Welcome to new and returning Graduate Institute students, friends, family, and colleagues.
Today you new students are beginning a program of study in which the books you read will challenge you to confront fundamental questions, to re-examine cherished opinions, and to analyze the very structure of your world. You are beginning this program in either the Literature or the Mathematics and Natural Science segment. Perhaps you see a vast gulf between the world of mathematics and the world of literature. No doubt, these categories reflect our sense of two radically different human possibilities. However, we at St. John's hesitate to pigeonhole our texts and we often prefer to leave it to you to disentangle the various threads of knowledge. Euclid's Elements is a constructed work of art and rhetoric as well as a progressive development of the logical consequences of certain axioms. Characters in Sophocles' Antigone propound reasoned arguments for and against the primacy of the state even as they embody the raging passions, flawed insecurities, and irrational loyalties of all-too-human individuals. So, in the Graduate Institute, the organization of "great books" into categories for our segments is done with a bit of a bad conscience; it is knowingly done more for the sake of practical considerations than as a serious claim regarding the boundaries of either the books or the issues therein.
In this spirit of modest challenge regarding the compartmentalization of learning, I would like today to reflect upon an underlying shared motif in our Mathematics and Natural Science and Literature segments: namely, imagination.
The role of imagination in literature—both on the part of the author and as necessitated in the reader—is universally acknowledged. Whether we view it with suspicion ("oh, that could only happen in your imagination") or with reverence ("in metaphor the imagination is life"—Wallace Stevens, Three Academic Pieces), we all experience a particular sort of consciousness when we open a novel or play or epic and begin to read. Immediately, we envision a scrawny knight on a skeletal steed accompanied by a rotund peasant, charging a giant/windmill with lance atilt. Or, guided by the well-turned phrase, we conjure up a gory battlefield littered with corpses where Greek and Trojan antagonists pause to exchange personal histories before swinging the bludgeon or thrusting the spear. Literature works upon our minds, quickening our productive powers and prompting us to construct a world, usually a little bit alien, populated with vivid, captivating images.
You may, on the other hand, be only beginning to appreciate the role of imagination in mathematics—as you linger over the definitions and postulates of Euclid's geometry. Unlike the austere equations of algebra that you will have experienced in a prior educational life, Euclid's formulations prompt, indeed demand, active visualization. True, the first definition, "A point is that which has no part" resists easy visualization. Nonetheless, in order to understand this definition, very quickly we begin trying to picture something which has no part. Usually, someone draws a dot on the chalkboard. However, we can literally see the specks (or parts) of chalk dust, so instead we represent the dot in our mind's eye; we strip away the specks of chalk dust; we isolate a speck; we zoom in with our mind's eye microscope and attempt to examine whether we have stripped away all parts and are left with a single, ultimate unity. Surely we are exercising our imagination in this process—a process we repeat with the second definition, "A line is breadthless length." Again, we must picture something, some stretching forth through the inner space of the mind; again, we must strip away any visible width from this image; we must repeat this process so long as any width is "visible"; we must try to isolate the direction until it is all but invisible.
This peculiar constructing of an image of Euclid's definitions certainly involves the imagination. Drawn images are, we infer, not the ones that Euclid intends. Why then does Euclid formulate his first postulate as: "to draw a straight line from any point to any point"
It is a puzzle. However, I might note that the Greek word translated as "draw" also means "lead" as in "lead the troops into battle." So perhaps this verb conveys more the sense of an energetic activity required of the student of geometry than a physical drawing. We must be able to summon forth, to "lead" a line out of one point to another. But what translators generally fail to capture is that Euclid uses not only an infinitive (which already mutes the sense of activity), but a perfect infinitive—something like "to have drawn." So, though Euclid points us to activity, he places us in an odd temporal relation to such activity. The activity has already been accomplished. Moreover, the infinitive is not a direct command to us, nor even a permission to us to perform this activity.
Taking into account Euclid's prefatory clause, "Let the following be postulated," we may construe Euclid to be begging for our permission. But again, translators have trouble rendering Euclid's exact grammar: he uses a third person perfect passive imperative. No wonder they have trouble. We have little experience of the third person imperative in English; the St. John's Greek text suggests that the playwright's "Enter the king" is an equivalent. Such an imperative does not directly address the reader, but instead some other, hidden being who presumably can implement the command. The passivity of Euclid's imperative further weakens any sense of our participation in the process. I too am inadequate to yield a faithful translation, but perhaps "Let it have been begged: to have led out a line from any point to any point."
What then does Euclid want us to do? His text is filled with action verbs such as "describe," "apply," "construct," as well as "draw." These verbs are almost universally governed by the aforementioned perfect passive imperative; nevertheless, they are powerful incitements to temporal processes. What is our relation to these processes and to the objects they animate?
Speusippus, Plato's nephew and his successor as head of his Academy, is reputed to have commented on Euclid's geometrical objects:
...it is better to assert that all these things are and that we observe the coming-into-being of these not in the manner of making, but of recognizing, treating the timeless beings as though they were presently coming into being....
The language Speusippus chooses, "as though they were presently coming into being," highlights, I think, the opposing aspects I've noted in Euclid's treatment. This phrase, "as though" also captures the imaginative aspect of his entire enterprise. While Euclid is not telling us that we must create a line, ex nihilo, out of nothing, he is encouraging us to reflect upon, to imagine, its having come into being, its genesis, its nature.
Similarly, every Euclidean proof launches us on a discursive and imaginative journey. First, the proposition states a universal truth, such as: "In isosceles triangles, the angles at the base are equal to one another." Next, some particular figure, here a definite isosceles triangle ABC, is called up for inspection—using the perfect passive imperative that lets us know the triangle connecting those points has previously been constructed. What follows is a step-by-step unfolding of a discovery of the relations establishing the truth. These relations permeate the object already—an object whose existence is antecedent to the proof itself. But we are presumably only now revealing them to ourselves. If someone goes to the board to reiterate the relevant steps, you begin to appreciate how integral the imagination is to understanding the proposition. Should the person at the board make the mistake of drawing first the entire finished diagram with all the enhancements to be added by the various steps, rather than allowing the drawing to take shape gradually, you will find it nearly impossible to understand, to see, to grasp the interconnected relations that exhibit the general truth proposed.
As we present each relation to ourselves in embodied form, i.e., drawn on the board, we use double vision of a sort to see through the drawn diagram to the perfected vision in our mind's eye (we do this almost effortlessly); simultaneously, we subject this perfected vision to the crucible of our critical reasoning (this requires some real effort and the assistance of Euclid's prose). Thus, our activity while doing proofs mediates between thoughtless receptivity of facts or sensations and timeless apprehension of timeless truths. And this mediation relies crucially upon imagination. Only through an imaginative encounter with the unfolding proof are we roused from mere passive sensation, inspired to an examination of the architecture of our inner space, and pushed forward and beyond our initial survey.
Certainly, unrestrained imagination cannot achieve the desired moment of dawning recognition that the proposition must necessarily be true. Imaginative constructions must be challenged, questioned, articulated, limited by the requirements of reason. But I would like to point out that reason needs some regulation also; Meno's slave boy is seduced by the attractive, echoing sound of the words, when he leaps to the notion that "double the line will produce double the square." Socrates uses a drawn image to help him correct his own mistake.
Here, I am speaking as though imagination and reason perform their functions in some clearly separated, disjunct fashion. But I think the case is much more complex. Even the perfected vision necessary to "see" a point or line or triangle is not clearly the domain of imagination alone. Can imagination truly achieve a depiction of partlessness or breadthlessness? Does imagination achieve the final leap from a dot to a point? We intuit that the repetitive visualization process, stripping away breadth from a ruler, say, could proceed ad infinitum; and this understanding gives rise to some shaped idea in us. But the understanding must partner with imagination to give us access to such an idea—an idea we are fairly confident is identical for each of us, a specific, articulable, unambiguous, and essentially spatial idea. An idea we access through spatial, visualizable images. Trying to understand geometry without the imagination may be like a blind person reasoning about colors.
This partnership between imagination and reason functions in the proof activity I described before as well. As we bring the diagram itself into being in front of us, we are simultaneously bringing our own understanding of the proposition into being. Here too, it is difficult to disentangle the vision of the understanding as it grasps the truth of each step and of the whole from the vision of the imagination as it peers into the diagram to see the claims themselves. Speusippus articulated this vision as a "recognizing"; the Greek word there is different from the one Plato uses in the Meno regarding recollection. However, I think both words strive to capture the aspect of the experience of knowing that feels as though it is a looking, a looking at something at once separate and other, yet immediately appropriated. I note that Socrates chose a geometrical example to illustrate his notion of recollection. No doubt there are many reasons for his choice, but I think one reason is that such an example highlights the indispensable role of images in coming to know.
Perhaps it seems extravagant to claim that images are necessary for knowledge. We at St. John's stress that reason is the arbiter in our discussions, and I certainly don't want to minimize reason's importance. But I am in good company when I emphasize images in our search for a glimpse of unchangeable, immutable truths. In De Anima, Aristotle asserts that "without imagination, there can be no thought." (427b16) One of Descartes' rules is "not to recognize those metaphysical entities which really cannot be presented to the imagination." (Rules for the Direction of the Mind, Rule XIV, p. 57) For Kant, imagination is called upon to put together even space and time, as well as every appearance of an object in space and time. He says:
Synthesis in general... is the mere result of the power of imagination, a blind but indispensable function of the soul, without which we should have no knowledge whatsoever, but of which we are scarcely ever conscious. (Critique of Pure Reason, B103)
To focus more intently on the power of the imagination as encompassing both our story-making capacity and our mathematical activity, I'm going finally to turn toward Plato. In fact, I'm going to turn toward an image that Plato uses in a book we've placed in neither Literature nor Mathematics/Natural Science but in the Politics and Society segment, namely the Republic. It is, however, a book that speaks to us as complete human beings.
Somewhere near the heart of that enormous work (509d-511e), Socrates proposes an image of a divided line as a representation of the entire cosmos, both the sensible world and the intellectual world. He uses the image to depict the relationships of various categories of things, to chart an ascent through these categories up to the idea of the good, and to discuss the human powers correlate to each stage of the journey. I'm going to isolate one layer of this rich and complicated image: the image is itself about images.
Socrates divides the line first to represent visible and intelligible things; then he divides each of those segments similarly. The visibles are further separated into images (shadows, reflections in water, mirror appearances) and the objects of which those things are images (animals, plants, artifacts). The intelligibles are divided into ideas approached deductively—for example, geometrical ideas—and ideas grasped without hypotheses. Socrates proposes (517b) that the visibles themselves reflect the same relationship to the intelligibles that the first subdivision does: visible objects are images of the intelligible objects. Note the reversal of our usual interpretation of the material world as the real thing from which we abstract vapory concepts.
Socrates uses the divided line to discuss four different human relations to the four categories of objects (comparison, trust, deductive thinking, understanding). Though an unmediated knowledge of ideas is held out as the proper intellectual engagement with the very highest category, the divided line itself is a result of Socrates' reluctance to speak "about what one doesn't know as though one knew." (506c) Socrates presents his interlocutors with the poetic/mathematical metaphor of the divided line when they press him to reveal his own (mere) opinions about the good and knowledge (506b). Apparently, at least this instance of image-making allows him to speak appropriately about what he does not know. Of course, the divided line has a patent character as an image; we are in no danger of being seduced into thinking the visible things actually are a line. But why is this image the right kind of speech for communicating Socrates' opinions (even if he must warn us to guard against any unwilling deception therein—507a)? It helps to remember that, for Socrates, opinions are never mere opinions; images are never mere images; they are waystations on the path toward truth.
I conjecture that the line itself reveals something even more general than Socrates elucidates; it elicits a fifth mode of thinking. The relationship each part of the line possesses to its neighbor is that of being an image. The philosophic ascent up the line involves understanding—seeing—the imaging relationship, seeing each image as an image (rather than imbibing it as a complete and finished story).
Inspection of an image, whether poetic or geometric, whether a living organism or a painting, is a proper activity of the philosopher. The realization that an animal or a poem or a circle is an image—both manifesting and hiding reality—is what moves us as thinkers up the divided line, what puts us at least at the point between two parts of the divided line. We become fully engaged with what's in front of us, asking questions, making judgments, winnowing claims—even making helpful images. Only thus do we reveal to ourselves the potential for something we might call recollection or recognition or learning.
"A point is that which has no part."