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Mind and Sign: Method and the Interpretation of Mathematics in Descartes's Early Work

Published online by Cambridge University Press:  01 January 2020

Amy M. Schmitter
Amy M. Schmitter*
Affiliation:
University of New Mexico, Albuquerque, NM87131-1151, USA
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Extract

Method may be second only to substance-dualism as the best-known among Descartes's enthusiasms. But knowing that Descartes wants to promote good method is one thing; knowing what exactly he wants to promote is another. Two views seem fairly widespread. The first rests on the claim that Descartes endorses a purely procedural picture of reason, so that right reasoning is a matter of proprieties of operation, rather than (say) respect for its objects. On this view, a method for regulating our reason would offer general rules of procedure, abstracted as much as possible from the content of particular problems. Second is the view that Descartes maintains what we might call an ‘intellectualist’ approach to method, one that restricts right reasoning to operations internal to the mind, and allows the use of external bodily resources (such as sensory perceptions, activities in the corporeal imagination, or written notes) only as initial inputs or as helpful props— convenient, but marginal to the procedure and readily eliminated from it.

Type
Research Article
Information
Canadian Journal of Philosophy , Volume 30 , Issue 3 , September 2000 , pp. 371 - 411
Copyright
Copyright © The Authors 2000

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References

1 This view is particularly pronounced in Taylor, Charles Philosophical Arguments (Cambridge, MA: Harvard University Press 1995); see esp. 5, 40, 63Google Scholar. There Taylor takes Descartes to be the instigator of ‘the’ modem epistemological tradition, which shows ‘a strong draw towards distinguishing and mapping the formal operations of our thinking’ (5-6). Taylor is somewhat unusual, however, in explicitly making the link between this approach of Descartes and contemporary preferences for computational models of thinking; see 6, 63.

2 I am tempted to call this a ‘partial notion of computability’ to distinguish it from the contemporary understanding of functions that are partially computable. But there is nothing partial in Descartes's notion, and as long as we bear in mind that this notion does not apply to particular sorts of functions, no confusion should result.

3 This is simplified version of the dating given in Weber, J.P. La constitution du texte des Regulae (Paris, France: Société d'édition d'enseignement supérieur 1964)Google Scholar, one suggested by Steven Gaukroger. For the full scheme he recommends, see Gaukroger, Descartes: An Intellectual Biography (Oxford, England: Clarendon Press 1995), 111–12Google Scholar. Pitte, F. Van De disputes the early dating of Rule 4B, ‘The Dating of Rule IV -B in Descartes's Regulae ad directionem ingenii,’ Journal of the History of Philosophy 29 (1991), 375–9Google Scholar. In contrast, John Cottingham simply says that the whole was written around 1628 or a bit before; see his ‘Translator's Preface’ in Cottingham, J. Stoothoff, R. and Murdoch, D. trans., The Philosophical Writings of Descartes, Vols. I and II (Cambridge, England: Cambridge University Press 1984), 7Google Scholar. For two quite different views, see Marion, Jean-Luc Sur l'Ontologie grise de Descartes: Science cartésienne et savoir aristotelicien dans les Regulae (Paris, France: J. Vrin 1975)Google Scholar, which proposes a ‘coherentist’ view; and Sepper, Dennis Descartes's Imagination: Proportion, Images, and the Activity of Thinking (Berkeley, CA: University of California Press 1996), 28-9, 37–8Google Scholar, which argues that it was composed in part in the early 1620s and abandoned perhaps as late as the early 1630s. Little in this discussion hangs on any precise dating of the parts, as long as we don't assume that terms are used consistently throughout, and acknowledge the change with Rule 12 - a point of widespread agreement.

4 Thomas applies this term to Descartes to describe ‘the (putative) subdiscipline within epistemology which attempts to construct a regulative, or prescriptive, theory of justification,’ or more generally, to provide’ agents with substantive advice about how they ought to conduct themselves as cognitive agents,’ see ‘Cartesian Epistemics and Descartes’ Regulae,’ History of Philosophy Quarterly 13 (1996),433-49. To tell the truth, I am skeptical that Descartes was ever much interested in most of the issues of epistemology proper, however much he was concerned with epistemics and a theory of error.

5 See Garber, Daniel Descartes’ Metaphysical Physics (Chicago, IL: University of Chicago Press 1992), 50Google Scholar. As Garber ably explains, the method that addresses ‘a series of discrete questions about the natural world’ is somewhat at odds with Descartes's announced project for the sciences, ibid., 31-44.

6 The general importance of method in this period is documented by Gaukroger, Descartes: An Intellectual Biography, 112–13Google Scholar.

7 AT VI 7, my translation. All quotations from Descartes's works are taken from Adam, C. and Tannery, P. eds. Oeuvres de Descartes (Paris, France: J. Vrin 1996)Google Scholar (henceforth AT); English translations not my own are from J. Cottingham, R. Stoothoff, and D. Murdoch (henceforth CSM).

8 This is syllogism of the form Barbara: ‘All A are B; All Bare C; therefore, All A are C.’ 9 See Gaukroger, Stephen ‘Descartes’ Project for a Mathematical Physics’ in Gaukroger, ed. Descartes: Philosophy, Mathematics and Physics (Sussex, England: Harvester Press 1980), 104–5Google Scholar, and Mancosu, Paolo Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (New York, NY: Oxford University Press 1996), 11Google Scholar.

10 The issue was first raised (at least in its Renaissance guise) in the 1547 work of Alessandro Piccolornini, Commentarium de Certitudine Mathematicarum Disciplinarum; see Mancosu, 12.

11 Mancosu describes the attempts of Giuseppe Biancani, among others, to do so; see 15-18. For instance, the mathematical use of definitions can answer to the notion of formal causation, while considerations of the relation of parts to wholes can provide material causes.

12 Ibid., 24-33. Most important for the Quaestio was the use of proofs by contradiction to prove geometrical converses of theorems, propositions which are logically equivalent. As such, little room is left for the priority of causes over effects, and the proof can appear entirely non-causal. Proofs by superposition were thought even more troubling, involving some sort of mechanical motion alien to demonstration.

13 This may have been the option exercised by many of Descartes's teachers at La Fleche; see Mancosu, 12-15, on the Jesuit teachers and texts who held this position.

14 Mancosu traces some attempts to do so; see especially chapters 1 and 2. The difficulties in reforming mathematical practices to meet the demands of scientific demonstration were indeed formidable.

15 As Gaukroger points out, the priority of the syllogistic form in general could raise issues of how new knowledge could be generated from purely deductive inference (Gaukroger, Descartes: An Intellectual Biography, 113)Google Scholar. Gaukroger further claims that the demand of proofs to reveal the process of discovery arose from a confusion of Aristotle's methods of presentation, the study of formally valid inference, with methods of discovery.

16 Cited in Mancosu, 63.

17 See AT X 376-7, CSM I 18-19; Descartes even suggests that the true method of the ancient mathematicians is ‘none other than the art that goes by the outlandish name of “algebra”—-or rather, a cleaned-up version of algebra.

18 Similar demands are made in the second set of Rules, where Descartes requires solving problems in a ‘direct’ way, although not according to Nardi's notion of ‘science.’ As my colleague Sergio Tenenbaum pointed out, Nardi's demand for direct proof foreshadows Kant's description of ‘the third rule peculiar to pure reason,’ namely that philosophic proofs be ‘direct or ostensive,’ combining ‘with the conviction of its truth insight into the sources of its truth;’ see B 817-8 in Smith, N. Kemp trans., Critique of Pure Reason (New York, NY: St. Martin's Press 1965), 625Google Scholar. For a discussion of Kant's views, see Mancosu, Chapter 4, Section 4.3.

19 Mancosu provides a great deal of material illustrating Cavalieri's use of ‘infinity’ in his procedure that might suggest that it could only be interpreted causally (i.e., in a material sense) at the cost of strong ontological commitments that would be anathema to Aristoteleans; see especially the debate with Guldin (Mancosu, 38-62). Also interesting is the issue of whether Cavalieri's nonfinitistic procedures could be supported by any available notion of syllogistic inference.

20 See Lachterman, David The Ethics of Geometry: a Genealogy of Modernity (New York, NY: Routledge, Chapman and Hall 1989), 149.Google Scholar

21 See Michael Mahoney, ‘The Beginnings of Algebraic Thought in the Seventeenth Century’ in Gaukroger, ed.

22 The importance of the notion of analysis can be seen in the title and first chapter of Vieta's Introduction to the Analytic Art, translated as an appendix in Klein, Jacob Greek Mathematical Thought and the Origin of Algebra, Brann, E. trans. (New York, NY: Dover Publications 1992), 315–53Google Scholar.

23 Cited in ibid., 259, n.217. Klein states that sixteenth century humanists, such as Vieta, ascribed the whole distinction to Theon.

24 Cited in ibid., 155.

25 Klein explains the distinction as resting on whether an analysis ‘is to be applied to the discovery of the proof of a “theorem” or to the solution (i.e., construction) of a “problem’” (155); see also Gaukroger, ‘The Nature of Abstract Reasoning: Philosophical Aspects of Descartes’ Work in Algebra’ in Cottingham, J. ed., The Cambridge Companion to Descartes (Cambridge, England: Cambridge University Press 1992), 106Google Scholar, and Gaukroger, Descartes: An Intellectual Biography, 125Google Scholar.

26 Cited in Klein, 156.

27 Cited in ibid., 260, n.218.

28 For Vieta's emphasis on analysis over synthesis, see ibid., esp. 268-9, n.235.

29 I again thank Sergio Tenenbaum for bringing this issue to my attention. In the ‘Second Replies’ to the Meditations, Descartes confuses the issue, by reversing the traditional understandings of a priori and a posteriori reasonings, attributing the former to analysis and the latter to synthesis (AT VII 155-6; CSM II 110-1). Lachterman, 158-9, has an interesting treatment of the passage, in which he suggests that Descartes may here be addressing causality in the sense of how fruitful each form of proof procedure might be-not the traditional sort of causality at all. And in the Rules, Descartes seems clearly to refuse to think of the distinction between analysis and synthesis in terms of any particular causal order.

30 This is a view expressed by everybody from the Rosicrucians (the goofballs) to such mathematicians as Vieta, see Gaukroger, Descartes: An Intellectual Biography, 112-13Google Scholar, and Mahoney, 147-51.

31 See especially Rule Seventeen, where Descartes declares that treating unknowns as if they were known ‘may enable us to adopt the easy and direct method of inquiry even in the most complicated of problems’ (AT X 460, CSM I 71).

32 One reason for the absence of many examples in the Rules may lie in the reputation Descartes had already achieved for remarkable success in solving problems; Descartes could rest on his laurels at that point. The unfinished, incomplete, and perhaps overly ambitious nature of the Rules may offer another explanation.

33 Gaukroger, Descartes: An Intellectual Biography, 99Google Scholar

34 Those who assimilate method to mathesis universalis include Sepper, 180, 192 and Foucault, Michel in The Order of Things (New York, NY: Vintage Books 1994), 52–6Google Scholar. Gaukroger warns against identifying the two terms (Gaukroger, Descartes: An Intellectual Biography, 106, 112)Google Scholar. Order is certainly important to the method, but mathesis universalis specifically takes order and measure as its object. But cf. Lachterman, 175. For general treatments of mathesis universalis, see Crapulli, Giovanni Mathesis Universalis: Genesi di un'Idea nel XI Secolo (Rome, Italy: Edizioni dell’ Ateneo 1969)Google Scholar, and Marion's, Jean-Luc translation of Descartes's Regles utiles et claires pour Ia direction de l'esprit en Ia recherche de Ia vérité (LaHaye: Nijhoff 1977), esp. n.31, 156-8, 302–9Google Scholar.

35 This compass and its use is described in the Cogitationes Privatae, dated January 1619 (AT X 234-242), although as Gaukroger points out, Descartes makes some careless mistakes there (Gaukroger, Descartes: An Intellectual Biography, 97)Google Scholar.

36 Claiming that the following solution procedure produces evidence will require appealing to our pre-analytic sense of evidence, not to the term of art found in early modern discussions of the nature of demonstration. Such an appeal is made by Lachterman when he treats Descartes's demonstration ad oculos (155). Another court of appeal might be provided by the psychologistic senses of evidence drawn from rhetorical and legal tradition, described by Gaukroger, Descartes: An Intellectual Biography, 121-4. My main evidence for the claim that Descartes's procedure provides evidence is inductive- the many ‘ahas’ the procedure elicits. But here the evidentiary proof is in the pudding: the reader is invited to supply her own inductive evidence.

37 That is, YB/YC = YC/YD = YD/YE ….

38 Descartes's procedure shares many of the desirable features illustrated by Gauss's proof for the sum of a simple arithmetical series, e.g., for 1 + 2 +, … n, S = n(n+1)/2. Gauss's proof shows that the sum is computable for any number n, but unlike proofs by induction, it thereby explains why the equation holds and illustrates how it was discovered - all without requiring any specialized knowledge. My thanks to Aladdin Yaqub and Sergio Tenenbaum for bringing this example of mathematical evidence to my attention.

39 Gaukroger, Descartes: An Intellectual Biography, 99Google Scholar

40 The view that different sorts of objects, different subject-areas, can nonetheless be treated by a single science and a single mental faculty is another touchstone throughout Descartes's career, and is clearly announced in Rule One of the Rules, see AT X 360, CSM I 9.

41 For a brief review of these claims as they relate to the issue of the priority of algebra vis-à-vis geometry, see Mancosu, 87.

42 In contrast, his claims in the Fifth Meditation that the objects of mathematics, particularly geometrical objects, provide the essence of material things suggest that he may there interpret magnitude geometrically.

43 For example, he seems to think that the same procedure could be used to solve an equation of the form xY= x+a, where a is a constant andy is any positive whole number whatsoever, not merely an odd number.

44 As I shall argue below, Descartes may well see limits to mechanical procedures of solution, but the ambition to ‘mechanize’ them as much as possible is still important.

45 On the other hand, in Rule 14, he may attack the validity of certain inferences performed on the abstractions of the ‘dialecticians.’

46 See AT X 406, CSM I 36. Gaukroger suggests that Descartes's Jesuit training had tended to associate logic with rhetoric, more particularly, to supplant the notion of validity with psychological persuasion, laying particular emphasis on the importance of ‘vivacity’ (Gaukroger, Descartes: An Intellectual Biography, 54, 119–24)Google Scholar. This offers another twist on how demonstration that merely forces assent lacks ‘evidence.’

47 See AT X 372; CSM I 16.

48 Sepper, 88, see also 90-1. Leslie J. Beck associates it with ‘esprit,’ and suggests that it can be used a little more widely than the ‘reason,’ which is naturally equal in all people. See Beck, The Method of Descartes: A Study of the ‘Regulae’ (Oxford, England: Clarendon Press 1952), 35.Google Scholar

49 See especially Rules 9 and 10, AT X 400-404, CSM I 33-35.

50 For the widespread emphasis on self-discipline, see Gaukroger, Descartes: An Intellectual Biography, 28-33, 42-3, 65–6.Google Scholar

51 Indeed, the problems with the formal models of the ‘dialecticians’ may come to much the same thing: as we shall see below, Descartes holds that these models can encourage careless inference, e.g., when false abstractions draw attention away from the contents on which we base our inference.

52 Deduction, as Gaukroger points out, is used very loosely at the time: proof, induction, justification or even just ‘narration’ of an argument count as deduction; see Gaukroger, Descartes: An Intellectual Biography, 115-16.Google Scholar

53 See AT X 389, CSM I 26.

54 Harold Joachim claims that ‘enumeration’ actually serves two purposes: one preparatory and another rather like the one I describe above, see Joachim, Descartes's Rules for the Direction of the Mind (Bristol, England: Thoemmes Press 1997), 59Google Scholar.

55 Here and elsewhere, Descartes's emphasis on proper order recalls some of the most common strategies of the ‘arts of memory,’ especially of the ‘logical’ variety introduced by Peter Ramus. But whereas these arts exploit techniques of ordering as an aid to recollection, Descartes treats them as a way to bypass reliance on memory. For discussions of the memory arts, see Yates, Frances The Art of Memory (Chicago: University of Chicago Press 1966), especially 2-3, 232–6Google Scholar.

56 This is an example given in Rule 6 and repeated in Rules 7 and 11. Rule 8 also considers simple inferences to our mental powers and faculties. Other examples of problems can be found in Rule 7, which remarks on some simple geometrical and word problems (finding an anagram of a name). To be sure, Descartes also treats rather more involved problems in the first part of the Rules, such as the finding of the anaclastic (Rule 8), or the investigation of the transmission of natural powers (Rule 9). Yet none of these is offered there as an example of either intuition or chain-like inference.

57 See especially Rules 3, 6, and 8.

58 This may be evidence that Rule Eight is indeed a pastiche composed at various times; see, e.g., Gaukroger, Descartes: An Intellectual Biography, 111-12, 152–3, 157-8, 434-5 n.18Google Scholar.

59 See AT X 418,420, CSM I 44, 45.

60 This would make us prone to what is for him a rather unnatural error: undue diffidence ‘when we think we are ignorant of things we really know.’ In contrast, we always have ‘complete knowledge’ of simple natures, although this does not mean that they are complete things. See AT X 420, CSM I 45.

61 Descartes may well think that for such directed attention to be focused it needs to heed something like the articulations a metaphysics could provide, but he lacks a metaphysical vocabulary to express articulations at the level he needs for the purposes of problem-solving. We might note, however, that the examples he gives of simple natures here are in large part the sorts of things he will later identify as modes: knowledge, doubt, ignorance, volition, on the one hand; shape and motion on the other hand.

62 I think that this can help explain what seems a sheer equivocation were Descartes's points to be interpreted metaphysically: first Descartes dismisses abstractions as candidates for truly simple natures, for a terms such as ‘limit’ does not apply univocally. But a mere paragraph later he talks about simple natures that are ‘common,’ in the sense that they ‘are ascribed indifferently, now to corporeal things, now to spirits’ (AT X 419, CSM I 44-5). The difference that makes a difference here seems to be the usefulness of isolating the object for the purposes of problem-solving. Mere abstractions produce pointlessly fine discriminations. See also Joachim, 90-3.

63 A case in point is Descartes's description of a ‘Tantalus cup,’ a cup designed to hold water until it reached the mouth of a figure depicting Tantalus, at which point it all runs out. Descartes warns us against attention to irrelevant considerations; the problem asks ‘how must the bowl be constructed if it lets out all the water as soon as, but not before, it reaches a fixed height.’ See AT X 435-6, CSM I 54-5.

64 This use of ‘idea,’ one of its first appearances in the Rules, takes ideas to be formations in the corporeal imagination, e.g., figures. This is a sense of ‘idea’ that Descartes uses up through The Treatise on Man; see AT XI 176, CSM I 106.

65 Gaukroger thinks that this description may have been motivated by Descartes's interest in a mechanistic physiology. See Gaukroger, Descartes: An Intellectual Biography, 167Google Scholar.

66 This is the exact inverse of holding that perceived color represents primary qualities.

67 See AT X 412-3, CSM I 40-41.

68 In this context, Descartes's caveat against assuming that we have faithful representations of external things, representations that exactly resemble them, makes perfect sense. Representation does not require perfect resemblance, as long as we pay due heed to how the representations operate. Descartes is not here making any metaphysical claims about the nature of external things or expressing skepticism about sense-perception.

69 The verb translated as ‘delineate’ above is actually’ designare,’ and the whole process compared to finding the relations holding among the meanings, or ‘significations,’ of words.

70 The Latin is telling: ‘the same [problem] is to be transferred to the real extension of bodies’ [eadem est ad extensionem realem corporum transferenda … ] (my emphasis).

71 Particularly important here is Descartes's statement that a ‘real idea’ can turn (or convert) the intellect to other features of the ‘subject,’ which it might otherwise overlook (AT X 445).

72 Innate ideas would help here, for they could provide necessary conditions for any magnitude that are not a matter of mere logical form, but need not be derived from things in quasi-empirical fashion. On this topic, see my ‘The Wax and I: Perceptibility and Modality in the Second Meditation,’ Archiv für Geschichte der Philosophie, forthcoming.

73 I use ‘sign’ in one of the senses suggested by Saussure, embracing both signifier and signified. A similar point is made by Klein, 197, also 3.

74 For instance, he refuses to give the geometric interpretation of 2a3 (saying that it indicates three relations only), and he prefers the use of constants to that of numbers in expressions of this sort because it keeps the relations straight. See AT X 455-6, CSM I 67-8.

75 See Gaukroger, ‘The Nature of Abstract Reasorung,’ 105Google Scholar.

76 I'd like to thank the audiences at the Midwest Seminar for the History of Early Modem Philosophy, November 1997 in East Lansing, MI, for their many helpful comments on an earlier version of this paper, and at the University of New Mexico's Philosophy Department Colloquium. Paolo Mancosu generously helped with a number of points, especially in sections II and III, and Roger Florka contributed insightful comments about simple natures. Two anonymous reviewers for this journal went above and beyond the call of duty in offering informative remarks and checking for errors. Extra special thanks are owed to Aladdin Yaqub and Sergio Tenenbaum, who went over this paper with the proverbial fine-toothed comb; their comments were models of intellectual clarity and their willingness to help a model of collegiality.