Arbitrary combination and the use of signs in mathematics: Kant's 1763 Prize Essay and its Wolffian background

Abstract

In his 1763 Prize Essay, Kant is thought to endorse a version of formalism on which mathematical concepts need not apply to extramental objects. Against this reading, I argue that the Prize Essay has sufficient resources to explain how the objective reference of mathematical concepts is secured. This account of mathematical concepts’ objective reference employs material from Wolffian philosophy. On my reading, Kant's 1763 view still falls short of his Critical view in that it does not explain the universal, unconditional applicability of mathematical concepts.

Keywords::

• Kant's philosophy of mathematics
• Kant's pre-Critical philosophy
• Christian Freiherr von Wolff
• formalism in mathematics

Acknowledgements

For very helpful comments on drafts, I am grateful to Emily Carson, Lisa Shabel and participants in the workshop “Algebra, Magnitude, and Continuity” at the Max-Planck-Institut für Wissenschaftsgeschichte, Berlin, July 2014. I owe special thanks to Vincenzo De Risi for convening the workshop and for his comments. I am also indebted to Al Martinich for his generous assistance in translating Wolff's Latin texts.

Notes

^1. On the importance of Kant's conception of mathematical method, see Shabel (2006) and Hintikka (1965, 1967 cited as reprinted in Hintikka 1974). On his theory of definition, see Capozzi (1981) and Dunlop (2012). Carson (1999) is a thorough account of the continuities between the Prize Essay and the first Critique.

^2. Despite the excellent studies cited in this paper, in particular the work of H. W. Arndt. See also Sepper (2012).

^3. See Basso (2011) and Zac (1974).

^4. Descartes writes in the Sixth Meditation that “if I want to think of a chiliagon, although I understand that it is a figure consisting of a thousand sides just as well as I understand [a] triangle to be a three-sided figure, I do not imagine the thousand sides or see them as if they were present before me” as I do the three sides of the triangle (AT VII 72/CSM II 50, emphasis added). Descartes proceeds to argue that whatever “confused representation of some figure” I might happen to “construct in my mind” would not be a chiliagon, because it would “in no way [differ] from the representation I should form if I were thinking of a myriagon, or any figure with very many sides”, and would be “useless for recognizing the properties that distinguish a chiliagon from other polygons”.

^5. In the third volume of John Wallis's Opera Mathematica (Oxford: Sheldonian Theater), 620–622. Cited by Wolff ([1751] 1983, 180).

^6. Leibniz attributes the “confusion” of perceptions to their resulting from “impressions that the whole [sc. infinitely complex] universe makes upon us” in §13 of “Principles of Nature and Grace”, which was available to eighteenth-century readers (as well as in §33 of the “Discourse on Metaphysics”, which was not.)

^7. See the classic discussion of this passage by Cassirer (1953, vol. 1, 130–132). On the necessity of symbolization for thought, see also Pombo (1987, Ch. 2 of Pt. II).

^8. Wolff might be thought similarly to confine perceptual experience to non-certainty-yielding functions (in mathematical reasoning) when he distinguishes “mathematical” from “mechanical” demonstration. In contrast to mathematical demonstration, mechanical demonstration involves the use of sensory evidence to estimate the size of lines and angles (Shabel 2003, 98–101). Kant (if not Wolff himself) is clear that the latter method cannot yield certain knowledge; see Shabel (2003, 103–104). However, Wolff seems to draw this distinction only with regard to geometrical reasoning; as far as I can tell, it does not figure in his general account of mathematical method. Hence, in this broader context, the involvement of the non-rational faculties does not have to be understood in terms of mechanical demonstration.

^9. On Locke's view, demonstration consists in finding ideas to agree by a series of comparisons with “intermediate” ideas, such that at each stage two ideas are seen (“intuitively”) to agree, and the ideas comprising the proposition to be demonstrated are the first and last of the series. His example is that in order “to know the agreement or disagreement between the three angles of a triangle and two right ones”, the mind “is fain to find out some other angles, to which the three angles of a triangle have an equality; and, finding those equal to two right ones, comes to know their equality to two right ones” (IV.ii.3; cf. NE IV.ii, 367). Here, the comparison appears to be with respect to observed size (of, in a “particular” demonstration, the angles contained in particular drawn figures). This reading fits with Locke's speaking of “comparing and measuring” ideas of angles at IV.xviii.4, and of the perceptibility of equality or difference of size at IV.ii.10. However, at IV.xi.6 Locke distinguishes demonstrating from “examining [figures] by sight”, claiming that the latter gives “the evidence of our sight … a certainty approaching to that of demonstration itself”. This suggests that the “agreement” with respect to which ideas are compared (in demonstration) is a more abstract relation, perhaps one grasped by the rational faculty. (Leibniz does not comment specifically on this paragraph, but remarks, seemingly approvingly, with respect to IV.vi that the “connection” of sensible things that constitutes truth “depends on intellectual truths grounded in reason” (NE IV.xi, 444).)

^10. See also “Letter to Queen Sophie Charlotte of Prussia, On What is Independent of Sense and Matter”: “the mathematical sciences would not be demonstrative and would consist only in simple induction and observation if something higher, something that intelligence alone can provide, did not come to the aid of imagination and senses” (1989, 188). To my knowledge, this text was not available to Wolff or his contemporaries.

^11. A reason to discount the Nouveaux Essais, on the other hand, is given by Robert McRae, who suggests that “it is probably for tactical reasons that Leibniz, in making his own case against Locke, is silent about his own conception of the essential connection between mathematics and the imagination” (1995, 184).

^12. Giorgio Tonelli argues persuasively that in general, “the works of Leibniz published prior to 1765 almost exclusively represented the general metaphysical point of view” that knowledge originates inside the soul, at the expense of his epistemological and psychological views of how truths of various kinds are known (1974, 438). This left Wolff, in particular, free to hold that epistemologically and psychologically speaking, experience plays an important role in generating knowledge (445).

^13. Here Wolff is explaining how symbolic [figürlich] cognition can attain clarity and distinctness. He contrasts symbolic cognition, in which something is represented by means of “words or other signs”, with intuitive cognition, which is representation of a thing itself or an image thereof ([1751] 1983, §316, 173). The contrast makes it appear that “before the eyes” is not meant literally, for Wolff associates intuitive cognition with sensory representation (§319, 77; cf. Ungeheuer (1983, 93)). Moreover, Wolff claims that by means of the “symbolic art”, concepts “brought to mere signs” are thereby “completely abstracted from all images of the senses and imagination” (§324, 180). But Wolff also holds that by means of ars characteristica combinatoria, cognition can be “turned around” [convertitur] from symbolic to as-it-were [quasi] intuitive (Psychologia rationalis §312, 226). A natural way to understand this is that relations between things can be as it were seen in, or read off from, the formulae of a sufficiently apt system of signs (cf. Pimpinella (2001, 281–282). Clement Schwaiger holds that Wolff departs from Leibniz precisely by allowing that a whole can be grasped intuitively (as when we grasp by means of symbols how elements are related) even when we lack distinct cognition of its various parts (as when we do not know the symbols’ denotation) (2001, 1180).

^14. Wolff's fullest account of this procedure is in a minor work of 1709. See Arndt (1965).

^15. See Anderson (2005).

^16. I explain more fully how Wolff intends it to be useful in Dunlop, (m.s.).

^17. See the Mathematisches Lexicon entry “Numerus polygonus” ([1716] 1965, 958–960).

^18. For an account of how Leibniz represents predicates and their negations by positive and negative numbers, see Mittelstrass (1979, 609–610).

^19. The ars characteristica combinatoria was to consist of the “universal characteristic”, by means of which each primitive idea is designated by a symbol, and a combinatorial syntax (whose rules are powerful enough to generate all admissible strings).

^20. (1754) 1968, I.18, 131–132. I follow the anonymous eighteenth-century translation Logic, or Rational Thoughts on the Powers of the Human Understanding (London: for L. Hawes, W. Clarke, and R. Collins, 1770, reprint Hildesheim: Georg Olms, 2003). Cf. Tonelli (1959, 53–54).

^21. For instance, by Capozzi (2011, 317) and Sutherland (2010, §4).

^22. Shabel quotes an eighteenth-century translation of Wolff ([1742] 1968) in which negative quantities are described as “absurd”, “wanting reality” and “not real” (1998, 600 n. 17).

^23. Geometrical representation of negative numbers was treated as a problem in the eighteenth century (Boyer 1956, ch. 7), and graphical representation of complex numbers was introduced (by Gauss) only in the nineteenth century.

^24. The noted scholar Daniel Sutherland, for instance, writes (in an essay focusing on Kant's relationship to Wolff) that the ars combinatoria “is conspicuously absent” from Wolff's work (2010, 168).

^25. Charles Corr argues that because Wolff regards the ars characteristica combinatoria as part of a larger ars inveniendi, he “pays little attention to [it] as a theoretical system”. It is introduced “only to supplement the basic logic in the critical instance of scientific discovery” (1972, 333).

^26. As Engfer also observes (1982, 198).

^27. As in the essay “Primary Truths” (1989, 30–34) and the letter to Herman Conring of March 19, 1678 (1969, 186–187).

^28.Cf. Engfer (1983, 54).

^29. Kant's denial that a mathematical concept has any “significance” apart from that conferred by its definition (2:291) can be taken to express the same view. Kant typically uses “significance” [Bedeutung] to mean relation to an object (as at A239-41/B298-300), but it can also mean the object to which a representation relates.

^30. In the first Critique, Kant “grants and presupposes” the definition of truth as correspondence with an object (A58/B82; cf. A820/B848). But he explicitly calls this definition “nominal”, meaning that it does not reveal the underlying nature of its object. He can reasonably be supposed to have taken this attitude (even) in the pre-Critical period, when he was sympathetic to the Leibnizian view that truth consists in the determination of a predicate in a subject (1:392).

^31. Rechter brings out the striking force of Kant's claim that correct use of rules in a system of arithmetical notation suffices to ensure that cognition established by the method cannot be false (2006, 25).

^32. Parsons appears to endorse this reading when he says the Prize Essay “suggests” a view not “compatible” with Kant's critical position, namely that “operation with signs according to the rules, without attention to what they signify, is” itself sufficient to guarantee correctness (1969, 138).

^33. Thus, Paul Guyer's account of Kant's change in view appears oversimplified. On Guyer's reading, the Prize Essay's claim that in mathematics “the definitions are the first thought which I can have of the thing to be explained” [erklärt] (2:281) forestalls, or sets aside, questions about the “truth or correspondence” of concepts “constructed” by arbitrary synthesis (1991, 37). The “supposition of 1762 that the starting-points of mathematics are merely arbitrary constructions” is then “rejected” in the Critique (44). Yet, the claims that mathematical concepts are based on “arbitrary synthesis” and are “first given through” definitions recur at A729-731/B757-9.

^34.Cf. Weir (2011).

^35. Detlefsen calls such advocacy “perhaps the most distinctive component” of the formalist “framework” (2005, 237; cf. p. 263).

^36. As Rechter notes (2006, 25).

^37. Among Rechter's reasons for denying that “the contribution of [the] ‘sensible’ character” of mathematical signs involves their “perceptibility” is that Kant claims words “are not ‘sensible’ in their capacity as signs” (33). But I take Kant's point to be that the structure perceptible in a word does not correspond to that of the concept it signifies, in contrast to the mathematical case. See Section 2.4.

^38. Rechter further observes that Kant does not provide an account of the notion of cardinal number or its application, which would be needed to explain how the reference of signs is verified by counting numeral tokens.

^39. It must be kept in mind that by “number” Kant means the natural or “counting” numbers. See Sutherland (2006).

^40. A reason to doubt that Kant relates arithmetic to geometry in this way in the Prize Essay is that he does not in the Critical period. In notes written in 1790 (responding to attacks published in J.A. Eberhard's Philosophisches Magazin), Kant states explicitly that “when the geometer says that a line could always be extended no matter how far one has drawn it, then this does not mean what is said in arithmetic about numbers, namely that one can always, and endlessly, increase them through the addition of other units or numbers” (20:419–420, quoted in the translation of Onof and Schulting [2014]). But in the context of the Prize Essay (§2 of the First Reflection), Kant seems clearly to want to provide a unified account of the use of signs in both branches of mathematics.

^41. That we could not see the adequacy of what Rechter calls a “formal symbolic alternative to the representation of [philosophical] concepts in natural language” complements her point that we would need to already “have access to the distinct concept even in order to recover” or correct the defects of natural-language designations (2006, 28).

^42. Tonelli notes (1959, 66) that the view that philosophy studies qualities, which are infinite in number, was widely held among Kant's contemporaries.

^43. See Carson 1999 and Guyer (1991, 36–37).

^44. “The procedure” of “mathematical and geometrical construction” is that “by means of which I put together in a pure intuition, just as in an empirical one, the manifold that belongs to the schema of a triangle in general and thus to its concept” (A718/B746).

^45. See Dunlop (2012, §4).

^46. In (2012), I argue specifically that the close connection between the schemata of mathematical concepts and our precise knowledge of their content (as expressed by their definitions) shows that the concepts are not arbitrary in the strong sense (of Section 2.1), as many of Kant's remarks would suggest.

^47. I elaborate this contrast in my (2012, 94–96).

^48. Capozzi (2011, 13) and Engfer (1983, 50) also see this as the outstanding difficulty for the view of the Prize Essay.

^49.Cf. Parsons (2009, 165–167).