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The Dynamically Imaginative Cognition of Descartes

Part of the Studies in History and Philosophy of Science book series (AUST,volume 33)


What imagination is cannot be answered apart from understanding its conceptual topology, the articulated framework of basic phenomena and concepts that govern our thinking about it. The Platonic ontology of the good (as well as the forms) imaging itself in all levels of reality and the Aristotelian psychology based on the claim there is no thinking without images exercised their influence for nearly two millennia, though chiefly in distorted forms that undermined their conception of imagination as dynamic. The third step in our historical investigation examines the work of René Descartes (1596–1650), who was strongly influenced by this heritage in surprising ways (e.g., the practice of spiritual exercises). His early work developed a universal method for using figures and images, both mathematical and nonmathematical, for representing the degree of similarity of things to natural forms. On this basis he developed techniques for resolving all kinds of questions and problems. His invention of analytic geometry was the most rigorous form of using images ever conceived. Marked positions in geometric figures are symbolically incorporated into algebraic formulas; algebra, conceived as a matrix field, allows manipulations of elements that are mapped back exactly into the geometric field. The biplanar character of imagination first discovered by Plato—that things on one level of reality embody and can be used to display relations on another level, and vice versa—was thus incorporated into, and became the basis for, the dynamism of modern analytical mathematics.


  • Pineal Gland
  • Analytic Geometry
  • Mathematical Truth
  • Algebraic Formula
  • Primitive Notion

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Fig. 6.1


  1. 1.

    This is an ugly neologism, to be sure, but no more so than “mathematization.” The thesis that modern rationalist-mathematical science has “disenchanted” the world, but that the disenchantment can somehow be reversed by “going back” to imagination, is hollow. There is no golden age to go back to, and the proposed remedy is just another symptom of the failure to understand reason, imagination, and their relation.

  2. 2.

    References to Descartes’s works are given by volume and page number in the 11-volume Adam and Tannery edition, Descartes 1964–1976, indicated by the abbreviation “AT.” All translations are my own.

  3. 3.

    The work is lost.

  4. 4.

    See McCracken 1983.

  5. 5.

    On Plotinus’s double phantasia, see Brann 1991, 48–50, and Warren 1966. The central Plotinian discussions of conceptual imagination are at Enneads I.4.10 and IV.3.30, and of sensible imagination at IV.3.23.

  6. 6.

    One should neither overemphasize nor minimize advances in medical knowledge in antiquity. If Aristotle was one of the first to recognize the need to incorporate ­techniques of dissection into the study of living things, he and his school did not take great strides in this direction, but in the following centuries Alexandrian researchers brought a greater sophistication to this work.

  7. 7.

    Even though historical research has made the story far more interesting and complicated, it is still basically true to say that Descartes gave impetus to a fundamental historical shift in the learned use of the terms idea (French idée) and cogitatio (pensée). I shall say more about this below.

  8. 8.

    See, for example, Eco 2002.

  9. 9.

    Once again, we see in Aristotle and Aristotelians the understanding that sensation is organized between extremes in a relevant qualitative field. In On the Soul the organ of sense is called a mean at 423b27 and 424b1, and sensation is called a mean at 424a25 and 426a27. See the brief discussion in the note to the passage marked “423b27 ff.” in Aristotle 1993, 112.

  10. 10.

    For example, in a recent modern edition of the Compendium (a French translation alongside the original Latin), excluding tables of terms there are 21 figures in 34 pages of Latin text; see Descartes 1987.

  11. 11.

    It is this shift from duple to triple rhythm that suggests the synthetic process of imagining is projectively conjectural and self-correcting. Whether the meter is double or triple is fully determined only after one has heard more than the first three beats.

  12. 12.

    But one can also certainly argue that the seed of the idea could have been spurred by combining the biplanar eikastic imagination of Plato and the Aristotelian-scholastic conception of the preparation of complex phantasms by the internal senses.

  13. 13.

    Geneviève Rodis-Lewis, who has done more than anyone to disentangle the confusing documentation, argues that Descartes matriculated in 1607 and left in 1615. See Rodis-Lewis 1998, 8–10. La Flèche is on the Loire river, halfway between Angers and Le Mans, about 220 km southwest of Paris.

  14. 14.

    See Risse 1970, 2: 14–47.

  15. 15.

    Sometimes contuition was the term used, the etymology of which suggests gaining a fundamental insight by bringing many things together in a single view. In any case, the –tuition terms both implied an intensive intellectual recognition of the unity of what was presented to the mind in a series of more or less complex phantasms. On the Victorines, see Sepper 2000.

  16. 16.

    Thus John’s “dark night of the soul” was not, as it is sometimes presented, a crisis of existential anguish, but a state preparatory to the vision of God. Metaphorically speaking, it is achieved by shutting off all the “lights” or appearances in the soul, including those of reason. Just as the light of the sun obscures the stars, the busy activity of sensation, memory, imagination, and reason hinder the apprehension of the light of God. All other appearances in the soul are to be “shut down,” so that God can become “visible.”

  17. 17.

    For an overview, see Adams and Harper 1992.

  18. 18.

    Notebook C was included in the effects sent back to France after his death in February 1650 in Sweden. That the notebook survived at all was close to a miracle, since the ship carrying the effects sank to the bottom of the Seine, and the chest containing the effects had to be salvaged. Leibniz transcribed notes from it during a visit to Paris in 1676. It is not clear what proportion of the notebook’s contents was ultimately preserved, although scholars have been able to reconstruct how it must have been organized. See especially Gouhier 1958, 11–18.

  19. 19.

    The increasing length of the vertical side of the triangle stood for speed, the increasing length of the horizontal side for time, and the increasing area for distance traversed. Beeckman recorded Descartes’s faulty solution and then offered a correct one of his own. For a detailed account, see Shea 1991, 35–60.

  20. 20.

    This is not to say that Descartes was the modern pioneer of such devices. Many others, including Galileo, had preceded him in thinking about and even building some. But this is to say that the natural approach or “tack” of Descartes’s mathematical, problem-solving imagination took this kind of course.

  21. 21.

    See also the discussion in Sepper 1989, esp. 383–384.

  22. 22.

    The work was translated and first published in Dutch, in 1684; the publication date of the original Latin was 1701. The title was assigned the work by its late seventeenth-­century editors. Descartes began work on it possibly as early as 1619, though most scholars place the start date in the middle or late 1620s. He apparently abandoned it around 1629. The conventional translation of the title into English is Rules for the Direction of the Mind, which unfortunately does not at all convey the specificity of ingenium, which is in fact one of the work’s key terms. Scholars have suggested alternatives like “native wit” and “native intelligence”; perhaps even better is the etymological cognate “ingenuity.”

  23. 23.

    Granting that the title of the work was provided by later editors, they did not choose badly; it is one of the most frequently occurring terms in the work. In terms of debates about method in the early modern period, Descartes was casting his lot with those who (especially in the rhetorical tradition) thought that presentation had to follow the dictates of invention rather than vice versa—and ingenium was the power that guided invention.

  24. 24.

    Intuitus and deductio in Latin. Etymologically intuitus is a power of looking clearly into a thing—not far removed from “insight.” The usual English rendering of it as “intuition” is less than perfect, especially if it is thought of as a rather mysterious power of anticipating things before there is evidence for them. I shall nevertheless use it here. Deductio etymologically suggests drawing or leading something down from something else, for which the English “deduction” is an adequate translation when it is a matter of drawing truth from other truths.

  25. 25.

    This is my phrase. It could be as simple as the difference between looking and then taking notice. You enter a room and look right where you see someone moving; you notice how large the room seems; you then remark that you are seeing your own reflection because the wall to the right is mirrored, and thus you adjust your assessment of the room’s size and furnishings. Those things all happened without a marked change in attitude. But now you can easily “change the take,” that is, take things in a different modality: for example, by considering the quality of the (image reflected by the) mirror. You have changed the level and context of the experience; it now plays out it in the phenomenon of mirroring, optics, fashion, etc.

  26. 26.

    He does discuss some problems of extended proportion. If we want a third number that is to the second the same as the second is to the first, and the first two are 3 and 6, it is simple to determine that the third is 12. The equation is 3/6 = 6/x; multiplying both sides by 2x gives us x = 12. It is a little harder, when given the first and the third, to find the second; that is, given 3 and 12, finding the geometric mean x such that 3/x = x/12 requires us to solve x 2 = 36. Even harder is being given the first and the fourth in a four-number series and then deducing what the second and the third must be (e.g., if the first and fourth numbers are 3 and 24, the values of x and y must solve the equation 3/x = y/24—there is no unique solution). But if we are given the first and the fifth as 3 and 48, determining the third (which is 12) turns out to be no harder than determining the 6 that comes between 3 and 12 (the second example of this note): we break the problem into parts, first finding the number x such that 3/x = x/48 (x = 12), and then we find the numbers y and z such that 3/y = y/x and x/z = z/48 (y = 6 and z = 24). Although the examples involve only simple arithmetic, the point is that the element or “nature” 3 is contained in each number of the series to a different but definite degree; because the series follows a rule and is well ordered, the exact degree of “participation” in three is easily determinable.

  27. 27.

    The phrase “clear and distinct” is rare in the Regulae; “clear” appears more frequently in association with “perspicuous.” Nevertheless, well before the phrase became a stock part of cartesian vocabulary, arriving at the clarified and the distinguished were essential goals of his method. As I shall argue below, it is a fundamental misconception to think that objects, ideas, or other things are clear and/or distinct per se; it is rather perceiving, conceiving, and portraying that are. This misconception has long encouraged excessively rationalist interpretations of Descartes.

  28. 28.

    The single best account of the real complexities of Descartes’s understanding of consciousness and how it might be compatible with (animal) body is Baker and Morris 1996.

  29. 29.

    See Nolan 2005, esp. 239–240.

  30. 30.

    “Remotion,” remotio in Latin, is based in a very simple logical procedure: one can take a positive attribute or predicate and negate it (in the sense of producing the contradictory term corresponding to the original). Thus good subjected to remotion becomes non–good, and just becomes non–just. But the former of each pair (good and just) is regarded as finite, so negation of it produces an “unlimited” or “infinite” term. More precisely, anything at all that is not characterized, or even characterizable, as good or just can be characterized as non–good or non–just. If the terms “just” and “unjust” apply to people and their actions, rocks cannot be either, but by that very fact they can be (in fact are) non–just, and, as it turns out, also non–unjust. Remotion proper emerges when the question is what attributes or predicates can be stated of God, who is infinite in every respect. If predicates like “good” and “just” apply the same, univocally, to God and finite things, then remotion does not enter into consideration. If finite predicates do not apply to God univocally, however, either they apply in some infinite but determinate proportion, and this way of applying the predicate is called analogical; or they do not properly apply at all, or at best they faintly, indistinctly, and indeterminately try to say something positive about God. Remotion at this level is a methodological principle, used in so-called negative theology, that aspires to a kind of knowledge about God through negation that is not available through ordinary, finite predication. God, not just or unjust in any finite human sense, is thus non–just and non–unjust, and the attempt to think this through, though not rational in a conventional way, may nevertheless allow for illumination and insight.

  31. 31.

    I use the miniscule or small letter to keep open the possibility of retaining “Cartesian” as an adjective meaning “genuinely characteristic of Descartes” rather than characteristic of his purported followers, the cartesians, who are of course cartesian but often not Cartesian.

  32. 32.

    The result was the third set of objections to the Meditations, to which Descartes provided a set of replies.

  33. 33.

    He points out in his reply to Hobbes’s fourth objection that if a Frenchman and a German can use different words to discuss a matter it is because the different words regard the same thing: “for if he admits that something is signified by the words, why does he not want our reasonings to be about what is signified, rather than about mere words?” (AT VII.179).

  34. 34.

    A continued fraction is generated by an unending recursive process that adds at each step a new term to the fraction. The well-ordered series of resulting formulas produces an imaginative pattern—and it is truly an imaginative pattern, because the series displays it, makes it appear.

  35. 35.

    Descartes declined to publish The World after hearing that Galileo had been condemned in Rome for advocating the heliocentric conception of the planetary system, which Descartes understood as a necessary consequence of the divisibility and motions of matter in his physics.

  36. 36.

    See the discussion of “natures” in rules 6 and 12, AT X.381–382 and 418–424. For a fuller account, see Sepper 1996, 195–197.

  37. 37.

    Strictly speaking, it was motion’s speed rather than velocity (which is speed in a definite direction) that was conserved. This was a principal source of the inadequacy of Descartes’s rules of motion: total speed was preserved in collisions, but direction was not. One consequence of the variability of the direction of motion was that it ultimately allowed the pineal gland to redirect spirit flows in the brain without violating the rules of motion.

  38. 38.

    Straight-line tendencies and inclinations do not typically lead to straight-line motions, however. Given the indefinite divisibility of matter, at any given moment there are huge numbers of impulses affecting every point of the continuum. The resulting motions of the total interaction will typically be curved rather than straight.

  39. 39.

    I hope that the reader recognizes that with this mathematics Descartes in effect anticipated what has only recently become possible with electronic computation (which is a much more powerful simulator than any he could build or even imagine). But even though computers have a much greater capacity to carry out the kinds of ­simulations The World and the Geometry foresaw, they are still finite means.

  40. 40.

    Platonic, Aristotelian, and Stoic traditions all shared this view, and it continues to have an effect, even in unexpected venues. For example, Peirce’s pragmaticism postulates that inquiry begins with an irritation and ends when an answer is found that brings the irritation to an end. But of course Peircean semeiosis is infinite, because ever new sources of irritation arise.

  41. 41.

    This is one of the most decisive respects in which he rejected Euclid’s Geometry as a model.

  42. 42.

    That is, the geometric curve expressed by algebraic formula.

  43. 43.

    Descartes knew that there were mathematically possible curves that could not be represented algebraically; he called them imaginary (as opposed to mechanical, that is, those that could be expressed by complex machines representable by algebraic formulas). Apparently he did not believe that they could be real problems, that is, could describe real situations in the physical world governed by God’s created mathematical truths. Unfortunately for the sake of his ambitions, many problems in nature cannot be expressed using algebraic equations, but at best only approximated.

  44. 44.

    If it is not rationalist, it is still rational. Rationalism conceives of pure reason set free from the limits of this mortal coil. By contrast, recall the elemental meaning of ratio explained in Sect. 4.7, n. 46, above: it is the putting of one thing into determinate relation with another. Mathematical proportions are rational in this sense, as also are propositions; and so is the imagination that imaginatively realizes a determinate possibility with respect to an imaginative field that is articulated by features that can be differentiated as more or as less.

  45. 45.

    Recall, however, that the Meditations describes a prolonged and often repeated process of arriving at insights, so one must constantly beware of assuming that one can pop ideas into one’s consciousness and immediately intuit everything about them. So part of the clarity and distinctness of the difference between imagining and cognizing the chiliagon is remembering that in contrast one can quite easily imagine a triangle or square, and that one can in comparison as easily conceive a one-thousand-sided figure as a three- or four-sided one. That does, however, also presuppose that one has learned arithmetic and geometry—so an infant cannot arrive at this insight of Meditation 6, although eventually he will be able to, even if he or she never actually does.

  46. 46.

    The understanding we are talking about here in each case is simply that the thing in question is a triangle, or a rectangle, or a pentagon, or in general an n–gon. The difference between imagining and understanding the same figure also becomes clearer when we reflect that even a good geometer understands far better than he imagines that an n–gon has the sum of its internal angles equal to 180° times (n–2).

  47. 47.

    The phrase is translatable in various ways: among others, without assistance/help/aid of imagination, without the power of imagination, without the resources/treasure of the imagination.

  48. 48.

    See Sect. 6.6, especially at n. 30, above.

  49. 49.

    Anyone who presumes to settle the question of what pure intellect means in Descartes has to resolve this question—and resolving it probably also requires having something more than just a historical understanding of the matter. This is a moment in Descartes’s thinking where his development of the conceptual topology of imagination and reason breaks down. It also prepares the way for a solution: the transcendental functions of imagination in Kant.

  50. 50.

    Prescinding and prescission are explained in Sect. 5.13, esp. n. 102. Prescission treats as absolute a difference that is actually relational. In the paragraph above I am arguing that Descartes’s method, insofar as it demands the maximal division of problems into parts and allows the inquirer to use artificial expedients when real ones fail, is based on prescission rather than abstraction, but a prescission which (in theory, at least) does not lose track of the relations in a falsifying manner.

  51. 51.

    It is almost universally pointed out that the familiar form of the cogito argument, “I think, therefore I am,” does not occur in the Meditations. That sentence, in particular its appearance of being a deduction from premisses, has nicely obscured Descartes’s manner of argument and meaning for more than three and a half centuries.

  52. 52.

    Descartes is nothing if not faithful to the method that he learned in the Regulae. Considering ourselves precisely as thinking beings is, in the Regulae’s terms, viewing a thing (ourselves) with respect to a nature (thinking). There are many true things about ourselves that we can thereby discover with respect to what we are as participating in thinking. Thinking of ourselves as sensing or imagining is to consider a thing (ourselves) with respect to natures other than thinking in the strictest intellectual sense, and so the truths we discover thereby will be different—even though (or especially because) it is also true that, when we examine the interparticipation of natures, sensing and imagining turn out both to be species of thinking.

  53. 53.

    Not, however, to all modern epistemologists. I will mention only Hume, who, in the demanding and original form of the Treatise (Hume 1739–1740; see esp. bk. 2, Sect. 2), understands all mental activity as having both a direct object and an indirect object (typically affects and passions, though one can say in general that the indirect object is the self—as long as one does not insist on too unitary a notion of self!). This dimension of mental activity is missing from the Enquiry (Hume 1748). It is the latter that has been more influential among professional philosophers.

  54. 54.

    Unfortunately it has often been treated as a kind of afterthought or simply reduced to a specialized application of mechanist reductionism. A major exception is the magnificent study by Kambouchner 1995. Kambouchner situates the work in the history of the theory of passions and shows how it resolves Cartesian problems and brings Descartes’s philosophizing to fulfillment.

  55. 55.

    21 May 1643, AT III.665–666. There are in effect four kinds of primitive notion introduced by this passage. First are “the most general [notions]—those of being, number, duration, etc.—which apply to everything we can conceive.” Then come those “as regard the body in particular” (extension, which entails shape and motion), followed by those “as regard the soul on its own” (thought, including intellectual perception and inclinations of will); “lastly, as regard the soul and the body together, we have only that of their union…” (AT III.665; for the continuation see the next paragraph of the main text). To recapitulate: there are four kinds of primitive notion: (1) the notions of the first kind, which apply to everything we can conceive, which includes soul, body, and soul and body together; (2) the notions that apply (a) some to body alone, (b) others to soul alone; and (3) those of the last kind, which belong to the union of soul and body. What are usually referred to by scholars as the three kinds of primitive notions are 2a, 2b, and 3.

  56. 56.

    “There being in the human soul two things on which depends all the knowledge that we can have of its nature, one of which is that it thinks, the other that, being united to the body, it can act and suffer with it; I have said almost nothing about this latter, and have exerted myself only to make the first well understood, because my principal design was to prove the distinction there is between the soul and the body; for which only the first could serve, and the other would have been harmful to it” (AT III.664–665).

  57. 57.

    New and unexpected in Descartes, at least in the most rationalist interpretations of his philosophy, but less original and unexpected insofar as they were traditional topics of imagination theory.


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Sepper, D.L. (2013). The Dynamically Imaginative Cognition of Descartes. In: Understanding Imagination. Studies in History and Philosophy of Science, vol 33. Springer, Dordrecht.

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