Abstract
This chapter returns to our narrative, explaining how all the projects of the young Descartes—physico-mathematics, universal mathematics and universal method—came to a climax and inflection point in the late 1620s. Working in the shadow of Marin Mersenne’s cultural battle against both radical scepticism and religiously heterodox natural philosophies, Descartes launched out, trying to realize his earlier dream of a methodologically sound ‘universal mathematics’. Riding on his physico-mathematical and more purely analytical mathematical results, and the confidence they fed into his dream of method, he worked himself into an intellectual dead end. This project, inscribed in the latter portions of his unfinished Rules for the Direction of the Mind, did not blossom into a magisterial work of method and universal mathematics. Rather, it collapsed in 1628, under its own weight of self-generating problems. From this point on, Descartes did not believe in his method, although he continued to exploit it for public presentation of his work. Descartes now entered upon a process of rapid change of direction of his intellectual agenda, and correlatively, his self-understanding and identity.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Between 1619 and 1625 Descartes travelled extensively, stopping in France briefly in the winter of 1622–1623. He settled in Paris in 1625 and remained there between trips to the countryside until late 1628, when he moved to the United Provinces and launched his projects in metaphysics and systematic corpuscular-mechanical natural philosophy.
- 2.
As we also know, some material in rule 8, dealing with the refraction of light and its law, can only date from 1626 or later; and as we shall see, the overall design of the text after the second paragraph of rule 8 also dates from 1626 or later. By late 1628 the entire enterprise had been abandoned for reasons we shall uncover in this chapter.
- 3.
Geometry, III,AT, VI, pp. 464–85. For a reconstruction of the path to the solution and its dating see Schuster (1977) 124–49.
- 4.
It is of course well known that in the years 1625–1628 Descartes associated with establishment literary figures like Guez de Balzac, an apologist for the Jesuits and fierce anti-sceptic and anti-stoic; religious apologists such as Silhon and Mersenne; and, with the neo-Augustinian fathers of the Oratory, including Gibieuf, Condren and, briefly, Cardinal Bérulle himself-the founder of the Order and chief figure in the French Counter–Reformation in that generation. See e.g. Adam, Vie de Descartes,AT, XII, 66–98; Sirven (1928) 313–37, Espinas (1906), Gadoffre (1961) ‘Introduction’, especially pp. xxff.
- 5.
Historians have frequently employed the term ‘crisis’ to deal with the period. Spink (1960) discerned the ‘crisis of 1619–1625’, which he saw primarily in institutional terms as a ‘repressive reaction on the part of the authorities, namely the Parlements’ against ‘libertinage’ and ‘free thought’. René Pintard, in his massive and now dated study of Le libertinage érudit(Pintard 1943), described ‘la crise de 1623–1625’, which was characterized as a breaking point when nascent scepticism and free thought began to meet increased resistance, mainly in the form of apologetical writings by thinkers as diverse as Mersenne, Garasse and Silhon. Thereafter, free thought became more a private affair of well-placed scholars such as Gassendi, La Mothe le Vayer and Naudé, rather than an aggressive public movement. Henri Gouhier (1954), starting with an assessment of the apologetical aims of Descartes’ metaphysics, described a ‘theological crisis’ in the ‘era’ of Descartes, fought out between Catholic proponents of ‘mystical’ and ‘positive’ theology. Popkin (1964) pointed to a generalized sceptical ‘crisis’ of the early seventeenth century, which came to a head in France in the 1620s and 1630s, and there elicited constructive new attempts at resolution, first in the form of Mersenne and Gassendi’s ‘mitigated scepticism’, and then in Descartes’ dogmatic metaphysics. A well-rounded account of the institutional and ideological conflicts of the period remains to be written to synthesize this set of largely single-factor accounts. For present purposes it need only be granted that there was a ‘common context’ of theological, political, ethical and epistemological turmoil in the period, and that different actors had differing perspectives on it. But, see more recently, Staquet (2009) and Torero-Ibad (2009).
- 6.
See, for example, Blanchet (1920), Gilson (1913), especially. Chapters IV, V; Gouhier (1924), especially. pp. 54–62 and Popkin (1964). One can, with Gouhier, stress the properly apologetical aims of the metaphysics, or argue that the main role of the metaphysics was to ground the mechanistic physics, which itself is to be seen as aimed at resolving the natural philosophical conflicts of the time. Alternatively, with Popkin one can stress a supposed sceptical core of the contemporary malaise and so identify the anti–sceptical thrust of the metaphysics as Descartes’ response to the situation. In any of these sub-theses, Descartes’ association with the Oratorians, with Bérulle, and with Silhon and Mersenne can take on special significance, in which one can stress the general apologetical aims of these figures: the special role of the Oratorians as proximate—if not sole or original—sources of Descartes’ neo-Augustinian leanings; the views of Mersenne on Voluntarist theology; and the anti-sceptical tenor of Silhon’s rationalistic apologetic. My thesis here is that no matter what precise position one takes on the apologetical role of the metaphysics and its intellectual sources, one crucial determinant of its aims, problems and content is the very failure of the ideological charged project of the later Regulae.
- 7.
On Mersenne see the immensely significant Lenoble (1943), which locates Mersenne’s interests as not simply scientific and anti-sceptical, but as apologetical in the sense of seeking the scientific refutation of apparently unorthodox natural philosophies, especially those of neo-Platonic, ‘Naturalist’ or Rosicrucian inspiration. Also, most importantly in the Anglophone literature, see Dear (1988), Popkin (1964) and Hine (1967).
- 8.
Rule 8, AT, X, p. 392 1.14 to p. 393 1.21.
- 9.
See above Sects. 5.6 and 5.7.
- 10.
Cf. Weber (1964) 88–103.
- 11.
Rule 8, AT, X, p. 393 1.22 to p. 396 1.25. Apart from illustrating the overall use of the method, the ‘cover story’ exemplifies in particular two points made at the beginning of rule 8: [1] Do not proceed where deduction cannot take one, as the pure mathematician can proceed only so far in the search for the law of refraction without physical premises about matter and cause. [2] Learn to know when your ‘enumerations’ need to be ‘complete’ and when merely ‘sufficient’, as in enumerating the types of ‘natural power’. On enumeration see above Sect. 5.6.
- 12.
See Sect. 4.9 for the several other issues about the path of discovery that Descartes’ method tale seeks to hide.
- 13.
The CSM translation (p.29) renders this as ‘the finest example of all’. I prefer here for dramatic effect the Haldane and Ross expression ‘most splendid example of all’, to underscore the import of this phrase, and shift in the text. Brunschwig in the Alquié edition (p.118) renders this as ‘l’exemple de tous le plus éclatant…’ See below Note 18 for a further preference for HR over CSM over another significant line in this part of the rule.
- 14.
Rule 8, AT, X, pp. 395–6; CSM p.30; HR, I, pp. 24–5.
- 15.
The second and third formulations occur at p. 396 1.26 to p. 397 1.3 and p. 397 1.26 to p. 398 1.5.
- 16.
Rule 8, AT, X, p. 398 1.10–25. See also Descartes’ enumeration of only three faculties of mind in the first setting (AT, X, pp. 395–6) and his enumeration of four faculties in the third setting of the example (AT, X, pp. 398–9).
- 17.
Rule 8, AT, X, p. 398 1.26 to p. 399 1.21. First Descartes promises to deal with the question of the faculties of mind in the succeeding proposition, but no such discussion occurs in rule 9. He proceeds to discuss the objects of knowledge on p. 399 but then pulls up short and introduces the second plan for a work in 36 rules divided into three books or 12 rules each. It is incorrect to think, as is widely held, that rules 13–24 were meant to pertain to mathematics and rules 25–36 to physics, or that the former were to pertain to ‘synthesis’ and the latter to ‘analysis’. Problems of physics occur in ‘book two’, provided they are ‘fully determinate’ and ‘book three’ could contain mathematical, as well as physical, problems in which the relevant terms and data have to be elicited from a larger body of raw material. In addition, as we shall see, the entire thrust of the procedure of universal mathematics is to reduce problems to solution in algebraic form, in which an ‘analysis’ takes on a deductive character. On ‘determinate’ sorts of problems see the Sect. 7.4below, on rules 14–18 and also AT, X, pp. 429–30.
- 18.
Rule 8, AT, X, p. 397 1.27 to p. 398 1.5; CSM p.31; HR, I, p. 26. Here again the HR translation is preferred, but not for any significant reason in the opening two sentences which are essentially similar in CSM. The issue is with the last sentence, which CSM render as ‘This is a task which everyone with the slightest love of the truth ought to undertake at least once in his life, since the true instruments of knowledge and the entire method are involved in the investigation of the problem.’ [p.31] (emphasis added) This loses the sense that the instruments of knowledge and method are going to be progressively and iteratively uncovered by means of pursuing this, and later inquiries, a significant, and I submit misleading watering down of Descartes’ statement. Brunschwig in the Alquié edition renders the last phrase as ‘…parce que c’est dans cette enquête que se trouvent les véritables outils du savoir, et la méthode tout entière.’ (p.120) The Marion (1977) translation of the Regulae, in contrast, has ‘parce que cette recherche contient les vrais instruments du savoir et la méthode toute entière’ (p.30). Now, the Latin verb in question is ‘continentur’amongst whose main meanings can be not only ‘involved in’ but ‘depend upon’ (according, for example, to several Ciceronian usages). Hence, again, taking the full contest into account we can construe Descartes intended message to be that, [ultimately uncovering] the true instruments of knowledge and whole method of inquiry depend upon [undertaking this investigation at least once in one’s life].
- 19.
Rule 7, AT, X, p. 392; CSM pp.27–8; HR, 1, p. 22.
- 20.
Between the second and third statements of the ‘example’ (AT, X, p. 397) Descartes introduces a telling metaphor in which the development of the method beyond the early rules, but on their basis, is likened to the origin of practical arts, in which first the tools themselves must be fashioned in a rough form before the art is practiced and perfected and its fruits produced. This might, in a different context, be read simply as one of Descartes’ broad and empty claims that the method consists ‘mainly in practice’. But coming here it indicates a consciousness of the fact that his project of method (read ‘methodologically grounded universal mathematics’) is going to be vastly deepened and widely articulated.
- 21.
It might be useful here to note the cash value of our proposed dating of the earlier rules (in Chap. 5) in the light of these findings about rule 8. One can now see that even if that dating proves untenable, the overall thesis of a change in aim and content of the text in rule 8 can be maintained, and the change can be dated from around 1626. One could even assume that rules 4A, 4B and 1–11 (excluding parts of 8 and with the caveat about the material following those passages in rule 8, given above in Note 2) were composed in Paris before 1626 or 1627. It would still be the case that universal mathematics was very likely first developed in 1619 and that universal method was even more probably initially worked out in the winter of 1619/1620. Rules 4B and 4A would still reflect at a distance the character of these projects. Moreover one could still demonstrate in rule 8, and then in rules 12–21, the very shift in aim and content which we have uncovered.
- 22.
The vis cognoscens willbe identified with the ‘understanding’ (intellectus)as used both earlier and later in the text. Context always indicates whether the term is used to denote the one spiritual faculty attending to purely intellectual matters (Descartes’ technical definition of the understanding), or whether it is attending to corporeal patterns in the brain.
- 23.
Rule 12, AT, X, pp. 415–16: ‘Atque una et eadem est vis, quae, si applicet se cum imaginatione ad sensum communem, dicitur videre, tangere etc; si ad imaginatione solam ut diversis figuris indutam, dicitur reminisci; si ad eamdem ut novas fingat, dicitur imaginari vel concipere ….’ (HR, I, p. 39; CSM, p. 42).
- 24.
Norman Kemp Smith (1952), 51–2, writes that what Descartes offers in a ‘quite unqualified way’ is: ‘an empirical realist view of the data available to the mind. The only ‘objects’ which he allows to the mind—all of them directly apprehended—are obtained, he [Descartes] holds, from one or other of two sources. (1) The self is aware of itself as thinking, i.e. as doubting, affirming, desiring etc …. (2) The self … is no less aware of the physical patterns which external objects, by way of their action on the bodily sense organs, imprint on the brain ....’ Cf. O’Neil (1967).
- 25.
See text cited in Note 23.
- 26.
In his mature metaphysics Descartes explicitly rejects the metaphor of the spiritual ‘helmsman’ in the ‘ship’ of the body, and that is a measure of the changes which overtake his epistemology in the wake of the difficulties created by the doctrine in the Regulae(see the discussion on the problems of perception in the Regulae, below Sect. 7.6.2) and also Discourse on Method, V,AT, VI, p. 59 (HR, I, p. 118); Sixth Meditation,AT, VII, p. 81 (HR, I, p. 192).
- 27.
Regulae,Rule 12, AT, X, p. 414; CSM pp.41–42; HR, I, p. 38.
- 28.
Ibid.
- 29.
Rule 12,AT, X, p. 414 1.17;cf. rule 14, p. 441 1.10 to 13:‘sequitur ex dictis ad regulam duodecimam, ubi phantasiam ipsam cum ideis in illa existentibus, nihil aliud esse concepimus, quam verum corpus reale extensum et figuratum’; Rule 14, p. 4501.10 to 11: ‘Quod attinet ad figuras, iam supra ostensum est, quomodo per illas solas return omnium ideae fingi possint …’ For further citations see Jean-Luc Marion’s translation of the Regulae, pp. 231–2. (Marion 1977).
- 30.
Ibid. AT, X, pp. 412–13; CSM pp.40–41;HR, 1, pp. 36–7.
- 31.
Ibid.AT, X, p. 414; CSM, p.41; HR, 1, pp. 37–8.
- 32.
Ibid.AT, X, p. 415 1.16 to 24; CSM p. 42; HR I, pp.38–39
- 33.
Dioptrique 1,AT, VI, pp. 83–6.
- 34.
Descartes does indeed introduce this material in a seemingly hypothetical tone. There is not space, he contends, to present all the material upon which the truth of the account depends; one need not believe ‘the facts are so’ unless one prefers to. Yet, despite the hypothetical tone, he also insists that his suppositions ‘do no harm to the truth’, that they ‘promote his purpose’ and that they ‘render the truth more clear’. He has already stated that the wax and seal offers an exact model for the impression of patterns onsense organs and their transmission through the nerves to the common sense and thence to the imagination. He also deploys the pen-analogy which derives from his seriously held mechanistic theory of light. Furthermore, he clearly implies that valid reasons could be advanced for the more detailed mechanical theories upon which the wax and seal model and pen-analogy trade (rule 12, AT, X, pp. 411–12; CSM p. 40; HR, I, p. 36). It seems likely, therefore, that Descartes wished the explicit physiological and psychological account to be taken as true in its main lines. Descartes also takes a decidedly hypothetical tone in introducing the idea that colors should be represented as figures and the difference between them taken as differences between figures (AT, X, p. 413; CSM p .41; HR, I, p. 37)This does not necessarily mean that the theory of mechanical sense impression is hypothetical per se,but only that any particular claim about the correlation of certain figures with certain colors must at present be conjectural.
- 35.
Working out the details in the Treatise of Mana few years later with the aid of some practical anatomical experience, Descartes devised a complicated mechanical account of the sensory and motor aspects of nervous function. But sense impression still depended upon the instantaneous passage of a mechanical impulse, now conceived to be conveyed along continuous filaments running in the centers of the nerves from sense organs (and sites of internal sensory excitation) to the central brain locus surrounding the pineal gland (AT, XI, pp. 141–6, 151–8).
- 36.
Rule 12, AT, X, p. 413; CSM p. 41; HR, I, p. 37.
- 37.
See the discussion of rules 14–18 in the following Section.
- 38.
- 39.
Kepler, Ad Vitelionem, (Kepler, 1938ff) Vol II, Chap. 1, Propositions I-V, XV, XVI.
- 40.
Ibid. Chapter V, Section 2.
- 41.
Dioptrique, V, AT, VI, pp. 114–29; Treatise of Man, AT, XI, pp. 133–4, 142–6, 151–60, 170–88.
- 42.
Acting in his ‘Beeckmanian’ style of mechanizing Kepler’s speculations, Descartes may have meditated about a mechanical theory of vision in 1620, upon reading Kepler’s optics. His notes from the time (AT, X, p. 243) contain some remarks on image formation which very plausibly derive from Kepler’s theory of vision and the new theory of image formation it entailed: a matter he clearly he did not keep in mind, or wish to be reminded about in the course of his discovery of the cosecant law of refraction! Alternatively, the mechanistic theory of vision may have awaited the discovery of the law of refraction and the formulation of a more precise covering mechanical theory of the action of light. In a sense the best evidence for Descartes’ possession of the theory in 1626–1628 is its implied role in the later Regulae.
- 43.
See above Note 34.
- 44.
Regulae,Rule 12, AT, X, p. 418 1.7 to 10; CSM p. 44; HR, I, p. 40. On the absence in the Regulaeof any of the specifically Cartesian metaphysical theses see, for example, Alquié (1950), 71 ff. and Gäbe (1972), 54 and passim.The present study differs from Alquié on the issue of just when Descartes’ characteristic mature metaphysical theses began to be developed, and it differs from Gäbe on the reasons for the abandonment of the project of the Regulae.
- 45.
Regulae,Rule 12, AT, X, p. 423 1.1 to 5, 13 to 16; CSM p.47; HR, I, p. 44.
- 46.
Ibid.AT, X, p. 423 1.1 to 7; CSM p.47; HR, I, p. 44. emphasis added
- 47.
Ibid. AT, X, p.423 1.13 to 20; CSM, p.47; HR, I, p. 44. Emphasis added. The translation combines elements of both CSM and HR.
- 48.
It is worth noting the ‘Mersenne-like’ and ‘Mersenne-transcending’ aspects of this position. That we know desperately little of the outside world for certain echoes Mersenne; what goes far beyond Mersenne’s proposals is that according to Descartes we have more than a piecemeal collection of reliable bits of knowledge, because we have a procedurally coherent, general discipline, universal mathematics, providing physico-mathematical knowledge of nature (as well as grounding the objects and procedures of all of mathematics).
- 49.
Regulae,Rule 14, AT, X, pp. 439–40; CSM p.57; HR, I, p. 55.
- 50.
Ibid.AT, X, pp. 440–1; CSM p.58; HR, I, p. 56.
- 51.
Boutroux (1900), 32, seems to have been the first to notice the ontological import of this passage. However, he did not see the justificatory aim, but rather stressed Descartes’ falling back on the use of imagination after an attempt to found a purely intellectual universal mathematics (p. 25). There is no evidence for this. On the interpretive conflations involved see below Sect. 7.7and the accompanying notes.
- 52.
Regulae,Rule 14, AT, X, p. 450; CSM, p.64 (HR, I, p. 63). The HR translation of ‘numerical assemblages’ is preferred to CSM’s ‘sets’.
- 53.
Ibid.AT, X, p. 452; CSM p.65; HR, I, p. 65. Descartes represents some discontinuous quantities in rule 15, but by rule 18 allmathematical operations are being carried out upon lines and rectangles, just as p. 452 1.22 to 26 suggests.
- 54.
Ibid.AT, X, p. 452; CSM, p.65; HR, I, pp. 64–5.
- 55.
Ibid.
- 56.
Ibid.
- 57.
Ibid.AT, X, pp. 447–8; CSM, p.62; HR, I, p. 61.
- 58.
AT X p. 448; CSM 62-63; HR I 61. Descartes may have had in mind dimensions measured in respect of conventionally selected co-ordinate frames in the solution of geometrical construction problems. See Geometry,I,AT, VI, pp. 382–3, 372.
- 59.
‘Unit’ is simply the element through which a given ‘dimension’ is measured. If a unit is not given for a sort of dimension involved in a problem, Descartes is perfectly willing to have the unit represented by any arbitrarily chosen magnitude of that type. Hence, he allows for units applicable to each type of figure which might be employed in a problem, whether, for example, collections of points, rectangular figures or straight lines, whose units would be a point, square or unit length respectively. Regulae,Rule 14, AT, X, pp. 449–50; CSM, pp.63–64; HR, 1, p. 63.
- 60.
Descartes gives the example of a triangle to be analyzed in terms of its ‘dimensions’, … ut in triangulo, si illud perfecte velimus dimetiri tria [dimensiones] a parte rei noscenda sunt, nempe vel tria latera, vel duo latera et unus angulus, vel duo anguli et area, etc; item in trapezio quinque, sex in tetraëdro, etc; quae omnia dici possunt dimensiones’ (AT, X, p. 449; CSM, p.63; HR, I, p. 62). See the treatment in terms of ‘simple natures’ in rule 12 prior to the transformation of his old ‘methodological’ terminology into the technical language of the new universal mathematics, rule 12 (AT, X, p. 422; CSM, p.46; HR, 1, p. 43).
- 61.
Regulae, Rule 13, AT, X, p. 431; HR, I, pp. 49–50. The HR translation is preferred for the following reason: CSM may be misleading as to Descartes’ intention and state of knowledge here, with their translation of the conditions on string C being ‘C is twice as long as A, though not so thick, and is tensioned by a weight four times as heavy.’ [emphasis added, p. 52] Surely Descartes idiomatic Latin was not meant to be conveying a mistake about Mersenne’s quite exact, and mathematically simple, results.
- 62.
For the time being we overlook the problem that according to Descartes’ account of perception in rule 12, sound also is delivered as a mechanical disturbance in the brain loci and is directly attended to by the vis cognoscens. See Sect. 7.6.2below.
- 63.
Lenoble (1943) 272–6, 313–17, 319–21.
- 64.
The general implication, not spelled out by Descartes, is that all relevant physical properties can somehow come to be expressed as geometrical extensions by means of sub-procedures constitutive of each of the ‘physico-mathematical’ fields subordinate to universal mathematics. So, expressed in terms of extensional measures, these properties can then become the objects of general analytical procedures, worked out in the corporeal imagination, according to rules given by the theory of equations.
- 65.
As we expect, given our interpretation of the universal mathematics of the later Regulaeas both articulating, and being constrained by, Descartes’ methodological ideas in the earlier strata of the text.
- 66.
It was Jacob Klein in his brilliant study (Klein 1968), who first attained the fundamental insight that Descartes was offering a mathematics expressed in and manipulated through line lengths functioning as operative symbols (pp. 198, 202, 208). Klein saw that in rule 14 Descartes was trying to ground his universal mathematics, a general science of proportions, in a symbolism consisting of real, concrete line lengths depicted in the corporeal imagination (pp. 197–8). Descartes wanted to realize, indeed materialize, abstract algebra in concrete, intuitively clear, objects and operations, and he wanted to show how a mathematical physics falls [actually a species of physico–mathematics, as we can now see] under the analytical procedures that algebra provides (p. 198). My only reservation with Klein’s reading arises from his tendency to say that Descartes intended the theory of mind and perception to give insight into the real structure of the world (p. 210). On my reading, Descartes is saying in ‘Mersenne-like’ fashion that we have access to certain aspects of the world, not that we have insight into the essential structure of it. To understand why Descartes later came to claim the latter through his metaphysically backed theory of matter-extension, one must comprehend the nature of the epistemological position in the Regulaeand the reasons for its demise (see Sect. 7.6below).
- 67.
Regulae,Rule 16, AT, X, pp. 454–9: the improved algebra aides memory by facilitating the recording of the results of the comparison and manipulation of magnitudes. All attention can then be directed to the comparison at hand. Second, the recording of the steps preserves the distinctions amongst the relevant quantities and reveals at a glance the operations performed upon them.
- 68.
Regulae,Rule 17, AT, X, pp. 459–60; CSM, p.70; HR, 1, pp. 70–1.
- 69.
Ibid.AT, X, p. 460; CSM, pp.70–1; HR, 1, pp. 71. Again as on occasion above, the HR translation is preferred to CSM, not on overall diction or accuracy, but for a matter of technical precision. CSM, whilst utilizing the word, ‘term’, earlier in the passage, before the portion we have quoted, nevertheless consistently render the above passage in terms of talk of extreme and intermediate ‘propositions’. In contrast HR keeps to the strongly implied technical mathematical context, and speaks of extreme and intermediate ‘terms’.
- 70.
Raising to a power and extracting a root are considered to be species of multiplication and division respectively. Difficulties arise from this in the case of root extractions. See Sect. 7.6.3below.
- 71.
Regulae, Rule 18, AT, X, pp. 464–5; CSM, p.73; HR, I, p. 73.
- 72.
Ibid.AT, X, pp. 465–6; CSM, p.74; HR, 1, pp. 74–5. One determines the line abby constructing a rectangle of area ab,one side of which is of unit length.
- 73.
Ibid. AT, X, pp. 466–7; CSM, p.75; HR, 1, pp. 75–6. The procedure as stated would assume the result is known beforehand. One can ‘reconstruct’ Descartes’ view of the operation as follows: represent the divisor by a line of length a; then normal to one end of a lay-off line b, the quotient, initially of unknown length. Box-off unit squares in the resulting rectangle until abunits, the dividend, has been obtained, thus specifying the actual length of b(and indicating any remainder). The complexity of this procedure weighs heavily against the notion that Descartes intended it as a practical aid to working calculations (see also below, Sect. 7.5): his aim was legitimatory.
- 74.
Ibid. AT, X, pp. 4678; CSM, pp.75–6; HR, I, p. 76. Descartes continues by asserting that these transformations between lines and rectangles can always be performed by geometers: ‘provided they recognize that whenever we compare lines with some rectangle, as here, we always conceive those lines as rectangles, one side of which is the length that we took to represent the unit. For if we do so the whole matter resolves itself into the following proposition: Given a rectangle to construct another rectangle equal to it upon a given side’. (AT, X, p. 468; CSM, p.76; HR, I, pp. 76–7.)
- 75.
- 76.
Noting again that in the physico-mathematical part of this universal mathematics, all inquiry remains on the macroscopic level of correlations of physical dimensions and problem solving about them. Strictly speaking, in view of these legitimatory ends and their supporting machinery, reduction to corpuscular–mechanical explanations is neither sought nor allowed, unlike the case in Descartes’ 1619 physico-mathematics, or in his recent optical work. As a result, problems lurk for the Regulaeproject, which we canvass in the next Section.
- 77.
- 78.
Geometry, I,AT, VI, p. 370: In Fig. 7.2BA is the unit; to multiply BD by BC, join points A and C and then draw DE parallel to AC, then BE is the product. For, by similar triangles:
(BC/BA) = (BE/BD); or BE = BC x BD. To divide BE by BD one reverses the process.
- 79.
He does this in rule 18, prior to introducing the logistic of extension symbols, AT, X, pp. 463–4; CSM, pp.72–3; HR, 1, pp. 72–3.
- 80.
See above Sect. 5.4.
- 81.
Descartes used his logistical machinery in solving a problem only once in his extant corpus of writings. This occurs in a report he gave to Beeckman in 1628 concerning his researches over the previous ten years (Algebrae Des Cartes Specimen Quoddam, AT, X, pp. 334–5). But in this case Descartes was illustrating the teaching of the Regulaerather than showing Beeckman how he ordinarily solved quadratic equations. Put bluntly, for the practicing mathematician familiar with the methods of arithmetic, algebra and geometry, the reconstruction of each step in terms of imaginative manipulation of straight lines and rectangles is heuristically otiose.
- 82.
Regulae, Rule 14, AT, X, pp. 442–5; CSM, pp.58–61; HR, I, pp. 57–60. This paragraph gives only the briefest sketch of this rich and significant material, in which Descartes displays a striking sarcasm toward the claims of the ‘naked intellect’ in these matters. See Schuster 1977, pp. 501–10.
- 83.
Ibid. AT, X, pp. 445–6; CSM, pp.60–61; HR, 1, pp. 59–60.
- 84.
Ibid. AT, X, pp. 446–7; CSM pp.61–2; HR, I, p. 60.
- 85.
Ibid. AT, X, p. 447; CSM, p.62; HR, I, p. 61. See Alquié (1950) 64. Once again the HR translation is preferred. CSM have ‘…our aim being to provide the easiest possible demonstration of such truth as may be found in arithmetic and geometry.’ (p.62)
- 86.
- 87.
Mersenne (1625) 226–7,and the tenor of the entire argument, in which Mersenne basically assumes and asserts that we are well advanced in acquiring a mathematical knowledge of appearances, regardless of sceptical doubts about the knowledge of essences.
- 88.
Of course concepts of natural philosophical provenance could be mathematicized (tending to the production of fields increasingly independent of the culture of natural philosophizing); and work in the traditional mixed mathematical fields could be read ‘physico–mathematically’ as bespeaking issues of matter and cause, hence of natural philosophical relevance. But, no sustained system of natural philosophy ever was, or could be, mathematical; that would be to mistake an instance of natural philosophizing for a bit of mathematical science.
- 89.
See Sect. 6.4 where we looked at this matter from the standpoint of ‘what, really, was involved when Descartes rendered corpuscular-mechanical explanations of phenomena such as magnetism—did his construction of corpuscular-mechanical discourse follow his method?’ Here we ask whether he could really treat magnetism under the precise protocols for the ‘science of dimensions’ outlined in the universal mathematics of the later Regulae.
- 90.
Regulae, Rule 12, AT, X, p. 427; Rule 13, AT, X, pp. 430–1.
- 91.
See above note 60, Rule 12, AT, X, p. 422; CSM p.46; HR, I, p. 43 and Rule 13 AT, X, p. 449; CSM, p.63; HR, I, p. 62.
- 92.
A ‘Whiggish’ ‘rational reconstruction’ of Descartes’ aims might suggest that he intended, in the light of his methodological ideal, to be a sort of Baroque Coulomb, applying measures of mathematically well-defined ‘force’ (determined through engineering applications of a science of mechanics) to fully determinate experimental conditions. This, of course, gets us nowhere, because Descartes could neither have conceived nor executed such a project, though it is precisely what the methodological ideal of universal mathematics demands in the strict sense. On Coulomb’s determination of the law governing the attraction and repulsion of electrostatic charges see Gillmor (1971) and King (1964).
- 93.
That is why we used this case in Chap. 6to illustrate the distance separating Descartes’ methodological story about magnet science from his actual discursive practices in constructing corpuscular-mechanical explanations of magnetism. In other words, there is a charitable reading of the relevant passages in the Regulae, according to which the ‘intermixture’ of simple natures might refer to a package of geometrico-mechanical properties to be ascribed to magnetic corpuscles, and that is how we proceeded back in Chap. 6: As he wrote out these passages Descartes may have intended such a gloss, for he may have been wearing his customary rosy-tinted methodological spectacles. This also clearly was part of Gerd Buchdahl’s incisive reading (Cf Sect. 6.4 Note 18 and text thereto) — a very persuasive reading, as well, provided one is not also factoring in a picture of the specific logistical and legitimatory machinery of the later Regulae, as we are here. Hence, on our own present strict reading of what the universal mathematics of the later Regulaeis about, we must conclude that its procedures will not really stretch so far, because the ‘dimensions’ then charitably in question would be neither observable, nor measurable, nor could the explanation take a properly mathematical form.
- 94.
For example, in the Discours, II,when he writes of the deductive inter-linking of all truths (AT, VI, p. 19; HR, I, p. 92) or, when he claims that parts of his physics were deduced from first principles (Discourse, V,AT, VI, p. 41, 63–4, HR, I, p. 106). Typical of such contexts are also: to Mersenne, 16 March 1640, AT, 111, p. 39; and Principles III,art 43 (but see the hypothetical tone of III, art 44), and IV, art 206 which makes strong deductivist claims but also wavers and waffles on their extent.
- 95.
See Chap. 6 note 19 for the consensus view on this matter in the literature. When dealing with this literature it is important not to slide into simply assuming that the young Descartes held the same sophisticated ‘probabilist’ position. That is, some may perhaps wish to say that Descartes already recognized in 1626–1628 the necessarily hypothetical status of his corpuscular models and was using the words ‘deduce’ and ‘intermixture’ in the loose sense (which he definitely adopted later) of ‘plausibly explain’. The problem with this is that there is no evidence for such an interpretation in the work of the younger Descartes in physico–mathematics, universal mathematics or method. Surely it is therefore preferable to say that Descartes only began to see the problem of the status and grounding of corpuscular-mechanical explanations in anything like his later fashion as a result of the inability of the later Regulaeto give him a physico-mathematical practice that was really mathematical and corpuscular-mechanical at the same time.
- 96.
See N. Kemp Smith, op. cit. pp.229–31.
- 97.
To Mersenne, 25 November 1630, AT, I, p. 182.
- 98.
Descartes had begun to compose Le Mondein the fall of 1629 (To Mersenne, 8 October 1629, AT, 1, p. 23; see To Mersenne 13 November 1629, AT, 1, p. 70). In November 1630 he termed the Dioptrique a ‘summary’ of Le Monde(ToMersenne, 25 November 1630, AT, I, p. 179), though neither text was yet ready.
- 99.
…nous remarquions qu’il est seulement question de savoir comment elles [patterns which are formed in the brain] peuvent donner moyen à l’ame de sentir toutes les diverses qualités des objets auxquels elles se rapportent, et non point comment elles ont en soi leur ressemblance.’ (AT, VI, p. 113) LLA translation p.113
- 100.
AT, VI, pp. 130–1, 137–40.
- 101.
Vous scavez bien que les paroles, n’ayant aucune ressemblance avec les choses qu’elles signifient, ne laissent pas de nous les faire concevoir, et souvent meme sans que nous prenions garde au son des mots, ni a leurs syllables … Or, si des mots, qui ne signifient rien que par I’institution des hommes, suffisent pour nous faire concevoir des choses, avec lesquelles ils n’ont aucune ressemblance: pourquoy la Nature ne pourra-t’elle pas aussi avoir estably certain signe, qui nous fasse avoir le sentiment de la Lumière, bien que ce signe n’ait rien en soi, qui soit semblable à ce sentiment? (AT, XI, p. 4.) MSM 3 (cf SG 3-4)
- 102.
For interesting articulation and reinforcement of the general tenor of the claims made in this section, compare Henk Bos’ informative recent study, ‘Descartes’ Attempt, in the Regulae, to base the certainty of algebra on mental vision—A Conjectural Reconstruction’ in Proceedings of the 13 th International Congress of Logic, Methodology and Philosophy of Science (Beijing August 9–15, 2007)forthcoming.
- 103.
Regulae,Rule 18, AT, X, p. 467; CSM p.75; HR, I, p. 76; HR translation preferred, emphasis added.
- 104.
Ibid.AT, X, pp. 467 1.17 to 468 1.6 (Division is mentioned explicitly in this passage).
- 105.
- 106.
Brunschvicg (1927) 292, held that Descartes’ metaphysics may be seen as an attempt to mediate between the increasingly divergent views of space which corresponded respectively to his newly extended abstract mathematics and to his mechanical corpuscular physics. As such, the metaphysics would have served to integrate the mathematics and physics on the justificatory plane, just as had been attempted in a more ‘scientific’, that is, natural philosophical, guise in the later Regulae.
References
Works of Descartes and Their Abbreviations
AT = Oeuvres de Descartes(revised edition, 12 vols.), edited by C. Adam and P. Tannery (Paris, 1964–76). References are by volume number (in roman) and page number (in Arabic).
SG = The World and Other Writings, edited and translated by Stephen Gaukroger (Cambridge,1998).
MM = René Descartes, The Principles of Philosophy, translated by V. R. Miller and R. P. Miller (Dordrecht, 1991)
MSM = Rene Descartes, Le Monde, ou Traité de la lumière, translated by Michael S. Mahoney (New York, 1979).
CSM(K) = The Philosophical Writings Of Descartes, 3 vols., translated by John Cottingham, Robert Stoothoff, and Dugald Murdoch, and (for vol. 3) Anthony Kenny, (Cambridge, 1988) References are by volume number (in roman) and page number (in arabic).
HR = The Philosophical Works of Descartes, vol I translated by E.S. Haldane and G.R.T. Ross (Cambridge, 1968 [1st ed. 1911])
Other
Alquié, F. 1950. La découverte métaphysique de l’homme chez Descartes. Paris: PUF.
Blanchet, L. 1920. Les antécédents historiques du ‘Je pense, donc je suis’. Paris: Felix Alcan.
Boutroux, P. 1900. L’imagination et les mathématiques selon Descartes. Paris: Felix Alcan.
Brunschvicg, L. 1922. Les Etapes de la Philosophie Mathématique,second edition. Paris: Felix Alcan.
Brunschvicg, L. 1927. Mathématique et métaphysique chez Descartes. Révue de Métaphysique et de Morale34: 277–324.
Clavelin, M. 1974. The natural philosophy of Galileo. Cambridge, Mass.: MIT.
Crombie, A.C. 1967. The mechanistic hypothesis and the scientific study of vision. In Historical aspects of microscopy, ed. S. Bradbury and G.E. Turner, 1–112. Cambridge: CUP.
Crombie, A.C. 1990. Science, optics and music in medieval and early modern thought. London: Hambledon.
Dear, Peter. 1988. Mersenne and the learning of the schools. Ithaca: Cornell University Press.
Espinas, A. 1906. Pour l’histoire du cartesianisme. Révue de Métaphysique et de Morale14: 265–293.
Gäbe, L. 1972. Descartes Selbstkritik. Untersuchungen zur Philosophie des Jungen Descartes. Hamburg: F. Meiner.
Gadoffre, G. 1961. Descartes’ Discours de la méthode, 2nd ed. Manchester: Manchester University Press.
Gaukroger, Stephen. 1978. Explanatory structures. Harvester: Hassocks.
Gillmor, C.S. 1971. Coulomb and the evolution of physics and engineering in eighteenth century France. Princeton: Princeton University Press.
Gilson, E. 1913. La liberté chez Descartes et la théologie. Paris: Alcan.
Gouhier, H. 1924. La pensée religieuse de Descartes. Paris: J. Vrin.
Gouhier. H. 1954. ‘La Crise de la théologie au temps de Descartes’, Révue de Théologie et de Philosophie,ser. 3, 4: 19–54.
Hine, W.L. 1967. The interrelationship of science and religion in the circle of Marin Mersenne.Unpublished Ph.D. dissertation. University of Oklahoma.
Kepler, Johannes. 1938ff. Gesammelte Werke, ed. M. Caspar. Munich: C.H. Beck.
King, W.J. 1964. The quantification of the concepts of electrical charge and electric current. The Natural Philosopher11: 107–125.
Klein, Jacob. 1968. Greek mathematical thought and the origin of algebra. Cambridge, MA: MIT Press.
Lenoble, Robert. 1943. Mersenne ou la naissance du mécanisme. Paris: J.Vrin.
Lindberg, D. 1976. Theories of vision from al-Kindi to Kepler.Chicago: University of Chicago Press.
Mahoney, M.S. 1971. Descartes. In Dictionary of scientific biography, vol. 4, ed. C.C. Gillespie, 56–57. New York: Scribners.
Marion, Jean-Luc. 1977. (ed. )and Trans. René Descartes, Regles utiles et claires pour la direction de l’esprit en la recherche de la verité.La Haye: Martinus Nijhoff.
Mersenne, Marin. 1625. La vérité des sciences. Paris: Toussainct du Bray.
Milhaud, Gaston. 1921. Descartes savant. Paris: Alcan.
O’Neil, B.E. 1967. Epistemological direct realism in Descartes philosophy. Unpublished Ph.D. dissertation. University of California, Berkeley.
Pintard, R. 1943. Le libertinage érudit,2 vols. Paris: Boivin.
Popkin, Richard. 1964. The History of Scepticism from Erasmus to Spinoza. Berkeley: University of California.
Schuster, J.A. 1977. Descartes and the scientific revolution 1618–34: An interpretation,2 vols. unpublished Ph.D. dissertation, Princeton University.
Sirven, J. 1928. Les années d’apprentissage de Descartes. Albi: Imprimerie Coopérative du Sud-Ouest.
Smith, Norman Kemp. 1952. New studies in the philosophy of Descartes. London: Macmillan.
Spink, J.S. 1960. French free thought from Gassendi to Voltaire.London: Athlone Press.
Staquet, Anne. 2009. Descartes et le libertinage. Paris: Hermann.
Straker, S. 1970. Kepler’s Optics. Unpublished Ph. D dissertation, Indiana University.
Torero-Ibad, Alexandra. 2009. Libertinage, science et philosophie dans le matérialisme de Cyrano de Bergerac. Paris: Honoré Champion.
Weber, J.-P. 1964. La constitution du texte des Regulae. Paris: Société d’édition d’enseignement supérieur.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media Dordrecht.
About this chapter
Cite this chapter
Schuster, J. (2012). Universal Mathematics Interruptus: The Program of the Later Regulaeand Its Collapse 1626–1628. In: Descartes-Agonistes. Studies in History and Philosophy of Science, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4746-3_7
Download citation
DOI: https://doi.org/10.1007/978-94-007-4746-3_7
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-4745-6
Online ISBN: 978-94-007-4746-3
eBook Packages: Humanities, Social Sciences and LawHistory (R0)