H. MATHEMATICS, IMAGINATION, AND MATHESIS

Chapter 1 introduced the medieval doctrine of the internal senses and argued that its durability was due not simply to the Aristotelianism of West-em Scholasticism but also to support from other traditions, especially medical theory. The understanding of the imagination as a power of sensibility dependent on localized brain functions and playing a significant role in cognition was therefore not specifically bound to the fate of Aristotelian

[24] It seems unlikely that this would accurately describe his situation in late 1619, though it seems more than apt for his intention to explore first philosophy at the urging of Cardinal Bérulle (perhaps in late 1627) and his actual execution of that intention during his first nine months in Holland (i.e., in 1629). One should note that Rule 4's autobiographical approach to the history of mathesis is not echoed in any of the notes from the period 1619-162l, apart from the narration of the dreams of 10-11 November 1619, which is an episode to be interpreted rather than a life to be narrated. The Studium bonae mentis, the only other early autobiographical piece, cannot antedate 2623, since according to Baillet's account it discusses events of that year, and it is possible that this no longer extant work was written years later. Once again the evidence points to a later rather than an earlier date for the composition of Rule 4.

or Scholastic tenets. The centrality of imagination was bolstered by the concurrence of other traditions that contributed theories of its importance. For example, the Stoics taught that the dominant power of reason, the hegemonikon, was located in the brain and there received the phantasiai from the external world. In the Neoplatonic Augustine, in contrast, memory was, along with intellect and will, a member of the highest triad of psychological powers, while corporeal imagination was part of the immediately subordinate triad of sensitive powers. Indeed any Neoplatonist is as likely as not to ascribe an important role to imagination. Although Neoplatonism is, "in general, inimicable to a constructive theory of fantasy," it

is capable at any time of taking a course which leads to the ennoblement of a mental capacity which its basic philosophy affected to despise. This paradoxical nature of Neoplatonic thought concerning phantasia and related powers can hardly be overemphasized: its idealism taught it to despise phantasies, its dualism found a place for them, its psychology, largely Aristotelian, taught it to study them, and its passion for the Timaeus led it to recognize them as God-given.^[25]

According to a recurrent Neoplatonic theme, the different levels of reality image or mirror one another, so that imagination can acquire a certain ontological and epistemological credibility and partially bridge the gap between the corporeal and spiritual realms.

With respect to mathematics, Descartes was not the first to ascribe a special role to imagination; indeed, already in late antiquity Neoplatonists interpreted the phantasia as providing the medium and matter for mathematical objects. In his monograph on the background to Descartes's use of the term 'mathesis universalis', Giovanni Crapulli traces the notion back to the Greek mathematician Euclid and the Neoplatonic philosopher Proclus (412-485). In the sixteenth century the content of a universal mathematics was discussed under various names, such as scientia communis, mathesis generalis or universalis, mathesis universa, and mathematica generalis. These names were intended to indicate the discipline common to all the mathematical sciences; the core of it was the Euclidean theory of proportion presented in the fifth book of the Elements of Geometry. Moreover, within these sixteenth-century discussions of mathesis universalis there was an influential strand of thought going back to ancient commentaries on Euclid that understood the medium or material of mathematics and mathematical objects to be the intelligible matter of phantasia. So, for example, in his commentary on Euclid's geometry, Proclus presented the notion of

[25] Murray Wright Bundy, The Theory of Imagination in Classical and Mediaeval Thought, University of Illinois Studies in Language and Literature, vol. 12, nos. 2-3 (Urbana: University of Illinois Press, 1927), 146.

kinesis phantastike (imaginative motion) as underlying the mental activity of the geometer; elsewhere he described the phantasia as a "form-giving intellection of the intelligibles."^[26]

This doctrine of the imaginative basis of mathematics gained renewed topicality in the sixteenth century, especially through Alessandro Piccolo-mini's Commentarium de certitudine mathematicarum disciplinarum. In his discussion of the common science (scientia communis) he argued that the mathematical sciences were considered by the ancients as intermediate sciences both because they were neither entirely intellectual nor entirely immersed in matter, and also because of their efficient cause, the phantasia: "a certain power of soul that holds an intermediate place between sense and intellect," which Piccolomini, following Proclus and others, identified with Aristotelian intellectus passivus (nous pathetikos), passive or potential intellect. The things of mathematics "are not at all sensibles in the subject [= substance] nor things inwardly liberated from it: those mathematical figures are found rather in phantasia itself, garbed nevertheless on the occasion by quantities found in sensible matter." The phantasized quantity (quantum phantasiatum ) did not constitute the matter of any particular kind of mathematics; rather, "one thing of great weight is to be noted, that when we have shown the phantasized quantity to be the matter or subject of mathematicians, this is said to be the subject not of geometry or arithmetic, which two are the first genera of mathematics, but of a certain faculty common to geometry and arithmetic." Piccolomini understood this quantum phantasiatum and its underlying faculty of mind as clarifying what other authors had "improperly" called intelligible material; according to him, it was not so much intelligible as imaginative.^[27]

In the Ethics of Geometry, a reflection on the different significance of construction in ancient and modern mathematics (with Euclid and Descartes the protagonists), David Lachterman suggests that Proclus's elevation of imagination makes him the first modern mathematician. His influence, in combination with an affinity for imagination among early modern algebraists that was encouraged by their adaptation of techniques from the arts of memory taught by Llull, Petrus Ramus, and Giordano Bruno, decisively shaped modern mathematics. Lachterman considers, in particular, the use of diagrams in ancient and early modern mathematics within the context of an ontological dilemma posed by imageability. He

[26] Discussed by Lachterman, Ethics of Geometry, 89-90. The quote, given by Lachterman, is from the Commentary on the Republic, 1.235.18.

[27] See the account of Piccolomini in Giovanni Crapulli, Mathesis Universalis: Genesi di un'idea nel XVI secolo, Lessico Intellecttuale Europeo, no. 2 (Rome: Edizioni dell'Ateneo, 1969), 36-38. According to Crapulli (p. 41), Piccolomini was regarded by most of his contemporaries and immediate successors as a mathematician of eminence.

interprets Aristotle, in comparison to Neoplatonists, as more sharply separating the knowledge of mathematics from imagination, in that the process of abstraction makes the image transparent to intellection so that the latter can know unchangeable, universal essences.^[28] In following Proclus rather than Aristotle, modern thinking betrays a practical rather than a theoretical orientation toward mathematics. "Simply stated, radically modern thinking about imagination takes its bearing from the phenomenon of productive arts, including especially those arts adept at fashioning internal mental images and then embodying these elsewhere, by design"; Descartes, in particular, is one of these radically modern thinkers, and his mathematics is a paradigm of the modern approach.^[29]

To settle these issues would require not only broaching the controversies between ancients and moderns but also—and more to the point—set-fling the nature and source of mathematics and mathematical thinking. Both tasks are beyond the scope of this book. The point in raising them here is different: first, to show, however sketchily, that imagination was traditionally and perhaps inextricably bound up with the early modern understanding of mathematics and mathesis universalis; second, to make clear that universal mathematics, far from being merely a mathematician's concern, inevitably raises questions about the means by which things are knowable and about what ontological status the things known have. Even if Descartes had intended to confine his interests strictly to mathematics, the context of the subject matter would have forced him to move into metaphysical territory.

Yet it is probably incorrect to deduce that Descartes merely stumbled onto this territory. For from the sixth rule onward Descartes recurs more and more insistently not merely to the faculties by which we know and to the processes and method of knowing but also to the status of the things that are known and to the question of whether they can be known apart from concrete imaginings. It is not just the psychology of imagination that

[28] The discussion is in Lachterman, Ethics of Geometry, 76-90. Lachterman's inclination is to discount the importance in mathematics of the phantasm per se, because this transparency makes it invisible; the mind transcends the phantasm and proceeds to unchangeable. universal essences. He considers the noetic status of phantasms to be the key difference between ancient and modem mathematics. Ancients gave priority to the purely theoretical, whereas the modem orientation is more practical. I am not entirely persuaded by Lachterman's effort to distinguish Aristotle from Descartes on the issue of phantasms, however, not only because I interpret Descartes as attempting to think through some of the aporiai about phantasms left by Aristotle and Aristotelians but also because I believe that Lachterman understates the essentiality and indefeasibility of phantasms in Aristotle's theory of knowledge. Nevertheless. we can only regret that he is not alive to enlighten us further on this and many other things.

[29] Lachterman, Ethics of Geometry, 82, 86.

is important, then, but also the ontological underpinnings of this psychology and its associated epistemology—or, to use a name truer to the teachings of the Regulae, the ontological underpinnings of its way of mathesis.

Where this ontological concern first emerges with consummate clarity is in the doctrine of natures. Where Descartes begins the exploration of this doctrine is in Rule 6, a rule that he calls the secret of the whole art.