Relationship between
Totality and Parts

Only after psychology and philosophy are definitively spearated can the question of totality and its parts take concrete form and give hope of a satisfactory reply. But before we call on psychological research, let us turn aside to a specifically aesthetic domain. [p. 205]

Esotericism
At the edges of philosophical and theological thought we have always found reflections that particularly interest us. Certain real or imaginary numerical relations between man and the world, or, as they were later termed, between microcosmos and macroscosmos, have richly inspired both profane and religious philosophy for more than two thousand years. In the realm of numbers secrets were suspected to exist that one tried to explain in a more or less esoteric manner.

The root of all these theories is to be found in a verse of the "Hieros Logos," or "Sacred Discourse": "Everything is arranged according to Numbers." This line, though attributed to Pythagoras, was in fact written much later. However that may be, the idea it expresses was intimately linked not only with aesthetics but also with science until relatively recent times. Even today, in modern mathematics, we hear a curious echo of it. We know that certain theories in modern mathematics, such as set theory, were considered typical examples of pure science--like a marvelous yet gratuitous proof of the capacity of man's rigorous thought--up to the day when physicists discovered that to explain certain phenomena of microphysics, the "new numbers" of modern mathematics were perfectly capable of being utilized. Naturally, however, the philosophical question remains: is the atom indeed organized mathematically, or is it rather our consciousness that has discovered a new tool, a new crutch? "When the Universe began to be organized . . . . God gave each element its features through the interaction of Ideas and Numbers," wrote Plato in the Timaeus. We have no communication with God, therefore we can take no position as to this delicate question, for lack of the requisite information.

But let us leave this task of unraveling the enigma of the philosophers. Our business is with aesthetics. From that point of view we can assert, without any risk of deceiving ourselves, that to the man gazing at a picture it is of no consequence whether the forms--or parts--of the picture are arranged according to numbers or not, or at least not when his gaze seeks aesthetic pleasure. [The situation is a quite different one if he is looking at the picture in order to study it.] Mathematics speaks to our understanding; the picture, to our sensitivity. We by no means want to separate the two domains. On the contrary, we know that feeling is greatly influenced by knowing, and conversely, our knowing by our feeling. Doubtless, however, every visual creation speaks directly to our affectivity. Furthermore, this principle is the chief criterion of all pure plastic art.

Since certain artists and certain modern theoreticians attribute such importance to these "mystic" figures as an organizing force in the totality of artistic creation, we must pause at this point.

The Golden Number
From Pythagoras to Le Corbusier, including Vitruvius, Vuillard de Honnecourt, and Dürer, all the mathematical or geometrical theories of beauty have been based on the Golden Number.

The Golden Number is a geometrical progression whose formula is: a+b/a = a/b and after a slight mathematical manipulation, we get the formula: À = 1 +/-æÌ5/2. [NOTE: View 'Document Source' to see this formula] [p. 206] This mathematical formula has a number of remarkable aspects. But it is far from being the only interesting formula: mathematics is very rich in curious formulas that play an important role in science.

From our point of view, however, it is immaterial whether the basis of the composition of a work of art is the Golden Number or some other numerical relationship, some other "harmonic series." The mathematical beauty of a work of art is not perceptible immediately but only after long reflection, and furthermore, without mathematical knowledge it is not perceptible at all. Not even a mathematician's head contains any built-in ruler or scale or calculating machine to perform the requisite operations fast enough. One's aesthetic pleasure fades during that brief lapse of time the psychologists call "the density of the present." What goes on in our head in that fraction of a second, we scarcely know. It is hard enough to imagine that we can manage a not very complicated calculation, when we know that the time needed for simple operation is more than the density of the present, and this even for a man highly trained in mental calculations.

No matter how many rational or irrational properties there are in a pentagon, it is not beautiful because of that. If it is beautiful, the reason for its beauty must be sought elsewhere. Of course, mathematics too has its beauty but we have to traverse a long road of reflection before we perceive it. We have to "understand" mathematics, in the strictest sense of the word. At the end of the road that we travel to understand it, we can find true joy, true feelings, that belong in aesthetic categories such as "superb," "monumental," "marvelous," etc. On the other hand, the beauty of a work of art must reveal itself, it ought immediately to impart itself, as we have said above. This, of course, does not mean that the beauty of a work of art cannot remain hidden as long as desired under certain conditions--longer, perhaps, than the sense of mathematics to a mathematician. Indeed, mathematics explains itself, conveys itself. The work of art, on the other hand, cannot be explained. Its beauty may or may not reveal itself, but it does not explain itself. It is possible to help someone become permeable to this beauty, but we certainly cannot explain it to him, for there is nothing to explain. [p. 207]

Science
We see, therefore, that if we do not want to shut ourselves up within a blind mysticism, subject ourselves to a dangerous illusion, we cannot a priori accept any mathematical modular that excludes man. Such a rejection of the priority of mathematical formulas over aesthetics by no means signifies an a posteriori renunciation of mathematical formulation. If the foundation is man, and his sensitivity alone, it is easy to slip into a subjective idealism. To be sure, art is the only terrain where one can eventually accept a subjective realism. But--to paraphrase C. Cherry, "Speaking language, and speaking about language"--to look at painting is one thing, and talk about painting another. When we confront a work of art in order to analyze it, we must have a scientific attitude, we have to distrust idealism. We thus avoid all danger of philosophism. Let us say that all the philosophical consequences [metaphysical, ontological, etc.] do not now concern us. We know that there are numerous problems in philosophy, but if we want to arrive at any concrete result we have to set aside these problems for the time being. According to the rules of scientific procedure, we must define exactly what we are talking about and avoid talking about what we cannot define. Let us abandon vain speculations, a priori theories, and turn to experimental psychology, where we find the problem of totality and its parts at the level of the study of perception. [p. 207]

Classical Psychology
It would seem that the psychologists, in undertaking the problems of perception, in accordance with tradition, have first tried to avail themselves of an especially fruitful heuristic method used in other sciences. They have tried to explain complicated psychological facts by simpler facts. But to explain perception by elementary sensations--that is, an atomistic psychology, as Locke imagined--soon clashes with the observed facts. Indeed, pure sensation has never been demonstrated. Sensation cannot be separated from perception. Pure sensation does not exist. We always perceive something. The red spot I see on a rug is inseparable from its wooly material, which I see at the same instant as the red spot. According to Cézanne, we see not only the color of the apple but also its savor, its fragrance, What we perceive in thunder is not merely thunder but "thunder-light--silence before--silence after," as William James said. [p. 208]

Gestalt
The same idea has been expressed and examined independently of James by the theory of form [Gestalt]. This theory considers form as a whole, indivisible. We cannot study a tree, it is said, by beginning with its roots, then its trunk, its leaves, etc., but rather it is the tree as a unity that must be the center of our attention. The forest is quite another thing, something more than the sum of the trees, as popular wisdom has long known. We therefore have to substitute for the idea of sensation the idea of form as the prime value. Furthermore, this substitution is in perfect harmony with the latest researches in physiology. The physiologists have increasingly come to substitute for the differences in cell formation the differences in circuits. Today no one looks any longer for that portion of the cortex where perception takes definite form, but rather for the circuit of the cortex that makes a particular perception possible.

The gestalt theory has had considerable influence on psychology. Its success is due in large measure to the fact that this theory agreed with the philosophy that was dominant between the two wars. Thus, according to the French philosopher, Merleau-Ponty, form takes on an important philosophical role. Watson, the leading exponent of the behaviorist school of psychology, reproached the Gestaltists for taking over the idea of immanence from Kant

The Gestalt school has, moreover, formulated laws that seem exact and that can serve for a scientific study of painting. Still, we cannot content ourselves with only general laws. It would appear that for a more precise understanding we have to return somehow to a psychology of detail. Here lies the contradiction of modern psychology: the theory of associational psychology [classical atomistic psychology] is unsustainable because it contradicts observed facts, whereas the theory of form--a more faithful mirror of reality--falls in the last analysis into philosophism. The psychological research of these last decades is based on the awareness of these contradictions. [p. 208]

The Problem of Form
Let us again consider what form is. Spontaneously, we believe that we know what form is. On closer view, we see it is nothing of the sort. The idea of form as an abstraction from the complexity of the real is a vague one that can take on various meanings, depending on the author, on the subject, etc. Let us envisage the simplest example, the perception of a geometric form. I immediately and unhesitatingly recognize a triangle. At the same moment I am shown it, I see one triangle, [p. 208] then another, and again a third. Yet these triangles differ: on the one hand, in size and in their angles; on the other hand, in the corresponding images they make on the retina. In view of the astronomical number of different forms we perceive as triangles, certain astonished philosophers speak of a miracle. Miracle there certainly is not, but we believe that even the psychologists cannot yet provide us with any satisfactory solution of the problem: by what process, what abstraction, do we perceive a triangle? It is hardly likely that we recognize a form after certain mental analyses. We do not say: there is a form whose angles seem to total 180 degrees, it has three sides, etc., therefore this form must be a triangle. On the contrary, we say intuitively, without hesitating, even without reflecting, this is a triangle. Form, therefore, appears to constitute a bridge between art and science.

But what is form? The theorists define it thus: a group of elements perceived in their totality, as it were, and not as the product of any chance assemblage. [I emphasize "perceived," for of course form in reality may well be the result of a chance assemblage.] To create a form thus defined would mean to assume a certain previsibility of this form. In the strictest sense of the word, previsibility, or Vorsicht, means to imagine a phenomenon in the future, in terms of the past. Let us imagine a circle we watch someone drawing. At any moment the designer can interrupt the design, while for my part I can at any moment hazard a guess as to what he intends it to be. If, at the instant he stops drawing, the circle was almost closed, I can be the surer that he wants to draw a circle than I can when he is just beginning to draw. Thus there obviously exists a degree of previsibility, a connection that is at least a statistical one between past and future, a correlation between what happened just now and what is going to happen in the immediate future.

It is this connection between the past and the future of a form explained statistically that Wiener has called "autocorrelation." Autocorrelation evidently varies between 0 and 1. It is 0 when the phenomenon is totally lacking in order and its behavior in future is therefore not previsible. As order appears, it tends toward the value of 1, or the autocorrelation of a completely ordered phenomenon--in other words, one that is indefinitely previsible. At first sight, we see that autocorrelation expresses very well the law of good form in the Gestalt theory. The nearer autocorrelation comes to 1, the better the form. However, autocorrelation differs in one essential from good form, since the later does not separate into parts. [p. 209]