… The right angle, the plane, the volume, their intervals and their areas, will be recognized as chaotic, dense, compact — again teeming with folds and dark hiding places.
… The birth of beauty never stops; Harlequin has never donned his last costume.
… If by the birth of geometry one means the appearance of an absolute purity on the ocean filled with these shadows, then let us say, a few years after its death, that it was never born.
This is from the essay ‘Mathematics & Philosophy: What Thales Saw . . .’ by Michel Serres (1982):
Hieronymus informs us that he [Thales] measured the height of the pyramids by the shadow they cast, taking the observation at the hour when the length of our own shadow equals our height.
– Diogenes Laertius============
… The point is to transpose some unreachable figure into a more immediate realm in the form of a miniaturized schema.
Accessible, inaccessible, what does this mean? Near, distant; tangible, untouchable; possible or impossible transporting. Measurement, surveying, direct or immediate, are operations of application, in the sense that a metrics can be used in an applied science; in the sense that, most often, measurement is the essential element of application; but primarily in the sense of touch. Such and such a unit or such and such a ruler is applied to the object to be measured; it is placed on top of the object, it touches it. And this is done as often as necessary. Immediate or direct measurement is possible or impossible as long as this placing is possible or is not. Hence, the inaccessible is that which I cannot touch, that toward which I cannot carry the ruler, that of which the unit cannot be applied. Some say that one must use a ruse of reason to go from practice to theory, to imagine a substitute for those lengths my body cannot reach: the pyramids, the sun, the ship on the horizon, the far side of the river. In this sense, mathematics would be the path these ruses take.
I’m skipping Serre’s arguments that “Vision is tactile without contact.”
… There is an instance of clear knowledge that is hidden in the workers’ hands and in their relation to the blocks of stone. This knowledge is hidden there, it is locked in, and the key has been thrown away. It is in the shadow of the pyramid. Here is the scene of knowledge, the dramatization of the possible origin, dreamed about, conceptualized. The secret that the builder and the rock-cutter share, secret for him, for Thales, and for us, is the shadow-scene. In the shadow of the pyramids, Thales is in the domain of implicit knowledge; on the other side of the pyramid, the sun must make that knowledge explicit in our absence. Henceforth the entire question of the relationship between the schema and history, of the relationship between implicit knowledge and the artisans’ practice, will be posed in terms of shadow and sun, a dramatization of the Platonic mode, in terms of implicit and explicit, of knowledge and practical operations: on the one hand, the sun of knowledge and the sameness; on the other, the shadow of opinion, of empiricism, of objects.
… The origin of knowledge acquired through everyday practice is on the side of the shadow; the origin of a practice acquired through knowledge is on the side of light. One could learn a great deal about the emergence of a theory by diligently asking oneself about its various realizations a posteriori and by reversing the analysis.
… The edifice [the pyramid] is a volume of volumes, a polyhedron composed of cut-out blocks of stone. Now how is one to study and learn about a volume if not by means of a planar projection? And how can one lay hold of it if not by attacking its surfaces? Thales’s geometry says this, and so do architectural technique and the mason’s daily practice. In each of the three cases it is a matter of studying a solid in terms of all the bits of information that have been gathered at the relevant levels: the secrets of an object’s shaded surfaces and its cast shadow. I know nothing about a volume except what its planar projections tell me. But a projection assumes a point of view and a drawing on a smooth surface, a surface without any shaded area and without any hidden fold. I can know a stone, a solid, even a pyramid, only by its contour described by the sun on the plane of the desert sand. The sun-subject writes a form in the sand, a form that is changing and infinite like the profiles of Ptolemaic perception, a form that describes a cycle of representation. Each moment of the representation: a stable relationship with the same shadow, at the same moment, of another object — with me, for example. Here the geometry of perspectival measurement articulates the invariant in the variations of representation. The cast shadows vary, the secrets are transformed, but they share among them a secret which remains constant and which is the unknown, the pyramid’s secret: its inaccessible height. As variable as representation may be, it still designates, suddenly, a portion of the real, a stability proper to the object, its measurement. Which is why, from this position, I can only know about the volume that which is said, written, or described by cast shadows — the bits of information transported onto the sand by a ray of sunlight after its interception by the angles and summit of an opaque prism. This geometry is a perspective (an architecture), it is a physics, an optics: the shadow is a black specter.
… However, the story begun in the Nile delta will soon be completed by a sudden and incredibly audacious coup d’état: the radical negation of interior shadows.
… The archaic Thales of mensuration gives way to pure geometry, pure because it is cut through by the intuition of transparency and emptiness. Then and only then can the pyramid be born, the pure tetrahedron, first of the five Platonic bodies. By this miracle the sun is in the pyramid: the site, the source of light, the object, all in the same place.
Beneath this new sun, solids no longer have a shadow or a secret; light passes through them without being interrupted, just as it glides along a straight line or a plane; the world they constitute is thoroughly knowable. One can understand the importance that Plato and his school constantly attribute to the stereometry of volumes. The open history of infinite explications is closed by this power move, by this stroke of lightning that rips away the veils of shadow; this history is reoriented toward the transcendency of forms. There is no more specter, or analysis; the three shadows (the one on the shaded area of the surface, the one cast, and the one buried within) are snatched away by the sun of the Good.
… Nevertheless, this power move is not exactly a revolution. Plato kills the hen that laid the golden eggs: by cutting through the solids he nullifies history; the eternity of transcendency freezes diachrony and the genealogy of forms. The future of the square and the diagonal is decided as much on the sand where we describe them through the language that names them as it is decided in the sky of ideas. The realism of transparent idealities is still immersed in a philosophy of representation.
… This form is pre-judged to be without shadow or secret, it exists itself and in itself, but it never hides anything that could exceed the definition one has fixed for it. It exists as an ideality, transparent to vision, transparent to noesis. It is a theoretical element known thoroughly, something seen and known without residue. Intuition is blinded by its existence, but intuition passes through it. Its identity guarantees that it is ubiquitously identical, and hence its perception is not interrupted. Vision and knowledge are white specters. Now, precisely when this pure geometry, inherited from Plato, dies, when it is no longer possible to assume intuitive principles, when the theater of representation is closed, the secret, the shadow, and the implication will explode again among these abstract forms before the eyes of dumbfounded mathematicians — explosions that had been announced before all these deaths throughout history. The right angle, the plane, the volume, their intervals and their areas, will be recognized as chaotic, dense, compact — again teeming with folds and dark hiding places.
… what are the relationships of a technique, of a myth, of a communication, and of a philosophy? Again, the idealities implicit in technology, mobilized in representation, dramatized by myth, and transported by a particular language are filled to the brim with an implicit knowledge. The birth of beauty never stops; Harlequin has never donned his last costume. The myth is perpetuated; representation is spread further and further; archaisms resound through the centuries and are ferried to our feet like alluvia. What Thales saw at the base of the pyramids (the sun, the homothetic edifice, the shaded surface and the cast shadow), what Thales did alongside the pyramids (the partitioning off and the measurement of similar triangles in the parallelism of two gnomens, one of which is our body), are the thousands and thousands of implications that the history of science is slowly developing and that the eternal geometers will see, without always seeing them, and will create, without always knowing it. These implications express nothing less than the obscure articulations of rigorous knowledge and the totality of other human activities, indefinitely abandoned to their obscure fate. If by the birth of geometry one means the appearance of an absolute purity on the ocean filled with these shadows, then let us say, a few years after its death, that it was never born.
My most recent previous Serres post is here.
-Julie
