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The goal of Chaps. 4 and 5 was to understand the topology of imagination and its functioning in a Greek master and his master student, who together established the founding topological structures of Western psychological theory. The present chapter and the next one intend to show imagination and related powers, in theory and at work, in two modern refounders, René Descartes (1596–1650) and Immanuel Kant (1724–1804). As with Plato and Aristotle, I will be interested in showing important connections between them. But unlike the founders, these moderns are heir to complex traditions of conceiving imagination and the human psyche, traditions that had been cultivated for two millennia. Moreover, Descartes and Kant were not master and student but figures separated by more than a century of newly vigorous thinking about imagination.

The theories of reason and imagination that came down to Descartes were largely conventionalized. They were distant descendants of the ancient texts, but the lines of descent had been crossed by too many other influences to bear a simple resemblance to the originals. If Descartes read the originating texts of Plato and Aristotle, he left no direct indication of it. That he was familiar with some version or versions of the conventionalized Aristotelian tradition is, by contrast, clear.

If what he came to understand of imagination involved little actual reading of Plato and Aristotle, it nevertheless had a great deal to do with coming to see unexploited possibilities offered by the conceptual topology they had produced. To expand on this figuratively: the two Greek masters had labored over a series of interrelated maps of a territory showing the populations, the political divisions, the flora and fauna, in general the “lay of the land” of the human soul and its relations. What their followers derived from it were variations, amplifications, and simplifications that often lacked the metaphysical, physical, and psychological amplitude and density of the originals. Simplified schemas displaced the originals and were accepted as adequate representations of the phenomena. Even when the relationship to Plato or Aristotle was more direct, the rich conceptual topology they had elaborated usually lost out to other concerns of interpreters. To recover a richer sense of the topology it took someone like Descartes, who found the intellectual maps he inherited so unrepresentative that he had to explore things for himself. Many of the things he discovered, as original as they in fact were, were consequences of his rediscovery of elements of the old conceptual topology.

From an early age Descartes did not merely theorize about imagination, he practiced it. After a youthful flirtation with the hope of Renaissance humanism that poetic imagination might reach even the highest spiritual things, he turned to the philosophic-cognitive use of imagination in mathematics and physics. If there is any truth to the claim that Descartes “mathematized thought,” it has to be understood more precisely as an imaginization of thought.Footnote 1 This imaginative mathematics and physics provided the conceptual and methodological structure of his first publication (in 1637), three scientific essays (on the optics of refraction, the physics of atmospheric phenomena, and analytic geometry) and the more famous writing that served as preface, The Discourse of the Method for Rightly Conducting One’s Reason and Seeking the Truth in the Sciences. That was not the end of his imaginative practice and theorizing, however. The meditational approach of the 1641 Meditations on First Philosophy was modeled on the religious and philosophical method of using memory, imagination, and intensive cogitation to arrive at fundamental truths, and the work itself probes the limits of imagination. And in his last publication, the Passions of the Soul (1649), he conceived imagination as an act closer to will than to intellection and provided imagination with an important new noncognitive task: to help manage the feelings, emotions, and passions, with the goal not of suppressing them but rather of learning to use them in order to “taste as much as possible the sweetness of this life” (AT XI.488).Footnote 2 In this perspective, there may be more than mere symbolism in the fact that the last work he composed before his death in February 1650 was a masque, a combination of dance, music, and drama presented as an entertainment for the court of Queen Christina of Sweden.Footnote 3

That this is not the Descartes presented by traditional historiography of philosophy should not trouble us. If Cornelius Castoriadis’s judgment about the history of imagination in the West is correct, we should expect gaps and occultations in historians’ accounts. Castoriadis argued that the followers of the most innovative thinkers homogenize and conventionalize fundamental insights, and that even the innovators rarely draw all the consequences they could. One should therefore not expect to find the profoundest insights in standard accounts or the philosophy of schools.

The conventional “truth” that Descartes mathematized thought impedes our access to a deeper truth. The mathematics he invented, analytic geometry, the unification of geometry and algebra, was the most rigorous and active use of the imagination ever conceived. Moreover, several other thinkers and mathematicians who were in close contact with his thought (Leibniz, Malebranche, and Pascal, to name the three most important) agreed that mathematics and natural science were preeminently imaginative undertakings. But by the middle of the eighteenth century, less than a century after Descartes’s death, the notion that mathematics is imaginative work was largely displaced by the opinion, the cultural commonplace, that mathematics was the product of rational intellect.

I say opinion advisedly. No theory of mathematical intellection was advanced, no effort to come to terms with and to overthrow the understanding of great mathematician–philosophers like Descartes, Pascal, and Leibniz was made. The opinion simply came to prevail without evidence or argument, and it has been with us ever since. There is an historical irony involved: one of the reasons that the agreement of Descartes, Pascal, and Leibniz could be so easily contradicted is that their plainest statements about it were in writings available only posthumously. Pascal, to take one instance, was strongly influenced by Descartes’s conception; some of his philosophical and religious writings contain profound reflections on the consequences. Pascal in particular emphasized the moral temptations imagination was subject to and condemned the pride it exhibited, even as he acknowledged its positive use in the sciences. The problem for us, however, is that most of the crucial relevant ­passages were not available to his contemporaries, and in fact many were not published until nearly two centuries after his death. Leibniz was clearer about the role of imagination in his private letters than in his philosophical and scientific writings; not until much later did the former come to light. With Descartes, the fundamental role of imagination in mathematics and science was most richly documented in his earliest notes and compositions. These writings were for the most part unknown to his contemporaries (Pascal, Malebranche, and Leibniz, who had access to some of them, were exceptions) and were not published until generations, even centuries, after his death. When they appeared they seemed to be little more than curiosities.

The mathematical and scientific role of imagination was clearly and publicly enough stated by Malebranche in The Search After Truth (1674–1678), but that came in the middle of a very long book arguing that, outside of mathematics and natural sciences, imagination was deceptive. When all is said and done, Malebranche in his own right had little mathematical or scientific authority, and his predominantly negative evaluation of imagination, in a book that was widely read and greatly influential, had far more effect.Footnote 4 That gives a decided edge to the notion that we are dealing with an impaired tradition that, had it been cultivated, might have had glorious successes. Instead it suffered occlusions and occultations; it lapsed and sank from view.

6.1 Imagination After Aristotle and Before Descartes

Although this is not a historical survey of Western theories of imagination, it will be useful to have in mind a schematic overview of the more than 1,900 years between Aristotle’s death and Descartes’s birth.

The influences of Plato and Aristotle lived on and became intertwined, especially in neo-Platonism, which was the most successful philosophical school of late ­antiquity. It was, in particular, respected by educated Christians and adapted by them to give philosophical support to Christian theological tenets. Neo-Platonists typically tried to show that Plato and Aristotle were compatible with one another. They regarded Aristotle’s Organon (which contains not just logic and rhetoric but also the basic elements of his theories of being and knowing) as an introduction to philosophy proper. His other writings they treated with respect, especially those, like his writings on nature, that had no equivalent in Plato’s corpus. The psychological theory of On the Soul and the shorter essays of the Parva naturalia were readily adapted to Platonic purposes, not least because the conception of the hierarchy of soul powers was similar. Moreover, insofar as Aristotle’s theory of intellection was predicated on the noetic power’s capacity to recognize intelligible forms in images, it offered a naturalistic account of the ascent from the sense-apprehended material realm to the intelligible realm of mathematics and ideal forms.

In more strictly materialist schools the image-bearer theory that we first saw in Empedocles continued to prosper (for example, in the notion of the eidōlon in Epicurus, which has come down to us chiefly through De rerum natura of the Roman poet Lucretius). The Stoic school, which in an important sense counts as materialist (rational soul was regarded as the finest and most mobile kind of matter, capable of directing the motions of grosser matter), played a crucial part in reconceiving the critical moment when the images conveyed from the physical world appear at the threshold of mind. There, in the vestibule of the brain’s organ of rationality they called the hēgemonikon, the appearances had not yet been internalized and so could be put on trial with respect to whether they were true or illusory; only the former would be allowed to enter the domain of reason. Since Aristotle had presented common sensation, imagination, and memory as physical functions preparing phantasms for intellectual purposes, the Stoic and Aristotelian theories could thus coexist in a very simplified topology of the human soul.

The tradition most immediately important for understanding early modern ­developments nevertheless remains Aristotelian. Aristotle’s conceptual topology of imagination and soul exercised its influence well into the seventeenth century. Basic conventionalizations of its conceptual topology were accepted even by those who were not particularly well disposed to his overall philosophy.

After correcting Aristotle’s preference for heart over brain as the location where all the special or proper senses are united in common sensation, the physician–theorists of Western antiquity easily adapted his philosophy to their concerns. Medical theorists pursued in more detail the proper localization of soul powers in the brain and other organs of the body; a few even contradicted Aristotle’s exemption of intellect from localization by assigning it a brain place. Philosophers who were not physicians resisted for the most part the physical localization of reason, but otherwise the ratification by medical doctors of Aristotle’s organic psychology reinforced the authority of his theory and encouraged philosophers to continue developing psychology within its framework. Many of the greatest commentators of Aristotle in the middle ages, Muslim, Jew, and Christian, were themselves doctors or engaged in the study of nature—for example, Avicenna (ca. 980–1037), Averroës (1126–1198), Maimonides (ca. 1138–1204), Albert the Great (ca. 1200–1280), and Roger Bacon (1214/1220–1292). Even though most of the works of both Aristotle and Plato were lost to the early middle ages, at least in the West, their reputation among the learned remained strong. If for centuries the natural philosophy of the soul was no more than lightly cultivated (at least before the great Islamic philosopher–physicians), it was typically done in a blend of Aristotelian “faculty theory” and Platonic metaphysics.

One briefly observed early case will serve as an example. Plotinus (ca. 204/5–270), the greatest of the neo-Platonists, placed an Aristotelian division of human psychological powers within a Platonistic metaphysics of emanation. Emanation, a spilling over of the being of the ineffable Oneness that was the source of everything, produced eight progressively lower levels. Matter (where evil is located) was the lowest of all, soul immediately above it, and intellect above that. This metaphysics radicalized and further elaborated the tendencies of the good’s communication of itself into all levels of being that we saw in Chap. 4’s explication of the Republic. As with Aristotle, soul was understood as a form distinguishing living things from nonliving matter. Its aim was to rise above its material, organic conditions and, through the purifying ascent to intellect, to move closer to the One.

One of the curiosities of Plotinus’ psychology is a duplication of certain lower soul powers. There is, for example, memory of what happens to us bodily but also an independent intellectual memory. Similarly for the imagination. One imagination simply reproduces what is acquired by the senses, the other is oriented toward the spiritual; the spiritual reflects and preserves the purely spiritual aspect of the lower kind of phantasms. This allows Plotinus to retain the Aristotelian principle that there is no thinking/intellection without phantasms without having to retain any direct connection to materiality. The phantasms of the intellectualized imagination and memory are nonmaterial, and the soul, in its ascent to intellect, is purged of all traces of matter (it retains the pure forms of what was learned in the lowest levels of experience). Thus Plotinus could assert the unqualified primacy of intellect over sense and bodily imagination and memory—indeed, a total separation from them. This would not be the last time that philosophers would turn their backs on matter and try to make a home among the pure forms of reason.Footnote 5

With the progressive recovery in the West of Aristotle’s works—beginning roughly in the middle of the twelfth century and essentially complete by the middle of the thirteenth—the kind of internal-senses doctrine that I explained in conventionalized form in Sect. 5.2 became a staple of Western theories of the soul and its powers. Albert the Great, who wrote commentaries on many of Aristotle’s nature treatises and himself engaged in first-hand study of natural phenomena, ­proposed a five-part theory of the internal senses (following an interpretation of Aristotle by Avicenna), whereas his student Thomas Aquinas (1225–1274) proposed four (favoring an interpretation by Averroës). There was no single theory or number of powers that could be called canonical for the Western middle ages, ­however. Aristotle himself had left the question open, and there was no little argument over how different places in or around the brain ought to be correlated with the internal sense powers—a question that Aristotle had also largely left open. Already in antiquity anatomists had identified not organs but hollows in the brain—called ventricles—as the places of common sensation, imagination, and memory. There was no unanimity about the exact enumeration of the ventricles; often they were conceived in a way that suited the theorist’s preferred number of internal sense ­powers. Positionally the ventricles could be easily enough divided into anterior, medial, and posterior, but more refined theories of internal sense functions could subdivide them as needed. For example, if common sensation was typically attributed to the anterior ventricle, the one closest to the eyes, nose, and tongue, one might still argue for a differentiation of its functions depending on whether the activity took place in the front or rear of that ventricle; similarly for the ventricles of imagination and memory. Furthermore, internal senses theories also eventually incorporated an important element from Stoic philosophy, a very fine and active form of matter called spirits filling the ventricles (and, as knowledge of the nervous system emerged, filling nerve fibers as well). It was thought that spirits could take on and transmit evanescent forms of appearance; thus they were used to further elaborate the physical theory of images/phantasms.

As I mentioned in the conventionalized version of scholastic theory in Sect. 5.3, the internal senses engaged in a kind of “phantasm processing” intended to explain purposive animal behaviors apart from reason; in the human being they exercised similar functions but also prepared the phantasm for abstraction by intellect. The increasingly complex anatomical and physiological processes hypothesized by internal sense theorists were doubtless encouraged by improved knowledge about the brain and nerves not available to Aristotle,Footnote 6 but it responded more directly to questions that the general topology of Aristotle’s psychology raised. Aristotle had gone so far as to claim that the organic processes of both remembering and imagining took place in the primary place of common sensation, but he did not even begin to formulate the physical complications of how to conceive them anatomically or physiologically (recall that he denied that common sensation was the activity of a specific organ). When he said that we grasp the forms in phantasms he did not ask whether this occurred the same or differently in immediate sensing, imagining, and remembering. When at the end of the Posterior Analytics he described how similar sense experiences could reach a “stop” in the inductive recognition of some kind of thing, he did not ask questions about the anatomy, physiology, and phenomenology of that stop.

Nevertheless, Aristotle had blazed a trail that led to the questions he did not ask, because in principle if not in detail he had conceived the powers of common sensation, imagination, and memory as part of a complex psychic and physiological system that organically coordinated and integrated phantasms. The topology he had established not only suggested but impelled later inquiries and conceptions. Seen in this light, for example, II.19 of Posterior Analytics suggests that, before you arrive at a concept by induction, you store up sense experiences and sensory forms in memory, which are reawakened by related future sensations; and one day, finally, you see the same kind of thing again and, suddenly, the present and the past experience are pulled together into a sharply focused grasp of what the perceived thing is. It becomes conceptualized, and a name can be assigned to it. Even if ­medieval ­doctors and philosophers could not spell out in detail how ventricles and spirits interacted in a way differentiated by sensation, imagination, and memory, in principle this psychophysiological approach allowed them to stipulate physical processing and sequencing of phantasms corresponding to psychological differences. And then, at the end, intellect could take these vigorously processed phantasms and derive—abstract—from them concepts, intelligible species (as Posterior Analytics II.19 had argued only phenomenologically).

Without reminding ourselves of this background, Descartes’s (and other early modern) attempts to coordinate nerve and pineal gland motions, spirit flows, and the like look arbitrary and idiosyncratic. They are nothing of the kind; they are extensions and radicalizations of the psychophysiological conceptual topology that had prevailed for nearly 2,000 years. All sixteenth- and seventeenth-century thinkers, of whatever traditions or intellectual commitments, were aware of, and usually schooled in, this topology, and new discoveries in anatomy and physiology were incorporated into it. Descartes in particular began his philosophical inquiries by trying to adapt a simplified version of the standard psychophysiology to his own methodological, physical, and mathematical inquiries. He would give radically new meaning to the “preparation of phantasms” by his invention of new, highly intensive uses of imagination for cognitive purposes.

6.2 Descartes’s Starting Point

If you begin with the extant earliest writings of Descartes, none of which were published in his lifetime, you discover that imagination was fundamental to his mathematics, his science, and his conception of method. It was fundamental precisely insofar as Descartes worked through and revolutionized the conceptual topology of imagination he had inherited. He was interested in the question not just of how we have images, but even more of what we do with and to them. The conceptions he formed of idea and thinking Footnote 7 were radicalizations of image and imagining, respectively, though in thinking through the problem of imagining he also came to recognize the limitations of images and imagination. Unfortunately very few interpreters have grasped the implication of these things or even noticed them.

Considering the relatively small number of writings that are preserved from the earliest period, which begins shortly before 1620, when Descartes was still in his early twenties, it is surprising how often imagination and images/phantasms are mentioned and discussed. In fact they show themselves to be central to his conception of the acquisition and the dignity of knowledge. The earliest evidence of this work is the most precisely dated. In the autumn of 1618 Descartes was in Amsterdam. One day he and another man both stood reading a poster advertising a contest to solve a problem in mathematics. He remarked to the man that he could solve the problem and many more like it. They struck up a conversation that quickly led to a years-long friendship. We know the story because the other man was Isaac Beeckman, a scholar–scientist who kept a voluminous journal of his ideas and experiences. He was an advocate of explaining nature by using corpuscular matter theory and mathematics, what he called physico-mathematics. He found in the Frenchman someone who was carrying on a similar kind of research. The 22-year-old Descartes was as good as his word in demonstrating to Beeckman’s satisfaction the sophistication of his problem-solving abilities.

As a New Year’s 1619 gift Descartes presented Beeckman with a work about music theory, the Compendium musicae (Compendium of Music). At the outset Descartes laid down a set of postulates that present the sense of hearing as governed by simple proportions. This was an old standby of aesthetic theory, in virtually all the major philosophical traditions.Footnote 8 Medieval scholastic thinkers in particular had followed Aristotle in asserting that the sense organs themselves were proportional means or middles between the extremes of their objects and that sensation itself was a determination of proportion.Footnote 9 Descartes was scarcely unconventional in including among his postulates the thesis that the quality experienced by the sense organ was a kind of proportion, and furthermore that the pleasure or displeasure one feels was correlated with the proportionality or disproportionality of the object to the organ (for example, a very bright light will be unpleasant because it is disproportionate to the organ’s capabilities) and of the various qualities to one another (in hearing, tones that have simpler proportional relationships are pleasing, like the octave and the major fifth, whereas dissonances are not). Nor was he breaking new ground in claiming that, even if harmonies were pleasing and dissonances displeasing in themselves, a perceiver would be wearied by hearing only harmonies. The listener is pleased best, he asserted in agreement with long tradition, by a variety of tones, harmonies, and disharmonies, so that the goal of music is not to provide harmony at every moment but rather a pleasing impression overall.

The Compendium is richly provided with geometric and other figures to express and summarize the proportions that hold between sounds. This, too, is not particularly original, although perhaps the degree to which such figures populate a relatively short work is unusual.Footnote 10 It shows that the young Descartes was already quite conversant with proportional geometry and arithmetic, and that he had mastered the art of compressing information into geometric figures and images. But the most striking feature in the work occurs in an early passage that explicitly attributes to imagination the function of perceiving a musical composition as a whole by joining part to part to part in a kind of calculation of the song’s proportions.

In the very first section after he lays down his postulates, Descartes explains how we recognize the time, or rhythm, of music. While we are hearing the present beat we recall what we have heard before in proportional relation to it, and we progressively hear our way through the piece, extending all the proportions that we have heard right up to the present moment. This is not merely sensing what is immediately present plus remembering what is past but rather actively and continuously synthesizing a whole, here and now, out of the present and the past and moving conjecturally into the future:

For then, when we hear the first two members, we conceive them as one; when [we hear] the third member, we now conjoin that with the first ones, so that the proportion is tripleFootnote 11; thereafter, when we hear the fourth, we join that with the third so that we conceive [them] as one; thereupon we again conjoin the two first with the latter two so that we conceive these four simultaneously as one. And thus our imagination [imaginatio] proceeds all the way to the end, where finally it conceives the entire song as one thing fused out of many equal members. (AT X.94)

Without this synthetic power of imagination we of course would be able to hear what was sounding now, we might recall hearing sounds in the past, and perhaps we might be able to expect a new note in another fraction of a second. But in order to have a sensibility for a song as a unified whole, we have to perceive a progressively expanding unity through the experience of the parts. That is the work of imagination. Although Descartes does not bring up imagination expressly when discussing relationships of pitch, there is no reason to believe that imagination does not perform a parallel though more complex function in unifying successive and simultaneous harmonies and dissonances.

It is hard to overemphasize what an extraordinary idea this is. One measure would be the boast that Immanuel Kant made more than a century and a half later. In his critical philosophy he argued that imagination had more to do than merely follow upon sense perception, it actually helped constitute perception. One of the so-called transcendental functions of imagination was to “unify the manifold of sensibility” in a way that connects moment to moment and articulates the potential chaos of impressions into a well-ordered experience to which we can apply basic conceptual categories. Most historians of philosophy and psychology would more or less agree with Kant that this notion is his innovation. In one of the footnotes to the work in which he announced it, the first edition (1781) of the Critique of Pure Reason, he remarked: “Probably no psychologist till now has thought that the imagination is a necessary ingredient of perception itself.” One can certainly argue that a couple of sentences in passing in a piece of writing not meant for the general public by a young man who had not yet made his mark on the world hardly counts as much of an exception, especially since the idea is confined to listening to music, indeed just to rhythm. Nevertheless, it is still an extraordinary thought. It indicates that, at the beginning of his philosophical career, Descartes recognized a possibility of imagination that had been no more than implicit in previous developments of its conceptual topology.Footnote 12

Even more important, there is every reason to think from other roughly contemporary writings of his that Descartes recognized analogous functions for imagination in other tasks. This holds even more strongly for his strictly mathematical thinking and his efforts to apply mathematics to physical problems. Sensation gives us the data of the present, memory gives us data of the past, but the ability to see relations in and between the data requires the synthesizing power of imagination, which sets the present situation against the background of the past and tries to generate new appearances necessary for grasping what is at issue, and ultimately for solving problems and answering questions of almost any type. This kind of thinking is not just temporal, synthetic, and projective, it is also fundamentally biplanar, to recall a term that we introduced in Sect. 3.8, above, and have used of both Plato and Aristotle.

Biplanarity would be present even if it were just a question of synthesizing a presently heard note with past notes: one is setting the present note against the past as background. A trained musician will be able to do this in a more nuanced and ample way than a novice will. The novice might be able to notice basic rhythm or key changes when they occur, but the experienced musician will also hear them (imaginatively) as part of the entire sequence of key changes that have already occurred and will, furthermore, anticipate other changes to come. As he listens, the musician can frequently shift the momentary focus of attention, to highlight now the rhythms, now the keys, now the relations between the tonal and rhythmic style of this piece and another by the same composer, and then between the style of this composer and some other. Thus perception is never without memory, and never without the constant ability to recontextualize, to change backgrounds, foregrounds, and middle grounds, and thus never without this contextualizing work of the imagination. For the young Descartes, then, there would be no perception or memory without imagination, and there is no imagination without the establishment of various planes and the mental ability to shift attention between them.

Descartes’s acquaintance with theories and uses of imagination should not be entirely surprising. Starting at about the age of 11 he attended the Jesuit Collège Henri IV at La Flèche.Footnote 13 As at all Jesuit colleges of the time, philosophical instruction was central, and the basis of philosophical instruction was Aristotle. Moreover, it is said that the Jesuit Fathers rather indulged young René. His health was delicate, so they allowed him to stay in bed till late in the morning. Because of his talents he was allowed access to books that were not part of ordinary instruction. What he actually read and when he read it is conjectural. But the Jesuits were receptive to new developments in the sciences—natural philosophy, as it was still called. Especially at their university in Coimbra (in today’s Portugal) they were hard at work producing extensive commentaries on Aristotle’s writings that simultaneously summarized current speculations and discoveries, in natural philosophy and all the other parts of philosophy. These would doubtless have been available in the library at La Flèche.

The pages of the Coimbran commentaries on Aristotle are laid out like commentaries on Sacred Scripture: the original Greek text on the left-hand page, a Latin translation on the right-hand page, with commentary forming a U-shape in the wide margins surrounding these texts, on both sides and below. The volumes were in essence compendia of primary texts plus brief discussions of major interpretations and criticisms, from ancient to modern. Anyone using one of them—for example, the commentary devoted to On the Soul, and in particular the passages about phantasia—would have gleaned not just a clarification and elaboration of the principal text but also introductions to alternative conceptions through the ages and up to the time of the commentary’s publication.

Historical studies have shown that the Jesuits in the sixteenth and seventeenth centuries emphasized the importance of human psychology to philosophizing; this affected their manner of interpreting Aristotle’s work, especially his logic and other writings included in the Organon.Footnote 14 One plausible reason for their psychologizing interpretations of Aristotle was the mandatory Jesuit practice of spiritual exercises, a practice that was guided by the work under that name written by their founder, Ignatius Loyola. The spiritual exercises of the Jesuits were founded on the long-­established ancient practice and medieval theory of meditation and contemplation. The medieval high point of theoretical development came relatively early: in the twelfth century school of the Abbey of St. Victor in Paris. Hugh of St. Victor (ca. 1096–1141) laid the foundations, which were further elaborated by his student Richard of St. Victor (d. 1173). The theory they developed observed a progressive discipline of the internal senses. Although the corresponding practices were often used to meditate on passages in Sacred Scripture and truths of faith, and sometimes to strive toward mystical illumination, the theory was not intrinsically religious. Rather, it was about the methodical use of human faculties for the recognition of truths that reveal themselves only to a properly prepared soul and mind. As we have seen, both a tradition following Plato (the Victorines were Augustinians, and thus their basic philosophical orientation was Platonist) and one following Aristotle (whose theory of the necessity of phantasms for thinking had become part of the common philosophical and medical understanding of inward senses) would acknowledge the importance of using images in order to arrive at deep truths. Accordingly, the Victorine theories had emphasized the need to work persistently with and through the forms of sense in order to arrive at intuition.Footnote 15 If in the last analysis the goal was to reach a profound, intellectually apprehensible truth, this could be accomplished only by mulling over again and again what one sensed, remembered, and imagined.

If the ultimate goal of this process had something mystical about it—a feature that came especially to the fore in the sixteenth century meditational practices of John of the Cross and Teresa of Avila, for whom the goal was to “silence” the busy activities of all sensitive and cognitive faculties so that one might apprehend God’s glory and be overwhelmed by His light and loveFootnote 16—the process itself required extraordinarily intense and active imagining. The goal of Ignatius’s spiritual exercises in fact went beyond cognitive results. The Jesuits were a “militant” order dedicated to activity in the world, with the aim of converting it. Their practice of the exercises intended to form the human will. One strove for a deep knowledge and love of God in order to bring about in oneself the resolution always to do what is pleasing to Him and thus to be directed solely by His will (in particular by obeying one’s Jesuit superiors).

There are many ways in which this practice might have influenced the young René, each somewhat speculative when considered by itself but overwhelmingly likely when taken in sum. Students at La Flèche, even though they were not destined for holy orders, practiced certain abbreviated forms of the Ignatian spiritual ­exercises. Even in the shortest forms of spiritual exercises (as opposed to the full 6 weeks presented in Ignatius’s Spiritual Exercises) René would have been taught to make intense, proliferative use of imagination. For example, in meditating on the passion and death of Christ, one is directed to imagine being pricked by a pin, and then to imagine what it would be like not just to have a single pinprick applied but a crown of long thorns pushed deep into one’s skull. One is supposed to feel and see the bloody red rivulets trickling down one’s forehead, into one’s eyes, the bloody salt taste on the tongue, etc., and the racking pain. But of course this was only the beginning of Christ’s passion, only the beginning of trying to supply sensory content to events narrated in the gospel accounts and thus to give proper amplitude to the significance of the words. Imagining these things was not the end of the exercises, but it was the constant means.

There was another relevant practice the Jesuits employed that might have influenced René’s conception of imagining. They were masters of the emblem, of producing and interpreting visual images that condense and symbolize information and doctrine. In a sense the emblem has always been an aspect of religious, mythological, and historical art, but it was cultivated with special intensity in the Baroque period, the art of which was flourishing in Descartes’s lifetime.Footnote 17 One place where the emblem played an especially pervasive role was in the frontispieces of books, which can often be seen as figurative representations of what the book is trying to convey. At the very least it would have provided a stimulus for exploiting the instructional value of figuration, even of a more mathematical kind (for example, the figures he used in the Compendium).

In the seventeenth century alone, tens of thousands of Jesuit-guided students around Europe and the world performed some version of the exercises without their developing a philosophy like Descartes’s. Why it might have taken so differently and uniquely in his mind and soul is unknowable in any final sense. But his earliest extant notebooks do provide possible evidence. The most immediately germane is a note in which he says that at school he was in the habit of picking up a book, reading its title, and trying to bring to mind what the content must be; the note remarks further that he was successful in the majority of cases. Even if we allow for self-flattery and the fact that book titles (and frontispieces) in the sixteenth and seventeenth centuries were far more elaborate than our own, the claim suggests that at an early age he had become accustomed to exercise formidable anticipative imaginative powers in all kinds of ways.

6.3 The Imagination of Notebook C

The Compendium of Music is only one piece of evidence about imagination in the early Descartes. The richest source is notes dating from 1619 to 1621, kept in a now-lost notebook designated C.Footnote 18 The still extant notes were published in 1859 under the title Cogitationes privatae. Many are scientific and technical, examples of what Beeckman called physico-mathematics.

For example, in order to solve problems about how far and fast a free-falling body travels, Descartes tries (incorrectly, as it turns out) to use a right-angled ­triangle to correlate speed, time, and distance traveled; small changes in each he correlates with adding incremental, proportionally related segments in order to progressively increase the size of the triangle.Footnote 19 In order to solve other problems, both physical and mathematical—and sometimes just to see what happens—he describes forming mental figures (or sketches them in the notebook) and then proceeds to add to or subtract from them, to vary them, to manipulate them.

For example, Descartes visualizes two-dimensional geometric figures as increasing or decreasing in size, as sliding through the plane, as rotating around a point, or as rotating around a line and thus producing a three-dimensional solid. He pictures and tracks processes of division or analysis that never end but still approach a limit. He looks for ways to construct from existing figures a unit of length in terms of which all other lines and figures he is using can be expressed as whole-number multiples. He devises sketches of simple machines consisting of sliding line segment sides and pivoting points, all the parts of which move in a well-regulated, interactive way when a force is applied to a single part. Many of these he conceives as imaginative versions of possible real-world instruments for calculating problems and constructing figures, the capabilities of which would far exceed those of comparable instruments available to ancient mathematicians.Footnote 20 He imagines pencils being attached to moving parts of these devices and considers the paths they would sketch out as the device is operated. He even begins to conceive of systematically employing marks, symbols, and other representative forms to stand for the information embodied in the figures he drew and imagined, and to use these representations to express proportions, equalities, and inequalities, and thus serve for finding new ways of manipulating figures to construct solutions to mathematical and physical problems. This work was the beginning of what he gradually transformed into analytic geometry. The term that he used to describe all this kind of activity was not intellection or reasoning but imagining.Footnote 21

Other notes of C go well beyond mathematical and physico-mathematical concerns. A number of them reflect a conviction that those searching for truth are brought into touch with higher things by their resemblance to lower ones; the instrument of this ascent is, once again, imagination. One of them appeals to a Renaissance, quasi-Platonic sensibility for the spiritual symbolism of nature: “sensible things are fitted to conceiving Olympian things: wind signifies spirit; motion over time signifies life; light signifies knowledge; heat signifies love; instantaneous activity signifies creation. Every corporeal form acts through harmony” (AT X.218). Beyond reinforcing the notion that there is an expressly poetic and aesthetic character to the young Descartes’s conception of imagination, it suggests that he entertained the possibility of there being a natural concordance and active metaphysical harmony between things accessible to the senses and things of the spirit.

One of the notes goes so far as to suggest that reason is unable to keep up with sensory imagination and intellectual imagination as they work on different levels of existence:

As imagination uses figures to conceive bodies, so intellect uses certain sensible bodies to figure spiritual things, like wind, [and] light: whence, philosophizing in a higher way, we can by cognition raise the mind into the sublime.

It can seem amazing that weighty meanings [are to be found] in the writings of poets more than in those of philosophers. The reason is that poets write through enthusiasm and the force of imagination: there are particles [or seeds] of science in us, as in flintstone, that through reason are drawn out by philosophers, [but] that through imagination are struck forth by poets and shine out more. (AT X.217)

This note is especially pregnant with consequence, not just because it tries to establish by analogies the very principle of analogy but even more because it presents imagination under a twofold significance that was central to Descartes’s early conception. The very first clause of the first sentence treats imagination as the power of making corporeal things conceivable through figures and images; this is a very general way of describing the imaginative power that his mathematical and physical problem-solving notes were investigating. Next he analogizes to it the capacity of intellect—or, given its function, it might more accurately be called intellectual imagination—for using corporeal things as figures of spiritual or intelligible things. There is a lower imagination, and there is a higher imagination. The lower one figures or images physical bodies, it is physical imaging; a higher one uses bodies symbolically to express spiritual significance or presence. There is more than a little influence here of the kind of topology seen in Plato. The last sentences of the note are particularly surprising insofar as they argue that imagination can bring us more directly in touch with spiritual and intelligible things than can reason (ratio). Reason is plodding; imagination strikes sparks and shakes free the particles of science toward which reason plods. Whether, without the anticipative capabilities of imagination, reason would even know in which direction to plod is doubtful.

In this note there are two notions of imagination: image- and figure-making imagination in the strict sense, and then the projective, poetic-cognitive imagination that uses things and their representations for higher, spiritual purposes. If the expressly poetic concern for the most part drops out of Descartes’s later philosophizing, the cognitively driven projections of imagination remain a constant. Even when we simply see a thing, we immediately desire to make more out of it; we mentally schematize or simplify the thing in a figure and use the figure to conceive the thing and its motions, actions, and possibilities. This might well be a further, visual development of the kind of synthesizing and conjecturally projective ­imaginatio that Descartes thought was at work in music listening. Perception is not passive, or at least it is not finished just by presenting us with something. If it does simply present us with something, still, in the very next moment, we can take hold of it in a new way. The first look can cause us to intensify our attention in a second look, or we can take the appearance, the image, the phantasm of what we saw a moment ago and proceed to portray it in a new medium—for example, we can ­mentally conceive it in a geometrical configuration. That geometrical configuration then becomes in its turn a new object of attention, though it still implicitly refers to the phantasm of the original thing. (We shall return to the implications at the end of this section and the beginning of the next.)

But there is more to the imagination of the thing than just conceiving it mathematically. Things are related to others, and they are signs and symbols of other things on other levels of being. Thus, with intellectual rather than corporeal imagination, one can symbolize spiritual and intellectual matters with corporeal images, and thus one can think about them by thinking in terms of their images—that is, by intellectual imagining. Here the Platonic heritage is strongly in evidence: for example, when Socrates in book VI of the Republic adumbrates the governance of a city by talking about who is the best person to pilot a ship, or in book X presents cosmic justice by describing a soul’s journey through Elysium and Hades.

Imagination can therefore be conceived in both narrow and broad ways. Image making in the narrow sense is the ability to form, divide, and recompose images of corporeal things. Something like it is at work already as we exercise sense perception (we see something and as we look upon it we conceive it figuratively) and explicit in productive and reproductive imagination, in memory, and in mathematical and other kinds of figurative representation. Imagination in the broad sense is the generalized power of using one kind of appearance in order to reconceive and understand something else “figuratively,” as we say. It is based on the human capability, emphasized by Plato, of seeing through things and images to other things overarching them.

Another note shows the degree to which Descartes had begun to conceive of the work of imagination not just as ad hoc but as capable of grounding a method of investigation. The note begins by saying he was reading a book on the art of local memory. We encountered local memory very briefly in Sect. 5.6 (n. 47). It is a technique for remembering complicated matters by producing striking images and symbols and mentally positioning them in a familiar, remembered place. Descartes then explains to himself how his own methods improve on the arbitrariness of the memory art.

Reading through [the book]…I thought that everything I discovered could easily be grasped by imagination: It occurs by leading things back to causes; when all those are finally led back to a single one, it is evident that there is no need of memory for all sciences. For whoever understands causes, will easily form anew in the brain by the impression of the cause the altogether vanished phantasms. This is the true art of memory and it is plain contrary to the art of this sorry fellow: not that his lacks in effect, but that it requires the whole space that should be occupied by better things and consists in an incorrect order: which [right] order in this is that the images be formed from one another as interdependent. He omits this—I don’t know whether advisedly—which is the key to the whole mystery. (AT X.230)

That is, if one understands causes, one can generate image from image from image using causal understanding, rather than spend one’s ingenuity coming up with laboriously constructed, arbitrary images according to the so-called art of memory (or the related Jesuit-Baroque techniques of constructing symbolic emblems).

Descartes concludes his reflection on his new art of imagination (as opposed to the art of memory) by hypothesizing yet another technique of cognitive imagining. One could take several related images and generate a new one, either common to all or otherwise generated from previous ones taken all together. In this way, he says, each would have a determinate relation to every other: “not only would there be a relation to the closest, but also to the others: so that should the fifth relate to the first by way of a spear thrown on the ground, the middle one [would be related] by stairs from which they descend, the second one by a dart projected toward it, and the third by some similar analogy [ratio]” (AT X.230). Although the exact course of his thinking here is obscure, it looks as though he believes that concrete symbols can be used to symbolize and even to generate an elaborately detailed, proportional correspondence between real things. Quite clearly Descartes had hopes that the imagination could be deployed in a far more cognitively active and productive way than the imagination of memory art did. The kinds of figures he had used earlier, in the Compendium of Music, to embody information were perhaps too modest: he wanted to be able not merely to summarize information in figures but to generate new relations, new determinate proportions, and correspondingly new knowledge.

Plato had of course understood that we can image a thing in different ways and at different ontological levels, and in at least one passage about the practical/ethical use of imagination Aristotle had explicitly argued that human beings (and perhaps some other animals) have the ability to form a new image out of many existing ones. Both of them had understood there to be an interdependence between the different sensitive and intellective powers of the soul, and Aristotle had presented imagination as a kind of proportionalized movement that allowed for its involvement in other psychological activities. Yet they never attempted to show imagination in its detailed psychological functioning, and there is no strong reason to think that, if they had done so, they would have ascribed to it the continuous dynamic processing for which Descartes argues.

Far more than the great Greek thinkers did, Descartes conceives the human mind to be constantly, energetically active, productive, and inventive. It takes in information through sensation and then immediately reforms it in re–presentations. It makes new images from existing ones by applying rules and relations, and thereby it gains new insights. It analyzes aspects and parts of appearances and synthesizes new appearances. It can take different appearances and extract from them what is common, and this leads to its ability to generate series of new appearances from familiar ones. It is hardly surprising, then, that its functions are capable of being applied methodically and systematically to everything that appears, because imagination is founded on and supported by the nature of images and appearances and the proportions that govern and hold among them. The question then becomes what the limits of it are. If imagination poses and solves problems and can even address the spiritual and intellectual realms, is there anything it cannot do?

6.4 Imaginative Representation and Manipulation

In the first and second parts of the Discourse Descartes tells readers that he was not happy with the way people applied their minds to understanding things. Although he says that he was educated at one of the best schools in Europe, it is clear that he was not satisfied with what he learned and how he learned it. And so he began searching for an alternative way, which culminated in his method and the discourse he wrote about it.

This “and so” conclusion is drawn too hastily, however, and it short-circuits understanding what is really at issue. The method was a response to a question, to a problem—the term that the Latin-speaking Descartes used was a traditional one in medieval thought, quaestio. What was the problem to which the method responded? It was that the intellectual culture Descartes found himself in was predicated on the existence of knowledge, yet precious few people knew how to show that they actually possessed it, much less knew what possessing knowledge means. To put the problem more pointedly: people claimed to have science, but what they really had was the art of disputation.

As Descartes says in both the early Regulae ad directionem ingenii and the mid-­career Discourse, someone who knows should be able to show others what he knows and what makes it knowledge. The only people who regularly, but not always, do this are mathematicians. Why? Not because mathematicians are better or smarter than anyone else, not because only mathematical things are knowable, but because mathematics deals with things that are easy to understand or can be reduced, which here means brought back, to easily understandable things. When a researcher arrives at what is easy to see, there is really nothing more she can do than “see the truth” of the thing. Anyone who can do this also has the key to leading others to see the truth. If someone lacks the power to see simple truths, however, then there is simply nothing to be done with him.

Throughout his career, in various ways, Descartes asserted that every person with the least bit of reason has the ability to see simple truths. But from very early in his career he claimed that not everyone is willing to do the work necessary for easy seeing. Heraclitus in Greek antiquity had declared that people prefer their private reason to publicly accessible logos; Descartes’s claim is a little bit more generous. Yet ultimately he came to believe that it was far easier to teach peasants to see easy truths than the already well educated.

The legacy of antipsychologism in the past two centuries has made Descartes’s claims even harder to credit insofar as it has thought that private intuition and ­introspection are suspect. A truth may be “clear and distinct” to you or to Descartes, the antipsychologist can say, but what if it’s not clear to me? There follows the counterclaim that truth ought to be public and conform to an objective standard. But “objective standard” really means “objectively verifiable standard,” and that raises the question of who does the verifying and how. Objectively verifiable standards are usually standards arrived at by training human beings to accept a common standard—but someone (presumably everyone who is trained) still has to see that the standard is met. The standard is intersubjective. There is no escape from the fact that ­someone—what we call a subject—is going to have to see that something is true, that something is this way and not that. And that repeats, with respect to antipsychologism in the twenty-first century, what Descartes had to say in the seventeenth with respect to most of his philosophic predecessors and contemporaries. The secret to mathematics was not that it was mathematics but that it had simple objects and simple standards to apply. If one could specify how one gets from ordinary experience to complex representations of it, find ways to analyze and simplify those complex representations, then judge them in light of the analysis, one could achieve something similar for nonmathematical experience as well. For human beings, the representations taken from experience in the first instance are images or phantasms of the experience. Once the thing is no longer in front of us we have no recourse other than dealing with the phantasms of it. So learning how to explicate the relationships of phantasms to one another and how to generate new phantasms from existing ones is likely to be useful in understanding all kinds of things in general. And something of this lesson from mathematics can be applied to virtually every thing of every kind, in particular when we can evaluate things in light of the more and the less, the larger and the smaller, the less intense and the more intense, etc. This is Descartes’s rediscovery of Plato’s conception of fields of experience, and of Aristotle’s notion that the fields are articulated by the more and the less between extremes.

At any rate, at some point between late 1619 and the mid-1620s Descartes began formalizing his early insights into imagination into a full-blown method. The first evidence of this is a work he never published or even finished during his lifetime, the Regulae ad directionem ingenii.Footnote 22 The intensive methodological reflection that Descartes began in this work eventually led to the 1637 Discourse and the three scientific essays to which it served as a preface.

As with the Meditations, which appears to speak slightingly of imagination, there is a “tradition” of selective quotation from the Regulae that allows imagination to be dismissed from further consideration. A favorite line to quote in this spirit occurs in rule 3, where Descartes says that the most basic way of knowing, intuition (intuitus, which is being defined in the passage), is “not the fluctuating faith of the senses or the fallacious judgment of a badly composing imagination” (AT X.368). This is the first mention of imagination in the work, and the long phrase certainly makes it seem that imagination (as well as sensation) is not to be trusted. Quite apart from issues of the immediate context, however, a major task of the work as a whole is to show how imagination can compose things accurately and well. A more subtle point is that the phrase “fallacious judgment of a badly composing imagination” leaves open where the fault for the fallacy lies. It seems clear that the imagination does the bad composing, but the fallacious judgment could be either imagination’s fault (the subjective genitive) or the fault of some other power that judges the badly composing imagination fallaciously (the objective genitive). If it is the fault of another power, then there are two points to be addressed: how and why imagination composes badly and how and why the judging power is mistaken. Descartes says later, in the middle of showing the proper uses of imagination, that “badly judging intellect” is responsible for error (see rule 14, AT X.443)—which, by the standards of those who selectively quote the earlier passage to dismiss the importance of imagination for Descartes, ought to lead them to dismiss intellect’s importance as well.

Another line of attack is drawn from the fact that in the mature account of method, the Discourse, imagination is conspicuous by its absence. As Leslie J. Beck pointed out in his book-length analysis of that work, however, even if the role of imagination is not given detailed attention, Descartes nevertheless does expressly mention it. It is just as central to the Discourse, argues Beck, as it was to the Regulae, and in the same ways. Immediately after stating the four rules of method near the end of part 2 of the Discourse, Descartes says that he had discovered that the best technique for applying them involved representing the parts and relations of problems by lines,

because I did not find anything simpler, nor anything that I could represent more distinctly to my imagination and senses; but, in order to retain them, or to understand several together, it was necessary for me to explicate them by certain symbols, as short as possible; and, by this means, I would borrow all the best from geometrical analysis and from algebra, and would correct all the defects of each by the other. (AT VI.20)

This is not a bad summary of the lesson of the Regulae. Thus, once one recognizes that for Descartes mathematical representation and problem solving is an imaginative activity, it is impossible to miss these clues. That they are still widely overlooked is undoubtedly an index of the degree to which our culture assumes without evidence that mathematics is basically a rational activity. Of course it is rational, because it produces rationes, the setting of things in determinate relations to one another. But, for the most part, such setting things in determinate relations has to be done in fields of images.

The Regulae presents an art of problem solving that is supposed to be especially well adapted to the human being’s psychological capacities. The senses, memory, and especially imagination are to be deployed in aid of the intellect. Most errors people make, Descartes asserts in rule 14, are due to the intellect, especially when it makes judgments without reference to an imaginable object (i.e., there may be thinking without images, but it usually goes astray because it is not thinking about anything in particular). Descartes not only follows the late Renaissance tendency to reduce the number of internal senses to a minimum (common sensation, memory, and imagination), he in essence reduces all of them to functions of imagination. In particular, the ingenium, the “ingenuity” or “mind” or “native wit” that is to be directed by the rules,Footnote 23 is defined in rule 12 as the knowing force (vis cognoscens) “when it at one moment forms new ideas in phantasia [the organ and place of imagination], at another applies itself to those already made” (AT X.416). But ideas in phantasia are images; and thus ingenium is the power of conceiving, recalling, varying, and developing images—as one does in his mathematics, but not just in his mathematics.

The Regulae was to consist of three parts, with twelve rules in each. The first twelve deal with the method of solving problems in general, chiefly by discussing the human powers that are best fitted for understanding and solving problems (this part is essentially complete, though a few rules seem to have gaps). The second twelve were to deal with “perfect problems,” those that are sufficiently well defined to provide everything needed for a solution (thirteen through eighteen exist in fairly complete form, nineteen through twenty-one have headings only, twenty-two through twenty-four are nonexistent). These rules were to show how to use figurative and symbolic representation of the givens of a problem and then to break them down and combine them in the course of problem solving. The figures were chiefly geometrical line segments and plane figures produced from line segments; the symbols were marks or names of points, segments, and figures. Part two as it was left hardly gets further than showing how to add, subtract, and multiply line lengths, and the proper use of algebra is hardly more than mentioned before the work breaks off. Part three, which was to consist of twelve rules regarding “imperfect problems,” does not exist at all.

What was Descartes trying to do? In rule 14 he says outright that all the problems being solved by the representational and manipulative techniques he introduces will be using imagination, because quite literally one will be making and transforming figures and images and generating symbolic representations that will track and anticipate the transformations. It appears that Descartes is trying to give a systematic account of the problem-solving imagining he had been using ad hoc in his earlier notes and the Compendium musicae. The clearest sign of this is that he advises using only points, lines, and plane figures in the representation of problem elements and calculations; that is, one should avoid three-dimensional figures. This advice is a response to the difficulty mathematicians had had since antiquity in conceiving the multiplication of more than three numbers. Advances in mathematics since the sixteenth century, to which Descartes himself contributed in no small part, introduced the techniques that we use today: you multiply two of the numbers, you multiply the resulting number by the third, that result by the fourth, etc., as many times as you need to. This is easy because we take numbers as absolute: they are defined in terms of nothing but themselves, and using any operation to combine two of them gives just another number.

Before modern mathematics, however, numbers were regarded as attached to what they were the number of. In pure mathematics, number was considered to be the measure of geometrical figures. The measure of a line segment was not the same kind of measure as the measure of a surface area. Whenever you multiplied two simple numbers you were understood to really be creating a plane figure. Multiplying m times n was taking a line m units long and another n units long and producing the rectangle that had the line of length m as one side and the line of length n as the adjacent side. The product of m and n was therefore an area measured in square units. In order to multiply three numbers you had to create a three-dimensional ­figure—a rectangular parallelpiped, to give it a name. And multiplying by a fourth number was technically impossible, because it would have required entering a fourth spatial dimension. Ancient mathematicians had developed “workarounds” for this, but they had no systematic justification.

It is no accident that Descartes’s earliest mathematical thought concerned problems that required shifting back and forth, from one to two to three dimensions and beyond, for example by making a triangular area representing distance out of two line segments representing speed and time, and using the figures to solve problems of proportional relations between the parts. The theory of proportionality that was so much on Descartes’s mind was the heart of the ancient methods of calculation. If you could not directly compare a square area to a linear length, you could set up a relation between two square areas that was proportional or equal to the relation between two line lengths (square area #1 is to square area #2 as length of line A is to length of line B); and indirectly (alternando, in the Latin terminology used for this kind of proportion) you could say that the proportional relation of the first square area to line length A was the same as the proportional relation of the second square area to line length B. To us, who are used to algebraic imagining, this is an overcomplicated way of saying that if you have the equation “area #1/area #2 = length A/length B,” then you can multiply both sides of the equation by “area #2/length A” to get the new equation “area #1/length A = area #2/length B.” Notice that what allows us to do this is the postulate that we can treat both area and length indifferently as absolute numbers. This violates the traditional sense that a number cannot be so cavalierly separated from what it is a number of. For the ancient and medieval mathematicians, only measures of the same kind could be compared directly.

The fact that Descartes proposed a not entirely uncomplicated alternative shows that, although he was not committed to the strict limits of the older methods, he was not quite ready to treat numbers as absolute. His alternative techniques allowed one to convert any line length to an area or any area to a line length, although they took line length as the authoritative or canonical form of measure because it was simplest. Thus if you needed to multiply four numbers you would represent each by a line length, combine two into a rectangle to get the multiplied area, apply the conversion technique to reduce this to an equivalent line length, multiply this new length by the third original length in a new rectangle, convert this rectangle to an equivalent line length, multiply this new line by the fourth original length in a third rectangle, then convert this rectangle to a line that (finally) represents the product of multiplying all four numbers! Addition and subtraction were comparatively easy, but division of two numbers and the taking of roots was complicated.

Even with our modern computational techniques there are remnants of this ancient problem that we still have to deal with. If you have twelve apples to distribute among six people you divide the twelve apples by the six people, get two, and proceed to hand two apples to each person. But what does it mean to divide apples by people? Giving it some thought, we see that we separate out the numbers from what they are numbers of and perform the division, then we recall that the answer is neither apples nor persons but “apples per person.” If this seems like a mere technicality, it is no mere technicality for natural scientists when they multiply acceleration by elapsed time to find out how much faster something moves after that number of seconds has passed. Acceleration is (say) meters per second per second, or m/s2, and the elapsed time is seconds, so the product is m/s, which is velocity (or speed, if we ignore direction—another complication that we don’t always attend to!). But the genius of modern as opposed to ancient arithmetic and algebra is that we can separate the reckoning of the numbers from the reckoning of the units that those numbers express, only to combine them again once calculations are at an end. Descartes was precisely on the cusp of the change from the old conception to the new.

All of this is dealing with ratios, and thus if any kind of thought deserves to be called rational it is this. Yet it is simultaneously all about routines for comparative imagining. The reader probably has a sense of relief that in order to multiply and divide we don’t have to worry about these complications any more—and because of electronic calculators we often no longer need to recall the algorithms for the operations on absolute numbers but can just punch the buttons. That is, we are very happy to be relieved of the need to think or imagine these things according to the older ways. What exactly the mental operations are that correspond to mathematical calculation more completely eludes us the more we use machines to do the reckoning.

The first thing to say about these mental operations is that, even today, there has to be some theory of what representation is of and what representation implies. That is, we need to deal with the ontology and epistemology of representation rather than focus on practical techniques of algorithmic imagining. The “we need” has to be taken a bit loosely, of course, since not that many people feel such a need. Descartes, historically alive to the conceptual topologies of his heritage, had a keen sense of all these concerns. Insofar as such questions are to us a dead letter, we are satisfied with imagining (rather weakly) that someone, somewhere actually understands and has justified what most of us do, so most of us can work in secure ignorance. But that means we have historically recapitulated the problem to which the method was a response. Analogous to medieval and early modern scholars, we use “rational” techniques that conceal our ignorance of or indifference to the specific character of truth.

6.5 The Dynamism of Imaginative Ingenuity

The first rule of the Regulae begins with the reflection that the light of reason shines on everything knowable and that there is a single method of knowing, in accordance with this light, whatever comes before the mind. The method has to acknowledge the nature and limitations of the human powers that make knowing possible and recognize what sorts of things are most knowable. Mathematics can provide a model for knowing because the things it studies are “so pure and simple that they make no assumptions that experience might render uncertain” (AT X.365). This does not mean that only mathematical things are knowable. Rather, what is simple is more knowable than what is complex, and when we are faced with complex objects we must face up to this difference and develop appropriate ways of relating what is complex to what is simple.

Descartes in rule 3 argues that there are only two acts of mind that are useful for knowing, intuition and deduction.Footnote 24 Intuition is the simple, clear, indubitable recognition of truth by “a clear and attentive mind.” The examples he gives are of a person’s intuiting that he exists, that he is thinking, that a triangle is bounded by just three lines and a sphere by a single surface. It is something that you can see all at once when you have properly prepared the view (about which preparation there will be much more to say in the later rules). When something simple is brought before the mind—whether it is an image or something more than an image—any person sensitive to the light of reason will see, will intuit, simple truths about it. Deduction is in essence a series of intuitions: we see first one truth, then a second, then a third, etc., and recognize that the last comes from the first, step by step. He compares this to knowing that the first link in a chain is joined to the last by inspecting, rapidly and one after another, each of the intermediate links. In rule 7 (AT X.388) he calls this movement of thought, from one thing to another in a continuous sweep, an act of the imagination. In a sense, the imaginative process that the Compendium of Music had analyzed as necessary for synthesizing the rhythm of song has been generalized to “seeing the truth” of any complex situation that we are able to articulate into a series of clear elements. In rule 16 he goes further: he suggests that at least in some cases what we originally come to know by deduction, in a fast but still step-by-step sweep of one intuition after another, could itself come to be apprehended by a single intuition.

How would one arrive at the point of being able to do all this mental activity as well as possible? Let us take advantage of the traditional name of the work, Regulae, rules, to point out that the work teaches how the imagination can be ruled and regulated (provided with a measuring stick) by the knowing power, by reason, by rationality; and that rationality works by setting one thing into a determinate relationship, into a ratio, with another. I believe that, rather than playing with words, this is an evocation not just of what is meant by Descartes but what is implied by the entire history of the conceptual topology of imagination and reason in Western thought.

Traditional logical demonstration, as well as mathematical proof modeled by Euclidean geometry, proceeds step by step. Descartes’s complaint was not with the stepwise advance, because it was characteristic of his own method in the Regulae as well. What he demanded was what they could or did not provide: the opportunity for insight, for seeing or intuiting the truth. Aristotelian-style logical demonstration was fine for reminding yourself of truths you already knew (all men are mortal, Athenians are men, therefore Athenians are mortal) but not for discovering anything new. His complaint about earlier and much contemporary mathematics was that it was filled with obscurities, both in concept and expression; even when what mathematicians presented was true, the clarity of presentation left a great deal to be desired. He countered that simple truths can be recognized by everyone possessing the least bit of rationality. He thought that people—especially educated people—neglected easy and commonplace truths in the pursuit of grand truths and mysteries, and thereby they impaired their ability to recognize more elemental and productive kinds of truth. The Regulae therefore counsels its readers to practice looking for truths in all circumstances of life, and especially advises looking for principles of organization and order no matter how humble or trivial they might seem. For example, they should examine the orderliness in the construction of a watch or a balance scale, the various weaves of fabric, and tactics useful for solving games and puzzles (see rule 9, AT X.401–403; rule 10, AT X.404; rule 7, AT X.391). The point is this: if you find a principle of order in which B or C follows A, the mind knows where to go when it comes upon A. Seeing that something is A and that it is followed immediately by B is one of the simplest acts that intuition performs, and the recognition of the organizational power of orderly series of things is the result of many intuitions put into series, which rule 3 calls deduction. The mind’s motion is thus ruled and regulated. In Aristotelian terms, the intellect sees the orderly forms in the phantasms.

It is no accident that rule 4 emphasizes order and measure as the principles of method, and it is no accident that, when rule 6 spells out “the whole secret of the art” of solving problems in an orderly way, it focuses on putting things into orderly series. The leading principle explained there is the degree or proportion in which a thing contains or participates in a nature. The rule, however, dismisses thinking of the nature as an essence that constitutes a thing, because such an approach is largely useless for solving problems. Some examples (not in the Regulae) will help illustrate what this means. A farmer planning his spring planting will not be assisted by reasoning from the essence of farming, the essence of plants, or the essence of spring. A general will never get around to organizing the battalions of his army if he has to reflect on the essence of an army or the essence of man. The essences of such things may not be totally irrelevant—for example, tactics require that soldiers accept their orders, and to accept and understand orders they must be rational animals—but they are ordinarily only of tangential relevance to what is being sought in the quaestio. It is not so much that essences do not exist as that they don’t usually help one find an answer to a given question. Essences can, for the most part, be taken for granted. A person solving a problem must, in the first instance, know enough to pare down the considerations to what is essential for solving the problem at hand.

Descartes first mentions specific examples of natures when he defines things that are called “absolute”: “whatever contains in itself the pure and simple nature that is in question: as [for example] everything that is considered as independent, a cause, simple, universal, one, equal, similar, right, or others of this kind” (rule 6, AT X.381). He contrasts to the absolute the “respective” (respectivum, usually translated as “relative”), “whatever participates in the same nature or at least in something from it, in accordance with which it can be referred to the absolute and deduced from it through some series,” as examples of which he gives what is “dependent, effect, composite, particular, many, unequal, dissimilar, oblique, etc.” (AT X.382). This second listing consists of opposites or contraries of the first list. What does this all mean?

We must recall that, in his earliest notes about imagination, Descartes conceived of taking something from experience and re–portraying or re–presenting it. The thing has already been received into mind: now we step back and take it in. Even if we only try to preserve the thing as received, something new will be added to the experience. There will be something dramatically newer if we “change the take.”Footnote 25 This not only creates the different levels or planes for imagining, it also means we are taking the thing in a certain respect, relative to a certain portrayable or imageable “nature” that we want to highlight. Rule 6 emphasizes not the representation or portrayal per se but the thing’s relation to the nature of which the experience is a specimen, and this allows us to put many things and their representations into a discrete series (the beginning of a matrix) in light of their “containment of” or “participation in” the nature. Once one has a nature in mind, one can put things or objects in a series with respect to the degree of participation in the nature, up to the extreme of wholly containing it. In any given series the “absolute” is the first member of the series, the thing that most participates in the nature, at least in one’s own experience; those that are more remote are called “respective.”

In rule 6 Descartes does not give many examples of what he means,Footnote 26 but it is not hard to construct a few plausible ones. A person considering a career change may be seeking autonomy; she will accordingly consider the options according to how much each participates in autonomy. If she is practical she will notice that the more autonomy a job has, the more initiative it entails, and perhaps also the more variability with respect to income. These are comparative relations that can be put into roughly parallel sequences, and to some degree they might be quantified and thus measured (this would, by the way, be an imperfect problem, in which the information is not complete enough to generate perfect solutions—the kind of problem part 3 of the Regulae was going to address).

If we are considering the purchase of a house and have visited ten, we may well actually put them order, from one to ten, according to their participation in various natures. First, there is the series of the asking prices; the nature being participated in might be called simply price, with a convenient unit of measure, the dollar ($328,000), or perhaps the unit will be 1,000 dollars ($328 K). But no house buyer will stop with that. For example, he will compare the houses according to their participation in the nature area—there is a convenient unit measure (square feet or meters) that allows us to put them in a very strict series. The real-estate agent will probably have informed him of the average price per square foot in each neighborhood visited (these can be put in a series); thus the buyer will have been provided with mathematical proportions that allow the production of a new series: what the houses should cost if they sold at the average square-foot price for their neighborhoods. This series (call it the expected price series) will probably be at least slightly different from the asking-price series. The buyer can create yet another series by subtracting the expected price from the asking price: this will give a comparison of how much above or below the expected price sellers are pegging their asking prices. This series might then be informally coordinated by the buyer or the real-estate agent to certain psychological traits of the sellers: pride in how well they have maintained the house, financial realism or irrealism, greed, etc.

Of course if we are good house buyers we are not yet finished. Whether a house is a real bargain depends on other factors beyond the discrepancy between asking price and average expected price—for example, we will create a series corresponding to refurbishing costs. Probably we will need to break down refurbishing into different factors, each of which will give rise to a new series—roof repair, HVAC replacement, painting—that will lead to ever more intricate comparisons. If we are worried about heating and air-conditioning costs we can compare the houses in volume; this series will not diverge much from the area series, except insofar as the houses have different ceiling heights. But volume is just one of the factors that enter into heating costs, so we might have to sketch out some other quick comparisons of window age and quality, insulation, blower capacity of the furnace units, etc.—and we might finally devise a formula that takes all these factors into account (contrariwise, we could just do an “eyeball” estimate). We will compare them according to the number of bedrooms or baths. We will compare according to less clear-cut features or natures, like expandability or brightness or airiness or comfort. We can easily imagine coming up with some kind of measure for expandability and brightness, but airiness is a little harder—it depends on feel, though it is probably also related to ceiling height, overall room proportions, and admitted light (which is related to brightness). Comfort is probably the least tangible of all, though one could probably specify certain factors that enter into it, like ease of movement through rooms and the appropriateness of the layout of the house to one’s preferred ways of occupying a living space. In each of the series we create, the absolute is the member of the series that participates most in the nature, and all the other members of the same series will be called respective or relative.

We should not think that by using the example of house buying we are trivializing the procedure, as though only solemn acts of scientific, mathematical, and philosophical reason are eligible for consideration. Descartes’s complaint is that people simply do not consider how many different types of order, nature, and measure are exhibited to them every day, in every way, and how the various natures and measures are interrelated. If they do not see it in ordinary things, they will not know how to find it in the complex. Descartes points out (in rule 12) that sometimes an artificial or fictitious “nature” can serve for ordering the materials of a problem. The key is less whether the nature is solidly real than whether, considering things “in respect to” the possibly fictitious (purely imaginative?) nature, it provides the ability to put the things into an articulated series.

Organizing things into series according to natures and dimensions is a kind of preparatory work. Each problem we face requires us to deal with such series. Moreover, every series we have created in the past serves as background information and knowledge for addressing future problems. They become part of our imaginative “tool kit.” Of course geometry and arithmetic/algebra problems are usually far more accurately solvable than real-world problems because the natures involved are already quite simplified. Geometrical things all participate in the nature extension or spatiality. One way to order them is according to the traditional dimensions of spatiality: points have no dimensions, lines have one, planes have two, and ­cartesian or euclidean space three. If these are the natures one is considering, then (for example) every closed plane figure wholly contains the nature of being two-­dimensional and closed. Another way they can be ordered and compared is according to their size in the appropriate dimension; that is, we put into series line segments by their measured length, two-dimensional figures by their measured area, solids by their volume.

Most series one forms are a means rather than an end, and often previous series do not have to be explicitly called back to mind and expressly presented in their full extent. Each serves as a limited but extendable topological field matrix with marked positions into which we can project a new problem or, more likely, some aspect of a new problem. So if one is posed a geometry problem that specifies three line segments, an angle, and an area, and then asked to construct from them a four-sided plane figure of the specified area with two of the line segments meeting at the specified angle and the third joined at one of its endpoints to the open endpoint of one of the other two segments, one may well not actually array the three line segments from shortest to longest, but instead one will try to remember orderly procedures one has used in the past to solve problems of the same general type. In this way one interrelates different series, and in effect creates a matrix of two, three, or more dimensions. One will also very likely apply some of the other techniques that Descartes presents in the Regulae. For example, one will use letter designations for the line segments (a, b, c) that are known, the unknown side will not at first be drawn but its length will be indicated by a special letter marking an unknown (x), and one will use various formulas (equations) one knows from geometry, trigonometry, and algebra to express the area of a quadrilateral consisting of sides a, b, c, and x. We will then be able to manipulate the equations to give us insight into the nature of the problem and its solution. Perhaps the equations will give us a direct solution in a single grand algebraic manipulation; more likely we will use only as much algebra as we need at a given moment, constantly correlating it with some aspect of the problem at hand. And even if we do arrive at a single grand solution, we will still need at the end to actually construct the solution to prove that the equational ­solution corresponds to something real.

Problem solving according to series ordered by the degree of participation in natures does not, in the process of arriving at a solution, have to respect the intrinsic essence of a thing—not even when the thing is mathematical (say, a circle or a triangle). Think about this for a moment in terms of our everyday methods of problem solving. When asked to divide twelve oranges among six people we for a moment just forget about the oranges and the people and calculate the numbers (twelve divided by two). If we were asked to divide twelve oranges among five people and realized it was 2.4 per person, we could apply an axe to the oranges. But it would at that point be wiser to remember something about the way oranges are (their essence?): they come naturally in sections. To solve the problem in certain circumstances (say, everyone wants to eat the oranges right now) it would be advisable to give two whole ones to each of the five people and then to peel the two remaining ones and divide them into sections. We might hope that those remaining two naturally divided into fifteen or twenty sections so we could give three or four to each person, but even if they did not, everyone would probably be satisfied by an approximation. If we were making orange juice, on the other hand, all this subdivision would be irrelevant to the solution. In the process of problem solving, we set aside the particular essences of things for the time being and think only in terms of the natures and aspects relevant to the problem at hand, and we do our calculations according to the series and measures we have established. This goes back to the basic situation of the Regulae: we have finite powers for understanding things, there is a limit to how many things we can consider simultaneously, we must break down complicated problems into parts or aspects for easier solution. We do keep track of what comes from what and what kind each thing is, but we do not actively consider these at every moment. This is the working situation of intelligent imagining.

Descartes is careful to note that, even in the case of expressly mathematical problems, we do not necessarily work the solution in terms of the kind of object we have before us. If the givens of a geometry problem are six regular solids with particular dimensions and we need to order them with respect to volume, for the sake of problem solving we may decide to convert the volume of each into a rectangular representation or into straight line lengths or pure numbers; the deciding factor is convenience. As I have already noted, Descartes recognized further that we could put a numerical symbol (or some other symbol) to stand for the line segment or the rectangle, and that, if we did not know a given factor but knew or suspected that it existed, we could represent it by a “dummy” representative, like the letters x or y or z—the classic representatives of unknowns in analytic geometry. Indeed, one can argue that his essential innovation was to treat unknowns in the process of problem solving exactly the way knowns are treated, the point being that you keep manipulating the formulas and the figures until you can directly determine the actual value of the unknown. In terms of equation manipulation, if you start with x 2 − 6x + 9 = 0, you want to be able to manipulate it so that it reduces to x = 3. Thus mathematics problems are not different from other problems: one chooses the way one will represent the knowns and unknowns of the problem not according to the nature or essence of the givens (for example, that one is dealing with squares or heptagons) but according to whether the way is convenient for getting at the desired solution. For any particular problem, it is possible—even likely—that there will be different sets of representations and/or different approaches that will provide one with the (same) solution.

In rule 12 Descartes points out that if we are dealing with colors we might want to represent the colors white, blue, and red, respectively, using a series of figures (see Fig. 6.1). This is a case of suggesting a fictional principle of ordering. Descartes is not asserting that this ordering–patterning is physically correct. What stands in its favor is that color is produced by light from an object striking the eye; we can easily imagine or conceive that some two-dimensional pattern might be impressed on the retina (which Descartes calls “the first opaque membrane”), and so colors might well be distinguished from one another naturally by such pattern differences. This type of pattern representation further suggests the possibility that there might exist a “system” of such patterns that would allow us to “calculate” or predict the result of adding different colors to one another. He is not arguing that this representation is the right one or will lead to such a system, but rather pointing out that, among the almost limitless number of possible two-dimensional patterns, some set might work in the way he suggests. In the long run, we are likely to discover a set of imaginative representations that allow us, in the process of calculation, to set aside thinking and experiencing the actual colors long enough to do the calculation. This is, in very rough approximation, how modern systems of color representation work. Clearly Descartes is presenting the question of two-dimensional patterns for colors not as a solution to a “perfectly understood” problem but instead as an example of how we can imaginatively approach a problem when we have ideas that are only plausible. We look at the phenomena we are interested in as a field (colors in general), we note their discrimination from one another (as hues), we try to find some other orderly field we are acquainted with (two-dimensional patterns), we note that in such a field we can create more complex patterns by combining simpler ones (a pattern of parallel vertical lines can be overlaid with a pattern of parallel oblique lines to yield a cross-hatched pattern), and we see whether we can mark more explicitly and determinately the orderliness of one (the hues within the field of colors) by representing them in the field of the other (the line patterns). If the system we come up with works, then we have a solution to the problem. It may be final, or it may need further refinement; it may be real, or it may turn out to be artificial. But when we next address the situation we are better off than when we started, because we have some principles of order, organization, and representation to call upon. Moreover, we may discover complexities in the colors that cannot be adequately represented by existing line patterns, and that can push us to advances in our understanding of lines patterns, eventually even apart from colors. And if some day it occurs to us that the patterns among hues in some respects resemble the patterns among tones in an eight-step musical scale, we might be tempted (as Newton was in his optics) to use well-understood mathematical representations of the latter to help order our understanding of the former. This is how Descartes’s biplanar or dual-field imagining works.

Fig. 6.1
figure 00061

Descartes’s hypothetical representations of white, blue, and red

Full size image

Finally it is possible to understand more clearly how intuition and deduction function in the method of the Regulae. Intuition, I said earlier, is a simple act but not necessarily directed to simple things. Seeing green may be an incredibly complex physical, physiological, and psychological phenomenon, but simply to see that something is green or looks green is a matter of seeing, and to ascertain such seeing is a matter of making it as clear as possible. If we are trying to determine the color of an object through rain or fog, if it is behind a screen, if the illuminating light is dim or colored, we will need to do some work so that the phenomenon becomes clearer. The apparatus of a mechanical watch or a computer may be complicated, but someone experienced in their construction can clearly and distinctly perceive how it (as a whole) and each part (as distinct from others) are functioning when the rest of us only see “a lot of things” there. Clear (or perspicuous) and distinct seeing and understanding can be trained and even taught. An interior designer as well as a color scientist can teach us things about how we can make colors stand out more sharply; the watchmaker or computer designer can clarify and distinguish for us the parts of a computer and how they go together.

Although seeing in the first instance appears to be just registering what is there, as soon as there is a problem to be solved we have to mentally reconfigure and recontextualize the thing, the parts, and their situation. The very act of taking a look at something, re–presenting it, and setting it into distinct relationship with other things is a work of clarifying and distinguishing, of making things clear and distinct. Each moment of clarified and distinguished seeing is a moment of intuition. Every time we take a step from one moment of intuition to another, we are engaged in deduction, or at least an attempt at deduction. This is the central work of imagining in Descartes’s method.Footnote 27

6.6 How the Knowing Power Recognizes Itself in Imagining

The two-imaginations note of notebook C had not described the exact relationship between figurative imagining and the intellective use of figures and images. By the time (presumably a few years later) that Descartes formulated the matter in the Regulae there is no doubt about it: as rule 12 explains, imagination is not separate from intellect or the knowing power, it is a special kind of highly active work of presenting, representing, organizing, and manipulating that the knowing power does in and through the medium of the organ of imagination in the brain. All by itself, this makes it evident that Descartes would have a harder time than Aristotle ascribing imagination in this sense to animals without also ascribing to them the intellective power that directs this kind of work. Descartes did, however, accept that what produces the physical impression on the eye sets off a chain of physical/physiological actions and reactions; he possibly even accepted that certain animals are able to perceive the hue in color, and that in combination with the activity of the organ of phantasia and memory locations in the brain this perception might produce a kind of sorting of experiences that would lead to an appropriate response.Footnote 28 None of that, however, could be deliberative or even conscious, at least in the sense that human beings can (for instance) perceive colors as colors. The animal would not be capable of consciously placing the experience at one level in relation to another (which in fact begins when one sees teal as a blue, and blue as a color; predication, stating that S is P, is a biplanar act). This is another way of saying that, for Descartes, animals are complicated stimulus–response devices, and thus any consciousness they might have is certainly not biplanar. Their sense organs and brains can acquire image–impressions, but they have no ingenial power of manipulating them. Only the human being has intellect, and intellect is precisely the power that can take impressions in the brain not merely as appearances but as images. Once intellect takes an image as an image, the world of re–presentation and re–imaging commences.

But then a new but also quite traditional question arises: is there any kind of mental activity for human beings that does not involve imagining—in Descartes’s mature conception, that does not involve the physical activity at the pineal gland? Is there for Descartes any thinking without imagination? The Regulae talks of “pure intellect acting on its own” but says little about it. It is nevertheless possible to infer something of what he means.

Pure intellect is required for thinking what is other than an image: to think what is not an image, an aspect of an image, or a “take” on an image. Images are positive appearances to mind. They can be considered, they can be changed, they can be re–conceived and re–presented. But the act of imagining per se cannot negate images. To put it in terms of the knowing power or intellect: imagining is the knowing power’s forming, holding, varying, and reconfiguring image presentations by means of intellect’s actions in or on the organ of phantasia (the “pineal gland”). What is image or of image or related to image is conceived by the knowing power in the form of images. But negation is different. It is the work of pure intellect. One is tempted say that, when the knowing power recognizes what is wholly not image or entirely unconnected to images, it has to move “up and out” of the gland’s presentations. When we say that God is not imaginable we mean that he is not presentable in any way in or by an image. So we cannot understand the truth of the assertion “God is not an image” by observing an image, no matter how complex or dynamic. Rather, that statement requires that we observe what imagining does and the kinds of things it works with, then recognizing that God cannot be that in any way at all. But note well: thinking this thought clearly and distinctly requires having (had) images in mind, as well as having the thought of God. You cannot distinguish God from the image realm unless you have brought both to mind and see/intuit a basic difference. Similarly, you cannot clearly and distinctly perceive that imagination and intellection are different without having brought both before the mind in their difference; nor can you say that body is really distinct from mind without presenting body and (self-)presenting mind and taking in the difference.

In rule 14 Descartes makes clear that even some truths about extension require more than imagination. In order to see the truth of the assertion that “a geometric figure is extended” one must present to one’s mind a representative geometric figure. In the very act of presenting a geometric figure of any kind one simultaneously presents something extended. Yet it is also true that “figure is not extension” (that is, figure is extended, but that does not mean figure is identical with extension), and to think that thought clearly and distinctly is not just a matter of having figure and extension clearly in mind. The figure that one has in mind is actually extended, it has or contains extension: but figure in its essence is not the same as extension in its essence.

This does not justify the conclusion, however, that this act of differentiation of figure from extension somehow steps completely out of the realm of the imaginative into the realm of pure rationality.Footnote 29 Grasping what happens in this differential, negational thinking is subtle. To think “figure is not extension” accurately and properly, one must first imagine something figural. Second, one must in a manner “step away” from the particular image, to see it not merely as a (specific) figure but as representative of all figure. (Of course one cannot do this as a newborn, one must have acquired a sufficiently ample experience of natures and series through which one sees the truth of this representation.) Third, one must look upon the first and second thoughts and “step back” again, to think them not as figural but as extensional. Fourth, one reflects that the “taking” of the same figure as “a figure,” as “figure,” as “an extended thing,” and the like is in each case a different taking: the same figure can be taken in many different ways. The sequence of thought here progressively moves further and further from the thing with all its original specific determinations; this moving away from the object of thought and the truths that become correspondingly evident by moving away—a phenomenon that in medieval philosophy was called remotion Footnote 30—is a power of intellect and only of intellect, according to Descartes.

By following the method of series making one learns to put a single object into different series according to its participation in different natures; the ways of thinking the same thing (also the image of the thing) are at least as numerous as the number and complexity of natures. By carefully attending to these experienced differences in the presence of the “same” image or figure, one comes to recognize that the knowledge of the nonidentity of figure and extension derives not from the presence of different images but rather from different ways of taking the same images. It is the actor or agent who makes the differentiation by recognizing that the imaginative “takes” on the thing are different. Descartes says that this truth is thought by pure intellect, but clearly he is not implying that thinking this truth annuls all images and imagining. Properly speaking, one has to start with something imaginable (figure), move on to see the imaginable thing in a different respect (extension), and notice that the two are not the same precisely insofar as one has the ability to compare them. Intellect recognizes this, not by having one figure or two figures in mind but rather by having a figure in mind, noting its extension, then taking that figure as representative of all figures, as such noting that all figures must contain extension, recognizing that taking a figure as figure and taking it as extended reflect a difference in the taking of the presence of natures, then seeing that the different ways of taking the natures in the same figure is due not to the figure but to the knowing power. Only the knowing power, the intellect, can perform this differentiation. Thus one clearly and distinctly intuits the difference between imagining and intellection and recognizes that intellection is not per se a forming and holding of an image but rather also the stepping away from any image as image.

Perhaps I am too much belaboring the point about the quantity and quality of activity involved, both imagining and intellective, in thinking what is not imaginal. But I do this because conventional conceptions of cartesianismFootnote 31 underestimate the degree of activity involved in the (clarifying and distinguishing) work of thought. The cartesians—that is, followers of Descartes—and the post-cartesians—that is, later philosophers who, in various ways, responded to Descartes—came to speak not of clear and distinct perceiving but clear and distinct ideas. Ideas are not, however, intrinsically clear and distinct. Rather—and to take very seriously the definition of idea that Descartes gave in his reply to the second set of objections to the Meditations—the idea is the form of what appears to consciousness, but the total appearance is form plus “matter.” Here that “plus” has to be understood as indicating the actual total appearance in consciousness, with consciousness understood as having some medium in which appearances vary from moment to moment; this medium or receptivity is like the “matter” of consciousness that can be, and is, constantly formed and reformed. The idea is thus not simply a static form but a formative agent in this medium of receptive consciousness.

The upshot is that one cannot simply “insert” into mind a “clear and distinct idea”—of extension, of thinking, of ego, of God—and without further ado think it precisely as such. It is certainly easier to think extension or self or God clearly and distinctly after one has done it before, but in every instance of such thinking it still requires preparatory work by the mind: the mind’s clarifying and distinguishing activity. Thinking for Descartes is not simply “having an idea.” What the mind thinks always occurs in a context. This is a generalization of his principle that “givens” are precisely what they are as givens of a problem. In one problem a line segment represents speed, in another it represents degree of pain, in a third it represents the unity of God. Thinking is seeing appearances in a context, against a background, taking different approaches to them, trying to vary them, trying to situate them against new backgrounds, etc. It is only in this way that they can become ever clearer, and it is only by being set in contrast to other things against a background that allows them to become ever more distinct. Most of this human work of thinking is in the imaginative mode. But it is only by working imagination as hard as is humanly possible that one begins to genuinely glimpse the possibility of a thinking that is other than imagining.

When Thomas Hobbes was asked to comment on Descartes’s Meditations,Footnote 32 this partisan of the notion that thinking is nothing but the having of sequences of images and assigning them names took the author’s “idea” as a synonym for “image.” Hobbes criticized in particular Descartes’s use of “idea” for the thought of God. According to his philosophy, we have no image of God but only a name. Over and over Hobbes argues that we have no idea of God, only a name; over and over Descartes responds that we do have such an idea, although it is not an image, and Hobbes’s use of the word “God” is a sure indication that, at least in some unclear way, Hobbes has the idea.Footnote 33

Whoever engineered this set of objections and replies into a fictional disputation produced the appearance of an increasing irritation of the philosophers with one another’s stubbornness. The debate, if it can be called that, would be more amusing than enlightening were it not for an aside that Descartes makes about why he chose the word “idea” in the first place (in the reply to Hobbes’s fifth objection). If you want “idea” to be used only for “the images of material things depicted in our corporeal phantasia,” then it is true that we do not have an idea of either angels or God. But especially in the passages Hobbes objects to, Descartes says he was careful to use the word idea “for everything immediately perceived by the mind, so that, when I will and I fear, because I simultaneously perceive myself to will and to fear, this same volition and fear are counted by me among ideas.” Then comes a surprising remark: “And I used this word because it was already commonly used by philosophers to signify the forms of perception of the divine mind, even though we recognize [there is] no phantasia in God” (AT VI.181). That is, “idea” signifies, by analogy with God, the forms of perception of the human mind; but strictly speaking the analogy works only if God has a corporeal imagination, which he does not!

Descartes does not draw out the counterfactual comparison any further, but it shows as clearly as could be desired that even if there are ideas that are not properly images, ideas are the divine analogue of human images, the forms of God’s imagination. Ideas are images raised to a higher power, and even if God does not have corporeal imagination it is not nonsensical to think of them as a higher kind of image. Since the divine ideas were also, under the influence of Augustine, understood as the exemplars according to which all God’s creatures were made, they have to be understood not merely as passive shapes of creatures but as dynamically formative. If images for human beings are distinguishable from ideas that cannot be imaged, the ideas are nevertheless conceived as imagelike. Both the idea and the image are re–presentations in a plane different from that of what they represent, and the being that they re–present is, in principle, fuller and more ample than the re–presentation actually shows. Yet even the re–presentations are actualizations in the appearance–potentialities of their proper planes of representation. They are less static views than generated and generative appearances. Neither thinking nor imagining is passive, because both operate within and between planes, even when they try to hold something constantly before the mind.

Forgetting the work of variation and recontextualization, of placing and replacing appearances to consciousness in different kinds of situations and explicitly recognizing these as occurring in and between different planes, is to entirely miss the nature and character of what Descartes conceives thought (and imagining as a type of thought) to be. To miss that is to miss the nature and character of cogitationes and cogitare. In the philosopher who is remembered as the proponent of the cogito, of the cogito–argument, that is to miss nearly everything.

For Descartes, the real actor in imagination is the knowing power, intellect. Even in the work of imagination we can recognize that imagining, having and working images, is not the same as intellection. That nonidentity is true even if there is never any thinking totally apart from images, because we can recognize that some of what we do mentally is not merely the having of images or the working of images or the having or taking or conceiving of images in some particular respect. What does the having, the working, the taking, the conceiving, is a power that exceeds images proper. The directed mobility of imagining comes from some other source than the images. If animals have images and a change or movement of images, it is nevertheless different from what happens in human beings, because animals cannot recognize the source of the mobility as other–than–image. The human power that works and holds images moves not only between images and image fields but also away from and out of all images. Even if it can never fully rise above them, it can clearly and distinctly recognize the limits of images and imagining.

6.7 The Limits of Imagination

Between abandoning the Regulae and publishing the Discourse with its scientific essays, Descartes’s thinking about imagination began to acknowledge another kind of limit. The Regulae operated on the basis of a presupposition: that method reflects reality. More precisely: it presupposed that the orderly method of organizing appearances according to their participation in natures, along with the notion that the natures somehow combine or compound in complex experiences, will eventually be shown to correspond to the way things are, physically and metaphysically (Rule 12, AT X.418). At the end of rule 4 Descartes in fact remarked that the purpose of writing down the Regulae was to secure its precepts as preparation for more difficult philosophical tasks lying ahead. Presumably these included not only addressing actual problems presented by the world but also understanding the ontology of physics and the metaphysics of creator and creatures. The Regulae presented a comprehensive theory of the orderly processes of intelligent imagining that regulate human ingenuity, which can image and reimage things and rework and represent them in accordance with principles of order and measure, in particular by employing the simplest figural representations of order and measure. The ultimate warrant or guarantee for this process of efficient, problem-solving representation according to the degrees and measures of participation in natures had to come from a different kind of inquiry, an inquiry into the nature of natures.

Rule 12 began to present—but did not, and probably could not, complete—the ontology of natures. Natures there are divided into three basic kinds: material, common, and intellectual (AT X.419–420). The material ones “can be known only in bodies, as for example figure, extension, motion, etc.” Of the common natures, which can participate in both material and intellectual things, Descartes mentions just a few examples: “existence, unity, duration, and the like.” The intellectual ones get a longer account:

Purely intellectual are those that, by a certain inborn light, and without the assistance of a corporeal image, are known by intellect: for it is certain that there are some such, and that no corporeal idea can be made that represents to us what knowledge is, what doubt, what ignorance, the same for what the action of will is that is usually called volition, and similar things; all of which, nevertheless, we truly know, and as easily as possible, for which it suffices that we be participants in reason. (AT X.419)

The intellectual natures are thus present and appear in the acts (or act–states) of intellect or soul. No image can per se enact doubting, although a human being can have doubt about anything that is an image or connected with one (for example, whether a particular painting of Christ’s postresurrection appearance to his apostles manages to convey the doubt of Thomas and its resolution). Not even a text (consisting of word–images) can enact doubt like that expressed in the Meditations of Descartes; it is only when a real human mind enacts the words represented on the page and thinks their objects that doubt occurs.

Rule 14 had argued that the ordered techniques of imaginative representation could be used of any problem that was subject to mathesis universalis. This is not quite to say that they can be used only of corporeal things, things that share in corporeal natures, precisely because the common natures participate in both material and intellectual things. There is, however, a limit to the representability by points, lines, figures, symbols, and equations when one is representing intellectual natures. Recall the use of line patterns to stand for colors: although Descartes does not believe his particular representation truly represents the differences between hues, he points out that light striking the eye must produce some actual patterns. So even if the particular representation he chooses is false, the true one is of the same general kind, a pressure– or impact–pattern. If, however, we try to represent one act of will by a line segment and another act of will by a second segment, we quickly run up against obstacles. Insofar as there is a unity (a common nature) in each act of volition, we can legitimately say that each participates in unity, and there is nothing false about using something that has unity to stand for it. But if we think that the lengths of the lines express something further about their nature, for instance their duration or their intensity, we are at serious risk of confusion. What, after all, is the duration of a volition? One might well be able to say, “At 2:55:40 p.m. I decided to go to the Renaissance Fair, and that volition remained equally active in my consciousness for 10 seconds, and then began to fade erratically until there was no trace of it left in my mind by 2:56:30 p.m.” But if consciousness of some kind is essential to making an act of volition, it is not clear that the volition lapses when one no longer has it actively in mind. If the Fair doesn’t start until 8:00 p.m., I do not have to consciously renew the volition repeatedly over the next 5 hours. Descartes would probably argue that there is a duration involved in volition, but it is not intrinsically measured by clock time. Perhaps volition as intellectual or spiritual is more akin to character traits: it does not make much sense to ask how long they last, either.

On the other hand, the intensities of acts of will look like they could be sufficiently well represented by the relative lengths of line segments. It does not seem wrong to say something like this: “My will to go to the Fair is not as strong as my will to please my family.” But we can’t go on to say that anything more than “stronger” or “weaker” is represented by the lines. With line patterns representing hues we expect that something in reality corresponds to the particulars of the pattern, its size, its orientation, and the like, but they need to be determined by future work. We do not have any such expectation with a two-inch segment representing one act of will and a one-inch segment representing another about half as strong. We don’t actually know much about measuring will intensity or what the proper unit would be. Although acts of will can come into conflict, it is not like forces or impulses in space; we lack a general theory of how and when volitions act and interact. We know that a very strong will (to eat healthily) can give way to what is a passing velleity (which induces us to gobble down half a pound of Belgian chocolate). In some circumstances volitions don’t interact at all (visiting a Renaissance Fair as part of one’s professional activities takes nothing away from one’s family per se), in others they can conflict mildly, moderately, or enormously (the last when I selfishly insist on going in violation of a solemn promise). What this illustrates is that, although both corporeal and intellectual natures can be represented by mathesis universalis, elemental universal mathematics, with corporeal problems we expect there to be some more extensive reflection of reality in the representation, whereas the representation of spiritual things is superficial and “figurative.” If, according to Descartes’s early two-imaginations note, corporeal things can represent spiritual things, like wind standing for soul, and if such poetic tropes can be strikingly insightful, in the last analysis (and that is undoubtedly the right word) this kind of imagining cannot be taken very far, at least not within its original terms. At some point one must simply focus on the intellectual or spiritual phenomenon as such. A painting may give us insight into a moment of doubt, but to understand the nature of doubt we must look to actual doubting rather than to images that try to express it.

There was another, more important reason to set clear limits to imagining, however, and that was the question of infinity. Though not mentioned in the Regulae, the theme appears in mathematical writings dating from the same period (the 1620s). In those writings Descartes attempts to conceive and manipulate unending processes that nevertheless arrive at a determinate mathematical result. He says, for example, that imagination can conceive a limit to an unending reiteration of a procedure or the reapplication of a concept (e.g., drawing a limitless series of parallels to an original line, AT X.75, or performing endlessly more refined divisions of space or time, AT X.73 and 75). In another mathematical work of the period, “Excerpta mathematica,” Descartes drops the use of words signifying imagination but shows that indefinitely expanding algebraic representations of series of sums and differences, displayed in increasingly complex tabular form employing continued fractions,Footnote 34 can be used to express the length of the side of any regular polygon inscribed in a circle with unit radius. By proceeding in this way, one can easily derive a series of ever more accurate approximations to the value of π, which is the proportion of a circle’s circumference to its diameter. (By extending the number of terms indefinitely one can determine the length of the side of an inscribed regular n–gon for arbitrarily large choices of n ; see AT X.285–297.) This kind of physical and mathematical work must have convinced Descartes that pattern-deploying imagination could quite easily handle (a countable) infinity by employing an orderly method.

Two letters to his Paris friend and correspondent Marin Mersenne from 1629 and 1630 (that is, very near to the time when it is thought Descartes abandoned the Regulae) signal an important shift in Descartes’s thinking about infinity. In a letter of 20 November 1629 he responds to Mersenne’s inquiry about an author (referred to only as “Monsieur Hardy”) who claimed to have devised a universal language. Descartes judges Hardy’s proposals to be less original and less useful than they at first appear. In conclusion he adds something that, he says, he is sure Hardy has not thought of, because it requires the true philosophy and depends on ordering all the thoughts that human beings can have. Just as one can learn in a day “to give an infinity of names” to numbers, one might give names to all other human thoughts.

And if someone had explicated well what are the simple ideas that are in the imagination of men, out of which everything they think is composed, and if that were received by all the world, I would dare to hope as consequence a universal language very easy to learn, to pronounce, and to write, and what is the principal thing, that would aid judgment, representing to it all things so distinctly that it would be almost impossible for it to be deceived; instead of which, to the contrary, the words that we have have almost only confused significations, to which the spirits of men being long since accustomed, is the cause that people understand almost nothing perfectly. (AT I.81)

Yet less than 5 months later, in a letter of 15 April 1630, Descartes complained of those who speak of God as though he were Zeus (a finite god) rather than attempt to understand his total infinity. This is the earliest evidence of the distinction he commonly drew in his mature work between “ordinary” infinities—like that of space, the counting numbers, or the divisibility of a line segment—and the infinity of God; speaking strictly, he would no longer call the former kind “infinite,” but “indefinite” instead.

These nearly contemporaneous passages do not necessarily contradict one another. Yet his earlier confidence that we might perfectly understand an infinity of thoughts according to their composition in imagination is displaced by the claim that he has learned something new about human beings and about God, and that he has found arguments even more persuasive for demonstrating metaphysical truth than mathematical proof (which he had been conducting with the aid of the regulated imagination, the subject matter of the Regulae).

The difference may well be explained, at least in part, by another metaphysical discovery Descartes announces in the letter of 15 April 1630, the mathematical truths (which Mersenne calls “eternal truths”; the name that scholars tend to use is “created eternal truths”). He had concluded that “the mathematical truths, which you call eternal, have been established by God and depend entirely on him, just as much as all the rest of creatures” (AT X.145). This announcement appears to be behind a new development that he mentions in the letter, that he had begun a new approach to physics. This approach would eventually culminate in The World, a work completed in early 1633 but not published in his lifetime.Footnote 35 In one sense at least The World is merely an extension of the project of the Regulae: it develops the seeds of knowledge about proportionalized relations that are native to, inborn in, our minds. The technique of imaging by using proportionalized series of the Regulae was the foundation of the elemental universal mathematics that Descartes had called there mathesis universalis. But the Regulae had bracketed (or omitted) several basic questions: whether mathesis universalis required a foundation—thus whether there was some more ultimate frame or horizon within which it functioned—and in particular whether what mathesis discovered about the relationships between conceived things and natures actually corresponded to the reality of the things, especially physical things.Footnote 36

6.8 Imagining the Cosmos

In principle the imagination can imagine in any way it likes. It can populate its spaces with the gods of Olympus, satyrs, demigods, fairies, Alpha Centaurans, or brains in vats; it can proceed in any direction (literally or figuratively) that the imaginer likes; it can jump backward and forward in time and space with little or no logic guiding the development. (One need only look ahead to Descartes’s description of dreaming in the Meditations.) However much Descartes knew of such imagining, it was not the cognitively useful, directed imagining of the Regulae. The Regulae implicitly recognized the continuous motions of imagining, but the method it proposed for regulating it occurred by discrete steps and by the patterns of order we already know. Continuous imagining might be approximated, or even in some cases achieved, by learning how to traverse the steps so rapidly that the imaginative motion became a continuous sweep. The goal of problem solving was not to set up a real-world scenario and then set it in motion (say, to set two trucks traveling in opposite directions at different speeds, one from city A, the other from city B, and then to see where their paths cross) but to analyze the givens of a problem, to find in them some pattern capable of being represented, to use the simplest kinds of geometrical figures or other images to represent the givens and patterns (thus as much as possible to keep the figuration to no more than two spatial dimensions), to label the representations for more convenient use, to put the labeling symbols into formulas expressing the patterns and proportions of relations equationally, and to manipulate and calculate the equations until an unknown is expressed totally in terms of what is given or derivable from the given. Once the form of representation was selected, no realism in the movements and manipulations of the representations or direct emulation of real-world activities and motions in the original problem and the things it was about was necessarily implied. But once Descartes started addressing metaphysical questions in Holland in 1629 he recognized that one could employ an alternative, more physically real form of imagining.

The Regulae never required a preliminary understanding of the world as a whole, only attention to whatever parts and aspects were immediately relevant to solving a problem. The overriding consideration was that all relations can be put into proportionalized form. After carefully posing the problem in a set of consistent terms in the plane or field of the real world, one would translate it into a field of figures by representing the measure and degree of nature–participation with lines and other images and then work out solutions by manipulating them. To assist in this one could also resort to a third field, that of calculation, by using symbols to stand for the figures and their measures and manipulating those symbols according to rules of algebra. One did precisely as much, or as little, representation and manipulation in and between as many fields as one needed to solve the problem.

The World, by contrast, asked from the outset about the entirety of physical reality. Unlike the Regulae, The World is predicated on grasping any particular problem as part of a world situation. The world is indefinitely extendable and indefinitely divisible. In order to solve a problem one must place it in the world situation and be able to track its circumstances and its evolution. The limits one imposes depend on how much of the world situation one needs to take into account. If there are physical impulses coming from afar that affect objects of interest, one has to take them into account and represent them. If all the motions and impulses of relevance are local or can at least be accurately represented in terms of their local effect only, one needs to picture only that immediate vicinity.

Although the element in which The World operates is still imaginative—the model of the world is constructed in “imaginary space,” as he says in part 6—Descartes assigns an even more decided and directive role for intellect as providing the fundamental parameters for cognitive imagination. The chief issue is that it requires intellect to recognize that all motions, whatever paths they actually take, are at every moment based on straight-line tendencies or impulses (the actual motion is the result of myriad such impulses being applied to the same bits of matter); furthermore, the total motion in the universe is a constant. The latter point is basically what we would call a conservation law: the total amount of motion is conserved, so is neither greater nor less from moment to moment.Footnote 37 The former, straight-line requirement is a fundamental restriction on how to represent the components of an object’s motion and the various forces affecting that motion. The two requirements together privilege understanding natural situations not according to step-by-step analysis but rather by imagined scenarios of continuously changing motions. It was a consequence of his conception of created mathematical truths: out of all the possible worlds that God could create, he created one with elemental truths that governed all events of the physical world and that could be known as such. As a result, in the spaces of our imagination we can set imagined representatives of the world’s objects going and imagine them moving exactly the same way.

The approach of The World was not entirely new for Descartes. Recall that in notebook C he had imagined manipulating, rotating, sliding, etc., geometrical figures in all sorts of ways. The relevant imagining was always determined ad hoc, however, for the sake of solving some individual problem according to the particular principles of ordering one recognized in it. In The World, by contrast, he places figural imagining into a larger and far more dynamic imaginative context. After stating and discussing the intellect-discovered rules of motion, he recommences in part 6 by creating “in what philosophers call imaginary space” a world that duplicates the real one in three dimensions and the dimension of time. That is, what he had done with individual things, according to the two-imaginations note of C—take them and conceive them according to an image or figure—he now does with the totality of things: in imagination one produces a simulacrum or model, potentially of the entire universe. The universe is not conceived as actually infinite, yet it is indefinitely extendable in any direction one likes. It is infinitely divisible, both spatially and temporally, but because of the way the rules of motion work one does not need to break it all down actually to infinitesimals. To solve a problem about real or really imaginable things, one needs to set up in this imaginary space the situation that holds at some moment and then let it evolve according to the rules of straight-­line impulses and straight-line tendencies to motion.Footnote 38 The perfect icon defined by the Eleatic Stranger in Plato’s Sophist, the image that perfectly reproduces all the proportions of the original, is realized in Descartes’s World.

This mathematical approach led almost seamlessly to the more sophisticated mathematics that Descartes began working on in 1632–1633 and that culminated in the analytic geometry of the Geometry (published in 1637 as the last of the three scientific essays accompanying the Discourse). Analytic geometry can in principle tell us the path that every point of this world’s stuff, matter, or extension will follow when it is subjected to any number of inclinations, impulses, collisions, and the like. It solves real problems by raising them to a conceptual space—or, rather, an imagined space—though now (unlike in the Regulae) the solutions are not developed step-by-step but evolved continuously from given starting conditions by imagining matter in motion, in duplication of real motions of real matter in real space.

This is not to say that the new method completely abandons principles of the Regulae. In analyzing and setting up a problem for solution we still need to apply the part-by-part, step-by-step comparison of givens (for the most part done just two at a time). Nor is it the case that every solution will be a simulation of a real-world, three-dimensional, temporal scenario. As always, when it comes to problem solving Descartes remains an opportunist. Some of the classic problems of ancient mathematics that cannot be solved by the toolkit of Euclidean geometry, the straightedge and the compass, can be solved by real or conceptual “machines” that he had been thinking about since notebook C. Just as much as in the Regulae, one analyzes the elements of the problem, represents them in simple labeled forms, generates formulas/equations for the sake of algebraic manipulation that can be translated back into figures, and so forth. One will abstract these elements from the original way that they are given and incorporate their measures into the imaginative devices one has conceived. There is therefore a great deal of artifice in setting up the solution. But then, when problem-solving time arrives, one takes that device—which may be mathematically equivalent to a second-degree algebraic curve sliding long a fourth-­degree curve and producing another curve by the moving point of intersection of a tangent to one of those curves and yet another line—and sets it into motion. With a real device that is a real-world motion: one that simulates a possible state of the world analyzed according to dynamically imagined mathematics.

The Geometry develops a mathematics that gives us the possibility of tracing the movements of things to any degree of analytical complexity we need. However, what human beings can actually imagine or do is restricted by the complexity of the problem, the finitude of human capacities, and the practical limits of time and resources available for working out details. Some motions are of a complexity that requires equations of algebraic order far beyond the human capacity to analyze them in actuality. Although de jure our techniques apply to knowledge of any complexity whatsoever, de facto what we know will have different degrees of certainty. God alone can track the infinitely fine divisibility of matter subject to limitless collisions and impulses throughout indefinitely extended space and duration. The mathematical truths Descartes announced to Mersenne in April 1630 do imply, however, that we can clearly and distinctly perceive that all motions can in principle be analyzed according to the mathematics of analytic geometry, even if there are motions—in fact an infinite number of motions—that no human being can ever come close to grasping according to all their particular causes.Footnote 39

Looking back to the Regulae, one might say that its regulated, step-by-step motion of thought was a less decisive break with the philosophical past than it at first seems. Previous philosophies had for the most part modeled truth on the apprehension of unchanging, eternal things. Practical and technical matters were degraded forms of knowledge because they concerned themselves with what was changeable. Because the truly knowable things were stable, coming to know them was comparable to arriving at a state of rest, because once one knew the truth one’s inquiry would cease.Footnote 40 The method of the Regulae is a kind of compromise between rest and motion. It produces a kind of knowing that “hiccups” from step to step, from stopping point to stopping point. One intuits a truth (stop); then one remembers to look to the problem one is trying to solve to search for the next place to look; and one keeps looking until something else is intuited (stop); and the cycle begins once more, until finally the solution to the overall problem is reached (stop).

However, Descartes’s claim that intuitus could learn to sweep through the steps of a deductio to become continuous suggests that even early in his career the goal was to transform stepwise thinking into a continuous flow of imagining, and that he recognized that step-by-step motion, no matter how rapid, gives no more than a poor imitation of continuity.Footnote 41 The Geometry provided real continuity, not a poor imitation. It made continuous motion of points and lines fully and accurately trackable by virtue of the translation of regulated motion into algebraic formulas. An algebraic formula does not per se have a beginning or an ending point: any substitution value is as legitimate as any other, and the “point” of any substitution is its continuous relation to nearby values. The true account of a motion is one in which the mind traces the evolution of the formulable curveFootnote 42 that is the continuous path of a moving point. The Geometry thus fully implemented the mathematical truths that Descartes had announced in 1630 and that were implied by the physics of The World, in a manner that made their dynamism fundamental both to world and to mind. The things of the world move in a way that is accurately describable by the mathematics that the mathematical truths, created by God, establish and regulate, and the mind can always understand and imagine these motions in principle, even when it is not possible to imagine them in full factual detail. Knowing was not achieving rest from inquiry but being able to track in space (real and imagined) everything that a ­formula–solution implicitly contains.

What this means in its ultimate development, Descartes thought, was that all mathematics and all physics could be represented by rigorously imaginable figures and their motions, and these motions could be correlated with algebraic formulas. There was a fundamental intertranslatability of the geometrical and the algebraic. This mathematical realm was the res extensa, the “extended thing,” conceivable as existing in imaginary spaces according to The World (AT XI.31–32) and described at the end of Meditation 5 as “the whole of that corporeal nature which is the subject matter of pure mathematics” (AT VII.71). Descartes thought (incorrectly, as we know)Footnote 43 that this approach would be truly comprehensive—that algebraic equations would be sufficient for tracking all actual motions and for solving all problems capable of solution. But because this mathematical and physical knowledge was grounded in the mathematical truths (because God had created them and sustained the world in accordance with them), we truly know that this approach to real, worldly things is correct, even when our human limitations keep us from analyzing all the detail. From the arguments in the Discourse and the Meditations we know that certainty about this correctness ultimately depends on our clearly and distinctly perceiving the difference—the real distinction—between extended things and thinking things on the one hand and between finite thinking things and the single, truly infinite thinking thing on the other. This is, of course, the reason that God becomes the focal point in the Meditations as the best known thing of all. He is best known, above all, because without him the rest of our knowledge is not truly knowable. The Regulae’s equal knowability of all truths had thus been radically and thoroughly displaced by the differential knowability of things represented in imaginative knowing and the ultimate but unimaginable knowability of God.

If we think this ultimate knowledge of mathematics and nature is solely rationalistFootnote 44 we are sadly mistaken, in Descartes’s view. This kind of problem solving is a complex activity involving the several levels of sensation, imagination, memory, and intellect. It is true that the intellect notices what is or is not the case in problem solving, but the objects to which it attends are not pure ideas but the elements of problems represented by images and symbols. Intellect guides the whole procedure by shifting its attention from one thing to another and noticing the proportions that hold between them, but it takes no steps without the help of images and symbols. There is no intellectual problem solving without imaginative figuration.

As we know from the Geometry, one of the three essays that accompanied the Discourse, the algebraic approach to geometry—analytic geometry as we call it—is predicated on the following insights. First, the rigorously interconnected motions of points, curves, and lines generate new curves and lines. The various aspects of these motions can be expressed by algebraic symbols and equations and can be used to solve any problem based in mathematical proportions. The mathematical figures and equations, in their turn, can be used to represent the motions of actual bodies in space. Analytic geometry is therefore the fulfillment of the hope that Descartes expressed in notebook C, that unlike in the memory art he would find a way to generate images from other images, once the principle of their causation and proportionality was evident. As much as had the method of the Regulae or of his early notes about imagination’s power, this ultimate expression of Cartesian method required great mental agility. The mind had to be able to move from plane to plane, from field to field, from space to space. Original problems had to be translated (literally “borne across,” “carried across”) into terms of analysis with shared or interrelated dimensions; bare lines standing for quantities of representable natures had to be carried over into a system of interrelated lines in the space of analytic geometry, marked positions of points and lines had to be translated symbolically into well-­formed algebraic formulas, the calculations of algebra had to be translated back into the movements of the points and lines in geometric space: back and forth and back and forth until the imagined solution could be rendered back into the original terms of the problem and its corresponding real-world situation.

In comparison, the Socrates who makes and analyzes lines, squares, rectangles, and triangles in the Meno had noticed just the tip of the iceberg of imaginative mobility. Descartes had not only rediscovered the existence and the virtues of imaginative fields and the ability of mind to move between them, he had discovered in this multiplicity of fields of imagining a flexibility, a dynamism, a cognitive power that had never before been conceived. What nearly four centuries of development more has shown is that this method retains its power even when used by those who can give accounts neither of rationality nor of imagination.

6.9 Imagination in the Meditations

Why, then, is our first inclination still to think that Descartes places human beings above and beyond imagination?

When it comes to imagination in Descartes, readers will first of all remember two passages in Meditation 6. One argues that human beings cannot properly imagine a chiliagon, though they can easily understand it clearly and distinctly. The other claims that understanding but not imagination is part of our essence.

Taken by itself, the first shows a difference between imagining and understanding with respect to both clarity (in that the vagueness in one’s imagining of a chiliagon is as evident as is the precision of understanding that a chiliagon has exactly one thousand sides) and distinctness (in that imagining and understanding the same thing, a chiliagon, are set against one another to show a sharp contrast).Footnote 45 If we ask ourselves how clearly and distinctly we imagine and understand (respectively) a triangle, it is likely that we will not notice a major difference. Indeed, one of the advantages of starting with the triangle is that we easily conceive the situation as follows: in both imagining and understanding the triangle, we begin by first imagining a triangle and immediately “seeing” that it has three sides. In the clear imagining of the triangle there is simultaneously a clear understanding of it as a three-sided figure.Footnote 46 Even if there might be a doubt or two here about whether we have expressed this quite rightly, it looks in first approximation as though understanding the triangle is either (1) nothing other than clearly and distinctly imagining it or (2) something that must immediately follow our clearly and distinctly imagining it. But if we repeat the experiment over and over, with a four-sided figure, a five-sided, etc., we will see that we were deceived. At some point we will find that it is harder and harder to distinctly imagine the figure we are trying to picture, whereas understanding that you are adding one to the number of sides is no harder when you move from nine hundred ninety-nine to one thousand than from three to four. Thus understanding an n–gon as an n–gon is different from imagining it. It does not require a clear and distinct image of the n–gon, and even when a clear and distinct image of a figure hovers before our mind there is still a difference between imagining as an act of the mind and understanding as an act of the mind.

The chiliagon passage is less definitive about imagination than it looks, however. First, in the Meditations Descartes tends to conceive imagination in a very narrow, physiological sense. In Meditation 3 he says that he uses “image” to mean the image or figure that is formed in the pineal gland, toward which the mind turns its attention (in imagining) so that it has an experiential idea of the imaged thing. The chiliagon example does not intend “image” and “imagination” so narrowly, since it is less about the figure formed in the pineal gland than about the experiential idea of the figure as it appears to us in ordinary geometric consciousness. The passage therefore does not justify the conclusion that we understand the chiliagon as such without any help from the imagination whatsoever. The argument is presented in comparative mode. It is not that there is nothing going on in the imagination when one thinks the chiliagon, but rather that the success or failure of the imagining is not essential to the understanding, no more so than if we were trying to imagine a myriagon (with its ten thousand sides). Imagining, we see, requires a harder effort of a different kind than does understanding (intellecting).

But doesn’t the argument show that understanding is independent—redundantly one might say “completely independent”—of imagination? Even the one phrase that is most suggestive of an intellection so pure that it has no need of imagination in any sense whatsoever is not decisive. It occurs in the following sentence:

If in fact the question were about a pentagon, I can indeed understand its figure, just like the figure of a chiliagon, without resource of imagination; but I can also imagine the same, viz. by applying the sharp edge of the mind to its five sides, and at the same time to the area contained by these; and here I manifestly notice there is need for a certain exertion of rational soul [animi] peculiar to me for imagining, which I do not use for understanding: this new exertion of rational soul shows clearly a difference between imagination and pure intellection. (AT VII.72–73, emphasis added)

The question is whether the phrase absque ope imaginationis Footnote 47 that I have emphasized can mean, purely and simply, that the imagination (or sensation) is totally irrelevant to understanding. Of course one can say “I know that a pentagon has five sides” as easily as “I know that a chiliagon has one thousand sides.” Yet, already in the Regulae, Descartes had stated that it is properly the intellect that makes mistakes, not imagination, and that the intellect is especially prone to error when it does not provide itself with an appropriate image. Here in the Meditations, just two paragraphs earlier at the end of Meditation 5, the meditator congratulates himself on coming to realize that all knowledge depends on God, then says that “now indeed innumerable things can be plainly known and certain to me, both about God and other intellectual things, and about that entire corporeal nature that is the object of pure Mathesis.” In conformity with his mature conception of the essential identity between the entirety of mathematical space and real space, in these concluding words of Meditation 5 he is dividing all human knowing into the purely intellectual and the purely corporeal. The latter realm, whether virtual (mathematical) or real (physical), is known by the imaginative methods of the Geometry. And of course Descartes had used the word mathesis 20 years earlier for the kind of orderly, imaginative, universal mathematics that he presented in the Regulae as the foundation of mathematics and scientific knowing. It is inconceivable that Descartes would almost immediately assert, with the chiliagon example at the beginning of Meditation 6, that the intellect in understanding something mathematical (a triangle, a chiliagon, a myriagon) could accomplish this in complete and total abstraction from extension, whether perceived or imagined. At the very least, the truth intellectually apprehendable about the thousand-sided figure must contain some reference to the fact that it is a geometrical figure, that it has straight sides, that it is plane and closed, etc., etc., etc. Insofar as it does not implicitly refer to any such things—that is, insofar as such things are not implicated, enfolded, in understanding it—the formula “a chiliagon is a thousand-sided plane figure” is vacuous and meaningless. Given all these considerations, it would be better to read the phrase absque ope imaginationis as meaning “without any special aid of imagination.”

It is of course possible that Descartes’s formulation indicates some shift of position. A later passage from one of his letters to Princess Elisabeth supports the notion that he is talking about an understanding totally devoid of imagining, when he says that “the body, that is extension, figures, and movements, can also be known by the understanding alone, but much better by the understanding aided by the imagination” (28 June 1643, AT III.691). That this is unintelligible in light of a never otherwise repudiated conception of mathematics as imaginative in essence should make us at least pause before conceding. If there is an ultimate solution, it seems to me it requires conceiving this understanding without imagination as the immediate result of a remotionFootnote 48 (negating motion) of thought like that involved, in the Regulae, in the pure intellect’s understanding that figure is not extension. That is, it is not that the imagination is put completely out of action but that the understanding is not intrinsically a matter of clearly perceiving an image or something that an image directly shows. To recognize that figure is not extension one must begin with something extended, see it in its figure and in its extension, and notice that being figure and being extension are not identical. This is intellect’s remotional recognition that figure is not extension. Understanding in such a case is neither having a specific image in view nor staring at a formula in the absence of any reference. Neither having the image nor repeating a formula is thinking about figure or extension. Analogously, having a particular image in mind or even any finite series of images is not all by itself an understanding of body. Understanding is something that the intellect brings to the experience of the world so that we can see what does not directly or immediately show itself as such.Footnote 49

The second passage in the Meditations that discourages ascribing importance to imagination is the claim that we would still be thinking things, res cogitantes, if we did not have imaginations (at AT VII.73). This, too, turns out to have less force than first appears. Though true, the claim needs to be seen in the context of the Meditations’ method and set against what Descartes says later in Meditation 6 about the total nature of the human being. Most of the inquiry of the Meditations is conducted by the thinking being precisely insofar as he/she/it is a thinking thing, prescindingFootnote 50 from any other possible character of that being and (at the beginning of Meditation 6) from the being of any other being besides God. Descartes’s entitlement to this kind of prescission is tightly tied to his method, which says (1) to accept nothing that is not clearly and distinctly seen to be true, (2) to divide problems as thoroughly as necessary, (3) to construct complex knowledge from simple in an orderly way, and (4) to ensure (at the end) that nothing of relevance has been left out. In particular, wherever one clearly and distinctly perceives some difference, that means that a division is possible in thought, and if it is possible in thought then it is possible for the infinitely powerful God to make things actually different in such a way.

This prescissionary method is precisely why the images and ideas of things can be divided from the actual existence of the things, because our experiences of being wrong about sensation, dreaming, remembering, imagining, etc., provide the differences that doubt exaggerates. Our senses show us one thing when another is true, we can dream an entire world that, when we awake, we see does not exist at all, and so forth. Descartes goes through these things in an order that corresponds to setting aside our confidence in, and where possible “turning off,” one after another, the external senses, the internal senses (in the forms of dreaming, memory, and imagining—including mathematical imagining), and then trying to do the same with intellect itself. The fundamental failure of each power, when it fails, is reflected in our inability to know that something real corresponds to what appears to us. Eventually the meditator expresses this difference in its most fundamental sense by making the terminological contrast between the formally real (to use the term Descartes settles on in Meditation 3 for “really existing,” “existing in a thinglike way”) and the objectively real (existing as present to mind in an idea–appearance). But with intellect, with my act of thinking, the arguments of doubt fail. The reason that the attempt against reason fails is that as long as I am conceiving, thinking, proposing the meaning of “I am, I exist” and trying as hard as I can to doubt it, I cannot be nonexistent, precisely as a conceiving, thinking, proposing, and doubting being. I may be nothing more than a thinking being, but the self-activated experience of thinking in the specific form of doubting leaves no doubt that, precisely as a doubting and therefore thinking being, I exist.Footnote 51

But this does not at all settle the status, relevance, or use of imagination for human beings. Throughout the Meditations, and even more emphatically in his replies to objections, Descartes is careful to say that the conclusions drawn in the course of the work are carefully qualified and often only provisional, so they may not be cited as unqualifiedly true unless they are said to be so at the end. The imagination words are used quite narrowly in the Meditations. Imagination per se is the ability of our minds to focus on a figure produced in the pineal gland. But if we have no bodies (as our doubting has led us to think possible), then we have no pineal glands and no figures produced there to be focused upon. Thus it is possible that I have the appearances that I call imagining without there being any real things corresponding to them. So I would be having thoughts that were like what people call images, but I would not be imagining. Without a pineal gland and images there I would still be the same being I am, a being that thinks. My essence as a thinking being is thus unaffected even if there is no such thing as imagining proper—precisely because I can imagine myself, picture myself to myself, as existing without imagination! Once again the meditation must engage imaginatively in remotional thinking in order to see a truth that directs our attention to what lies beyond our imagining, even if we never manage to think it with our imagining totally annihilated.

As I will point out momentarily, this conclusion about imagination is not the last word. Understanding this requires that we go beyond the question of our essence as thinking beings (which in fact is just an aspect of our being) to the question of our human nature (which is about our total being).Footnote 52 And, as things turn out, this means that in the last analysis, in order to understand imagination in Descartes we have to go beyond even considerations of the Aristotelian internal senses, beyond our merely cognitive powers, whether sensitive, imaginative, or rational, to an even ampler and more fully adequate understanding of what it is to be human.

In Descartes’s Imagination I argued at length that the entire Meditations is meditational, and that means that it is as centered in imagining as its medieval and early modern forebears (meditations and spiritual exercises) always were. I even claimed that the cogito–proof and Meditation 3’s proof of God’s existence required both positive and negative-remotional uses of imagination (in the sense of “remotion” explained in Sect. 6.6, above). For our purposes here it is not necessary to go quite so far. Meditation 1 is, by any measure, the work of insistent, repeated, concrete imagining (of course, as always with Descartes, at the direction of intellect). We bring to mind different situations in which we have put trust in the senses but been deceived, we think about what madmen claim to experience, we think about what we take to be real in our dreams and see it can be every bit as mad as what madmen say, we wonder about ways of dividing up our experiences into components so that, even if how everything appears all together is not right, we might nevertheless find that the components are real and true, we see how confidently we assert and see the truth of mathematical claims but then recall that we make many mistakes, we come around again to assert that at least the simplest elements of mathematics must be true, then we realize that we do even make simple mistakes of counting and addition and then wonder fearfully whether our minds might not always “slip a cog” when we try to add simple numbers or count the sides of a figure, and so forth. When the Meditation 1 doubt thereby reaches the threshold of purely intelligible things (e.g., God’s being, in particular whether he could be a deceiver) the meditator pulls up and changes tactics, because the grounds for legitimate doubt become less clear than before. The meditator at that point has to devise an expedient, the evil genius dedicated to deceiving the meditator in every way possible. This is, of course, an imaginative device, and in particular it is motivated by a resolution of will. The meditator near the end of Meditation 1 has noticed that the earnestness and success of doubting fades as one becomes fatigued, and after resting one has lost the vivid sense of doubt one had good arguments for earlier, so that after a short while one is ready to accept again, without reason, what one has found reason to doubt (in particular the evidence of the senses). This is one of the first indications in Descartes’s published writings that imagination is somehow connected with will more than with intellect, but not until the 1649 Passions of the Soul does Descartes explain in detail what this means.

The Meditations is more centrally concerned with the relationship between will and intellect than between either of them and imagination. Intellect, it turns out, is finite, or rather one might say indefinite: although there is no limit to how much we can know there is always indefinitely more that we do not know. Moreover, the fact that intellect can be easily misled shows that it is imperfect. Of will, however, the meditator says that it is perfect in its kind, that is, precisely as will (in Meditation 4, AT VII.56–57). Our knowledge about many things is and remains obscure, and since perceiving is the chief characteristic of the knowing power, obscure perceiving is a flaw. But any object that can come into our consciousness in any way or to any degree, obscure or clear/distinct, is equally well a potential object of our volition. We might greatly desire something we do not clearly see or understand because we do not grasp it clearly; if it were seen clearly we might see it as problematic. In such a sense will might almost be called genuinely infinite in us, and, insofar as at the end of Meditation 3 the meditator invokes the Judaeo-Christian theological doctrine of man’s being made in the image and likeness of God, it is more with respect to will that man resembles God than with respect to intellect. This supremacy of will is a theme that is further developed in the 1644 Principles and the 1649 Passions: in particular when Descartes argues that thinking consists of both perceptions and volitions. Perception is a passion of the soul, whereas volition is an action; every perception is the passive side of the volition to know; and since things are more properly named in terms of their actions than their passions, the thinking thing is more properly understood as the willing thing than as the perceiving/understanding thing. Insofar as imagining is preeminently a volition—something clearly stated and argued in the Passions (AT XI.342–343)—that suggests that, contrary to what the Meditations says, imagining may be closer to the essence of the thinking thing than is intellectual perceiving.

The Meditation 4 reflections on intellect and will conclude that the disproportion between them is the main cause of our errors. We want more things to be true than we see to be true. This is important to Descartes’s conception of both truth and error, since he distinguishes (clearly and distinctly, he doubtless would say) between what we perceive things to be and our affirming (by will) that things are the way we perceive them. Thus wrongly seeing things does not force us to commit errors; it is, rather, the fact that our will wants them that way (for example, in cases when our intellect does not perceive clearly and distinctly). But since the thinking thing that is the human being can train intellect and will to become differently balanced—that is what the meditator achieves by persisting in the full course of his meditation, his spiritual exercises—God bears none of the fault for our errors.

In the course of these considerations Descartes entertains the possibility that God could have made us to perceive things quite differently than we actually do. We might have had almost perfect clairvoyance about some or all of what we direct our attention to, for example, or an intelligence like that of some other kind of thinking being, or one that operated in a way we cannot even begin to conceive or imagine. Not far from the surface is the contention that God could have given us perception and will in many different particular ways, even, for example, with a kind of knowing that depended not at all on imagining or sensing (like the angels of Christian doctrine), or on a radically different kind of imagining and sensing. As perceiving beings we would still be essentially the same, only the objects and typical certainty of perception would be different. This is the substance of Descartes’s statement that, precisely as thinking beings, God might have made us without imagination, and in that sense imagining and sensing are not part of our essence as thinking beings.

But that is not the end of the matter, when in Meditation 6 Descartes turns to the question of what our total nature is as human beings. It is at that level that God gave us sensation: not for the purpose of cognition, but for the purpose of staying alive—or, as the meditator puts it, maintaining the unity of thinking and extension, the unity of our soul and body. That unity and everything that provides for and maintains it is our nature. Intellectual perception is made for cognition, but sense perception is for self-preservation. Willing is superior to perception, so human beings are made not just for knowing but even more for acting properly in accordance with volition, whether those actions are thoughts or real-world actions.

In modern philosophy there is a tendency to treat volition as a question of ethics and politics rather than as part of fundamental anthropology, psychology, and epistemology, much less as part of metaphysics—in fact will is virtually irrelevant to modern epistemology.Footnote 53 Descartes’s discussion, however, is more strongly related to medieval discussions that asked which of the two faculties, intellect or will, is nobler. The answer depended, to a great extent, on the thinker’s conception of intellect and will in God. Of course the medievals understood these (and all other positive powers and qualities) as unified in God, who was conceived as radically one and simple. Like all other creatures, human beings were more diverse than God, so that one could at best think of the existence of intellect and will in the human being analogously to their radically simple and unified existence in God. Some theologians asserted the primacy of will, especially insofar as love/charity was conceived as the culminating act of human existence; others argued for intellect, not least because the second person of the Trinity, the Son, was understood (following the opening sentence of the Gospel of John) to be the creative word and wisdom of God. But both sides in the debates recognized that every act of intellect implied an act of will and vice versa. In either case, the answer was not just a matter of psychology but also of anthropology, and not just of both of these but also of metaphysics, insofar as the existence of such powers in the human being implied an orientation toward their ultimate fulfillment in the metaphysical destiny or telos of human beings: loving God with the will entirely turned toward him, and turning toward God in proper apprehension and understanding of what precisely he is and thus being filled with the infinite intelligible species of the divine—something not accessible to human beings in their earthly state.

6.10 Willing, Images, and Passions

Descartes’s regulation of imagination for cognitive use culminated in understanding geometrical space as identical to the essence of matter and opened the way for an apparently thoroughgoing reduction of physical reality to the mathematically imagined mechanics of motion. His last work, published less than a year before his death, The Passions of the Soul, takes a different, indeed quite surprising tack. As I have already noted, after dividing thinking into its active side, volition, and its passive side, perceiving, he defined imagination as an action or volition of the soul, rather than a passion.

The Passions gives Descartes’s most detailed published account of human psychophysiology. It is not only important for imagination or psychology more generally but also deserves fully canonical status in presenting Descartes’s philosophy and assessing his achievement.Footnote 54 Helpful in interpreting its significance is his correspondence with Princess Elisabeth of Bohemia (1618–1680; also known as Elisabeth of the Palatinate), since it was in the course of addressing her questions that Descartes worked out what was essentially a first draft of the Passions.

Elisabeth appreciated the metaphysical, scientific, and mathematical implications of the Meditations and the Discourse and accompanying scientific essays, but she had many questions about them as well as political, ethical, and anthropological concerns she did not find addressed. In her very first letter to Descartes she expressed dissatisfaction with his account of the relationship between mind and body. She could not understand how a physical account in terms of extension could be reconciled with the acts of a soul entirely lacking extension. She asked for more precision, and in particular for a definition of the substance of the soul apart from its actions (thoughts) and a more rigorous account of causality between mind and body. In his response Descartes introduced the theory of “primitive notions.”Footnote 55 By primitive notion he meant something that could be experienced and known only through itself, and not by trying to divide it up into simpler components out of which the whole would supposedly be grasped.

Two kinds of primitive notion are hardly unexpected, for they correspond to his division of things into thinking things and extended things. There are those that belong to the soul alone, then those that belong to the body alone. These and their difference, he explains to Elisabeth, are what his earlier works had distinguished. What is more surprising is the last kind he identifies, notions concerning soul and body together, for which “we have only that of their union, on which depends that of the force the soul has to move the body, and the body to act on the soul, in causing its sensations and its passions” (AT III.665). It would appear at first glance that the third kind of primitive notion must and should be analyzed into its two components, soul and body, that is, into the other two kinds, and then understood from their combination. But that is precisely what Descartes says is not possible. No matter how clearly and distinctly we understand thinking and body separately, there is nothing in that understanding that explains, articulates, predicts, or otherwise accounts for the how and the why of their being together. Perhaps this should not be surprising after all, however. Thinking and extension/corporeality share no trait in common. There is no reason to think that perceiving them clearly and distinctly, each apart from the other, could lead to understanding how they are united.

Just before introducing the primitive notions Descartes tells Elisabeth that she is assuming he has already tried to explain in his works how body and soul go together. Almost everything he has written hitherto tries to show the real distinction between body and soul, he says, whereas with respect to how they “act and suffer” together he has actually said almost nothing at all!Footnote 56 Is the Passions of the Soul the work in which he finally treats them together?

Reading Descartes’s remark in the introduction to the Passions that he composed it en physicien, “as physicist” (given seventeenth-century usage, one might even expand that to “physicist, physiologist, and natural philosopher”), it seems more likely that he was viewing the passions solely in their corporeal aspect. On the other hand, at the outset he delves into an abbreviated account of prerequisites from his metaphysics and psychology, and throughout the work he parallels the account of what happens physiologically in the various passions and emotions with psychological descriptions and occasional pragmatic and ethical considerations. As Kambouchner suggests, we need not take Descartes’s words to mean he is writing solely as a physicist, but rather understand that he believed his distinctive contribution to the tradition of philosophical inquiry into the passions was to coordinate psychology very tightly with physiology.

After familiarizing ourselves with the whole work and engaging its explanations we notice three things: the physical and psychological particulars are often false and sometimes even comical to our twenty-first-century sensibilities; the explanations he gives of nerve, spirit, and other physiological activities indeed represent an application of the kind of mechanism that he had begun undertaking in the 1630s; and the causation he appeals to seems to work sometimes from the physiological to the psychological, sometimes the reverse. Despite all the shortcomings, Descartes is attempting an account that is recognizably like later scientific approaches. If he talks about spirit flows and motions in the nerves, to and from the pineal gland, we do something similar when we talk of ions flowing across synaptic connections and electrochemical impulses moving along neuronal axons to and from the brain. Descartes is also attempting to connect the psychological features of passions with the various physiological activities that help define them, just as a contemporary neurophysiologist might try to explain different aspects of an appearance (say, horizontal boundaries in vision) according to where the nerve signals are processed in the brain. And, looking backward, one might say that he is trying to give Aristotle’s psychophysiology, or more exactly the conceptual topology that presents the soul as the activity of a body divided into organs, a revolutionary new basis. If in other respects Descartes saw himself as refuting Aristotle’s philosophy, as practitioner of theoretical psychophysiology he was in essence continuing a kind of research that Aristotle had first defined and started to put to work.

Descartes’s mechanistic physiology of the 1630s had been concerned to lay down a broad outline of the physiology of sensation, imagination, memory, and motor activity. Nerves were hollow tubes with a fiber running down the center. When the nerve in a sense organ was stimulated by an external object the resulting motion would be conveyed along the fiber, all the way to the chamber (the center ventricle) in the lower middle of the brain where the pineal gland was suspended. The hollow of the chamber was filled with animal spirits (the liveliest spirits that had been filtered from blood and food). When a nerve motion from a sense-organ nerve arrived at the periphery of the chamber it was translated into spirit pressure directed toward the gland. According to God’s institution of basic correlations between motions of the pineal gland and what is experienced (i.e., ideas, as part of the human being’s total nature)—an institution that could be partially modified by the person’s life history—the thinking thing would see, hear, taste, etc., and then respond in some way (perhaps in an act of will, perhaps automatically). The response would be translated into a pineal gland motion that caused new spirit flows, some of which might communicate with places in the brain where traces of previous experience were preserved, others of which would be communicated to the muscles, through the opening and closing of the ends of the nerve tubes reaching into the spirit chamber around the pineal gland, in order to produce physical motion. This is a system predicated on rapid and efficient response and action—and, apart from the motions of the pineal gland initiated by thinking, it works similarly in human beings and animals.

Despite his practical and theoretical emphasis on imagination in his earliest mathematical and scientific writings as well as in the Regulae, in his 1630s productions regarding anatomy and physiology Descartes did not give much attention to how imagination fit into them. Yet, oddly enough, he did retain images as a central part of the descriptive and explanatory apparatus of his psychophysiology. It is odd precisely because of the nature and history of mechanistic explanation. Galileo, for instance, had argued that nothing about the process of moving a soft feather tip back and forth across the skin was the feeling called tickling. Sensible qualities like color, sound, smell, and taste do not resemble their actual causes, which are the size, shape, quantity, position, and motion of the parts of bodies. For Descartes, once light stimulates a nerve ending in the eye, that stimulus is translated into nerve motion, which is not an image. If we conceive the same thing happening in tens of thousands of nerves across the retina, what we get is tens of thousands of nerves transmitting their motion, each independent of the others, to the brain. At the periphery of the central brain chamber the arrival of the nerve motion affects the opening of the nerve tubes and changes the pressures and the flows of spirits in the central chamber; when those pressures and flows finally reach the pineal gland they change its position and shape, if only slightly. None of this mechanism involves images per se. Although you might say that an outline of the viewed scene is thereby impressed (quite literally as pressure) on the surface of the pineal gland, there is no immediate reason (without knowing a great deal more about the arrangement of the nerve endings and the fluid mechanics of spirit flows in the chamber) to assume that there is a particularly good or accurate “image” of the scene there. And there is even less reason to speak of images when some other sense than vision is in question.

Descartes nevertheless continued to speak of images. For example, in the Meditations Descartes calls what is physically/physiologically formed in the brain a “corporeal image”; he carefully distinguishes it from the idea that is psychologically experienced. He says that the latter derives from the mind’s attending to or being directed toward the corporeal image. Clearly he is not invoking the so-called homunculus, as though a little man in the brain directs the little gaze of his little eyes toward the little image formed on the gland. The mind attending to the corporeal image may be an expansion of the schema that he had tentatively introduced in rule 12 of the Regulae, when he suggested conceiving the differences between colors as corresponding to differences between patterns of two-dimensional figures. That is, the physics of the cosmos and the physiology of the body lead to the production of some kind of patterned impression on the pineal gland. Descartes does not want simply to say that such a pattern will automatically produce a given color (say red) in the mind, because he frequently invokes the phenomenon of human consciousness in which our mind or attention is elsewhere than on what lies before our noses and eyes—so that we do not notice the red until our attention is called back to our sensory experience. Thus even though he claims that God has instituted certain correspondences between gland motions and ideas as part of our nature, Descartes does not want us to think of this simply as what we might call a stimulus–response or reflex theory. In any case, the stimulus is a complex, imageable pattern, and his deep scientific conviction is that the methodically directed imagination of scientific research can correlate such physical and physiological patterns with patterns of perception. The perceived pattern does not, however, have to be of the same quality as the stimulus pattern—if the pressure pattern on the pineal gland corresponding to red is “cross-hatched,” that does not mean that anything at all about perceived red will be cross-hatched.

Perhaps these correlations between impressed patterns and perceived images are uneasily reconciled remnants of an older way of thinking—or perhaps they are positions embraced within the underlying conceptual topology. At any rate, they appear to have encouraged Descartes to look for new possibilities and consequences in the old nerve–and–spirit theory. Some pineal gland activity would be produced directly by thought, but insofar as memory was involved, there would have to be spirit flows communicating from the pineal gland to memory locations. The activity of new imagining would induce other changes in the gland and in the surrounding spirits, not to mention the effects of any continuing acts of sense perceiving coming from the sense organs directed toward the changing vista of the outside world. If there are spirits in all the nerves communicating with all the organs of the body, and a sea of spirits bathing the parts of the brain, there are almost limitless possibilities for eddies, flows, and currents not directly connected with imagination, memory, sensation, and motor activity. Do these have any psychological effects? In the 1640s, in the Passions, Descartes gives a clear, affirmative answer.

The Passions in fact conceives all the parts of the body in contact with the spirit system as capable of inducing motions and impressions in the spirits. Thus virtually the entire body (and especially certain privileged areas, like those around the heart, liver, and stomach) is connected with the psychologically crucial, central spirit chamber and so can contribute to “spirit flow turbulence”; this turbulence is translated into appearances at the pineal gland in ways less definite and determinate than regular sensations. These appearances are feelings, emotions, passions, stray incipient images of dreams, daydreams, hallucinations, and the like. That is, in the Passions Descartes expanded the realm of psychophysiology beyond determinate sensations, images, memories, and motor activities to begin accounting for a fuller range of psychological and physiological phenomena and their essential interaction than he had before.

Already in the 1620s Descartes had conceived the knowing power’s relationship to the body as always involving the organ or gland of phantasia: imagination is the knowing power at work on the gland, memory the knowing power accessing brain memory locations through the gland, sensation the knowing power extending to the sense organs by way of the nerves that arrive at the gland, and motor activity the result of automatic and deliberate (i.e., produced by the knowing power) impulses issuing from the gland out to the body’s muscles. The Passions does not abandon this schema, but rather more expressly recognizes that everything involved with sensation, internal sensation (as the medievals called it), and motor activity is part of a single complex system of nerves and spirits. Directed imagination, which can be used for cognitive purposes, is in the first instance a question of will’s forming images in the gland; but there is also a kind of incipient imagination—call it para-­imagination—that is largely a byproduct of the physics and physiology of motions in the spirits and nerves as they are affected by the other parts of the body. Much of this does not lead to clear and definite images but rather vague and transient ones, and some of the spirit flows do not produce object–images at all but rather establish a background or foreground of feeling with greater or lesser duration.

Perhaps the most remarkable thing of all is that Descartes does not treat all of this as physiological and psychological “noise” disturbing the rational processes that are the philosopher–scientist’s central concern. Nor does he give in to the impulses of stoicism that would demand the suppression of feeling and passion. In response to the rationalist temptation he countered that the passions of the soul produce a system that naturally directs our minds to objects and holds our attention on them. Wonder is a passion that sensitively responds to what is new in experience and holds our attention long enough for us to begin intelligently dealing with it. Love, hatred, joy, fear, and desire, the other five primitive passions, keep us focused on objects of relevance to actions. All six primitive passions, in their myriad combinations, give rise to other particular passions that further diversify our attention and the character of our lives. Causality in this realm is bidirectional and biplanar: the physiological changes associated with a passion produce the psychological experience of the passion, and psychological reactions tend to prepare, sustain, change, or suppress some of the physiological responses.

This conception provides imagination with a new and unexpected function.Footnote 57 Descartes noted that the will does not have a direct effect on physiology: we cannot stop being sad simply by willing it. But the will can form images in a directed way; thus by volition we can choose to saturate our pineal gland with gladdening images that tend to relieve sadness and bring joy. We can think pleasant images and scenarios, we can read amusing books, we can view comedies; the thoughts they produce will, outward from the pineal gland, induce new impulses in the eddies and currents of the animal spirits and counter (if not always overcome) the physiological conditions that have produced the undesirable passion. Of course our intelligence is also at work in this process: we are, according to Descartes, supposed to determine the will to what is best, and it is intelligent perception, the power of comparing things to one another and seeing the appropriate ratio that holds between them (typically with the help of images, of course), that provides the volitions with rationality. But human beings are not just intelligence, not just will, not just imagination, not just memory, not just sensation, not just motor activity, not just passion, emotion, feeling: they are all of these together, in a psychic economy instituted as the whole nature of the human being by God. The thinking part of us has its essence precisely as thinking—though thinking begins with the activity of volition rather than with the passion of perceiving, even intellectually (recall that perception is a passion, not an action)—but our total natural being is precisely as complex as God created it. He could have made us otherwise, of course, for example as pure rational beings. But he did not. We are the way we are, and our task is to live well as the beings he made us.

Thus the Passions’ accounts regarding interactions of will, imagination, and passion directed toward what intellectual perception determines is best are not just another ruse of a rationalism claiming to be the master faculty in rational passion management. What these complications show instead is that, by the time of his death in 1650, Descartes had begun to come to terms with the entire economy of the psyche in its physiological incarnation. It gave him a way to show in detail that human life is not all about knowledge, that human beings are not created simply to be knowing beings. Perhaps it was just his Jesuit training coming out: the spiritual exercises of Ignatius exercised the imagination and reason in meditations in order to arrive at a resolution of the will to live as best one could in accordance with God’s will. The aim of living is to live well. The intellect and all its helps exist not primarily for the sake of proving the existence of the self and God or doing mathematics and science but for keeping body and soul together (literally!) and for always willing, and thus trying to do, what the intelligence determines is best among all the appearances the mind–body system shows.

If any proofs are needed, there are two strong supports for this “humanistic” interpretation, one from Descartes’s letters, the other from the conclusion to the Passions. In discussing the primitive notions Descartes pointed out that soul can be known only by intellect, that extension can be known by intellect but much better by imagination, and that soul and extension together in the incarnate human being are known by sensation. This would seem to settle the matter of their relative value, seeing that sensation has always been the lowest of the faculties of soul. But he then advises his reader (Elisabeth) to spend no more than a few hours a year in metaphysical-­intellectual reflection and no more than a few hours a week in scientific and mathematical speculations that occupy intellect and imagination together. These activities are very troublesome and fatiguing, he points out. The vast majority of one’s time should be devoted to conversation and other pleasures of the senses (AT III.692–693). If this sounds very uncartesian, perhaps it is because we misjudge Descartes: we think of him as though he were merely a follower of the Descartes of our interpretations—the Descartes that cartesians made of him—rather than a thinker who directed his attention to the sources of things. If we think that Descartes did not follow his own advice insofar as he worked on metaphysics, mathematics, and natural science, perhaps that means only that we do not know enough about the facts of his day-to-day living or notice that he does not seem to have spent much time after the 1644 Principles on metaphysics and physics/physiology, nor much time at all on creative mathematics after the 1637 Geometry. It is lost to us, but we know that the last work he composed, at the court of Queen Christina of Sweden, was a play, an entertainment (technically, a masque).

Perhaps Descartes meant exactly what he said at the end of the Passions. In the very last words of the concluding article 212 he writes:

As for the rest, the soul can have its separate pleasures; but as for those that are common to it with the body, they depend entirely on the passions, so that the human beings they can move the most are capable of tasting the most sweetness in this life. It is true that here they can also find the most bitterness when they do not know how to employ them well and if fortune is against them. But wisdom is principally useful in this point, that it teaches to make oneself so much a master of them and to manage them with so much address, that the bad things they cause are very supportable, and even that one draws joy from all. (AT XI.488)

The power of imagination, which in the early notes could strike sparks of poetic insight that philosophers could reach only by plodding, had gone through a long period of discipline that culminated in the mathematics of analytic geometry and the cosmological science of universal physics. It seemed to go into eclipse in the works of his metaphysical maturity, but that was more appearance than reality. In the last analysis—an analysis that began in his correspondence with Elisabeth of Bohemia and came to fruition in the Passions of the Soul—imagination was recast as the will-­directed art of entertaining and managing all the appearances that redound to the human being who is soul and body together. The originals of appearance for the most part come to us through sensation, feeling, and emotion; imagination is the power by which we can “pull back” from the immediacy of the appearances of the moment and play with their possibilities. Through this play of imagination the best (the goal of will) can emerge and appear to us (in intellectual perception) and thus help us live well and experience all the sweetness that the God-given passions afford us.

But this imaginative vision of a good life vanished in the hearts and minds of those who came after; it went into nearly total eclipse. Many of the best of his followers overlooked it in favor of the rationalism they took his philosophy for, and some of those who recognized its consequences took steps that undermined it. In the history of Western thought, imagination was never quite the same. The next chapter will try to explain why and how.