, or else left as part of their implicit content.24 Since containment relations are not typically explicit in the compositional syntax of concepts in that way, they can only be conveyed by relative position in a hierarchy that adheres to rules of division. So in the end, those rules, which guarantee that every relation of concepts is one of containment or exclusion, arerequired to mark off containment relations in a clear and general way. In this sense, they are the minimum sufficient conditions for a notion of analyticity precise and general enough for logical use. These division rules turn out to have important consequences for the expressive power of analytic truth.

The notion of concept containment provided the primary explanatory model for logical relations going all the way back to Ari~totle.~’ Syllogisms, for example, were seen as a way of connecting two concepts which lack immediate relation, by finding an indirect relation through a “middle term.” Concept containment then served as a unifying principle for the validity of such inferences, characterizing the sort of connections among terms that must be present in correct inference. Leibniz, for example, insists that “the whole theory of the syllogism could be demonstrated from the theory of de conti- nente et contento, of container and contained” (NE 486).26 Containment thus possessed canonical status within early modern logic.

Notice that if the second option is taken, and incompatibilities are not explicitly specified, Quinean worries resurface about whether there is any longer a clear notion of analyticity at all. Whether or not there is analytic containment then depends on unconstrained “intuitive” judgments of particular speakers about what is and is not implicit in a given concept. Leibniz does have a reply to such doubts, since in his system, incompatibility information is imbedded in the syntax of concepts-eg., ABC is incompatible with all and only those concepts that contain not-A, not-B, or m - C . This still leaves r m m for complaint about explicitness, however. In such a system, we could evaluate claims of the form ‘A is not (contained in) B’ only via complete analysis of the concepts, which, for a Leibnizian, would often have to be infinite, and so beyond our capacities. Aristotle explains the force of inference in terms of the containment relations among terms at Pr. Anal. 25b32-26a2,24b27-31, etpassim. The Port Royal Logic offers a typical early modern discussion of the two kinds of containment (which Arnauld called “com- prehension” and “extension,” Arnauld and Nicole 1996 [1683], 39-40), and also appeals to containment to explain inference (162, 163). In spite of the standard character of this notion of containment, numerous puzzles could be raised for the picture painted here, which I must leave aside for now. (Some discussion is available in Anderson, forthcoming). Leibniz’s logic was innovative in its search for a more systematic and abstract application of this idea than had been previously attempted (see Rescher 1954), but the basic idea was widespread. Aside from its appearance in Aristotle and Port Royal, noted above, Kant’s Reflexion 2894 (Ak, 16: 565) suggests that concept containment serves as the ground for the dictum de omni er nullo, which was recognized as a basic principle of syllogistic inference.

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B. Interlude: Worries about Intensionality

Despite its standing in the contemporary context, current readers of Kant, laboring under Quinean influence, might still worry that the present account fails to clarify analyticity in a substantial way. After all, it assumes fully intensional concepts with Aristotelian definitions, and thereby appears to presuppose notions like definition, essence, meaning, synonymy, and the like, which Quine famously located in the tight family circle of interdefinable ideas associated with analyticity. It was this circle in its entirety against which the Quinean skepticism was dire~ted.2~

Two kinds of consideration could motivate such a reaction. The broader thought would be that intensional concepts are intrinsically suspicious- e.g., because there is a philosophical puzzle about the possibility of the intensional as such. More narrowly, one could simply doubt that there are sufficiently concrete and explicit principles to allow rigorous identification of intensional contents, or precise characterization of their relations. The first, more global form of skepticism raises a clear problem for the account I have sketched: if it is accepted, only the strictly extensional content of a concept (specifying the individuals it applies to) counts as clear. Under extensional assumptions, a concept hierarchy would fail to elucidate analyticity, even if it did conform to the rules of division, since it could not separate class inclusion relations arising out of the meaning of the concepts (e.g., the mammals are included among the animals), from those due to accidental (or lawlike, but non-conceptual) matters of fact (e.g., the animals with kidneys are included among the animals with hearts) (cf. Quine 1961 [1953] 30-1).

On reflection, however, the underlying worry here is clearly not one that Kant himself seriously entertained. As we saw, his conception of the logical subject matter is intensional throughout. For Kant, even the extension of a concept (in the logical sense) is understood to be the group of intensional concepts contained under it, rather than the individual objects it applies to.28

*’ ** Different versions of this objection were pressed on me by Ken Taylor and Kit Fine.

The canonical statement of Kant’s view on logical extensions is the theory of the concept presented in the Jasche Logic (Ak. 9: 91-100; see esp. $1 7-16, pp. 95-100). The extension of a concept in the specifically logical sense involves the strong reciprocity of content and extension, following from the two “universal rules’’ that hold “in regard to the logical extension of concepts” (quoted above; Logic, Ak. 9: 98). The same view is implicit throughout Kant’s work, both in prominent arguments (e.g., at B 40), and in otherwise puzzling doctrines (e.g., the doctrine that the subject concept of a singular judgment has no extension (A 71/B 96), i.e., no general concepts under it; or the rejection of any infima species and the related logical law of specification (A 655-6/B 683-4)).

A qualification is also necessary here, however. Aside from this purely logical sense of extension, Kant also deploys a broader sense of conceptual extension in the service of his epistemological theory of synthetic cognition. (We will see the crucial role of such non-logical extensions below.) In that broader sense, individual objects or intuitions can be said to fall under a concept, or be comprised in its extension. Therefore, as Longue- nesse (1998, esp. 50,47, etpassim) has observed, Kant operates with two distinct senses

5 12 R. LANIER ANDERSON

Kant assumed that such concepts were objective, and that their intensional character posed no special problem for our logical knowledge of them and their relations; on the contrary, he took the theory of the concept to be the most basic and straightforward branch of logic, on which the others depended. Current philosophers may have reasons (tied to a strictly extensionalist point of view) to doubt our access to intensional contents, or our justification for positing them, but such thoughts are sufficiently foreign to Kant’s basic assumptions that they are unlikely to help us understand his notion of analyticity. I therefore leave them aside here.29

The second, narrower approach is more interesting, in that it motivates extensionalist skepticism about intensional contents on grounds Kant himself could recognize, by arguing that, even if we admit there might be intensions, there is no way to specify explicit rules for identifying them and handling their relation^.^' In the absence of such rules, perhaps the circle of intensional notions Quine attacked cannot be given clear sense. Something like this thought lies behind the classical complaint that what is “contained in” a concept could only be revealed by introspection, whose results would have to be either (a) potentially idiosyncratic (if it is just a matter of what I think when I use the concept), or (b) mysterious (if introspection is thought somehow to reveal the “concept itself’). The criticism goes all the way back to M d ’ s (1789) attack on the analytickynthetic di~tinction,~’ and I think it is the real motivation of most Kant scholars who doubt the clarity of the official containment definition of analyticity. It is clear, of course, that Kant thought of judgments and concepts as objective logical contents, not idiosyncratic states of individual cognitive agents (indeed, Kant’s disciple Schultz made just this reply to MaaB). That move, however, simply sharpens the question of how the relations among intensional contents can be objectively determined. The

(or kinds) of concepts and of extensions. (Thanks to Longuenesse for helpful exchanges on the point.) I would add that this difference of sense is intimately related to the analytic/synthetic distinction-a fact which emerges from a group of logical RejZexionen in which Kant’s distinction begins to emerge. There, Kant distinguishes between analytic and synthetic marks, where analytic marks are contained in or under concepts in the standard logical sense discussed here, and synthetic marks are contained under concepts in a separate way, which permits the possibility of essentially synthetic judgments. Intui- tive content may be included among the synthetic marks, which explains the possibility of intuitive marks, recently noted by Sni t (2000). The relevant Reflexionen include R 2286, R 2289, R 2291, R 2293, R 2357-8, and R 2363 (Ak. 16: 299-300; 300-01; 301; 302-3; 331; 332). I suspect that orthodox Kantians would not be much moved by a blunt, a priori invitation to global skepticism about the intensional (a la Quine). Likely, they would simply refuse the invitation, insisting that there is nothing unintelligible about the thought that in fact, the way concepts represent their non-logical extensions (i.e., the individuals they apply to) is indirect, by means ofhaving some intensional content (e.g., a sense, a rule, etc.). Cf. Quine’s discussion of the difficulty of specifying “semantical rules” adequate to underwrite a clear and general conception of analyticity, at Quine 1961 [1953] 32-7. MaaE 1789. esp. pp. 187-90. See Allison 1973,41-5 and 174-5 for discussion.


interesting worry, then, is that without explicit rules for establishing these relations, the identity conditions for intensional concepts are fixed by unconstrained “intuitive” judgments, which are worthy targets of Quinean suspi- cion.

Just here, however, the rules of division offer substantial help. They show how, beginning from a given intensional content (genus) and an appropriate stock of differentiae, a system of intensional concepts with transparent containment relations can be generated under the constraint of explicit logical rules (“From the highest concept ... the lowest ones are determined”; Ak. 24: 912). Thus, if we can grant initial intensional contents at all- on credit, as it were, against the promise that a tolerably clear system will be forthcoming- then the division rules cash the promissory note, by supplying an explicit procedure for generating a system of analytic containment relations (start from a genus, and divide by adding differentiae according to the rules). Since the whole system is intensionally understood from the start, its results specify genuinely conceptual containment- not merely inclusion relations among extensional classes- and it yields Aristotelian definitions for each concept. Kant’s thought, then, was never to settle questions of containment by appealing to contingent beliefs enshrined in the mental museums of individuals, but by recourse to a rule-governed objective structure of concepts. Such a structure counts as specifying analyticities just in case it conforms to clear rules that settle containment questions. I have argued that the rules of division do this work. As long as one admits intensional concepts at all, then, logical division determines their containment relations, and thereby settles which truths owe their force to conceptual contents alone. Put another way, once we consider the resources offered by the traditional logical context, it emerges that the cost of maintaining the Quinean doubts about analyticity is full scale skepticism about intensional contents.32

To conclude, the traditional logic characterizes a clear notion of concept containment, which provided the framework in terms of which logical relations were understood. Thus, Kant’s notion of analyticity has a straightforward basis. Indeed, once we take account of this context, Quinean criticisms of analyticity suffer an ironic twist: given the traditional logic, analyticity is perfectly clear; it is synthetic judgment that seems difficult to understand, for in the synthetic case, the basic logical relation between concepts- contain- ment- does not (per hypothesis) obtain.33

32 My answer to the second, more interesting group of considerations thus bears a family resemblance to the defense of analyticity offered by Boghossian (1997), who argues that Quine’s skepticism about analyticity is sustainable only if one also accepts a more radical skepticism about meaning in general, embodied (for Quine) in the indeterminacy of translation thesis. In this sense, Kant’s distinction gives rise to a fundamental problem about synthetic judgment, which Kant attempted to solve by deploying the theory of cognitive synthesis that

33


2. Kant’s Target: the Concept Hierarchy in Wolffian Metaphysics

In a long paragraph just prior to the official introduction of the analytic/synthetic distinction (A 3-6B 7-10), Kant sketches the general research program against which his distinction is directed. That program sought an a priori science of metaphysics by emulating the “splendid example” of mathematical knowledge (A 4 B 8), and highlighting our copious rational knowledge through “analyses of concepts,” thereby creating an “apparent thoroughness” (A 5/B 9). This agenda, as will become clear, is certainly that of Christian Wolff and his followers. Like the pre-critical Kant, Wolff is committed to the predicate-in-subject doctrine. Interestingly, Wolff does not simply define judgment itself in terms of concept containment, as Leibniz had.34 Nevertheless, he advocates the closely related idea that the principle of contradiction is the basis of all knowledge: “not only do inferences have their certainty from it, but it is also through it that a proposition that we experience is placed beyond all doubt” (Wolff 1983 [1751], 6; 5 10). Wolff does treat experience and the principle of sufficient reason as identifiable sources of knowledge alongside the principle of contradiction, but ultimately they rest on that logical principle (cf. Prol. 4: 270):

The certainty of reason therefore grounds itself in the certainty of inferences. But I have already shown in the [Deutsche Lo&] that the certainty of inferences depends on the ground of the principle of contradiction ( 5 10). Since now experience, too, ultimately has it [the principle of contradiction] to thank for its certainty ( $ 5 10, 330), so all certainty of cognition derives from it. [Wolff 1983 [17511,239; 5 3911

Wolff‘s arguments purporting to derive truths based on experience and sufficient reason from the principle of contradiction alone are circular,3’ but what is important for us is just his conviction that they could be so derived on conceptual grounds alone.

Wolff‘s position sounds strange to modern ears, but it harmonizes with two of his central views on the proper shape of philosophical knowledge. First, analysis is the characteristic method of philosophy. Philosophy seeks a complete understanding of things from the grounds of their possibility (Wolff 1965 [ 17541, 115. 110). Complete understanding, in turn, comes from

generates the core positive metaphysical results of the Critique of Pure Reason. I offer a sketch of Kant’s theory of synthesis in Anderson 2001. Wolff s caution is understandable. Leibniz’s view gave rise to significant metaphysical difficulties, which were a major topic in the LeibnidArnauld correspondence, where a good deal of the Leibnizian metaphysical apparatus is required to avoid the principle’s apparent necessitarian consequences. It was therefore reasonable for Wolff not to build the idea into the very definition of judgment. Beck (1969, 2645; 1978, 88-92) emphasizes this reticence, which leads him to an reading somewhat different from the one presented here. Nevertheless, the fundamental role of the principle of contradiction in Wolff commits him to the analyticity of knowledge claims just as surely as a Leibnizian definition of judgment would. Beck acknowledges the point (1969,264-5). For discussion, see Anderson, forthcoming.

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IT ADDS UP AlTER ALL 5 15

“adequate” concepts, for which we know the marks “distinctly,” in that we clearly represent their component marks, and the marks of those marks, and so forth (Wolff 1965 [1754], 129-31). Adequate concepts are thus delivered by analysis, which makes explicit the interconnections among these serially nested conceptual marks, and thereby locates them in a hierarchy. A systematic hierarchy of concepts would offer the wanted knowledge from grounds, since a higher concept provides a ground for its lower species, when taken together with the differentia that defines the species. The resulting judgments form a system of containment analyticities known by the principle of contradiction.

Second, Wolff grants syllogistic inference an unusually prominent cognitive role, compared to other early modern thinkers. Since genuine science is knowledge of things from their grounds, Wolff insists that all bona fide theoretical argumentation can be reconstructed in privileged form as a series of syllogisms, which reveal the grounds of the conclusion. It is precisely here that mathematics is to be the model, and the Deutsche Logik illustrates the point by reconstructing Euclid’s proof of the angle-sum property of triangles (Bk. I, Prop. 32) as a series of syllogisms (Wolff 1965 [ 17541, 173-5).

In support of the privilege for formal inference, Wolff offers some of the reasons later made familiar by Frege- e.g., that syllogistic reconstruction can reveal undefended assumptions, expose gaps in argumentation, and make it easy to uncover mistakes (Wolff 1965 [1754], 178-9, 243). But he goes further, claiming that syllogistic inference is a true method of discovery- the implicit path through which results like the angle-sum property are attained in the first place. If we could only bring this path to full explicitness, we would render the pursuit of knowledge reliably fruitful, systematic, and immune from dependence on mere lucky hits. The conclusion fits with the picture of the true system of philosophy as a perfected hierarchy of concepts. Traditional inference reveals an indirect connection (based on containment) between two concepts, through a middle term. Thus, the conceptual relations uncovered by inference-based inquiry should conform naturally to a regimented scientific concept hierarchy, which represents containment relations.

In both cases, then, (i.e., for analysis and for inference), the core com- mitments of Wolff‘s rationalism make sense in the context of a vision of the true system of philosophy as an adequate hierarchy of concepts organized by genus/species relations.36 On Wolffs picture, the progress of inquiry is a

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Perhaps surprisingly, the same ideas offer a natural explanation for the prominent role of experience in Wolffs theory. Wolff insists that empirical arguments, too, should be guided by syllogistic inference (Wolff 1965 [1754], 176-8). In such cases, experiences serve as unprovable premises, which enter the argument in the same way as definitions. Moreover, Wolff claims (1965 [1754], 134-5) that we often achieve full distincmess in our concepts only through experience, which thereby helps us discover the true concept

R. LANIER ANDERSON

pursuit of the correct hierarchy of concepts- one that makes explicit the logical relations among marks that were implicitly contained in the confused concepts from which we begin our quest for genuine scientific knowledge. If we construct it correctly, so that it represents logically proper containment relations, then the hierarchy will encode analyticities.

The function of Kant’s analytic/synthetic distinction was to define the expressive limits of the logical system appropriate to this Wolffian metaphysics. As we saw, Wolff treated mathematics as a paradigm for philosophical cognition (Wolff 1965 [1754], 115). Its certainty and deductive flavor make it the most plausible case for his project of reconstructing scientific argument in syllogistic form. Thus, when Kant set out to defend his claim that our substantive knowledge is synthetic, not analytic, mathematics was the natural test.

3. Mathematical Truth as Synthetic Judgment

We can now formulate in detail the argument I sketched at the outset, specifying the limitations of traditional logic that prevent it from adequately representing mathematical knowledge. The argument will also offer some indication of what allows intuition to transcend those limits, and give us guidance in understanding Kant’s blunt pronouncements about what is “thought in” mathematical concepts, particularly in the case of arithmetic.

As I noted, Kant’s explanation of the syntheticity of arithmetic seems frustratingly opaque. He can apparently offer only the curt insistence that “no matter how long I analyze my concept of such a possible sum [<7+5>], I will still not find twelve in it” (B 15)?’ The sense of analyticity outlined above, however, reveals how Kant’s treatment of ‘7+5=12’ points toward a persuasive argument for his thesis. Analysis of concepts sensu Kant should not be viewed as mere introspection, but as exploration of an objective concept hierarchy, constrained by specific rules of the traditional logic. Thus, Kant’s invitations to consider what is “thought in” the concepts of numbers and sums should be understood as demands to consider carefully how the concepts could be related to one another as genera and species.38 On this reading,

hierarchy (by showing what was obscurely contained in (!) our originally confused concepts). Compare, along the same lines, ‘‘I do not think the number 12 either in the representation of 7 nor in that of 5 nor in the representation of the combination of the two” (A 1 W B 205); and likewise, “the concept of the sum of 7 and 5 contains nothing more that the unification of both numbers in a single one, through which it is not at all thought what this single number is” (B 15). Occasionally, Kant himself talks as if analysis were based on introspection. (See, e.g., his 25 November 1788 letter to Schultz, Ak. 10: 556, where Kant writes that it is “contrary to consciousness” that the concepts <3 and 4> and <I2 and 5s have the same content.) Surely, however, this is just a manner of speaking. Kant does not mean to endorse the view that questions of concept identity turn on what mental states individuals happen to

37

38


Kant’s denial that the concept <12> is contained in the sum concept <7+5> amounts to the claim that a hierarchy conforming to the rules of logical division cannot express the truth that 7+5= 12 in terms of containment relations between <12> and <7+5>. As it turns out, the division rules do block the construction of an appropriate hierarchy, and so arithmetic must be synthetic.

The key consideration driving the conclusion is the strong reciprocity of logical contents and extensions presupposed by analytic relations among concepts. Such reciprocity places quite a restrictive demand on the relations among concepts, and that demand is violated by arithmetic relations among numbers, which, as a consequence, cannot be assimilated to analytic genus/species relations based on concatenation of conceptual marks. Note, the relevant reciprocity of content and extension is reflected in analytic hierarchies by their permitting affirmative containment relations only along a single dimension, the abovehelow (genudspecies) relation. Other overlaps are blocked by the exclusion rule, so a hierarchy can only connect <7+5> and <12> as genus to species- making one a species of the other, or else plac- ing both at the same node, as convertible concepts. It turns out, however, that an arithmetic judgment like ‘7+5=12’ expresses two basically different and orthogonal kinds of relation among the numbers 7, 5, and 12, each of which must be separately specified in order to capture the proposition’s content. This exceeds the expressive capacities of the one-dimensional hierarchy, and violates the reciprocity of content and extension mandated by the division rules.

Given the limited possibilities made available by the genudspecies relation, we can see the point through an argument by elimination, which shows how the various types of hierarchy that might pretend to represent arithmetic relations run afoul of the constraints required for analyticity. Perhaps the most plausible account of ‘7+5=12’ as an analytic relation of genus to species would treat <12> (the predicate) as a genus contained in the subject <7+5> (its species). Consider the shape of the resulting hierarchy. It is natural to locate the number concepts (<5>, <7>, <12>, etc.) at the same level, e.g., as species dividing the concept <number>. Then we could try to obtain the wanted genus/species relation by introducing sum concepts like <7+5>, <9+3>, etc. (along with infinitely many other operation concepts, e.g., <3*4>, <15-3>, <&>) among the subspecies under <12>. While <7+5>

have (as MaaB would later charge, to Kant’s great impatience), any more than Berkeley means to suggest, by his frequent remarks that he cannot find any idea of material substance in his own mind, that really it is a subjective matter whether there is anything to the notion. Rather, Kant is trying to get Schultz to reflect carefully on how the concepts in questions should be understood in relation to potential genus and species concepts. In my view, all such talk is ultimately to be redeemed in terms of whether the analytic relation in question can be represented by a concept hierarchy conforming to the rules of division.


does not seem to be a species of <12> in the way being a mammal is a way of being an animal, still there is at least some sense in which 7+5, 3*4, and the like, could be taken as different ways of being, or of “getting to,” 12. Since a division hierarchy needs some genus/species relation, let’s assume, for reductio now, that the analyticity of ‘7+5=12’ could be represented along the proposed lines as follows:

<number>

c7t5> t9+3> <3*4> <48/4> t14-2> others

Kant would likely have balked immediately, because the hierarchy posits divisions with infinitely many species. A division-based hierarchy captures analyticity because it specifies a precise content for concepts, explicitly pre- senting each as the conjunction of a definite (and thus finite) set of marks. This is one reason Kant insists (contra Leibniz) that “no concept ... can be thought as if it contained an infinite number of representations within i t - self’ (B 40). Kant admits, though, that a concept may contain an infinite number of representations “under itself” (B 40). Clearly, he did not actually intend to countenance a single division with infinitely many species, as in the current example; rather, the thought was that a concept’s species (and subspecies, etc.) may always be further divided into ever more particular subspecies concepts (since, qua concepts, they are all general). Nevertheless, the present proposal is another way of representing infinitely many concepts under a given one, which Kant officially allows.

One might still worry that it fails to conform to the ideal of explicitness built into the division rules. The demand that division exhaust the content under the divided concept is not met if the members cannot be enumerated. Kant’s Reflexionen show that he was moved by such worries. There, he sometimes distinguishes between analytic and synthetic marks, based on the idea that analytic marks stand in subordination relations terminating in a highest concept (Logic, Ak. 9: 59, 97; R 2293, Ak. 16: 320-3), whereas synthetic marks are coordinated in potentially infinite series. In both cases, marks are “partial representations” (R 2286; Ak. 16: 299), but for analytic marks, the whole to which they belong is “actual” (R 2289; Ak. 16: 300), because the concept to be analyzed is given, and the series of mutually subordinated marks contained in it terminates. By contrast, synthetic marks must be understood as parts of a merely “possible” whole, which they form only together with further marks that are not yet actually given and must be speci-


fied by synthetic means (R 2289; Ak. 16: 300-1); their series, in which they are not subordinated but merely coordinated with one another, does not termi- nate (R 2293; Ak. 16: 302).

In retrospect, this complaint looks deep, for reasons advanced by Friedman (1985; 1992, chs. 1-2). Traditional logic traces all logical form to relations among concepts that are themselves non-relational. Therefore, it cannot force an explicit representation of the infinite: such concepts would be rendered by one-place predicates, and a monadic predicate calculus cannot exploit quantifier dependence to create formulae with only infinite models.39 As Friedman notes, Kant himself could not have formulated the limitation just this way, which depends on later technical developments. But that is not the crucial point. Kant could well have sensed the underlying logical difficulties about representing infinity, and sought an account of mathematics responsive to them. Indeed, Friedman shows in impressive detail how key elements of Kant’s positive theory of mathematics are responsive to these issues: for Kant, mathematical argument essentially involves procedures of construction in pure intuition, whose indefinite iterability provides the needed representational power.

Despite the depth of Friedman’s analysis, though, I am not persuaded that these worries were among Kant’s primary motivations in his initial argument that elementary mathematical truths are non-analytic (as opposed to his full story of how synthetic a priori mathematical knowledge is possible). If they had been a central concern, it seems likely that he would have chosen mathematical results more interesting than ‘7+5= 12’ to illustrate the point.40 Moreover, Kant does not argue that the number or sum concepts must be (or must involve) essentially synthetic marks, standing in infinite, coordinated series of the sort mentioned in his notes. Instead, he tries to make his case on the basis of the singular arithmetic formula ‘7+5=12,’ an example ill-suited

39 For detailed discussion of the argument, see Friedman 1992, chs. 1-2. For example, many proofs in Euclid’s theory of numerical proportion rest on reasoning of the form Friedman emphasizes is impossible within the monadic predicate calculus (because it depends on quantifier nesting). Euclid VII, 2, for instance, offers a procedure for determining the greatest common factor (or measure) for any two numbers that have one. (That is, it shows thatfor any such numbers, a and b, there b some number n, such that n measures a and n measures b, andfor any number m that also measures a and b, m I n.) Moreover, Euclid deploys such reasoning in the contexts Friedman envisions, for example in the famous demonstration (Euclid IX, 20) that there are infinitely many prime numbers. Surely Kant was aware of such theorems, which are central to the Euclidean theory of numerical proportion (the very theory Kant cited in his mathematics lectures as the key historical source for elementary arithmetic; Ak. 291111: 52). Thus, he could have mentioned them, had his theory of arithmetic been motivated by the issues identified by Friedman, concerning infinity and, implicitly, its representation via quantifier nesting. The fact that Kant rests content with the banal ‘7+5=12’ seems to me evidence that his view was actually driven by the logically simpler (albeit mathematically less interesting) considerations I explore in the text below. Thanks to Solomon Feferman for discussion.


to support any such arg~ment.~’ In this context, it remains unclear that Kant’s generalized worry at B 40 is sufficient to rule out a hierarchy like the one above. A Wolffian might insist that even if the proposed divisions cannot be completed, the hierarchy nonetheless succeeds to represent analytic relations between the concepts it does include, notably <7+5> and <12>, and the same pattern could always be extended to arbitrary truths of elementary addition. Why, the Wolffian might demand, is this not explicitness enough?

Let us set aside worries about infinity, then. Even so, there remain problems of a logically more elementary sort showing that the proposed hierarchy fails to represent arithmetic as analytic. The point of a proposition like ‘7+5=12’ is to assert a relation among the three numbers, 7, 5 , and 12, but our hierarchy represents no such relation!* We take ‘7+5=12’ to make the relation explicit, notice, because we take the sum concept <7+5> to be composed out of <7> and <5>. (Kant glosses its content as “the unification of both numbers in a single one” (B 15).) But so far, our hierarchy represents no explicit relation between <7+5> and <7> or <5>. <7+5> is introduced as a new concept, and until its relation to <7> and <5> is specified, its place as a species under <12> establishes no connection among <5>, <7>, and <12>.43 We could try to remedy the defect by explicitly connecting <7+5> to <5> and <7>, as in the following hierarchy:

4’ It is unclear, for example, how Kant could sustain the claim that <7>, <5>, <7+5>, or <12> are “merely possible” wholes, in the sense of R 2289, within which essentially synthetic marks must be coordinated, rather than actual wholes which could be analyzed to reveal analytic marks. It is clear that Kant sees this point. It provides the force behind his insistence (at B 15) that even though “the concept of the sum of 7 and 5 contains ... the unification of both numbers in a single one,” the thought of such a “unification” is such that “through [it] it is not at all thought what this single number is” (my ital.). That is, the content of the proposition includes explicit expression of all three numbers; but since the sum concept serving as subject contains “nothing more than the unification” of 7 and 5 , and not yet what third number the sum yields, the predicate must add something not present in the subject, which brings the thnd number concept to the table for the first time. (Incidentally, this also seems to me to be the force of Kant’s somewhat perplexing claim further on (B 16) that his claim is more obvious for sums involving larger numbers. In those cases, we are less tempted to think that a clear understanding of the unification of some large x and y immediately contains, and therefore already brings forward, a representation of the third number that is their sum.) The content of (intensional) concepts is fixed by their place in the hierarchy, not by the shapes of the symbols that make them up. Without newly introduced, explicit links, <7+5> has no more relation (in OUT hierarchy) to <5> or <7> than <numbrous> would have to <numb>.

42

43’


-/I\- others 4> c6> d> c8> &= <lo> <11> <12> others

\/:\ ~ 7 + 5 > <9+3> <3*4> <48/4> <14-2 others

As soon as we do, however, we have violated the division rules. Now the extensions of <5>, <7>, and <12> overlap in the concept <7+5>, and the division of <number> is no longer exclusive. Just here, the one-dimensionality of the analytic hierarchy becomes fatal. The hierarchy can express only the reciprocal contained idunder relations which make <7+5> a species of <12>. But <5> and <7> cannot be contained in <7+5> in the same way <12> is; that would make them sums of seven and five. They would have to be contained in c7+5> in some other way . It is precisely this “other way” that the concept hierarchy cannot express.

No amount of fiddling with the details can remove the basic limitation. Since <5>, <7>, and <12> are members of a division, any relation connecting them all to the same concept below them would create an overlap of extensions, and thereby violate the division rules, and the strong reciprocity of conceptual content and extension they guarantee. That is, representing ‘7+5=12’ as a relation between <7> and <5>, on the one hand, and <12>, on the other, requires the suspension of the very rules that make a hierarchy an explicit and systematic representation of analyticities. Parallel considerations show that hierarchies which located arithmetic operation concepts (sums, roots, etc.) above the number concepts, or which treated them as convertible with the number concepts themselves, would be no improvement. The hierarchy would still permit only one type of relation between concepts, and arithmetic propositions like ‘7+5=12’ demand more than one.”

The first of the two remaining possibilities indicates a hierarchy something like one hinted at in Kbtner’s (1758) Anfangsgriinde der Arirhmerik (though it is not presented there as an adequate expressive system for arithmetic truths). Kastner proposes that the operation concepts are “the species of arithmetic” (Kastner 1758, 25; Kant makes the same claim at Ak. 29/11]: 52). One might therefore take them as species of the arithmetic concept of number. One could then have a hierarchy in which the concepts <sum>, <root>, <remainder>, etc. appeared as species of <number>, and had the particular operation concepts (e.g., <7+5>), as their subspecies. The number concepts (e.g., <12>) would then be subspecies of the operation concepts. Here, just as in the earlier example, the extensions of concepts overlap in a way prohibited by the rules of division (e.g.. <12> falls under many different members of higher divisions).

The last possibility treats <7+5> as a kind of definition, located at the same node as <12> itself, and similar difficulties resurface. Simply introducing 4 + 5 > as an alternative expression for <12> establishes no conceptual relation between 4 2 and <7>, and the concept <7+5>/<12>. If we insist that the content of <7+5> does depend on the concepts <5> and <7>, that dependence is impossible to specify with the limited resources of the hierarchy. The <7+5>/<12> concept is clearly not identical to <5> or <7>, nor is it plau- sibly construed as a species or genus of either.

44


Perhaps, however, all the trouble arose from the assumption that the number concepts are members of an exclusive division. The resources of concept containment do offer one alternative to that supposition: we cculd represent the numbers themselves above and below one another, as a successive series of genus, species, subspecies, etc. In some ways, the proposal has more promise. If we restrict attention to the natural numbers for simplicity, we could posit <I> as the highest genus, <2> as species, etc. Each number would be contained in (above) its successor, and thereby in all subsequent numbers as well. And in fact, smaller numbers are “contained” in larger numbers, at least in the sense in which, e.g., any group of seven contains within it a group of five.

But in the end, the idea makes no progress on the core problem of one- dimensionality. The only plausible way to introduce <7+5> into the successive hierarchy of number concepts is to make it convertible with <12>. But then there is no way to specify its relation to <5> or <7>. The proposed hierarchy, it is true, does represent <5> and <7> as contained in (i.e. above) the <12>/<7+5> concept, but not in the right way. For consider, <9> and <lo> are above <12> just as <7> and <5> are, yet they do not sum to twelve.45 Moreover, there are arithmetic operations equivalent to twelve whose component members are not above <12> (e.g., <27-15>, <a>). In the final analysis, then, the one-dimensionality of analytic conceptual relations prevents the formation of a hierarchy capable of representing arithmetic truths like ‘7+5=12,’ and Kant’s thesis is established.

This argument by elimination may be thought to underestimate the resources of Leibniz’s strategy for deriving arithmetic formulae from definitions, known to Kant from the New Essays ( N E 413-14).46 Leibniz proposed to prove ‘2+2=4’ from an axiom permitting substitution of equals for equals, plus the following definitions of numbers: 2 =def 1+1; 3 =def 2+1; 4 =def 3+1. The argument is straightforward: for the subject ‘2+2’ we may substitute ‘2+1+1,’ by the definition of <2>; then, by the definition of <3>, we substitute ‘3+1’ for ‘2+1+1,’ which concept (<3+1>) is identical to <4>, by the definition of <4>. In Leibniz’s proof, all these concepts (<4>, c2+2>, <2+1+1>, <3+1>) are convertible, so from the point of view of concept hierarchies, they would all be located at the same node, and- since <4> thereby implicitly contains <3>, <2>, and <1>- the number concepts

45 Notice, in fact, that what is lacking now is precisely the mutual exclusion of number concepts that created our problem before. The key point is that an analytic hierarchy cannot simultaneously express both the exclusion of one number from another, and their connection via an arithmetic operation. I return to the issue below. I am indebted to Michael Friedman for discussion of the issues arising from this possibility. It should be noted that in lectures, Kant himself offered proofs of arithmetic propositions along the same broad lines sketched by Leibniz, though not with any suggestion that they might establish arithmetic truths on the basis of containment relations among concepts (see Mathematik Herder; Ak. 29/11]: 57).

46


would best be treated as a genuslspecieslsubspecies series, in the fashion of the previous paragraph. The definitions are then wheeled in to unpack the surprising structure of each number concept.

On the view developed here, however, Kant has immediate grounds to reject the analyticity of the Leibnizian definitions themselves: ‘ 1+1=2’ is synthetic just as much as ‘7+5=12,’ and for the same reasons. Recall, expressing the content of ‘7+5=12’ demanded explicit recognition of one sort of relation between <7+5> and <12>, and an orthogonal relation of <7+5> to <7> and <5>. Since containment can forge affirmative connections between concepts only along a single dimension, the proposition cannot be a containment truth. (That is, however we characterize containment, it will capture at most one of the two relations we need, while excluding representation of the other.) In parallel fashion, if ‘1+1=2’ is to be a containment truth, then there must be some relation of containment between <1+1> and <2>. But if the proposition is to define <2> in terms of <1>, then < 1 + b must also contain <1>, in some sense. These two relations will be orthogonal, prevent- ing their simultaneous representation in an analytic hierarchy. After all, if arithmetic is analytic in general, then ‘l+l#l’ must be a containment truth just as ‘1+1=2.’ Therefore, we require one sense of containment in which <1+1> contains (or is contained in) <2>, but excludes <I>, and another sense in which <1+1> contains (or is contained in) <1>, while excluding <2>, since l r2 . Given the strong reciprocity of content and extension, and the corresponding one-dimensional character of hierarchies, the analytic containment relation just cannot do both jobs at once: each of the two needed kinds of containment rules out the other.

It is no help to suppose that the Leibnizian definitions are simply containment truths by stipulation. For what meaning is <1+1> stipulated to have? It could count as an illuminating analytic definition of <2> only if it is understood to be formed from e l> , by concatenation of the conceptual marks < I > and <l>.47 But then it will also have to be simply identical to <1>, and the definition fails to identify a new number, 2. For consider, concept concatenation begins from a given concept and adds some further mark, true of everything which falls under the new concept that results. The further mark, insofar as its content differs from that of the initially given concept, restricts the scope of the new concept to a subset of the initial extension. But if I add the same mark to itself (e.g., <1> and <1>), I do not specify a narrower extension, and the new concept is therefore convertible, or equivalent,

47 Someone might propose instead that <1+1> is simply a new concept, whose content is given via the definition itself. This move threatens to leave us with an account that defines <I+]> in terms of <2>, rather than the other way round, and in any case the reading offers no progress. Just as we saw in the case of <7+5>, the relation of <1+1> to <1> here remains unspecified, and when we do specify it, we see (as above) that <1+1> will have to exclude <1>, if it is to be identical with <2> as the definition claims.

524 R . LANIER ANDERSON

to the first, having the same content and extension (in our case, the concept is just <l>).48 Thus, the Leibnizian definition fails to represent containment relations of the needed sort among <1>, <1+1>, and <2>. Kant could nevertheless accept Leibniz’s definition, of course: he would simply insist that it, like all definitions of mathematical concepts, is essentially synthetic. What it offers is not a containment truth clarifying <2>, but an immediately evident construction of an object for the concept <2> itself, out of items that fall under a different concept, <1> (see A 727-3uB 755-60).

It is tempting to trace the syntheticity of the Leibnizian definitions to the role of the addition operator, which does have greater expressive power than analytic concept concatenation (see section 4, below). From that point of view, no proposition that includes any sum concept could be analytic. Kant, however, apparently did not see the matter in these terms. At times, he expresses perfectly standard concept concatenation via the sign for addition ( ‘+’)!9 More crucially, when arguing against the analyticity of arithmetic

~~ ~ ~~~~ ~

48 For example, if I concatenate <number> and <divisible by two>, I reach a concept true of the numbers which are also divisible by two, i.e., <even>, but if I concatenate <number> and <number>, or <1> and <1>, then I get only convertible concepts which are true of numbers, or of 1 (<number>, <I>). This tendency is most prominent in abstract discussions that use letters to stand for the marks to be combined. For example, Kant represents the matter this way in the official presentation of the distinction between analytic and synthetic judgments in the Logic:

49

An example of an analytic proposition is, To everything x, to which the concept of body (a+b) belongs, belongs also extension (b). An example of a synthetic proposition is, To everything x, to which the concept of body (a+@ belongs, belongs also attraction (c) . [I 36; Ak. 9: 1 111

(Jische drew this presentation from R 3127, Ak. 16: 671.) Another example is Kant’s handling of the proposition that the whole is greater than the part, which he takes to be analytic:

To be sure, a few principles that the geometers presuppose are actually analytic and rest on the principle of contradiction; but they also only serve, as identical propositions, for the chain of method, and not as principles, e.g., a=a, the whole is equal to itself, or (a+b)>a, i.e., the whole is greater than its part. [B 16-17]

This treatment of (a+b)>a poses a problem for my overall reading, since it seems just as hard to express in a concept hierarchy as ‘7+5=12.’ Kant’s glosses make it clear that he takes these propositions to be analytic in virtue of being direct instantiations of the principle of identity itself (first in its affirmative, then in its negative form, see NE sec. 1; Ak. 1: 388-91). which are therefore presumably supposed to rest on the identity and exclusion relations among the conceptual contents of <whole> and <paru. Admittedly, even with the gloss, the case remains problematic, because of the key role played by the relational concept <greater>, which raises serious problems for monadic hierarchies of the sort considered here (see sec. 4, below). As I argue in the text, however, Kant seems not to have focused on the relational character of mathematical concepts as the key source of syntheticity. It is thus plausible to think that he might have (erroneously) taken that (a+b)>a to be analytic in his sense, on the basis of reasoning something like Leibniz’s:

‘The whole is greater than the part’ ... [is] something easily demonstrated from the definition of ‘less’ or ‘greater,’ with the addition of the primitive axiom ... of identity.


Kant often assumes that the analytic content of a sum concept is given just by the standard concatenation of its two number concepts: that is the content “through which it is not at all thought what this single number is which comprehends the two” (B 15)’’ It seems, then, that in Kant’s mind, a definition like ‘1+1=2’ is synthetic not because it involves addition as opposed to concept concatenation, but because of something about the conrents that are combined. I believe Kant viewed it this way: if we are to arrive at 2 from 1+1, we cannot take the occurrences of ‘1’ in ‘ l+ l ’ to stand for concepts. The two occurrences of ‘1’ are identical as to type, so the second adds no conceptual content to the first. (Combining the two can lead to something other than <1> itself only because ‘1’ stands for some other sort of representation, which, for Kant, will turn out to be intuition.) If this is right, it follows that in ‘1+1=2’ the terms on either side of the identity sign cannot have the same conceptual content- the content of ‘ l+ l ’ is given in terms of <1>, while 2 falls under <2>. Naturally, though, they must have the same extension; otherwise the equation would fail to hold. It then follows that conceptual content and extension here come apart, in violation of the requirements for analyticity.

There is telling evidence that Kant’s ultimate position on the non-analyticity of arithmetic was driven by just this sort of consideration. It emerges from a crucial letter to Schultz (25 Nov. 1788; Ak. 10: 554-8), in which Kant defended the syntheticity of arithmetic to his disciple, who had been about to deny it in print.5’ Kant’s main argument in the letter rests on the idea that “I can make a concept of just the same magnitude through various

50

51

526

For the less is that which is equal to a part of the other (the greater) ... . Hence there is an argument of this sort: the part is equal to a part of the whole (it is, of course, equal to itself through the axiom of identity...), and what is equal to a part of a whole is less than the whole (from the definition of ‘less’). Therefore, the part is less than the whole. [AG 311

See also A164/B 205, where Kant insists that ‘7+5=12 is not analytic because “I do not think 12 either in the representation of 7 nor in that of 5 nor in the representation of the combination of the two”: that is, the content of the sum concept is given by the concatenation, or combination, of <7> and <5>. (Kant’s next sentence makes it clear that ‘combination’ here is meant to be distinguished from d i t i o n . ) The same usage is especially clear in Kant’s letter to Schultz, discussed below, where he says, for example, that when considering the conceptual content of <3+4> or <12-5>, “I think 3 and 4... [or] 12 and 5” (Ak. 1 0 556; see note 53, below). Thanks to Michael Friedman for pressing me to account for this usage, and for discussion of the letter. The letter is thus an especially good source for assessing the ultimate motivations for Kant’s thesis of the syntheticity of arithmetic. Schultz had sent Kant a draft of his PriiJimg der Kantischen Kritik der reinen Vernunfr for comment, wherein Schultz claimed that arithmetic (though not geometry) was in fact analytic, contra Kant. In his return letter, Kant attempted to marshal his best arguments for his position, in the (eventually success- ful) effort to persuade his disciple to amend the Prufung, so as to advocate the official Kantian stance. (N.B.: I cite the pagination of the more recent (1922) edition of Ak. 10; readers using the older (1900) version will find this letter on pp. 528-31.)

R. LANIER ANDERSON

kinds of combination and separation” (Ak. 10: 555). Different “kinds of combination” have different conceptual content, Kant insists, but if arithmetic were analytic, such conceptually different routes to the same quantity should not be possible. Since we are treating equations, the subject and predicate must each be contained in the other, and thus be convertible, or “reciprocal concepts,” as in a definition (Ak. 10: 529). But in fact, Kant argues, even though the two terms in an equation are “objectively identical” (Ak. 10: 528), in the sense that they have the same object under them, they are not subjectively identical, in that they are thought through different conceptual marks contained in themselves. That is, the same extension falls under different conceptual contents, breaking the strong reciprocity of content and extension. For that very reason, the judgment is not analytic:

I can make a concept of just the same magnitude through various kinds of combination and separation ..., which [concept] is objectively identical (as in every equation), but is subjectively very different, according to the kind of combination that I think in order to attain the concept .... So, I can attain the determination of one and the same magnitude = 8, through 3+5, 12-4, 2*4, or 2’. But in my thought 3+5, the thought 2*4 was not contained at all; just as little, therefore, was the concept of 8 contained, which has the same value as both. [Ak. 10: 5551

In an analytic definition like ‘Bachelors are unmarried men,’ the predicate and subject have the same content- they contain the same conceptual marks. By contrast, ‘3+5=8’ is true not because <3+5> and <8> have the same content above them (they don’t), but because the two concepts “determine the same object” (viz., the magnitude 8), which falls under them.52 The necessary failure of reciprocity between content and extension becomes fully transparent in an equation like ‘3+5=2*4,’ where it is obviously wrong to attribute the same content to the concepts on either side.53 The two terms are not conceptually identical; yet the equation still holds, because the two syntheses named by the terms- characterized though they were through such different concep-

~~~~ ~~ ~~ ~

52 To determine a concept, for Kant, is always to transform it, qua general representation, into a more specific representation, by choosing some “route down” through a hierarchy articulating the various, specific ways of having that general concept. Thus, all concepts, qua general, are determinable, but only objects or intuitions are fully determined, for Kant (see A 571-UB 599-600). So when Kant here insists that the various concepts connected in an equation (e.g., <3+5>, <2*4>, <8>) “determine the same object,” that just means that they overlap in their extensions (i.e., what is contained under them). The two concepts involve different operations on different numbers, and so share no common content. Kant returns to the same idea in restating his key argument later on in the letter:

53

Supposing it [‘3+4=7’] were an analytic judgment, then I would have to rliink exactly the same thing by 3+4 as by 7, and the judgment would only make me more clearly conscious of my thought. Since now 12-5=7 yields a number = 7, by which I actually think just the same thing which 1 previously thought by 3+4, so ... when I think 3 and 4, I would at the same time think 12 and 5, which is contrary to consciousness. [Ak. 10: 5561


tual formulations- nonetheless generate the same magnitude: thus, “the judgment goes beyond the concept I have of the [first] synthesis, in that it substitutes another kind of synthesis ... in place of the first, which nonetheless always determines the object in just the same way” (Ak. 10: 555). Since the identity in such arithmetic formulae emerges only under the concepts, in the magnitude they apply to, the judgments in question cannot rest on reciprocal containment relations among the concepts.

The structure of this failure provides an indication of why intuition is able to achieve what mere concepts cannot in the domain of representing arithmetic. Analytic relations of concepts prevent logical extension and conceptual content from coming apart in the way Kant’s argument shows to be inevita- ble in arithmetic equations. To represent such truths, then, we cannot rely on the concepts alone, but must have some other, direct access to what falls under the concepts- i.e., to their extensions, understood now in a non-logical sense that goes beyond the concepts analytically contained under them.54 Intuition does this work for Kant, and the contrast with concepts can help us see how.

We saw that the containment relation could be taken to impose a successive order on the number concepts, if we took them as a genuslspecieslsubspecies series. Viewed as an arbitrary series of terms, the ordered concepts could even serve as a model for the numbers, in the same way the numerals can. But this does not yet express truths of arithmetic, because the number concepts do not explicitly represent their own composition out of units. By contrast, consider a series of terms based on a common distinguished unit, say a stroke, where the first member of the series is the unit, and each subsequent member is constructed by concatenation of an additional token of the unit with a copy of the previous member.” Now that we have an explicit representation of the units, a given stroke symbol from this series, say ‘1111111’, is represented as containing another, like ‘11111’- but in a completely different sense from that in which a concept like <mammal> contains another, like <animal>. <Mammal> contains <animal> as one of its marks; i.e., the mammals are one kind of the animals. By contrast, ‘lllllll’ contains ‘11111’ as a proper part. There is no sense in which a 1111111 is “one kind of’ the IIIIIs, or a “way of being a 11111.”

54.

55

528

For discussion of the difference between logical and non-logical extensions, see note 28 above. The contrast is helpfully treated by Longuenesse (1998, 50, 47, et passim), and is also noted by Sutherland (unpublished). Thus, the series would be of the form: 1, 11, 111, 1111 ,... This would be parallel to the understanding of the numeral series as generated by the iteration of the successor function-0, SO, SSO, SSSO ... . The ensuing discussion is generally indebted to the illuminating treatment in Parsons 1983, 110-49 (see esp. pp. 135-49), though I depart from Parsons in certain respects.

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Precisely the purely extensionally mereological form of their containment relations enables stroke symbols to represent arithmetic, where analytic containment cannot.56 For example, in considering ‘1111111 + 11111 = 111111111111.’ we can read ‘111111111111’ itself as composed, or constructed, out of a group of seven strokes and a group of five, according to the addition rule. The rule tells us to enter, in the symbol to the right of the identity sign, one stroke to match each stroke of each of the stroke symbols connected by ‘+,’ to the left of ‘=.’ On that reading of the symbol ‘111111111111’. to the right of the identity sign, each of its first seven strokes corresponds to one of the strokes from ‘1111111’, on the left, and each of its last five strokes corresponds to one from ‘11111’, on the left. Thus, the stroke symbols represent ‘7+5=12’ as true, when its subject concept is understood as “the unification of both numbers in a single one” @ 15). For note, the concepts <5> and c7> apply to exclusive and exhaustive proper parts of the stroke symbol ‘IIIIIIIIIIII’, and <12> applies to the whole. The constructive representation therefore expresses the proposition’s truth by showing how the conceptually different subject (<7+5>) and predicate (< 12>) each apply to the same thing contained under them both (the stroke symbol ‘111111111111’). There is complete overlap in their (non-logical) extensions, even though not in the differentiae contained in them. In this respect, again, ‘7+5=12’ violates the reciprocity of content and extension typical of analyticity, and thus conforms to the typical pattern of synthetic judgment.

For a positive explanation of the possibility of such judgments, Kant turns to syntheses that combine intuitions to construct the magnitude in question. What is crucial for non-analyticity itself, though, is just that the <7+5> synthesis and the <12> synthesis construct the magnitude in different

~~~ ~ ~

56 Here, of course, ‘extensional’ is meant in Kant’s non-logical (= our modem) sense, and not in the restricted logical sense of ‘extension’ Kant uses to characterize merely analytic relations of “containment under.” For discussion of some general limitations that extensional assumptions place on the representation of part/whole relations, see Simons 1982. Daniel Sutherland (unpublished, and also forthcoming) has offered a penetrating analysis of the connection between the strict homogeneity of mathematical intuition and its ability to express extensionally mereological relations of the needed sort. It is worth noting that the partlwhole relations set up by analytic containment are not extensional in the relevant sense. One way this emerges is that the associative law-which, as Frege (1980 [1884], 7-8) notes, is essential to Leibniz’s proof of ‘2+2=4’-fails for the relation of analytic concept concatenation. For example, in the domain of my closet, the concepts <long pieces of clothing with s t r ipes and <pieces of clothing with long stripes> are not equivalent. (The former concept covers a tie bearing a pattern of small boxes tilled by very short stripes, but the latter does not.) Concept concatenation is thus not order-indif- ferent (in ow example, the order in which <long>, <stripes>, and <clothing> are combined matters), and this is precisely because the concepts to be concatenated are not strictly homogeneous, in the way discussed by Sutherland, and so do not admit of extensional treatment. As Sutherland also notes, Kant was aware of something like this point: it serves as the basis for his argument that comparison of how much is contained in (or under) concepts is possible only for cases in which one concept is fully contained under another (see Logic Ak. 9: 103; R 3036, Ak. 16: 627).


ways (by adding five strokes to seven, and by counting strokes up to twelve). For that reason, the two syntheses must be represented by concepts with different content, despite their identity in extension (Ak. 10: 555-6). And that is why, to arrive at the arithmetic truth, “one must go beyond these concepts, seeking assistance in the intuition ..., one’s five fingers, say, or (as in Segner’s arithmetic) five points” (B 15), thereby discovering the identity of the underlying magnitude.57

4. Synthetic Truth and the Representation of Relations The argument of the previous section revealed three interconnected points, which demonstrate expressive limits that prevent explicit representation of elementary mathematical truths by means of analytic conceptual relations. First, the analytic hierarchy is one-dimensional; it represents only one kind of affirmative relation among concepts, whereas adequate representation of arithmetic truths depends on the ability to express orthogonal relations simultaneously. Second, the argument points toward the character of analytic relations among concepts, by contrast to the more complicated, synthetic relations involved in arithmetic. Analytic containment represents one thing as a specific “way of being” another; for instance, being a mammal is a way of being an animal. The one-dimensionality of that conceptual relationship allows it to be fully represented by the hierarchy. By contrast, in arithmetic we do not represent a sum concept like <7+5> as a way of being its sum, <12>, but as a “way of making” or generating it. As Michael Friedman (1992, 83-9) has shown, the operations we use to capture such relations are expressively stronger than anything that can be represented within the traditional logic. Indeed, it is precisely to capture this idea, that we need to represent two different kinds of relations to the sum concept <7+5>: <7> and <5> are inputs to the operation, and the magnitude 12 is its output. For Kant, the explicit representation of such “generating” comes via intuition, e.g., through the process of mathematical construction.

The first two points together rest on a third- the strong reciprocity between conceptual contents and logical extensions, which we have seen to be assumed by analytic containment. As I noted, the genudspecies character of

57 Thus, intuition can represent truths that analytic containment cannot. By contrast, diagrammatic systems can be devised which exploit spatial part/whole relations to express the analytic logical relations among concepts. The thought is simply to deploy partlwhole containment relations that are isomorphic to the logical containment relations among concepts. Along these lines, syllogistic inferences can be captured by the device of Venn or Euler diagram (see, e.g., Salmon 1989, Shin 1994). Such systems are based on the thought that for valid syllogism, the diagram of the premises contains or includes the diagram of the conclusion. As I point out in note 61 below, a diagrammatic representation of the analytic concept hierarchy would be significantly weaker than even the theory of the categorical syllogism, and thereby weaker than systems like Shin’s Venn-I. Thanks to Joel Friedman, John Perry, and Lisa Shabel for discussion.

530 R . LANIER ANDERSON

the “way of being” relation tracks this feature, and within the resulting framework, there is a unique way of building up each concept out of its component marks. In the case of the “way of making” relation, however, conceptual content and (now, non-logical) extension come apart, because there can be wholly different ways of generating the same thing, using different components. Thus, we saw Kant point out to Schultz that while <3+5> and <2*4> deploy different inputs making up their content, they nonetheless overlap in extension, since both apply to the same (output) magnitude, <8>.

The broad way-of-beingway-of-making distinction indicates that Kant’s underlying point has application well beyond his specific attack on Wolff. A wide variety of ordinary and scientific concepts fall into the “way of making” class, and the resulting asymmetry between content and extension, which undermines the possibility of analytic representation, can run in both direc- tions across the way-of-making relation. To stick with simple-minded exam- ples, I can make (what is in every important sense) the same cake in two different ways, e.g., by substituting baking soda and cream of tartar for the baking powder in the original recipe. Conversely, by varying my method of preparation, I can make two quite different sauces- a marchand de vin sauce, and a red wine variant of beurre blanc sauce, for example- from the same input (shallots, red wine, salt, pepper, butter, and herbs). The ubiquity of such “way of making” relations in our representational toolbox indicates that we can hope to capture the structure of experience only via a theory that includes essentially synthetic truths. In particular, Kant is keen to empha- size the important role played in any account of nature by causal judgments capturing the way of making, or producing, a given effect, and to hold them up as paradigmatically synthetic claims. Since we clearly cannot do without such judgments, the rationalist dream of an expressively adequate, but strictly analytic, system of metaphysics is doomed.

The expressive limitations of the analytic concept hierarchy, which Kant identified in his philosophy of mathematics, are connected to a widely recognized and obvious limitation in the apparatus of the traditional logic - viz., the lack of any explicit device for the representation of relations.” Traditional logic focused on categorical (‘S is P’) judgment, which attributes a one-place predicate to a subject. No special forms were introduced to capture relations among distinct objects, of the sort that become salient when we want to consider the relations of inputs and outputs. Instead, relational structures were simply compressed into complex one-place predicates; for instance, ‘Every saint is a friend of God’ would be analyzed as the attribution to saints of the

58 This particular limitation was famously emphasized by Russell, in connection with Kant, as well as with other early modem logicians (Russell 1903, 5 4 3 4 and 1920, 145). For a clear statement of the idea applied to Leibniz, see Russell 1937, 13-15. Friedman (1992, 80-1, and more generally, ch. 1) makes a similar point, though in a context more sympa- thetic to Kant’s insights into his contemporary mathematical practice.


concept <friend of God>, rather than as the expression of a relation (fiend- ship) between saints and Given a standard metaphysics of substances and attributes, this logical apparatus expresses exactly what is definitively real, viz., the possession of intrinsic (non-relational) attributes by substances. Indeed, in the tradition founded by Leibniz and Wolff, relations were often treated as merely an imperfect way of looking at a basic, underlying reality of intrinsic properties possessed by independent substances- a reality fully expressible in categorical ( ‘ S is P’) propositions.

The distinctively Wolffian commitment to the ideal of an expressively adequate analytic hierarchy makes this last metaphysical position unavoid- able, and exposes the significance of the lack of dedicated tools for representing relations. The hierarchy’s only device for representing a relation between individuals would be a concept contained in both relata, which, given the rules of division, could only be a common genus, or intrinsic property. Many relations cannot be handled in this fashion, because the rules of division inter- fere with the standard traditional strategy for compressing relations into one- place predicates. Consider a proposition like ‘Antoine Amauld, the great theo- logian, was the son of Antoine Amauld, the lawyer.’ On the standard approach, the judgment attributes the predicate <son of Arnauld, the lawyen to the individual subject concept <Arnauld, the great>. Note that predicate’s important logical features depend on its composition out of the relational concept <son of x> and the individual concept <Arnauld, the lawyer>: only that composition would have any hope of explaining why it should follow from our model proposition that <Arnauld, the great> also contains concepts like <brother of AngClique Amadd>, <grandson of Amauld, lord of La Mothe>, etc. Such composition, however, cannot be represented in the analytic hierarchy, for reasons parallel those we saw above. Just as <7> and <5> contribute to the content of <7+5> as inputs- not generic marks- so the individual concept <Amauld, the lawyen is an input toward, not a generic mark of, the complex concept <son of Amauld, the lawyer>. As we saw, the one-dimensional analytic hierarchy can only represent generic mark relations; it cannot simultaneously express the composition of complex concepts like <son of Amauld, the lawyer>, and their containment relations.@’ In the con-

59 The proposition is used in Port Royal as the major of an example syllogism: ‘Every saint is a friend of God Some saint is poor; therefore, Some friend of God is poor’ (see Arnauld and Nicole 1996 [1683], 164). Arnauld’s explanation of the syllogism demands the ‘S is P analysis of the major, as described in the text. According to him, the syllogism depends on the containment of the concept <friend of G o b in the concept csainb. The relation between saints and God never enters into the logic at all. In addition, as Leibniz already noted, individual concepts like <Arnauld, the lawyer> carry radically particular content, and consequently must be treated as infirnu species; they form the bottom level of any hierarchy, because there is no more specific thing of which such a concept could be the genus. It would therefore be impermissible in any case to introduce CArnauld, the lawye= higher up, as part of a generic concept, as we


text of the special restrictions proper to an analytic hierarchy, then, the monadic character of the traditional logic leads to expressive limitations that are quite severe indeed, extending even to expressive failures involving the very simplest relational concepts, like <son>, <grandson>, <brother>, and the like.

It should now be apparent that the logic whose limits Kant charts with the analytichynthetic distinction is very weak indeed. It is only a fragment even of the traditional logic, because the requirements imposed by the rules of division prevent the formation of certain inferences treated in syllogistic logic beyond the first figure, and in modes other than Barbara (e.g., those that include accidental predications, or exploit the form of the hypothetical syllogism).61 Nevertheless, this fragment of the traditional logic is of central theoretical interest, due to the important role of the concept hierarchy in the eighteenth century rationalist understanding of a proper metaphysical system. The expressive limitations identified in Kant’s philosophy of mathematics are devastating for the Wolffian program. After all, Wolff presented elementary mathematics as the paradigm case of strictly logicalkonceptual knowledge, with which the rest of science was supposed to be brought into line, as the true metaphysics was achieved. Kant shows that the Wolffian program fails even in the best possible case. It claims expressive power for the pure intel- lectual faculties, through the mere logic of concepts, which that logic cannot sustain.

Of course, Kant did not see the failure of Wolffian metaphysics as any ground for skepticism about the possibility an adequate system of philosophy. On the contrary, to his mind, the expressive limitations of the tradi-

would have to do, in order to use it to saturate <son of x>, yielding the generic concept <son of Arnauld, the lawyer>. Indeed, we can see Kant’s point about the lack of expressive power of the analytic hierarchy, along the lines of note 57 above, by contrasting it again to the case of diagrammatic reasoning. Consider, for example, the relative expressive power of different diagrammatic systems, equivalent to various fragments of first order logic. A diagrammatic system equivalent to the analytic hierarchy could use a system like the Euler circles, but would have rules (parallel to the rules of division) prohibiting the formation of partially overlapping extensions. So all circles would be either non-intersecting, or contained one within the other. For obvious reasons, this is a very restricted system. By contrast, Shin (1994, 41-110) shows that a system of Venn diagrams (her Venn-I), enhanced by Peirce’s system of x’s for representing the non-emptiness of sets, is a sound and complete logical system equivalent to the theory of the categorical syllogism. She introduces a special rule requiring padal overlap of the Venn circles (1994, 57-60) precisely in order to avoid expressive limitations proper to a system of the sort imagined for the analytic hierarchy. Even her Venn-I is seriously limited, however. In order to represent non-categorical syllogisms, she needed to add further syntax in order to express certain disjunc- tive information (e.g., disjunctions of facts about set emptiness), resulting in her Venn-11. Systems for the diagrammatic representation of mathematics, like Euclidean construction, are richer still, since they are not limited to the expressive power of the monadic predicate calculus at all, as shown by Friedman 1992, chs. 1-2. Thanks to Joel Friedman and John Peny for ideas contributing to this note.

6’

IT ADDS UP A R E R ALL 533

tional logic showed not that logic and reason are inadequate, but simply that much of our knowledge of the world is non-logical, or synthetic. The central task of the critical philosophy, therefore, must be to explain how such synthetic knowledge is possible.62 Kant’s core arguments in the Transcendental Analytic develop a theory of cognitive synthesis in the effort to answer that organizing question, and significantly, the procedure of mathematical construction, which explains synthetic results in the case where Kant first definitively demonstrates non-analytic knowledge, itself serves as an important guiding paradigm for the general theory of synthesis (see Anderson 2001).

6. Conclusion

A full discussion of synthesis must await another occasion, but we can already see how the positive account of the Critique of Pure Reason was intended to replace the dream of the Wolffian metaphysics- the dream of an analytic system of adequate concepts matching the essences God chose from among those resident in His intellect, and then actualized in creating the world. In Wolff s vision, such a system would not only have captured the truths about the world thereby created, but would simultaneously have illu- minated its distinctively rational structure: the hierarchical ordering of concepts would make their inferential relations transparent, and reveal every truth as a conceptual one. The Critique’s analytichynthetic distinction marks Kant’s awakening from the Wolffian dream, to the realization that our knowledge must have non-logical sources.63 The core insight was that the traditional logic of concepts lacks the expressive power even to represent key bodies of knowledge, like elementary mathematics, which are essential to any adequate theoretical system.

It can seem, however, that the particular shape of Kant’s distinction is of merely historical significance. No one any longer accepts that the fragment of the traditional logic captured by his notion of analyticity is logic simplic- iter, so his demonstration that mathematical truths are synthetic, in his sense, is not even sufficient to show that they are non-logical in our sense. And indeed, later philosophers have expanded the notion of analyticity to include logical truths (and other “truths by meaning”) much richer anything expressible through an analytic concept hierarchy.@

Just such expansions, though, threaten to open the door to Quinean criticisms that the boundaries of the analytic are no longer precise. There is a general trade-off here: the more expressive the analytic truths are allowed to

62 Kant insists repeatedly that this is the central organizing problem of the critical philosophy, on the solution of which “metaphysics stands of falls” (B 19). See also A 1 0 Prol. 278,377; etpassim. In the pre-critical period, of course, this Wolffan dream was Kant’s, as well. See especially the discussion at Ak. 2: 60-1. See Shin 1997,5-12, for illuminating discussion.

63

@


be, the more difficult it is to make the concept of analyticity precise. One instance of the point arose above, with the question whether strict rules of division are r e q u d for an analytic hierarchy. By relaxing the rules of division, more and richer judgments could be counted as analytic, but if divisions are not exhaustive and exclusive, the hierarchical representation itself does no work to specify relations of conceptual exclusion and containment. As a result, prior (and unconstrained) intuitive judgments about concept containment, or meaning inclusion, must conversely perform the function of shap- ing the hierarchy. Quinean doubts about the precision of analyticity then arise, for we lack clear criteria for a complete analysis, by which questions of concept containment could be settled. By contrast, as long as the rules of division are observed, even a partial hierarchy can specify analyticities. The hierarchy determines a containment or exclusion relation between any pair of its concepts- and even between its concepts and arbitrary further concepts, if we assume (with Wolff) that all analyticities would ultimately be expressed in a single hierar~hy.~’

Of course, the fragment of traditional logic that expresses analyticities in Kant’s sense is not the only logical system whose expressive resources can be precisely characterized, and identified as analytic. Many contemporary philosophers might count all of first order quantificational logic.% Some might accept more powerful systems of logic, set theory, and so on. Here again, however, we meet the trade-off: the more powerful the system, the more likely there will be controversy over whether such truths are really strictly conceptual in nature. Once seen in light of the traditional logic, Kant’s own definition of analyticity has the striking advantage, unexpected by so many of his twentieth century readers, that its characterization of the class of analyticities is perfectly clear and precise. The yield of that clarity is a compel-

6s If there is ultimately a single concept hierarchy, then any additional concept we might compare to a given partial hierarchy could occur only 1 ) as a genus above or species below it, in which case it will have definite relations of containment to the given concepts, or else 2) as a member belonging to a part of the total hierarchy that is excluded from the given partial one. By contrast, if there is not a single hierarchy, it might be impossible to specify an analytic relation between the concepts of a partial hierarchy, and another arbitrary concept. For example, perhaps different extended hierarchies could be “plugged in” under a particular genus “one at a time” (as it were), without any analytic relation between them being thereby specified; for example, the concept <human> might be alternately divided by <learned>/<unlearned>, etc., and then by <virtuous>/<vicious> and their subspecies. These are precisely the circumstances where, Kant would insist, we can represent only synfhetic connections between the concepts. That is, insofar as the uniqueness of the hierarchy is compromised, the scope of analyticity runs out. This note benefitted from a line of questions pressed on me by Agni- eszka Jaworska. Some would resist even this much. Hintikka suggests that existential instantiation, standard in first order logic, should not be counted as analytic, and he advocates the view not only as a matter of Kant scholarship, but as part of an effort to develop an appropriate conception of analyticity for current philosophical purposes. See Hintikka 1965 and 1968.

66


ling argument that mathematical truth is synthetic, in Kant’s understanding of the term.

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