Michel Serres

Modernity vs. the Oedipal Incest of knowl-Edge

Serres wrote:

"The bridge is a path that connects two banks, or that makes a dis­ continuity continuous, or that crosses a fracture, or that patches a crack. The space of an itinerary is interrupted by a river; it is not a space of transport. Consequently, there is no longer one space; there are two without common boundaries. They are so different that they require a difficult, or dangerous, operator to connect their boundaries-difficult since at the very least a pontiff is necessary, dangerous since most of the time a devil of some sort stands watch or the enemies of Horatius Cocles stand ready to attack. Communication was interrupted; the bridge re-es­ tabiishes it vertiginously. The well is a hole in space, a local tear in a spatial variety. It can disconnect a trajectory that passes through, and the traveler falls in, the fall of the vector, but it can also connect spatial varieties that might be piled upon one another : leaves, layers, geological formations. The bridge is paradoxical : it connects the disconnected. The well is more paradoxical still: it disconnects the connected, but it also connects the disconnected. The astronomer falls in (Thales); the truth comes out. The killer dragon lives there, but one draws the water of immortality from there.

And suddenly, I am speaking with several voices; I can no longer draw the line between narrative, myth, and science. Is this bridge the Konigsberg bridge where Euler invented topology, a bridge over the Viorne or the Seine in the Rougon-Macquart cycle, or the whole group of bridges revealed in mythical discourses? No, I no longer have the choice, and it is the same bridge. Is this well a hole in Riemannian spatial varieties, a well of potentiality in which, at its lowest ebb, appears the germinating point, as in Thorn, or the Plassans well, or Jacob's? No, I no longer have the choice, and it is the same well. In every case, and so much the worse for classification, connec­ tion and non-connection are at stake, space is at stake, an itinerary is at stake. And thus the essential thing is no longer this particular figure, this particular symbol, or this particular artifact; the formal invariant is some­ thing like a transport, a wandering, a journey across separated spatial varieties. Circumnavigation of Ulysses or of Gilgamesh and topology.

My body (I cannot help it) is not plunged into a single, specified space. It works in Euclidean space, but it only works there. It sees in a projective space; it touches, caresses, and feels in a topological space; it suffers in another; hears and communicates in a third; and so forth, as far as one wishes to go. Euclidean space was chosen in our work-oriented cultures because it is the space of work-of the mason, the surveyor, or the architect. Hence the cultural idea of the practical origins of geometry that is a tautology, since the only recognized space is preciselv that of work,_of tranport. My body, there­ fore, is not plunged into a single space, but into the difficult intersection of this numerous family, into the set of connections and junctions to be established between these varieties. This is not simply given or is not always already there, as the saying goes. This intersection, these junctions, always need to be constructed. And in general whoever is unsuccessful in this undertaking is considered sick. His body explodes from the discon­ nection of spaces. My body lives in as many spaces as the society, the group, or the collectivity have formed: the Euclidean house, the street and its network, the open and closed garden, the church or the enclosed spaces of the sacred, the school and its spatial varieties containing fixed points, and the complex ensemble of flow-charts, those of language, of the factory, of the family, of the political party, and so forth. Conse­ quently, my body is not plunged into one space but into the intersection or the junctions of this multiplicity. Again, whoever fails or refuses to pass like everyone else through the crossroads of these multiple con­ nections-whoever remains in one of these spaces, or, on the contrary, refuses all of them -is treated as ill-adapted or delinquent or disoriented. Such is the case, for example, for whoever remains frozen, hung up, in the family tree, whoever fears leaving a closed paradise between two branches of a river, or whoever wants to tear apart the network which he endures as he would a prison or slavery's iron shackles. This brings us to the beginning. The fact is that in general a culture constructs in and by its history an original intersection between such spatial varieties, a node of very precise and particular connections. This construction, I believe, is that culture's very history. Cultures are differentiated by the form of the set of junctions, its appearance, its place, as well as by its changes of state, its fluctuations. But what they have in common and what constitutes them as such is the operation itself of joining, of connecting. The image of the weaver arises at this point: to link, to tie, to open bridges, path­ ways, wells, or relays among radically different spaces; to say (dire) what takes place between them; to inter-dict (inter-dire). The category of be­ tween is fundamental in topology and for our purposes here: to inter­ dict in the rupture and cracks between varieties completely enclosed upon themselves. "Enclosed" means isolated, closed, separated; it also means untainted, pure, and chaste. Now, that which is not chaste, incestus, can be incest. The incest prohibition (inter-diction) is, then, literally a local singularity exemplary of this operation in general, of the global project of connecting the disconnected, or the opposite, of opening what is closed, or again the opposite, and so forth.

Oedipus wanders and journeys, having set out from the palace of King Polybus to seek counsel from the Delphic oracle. At a crossroads he en­ counters Laius, his father, and Polyphontes, his father's herald, whom he kills. The crossroads is precisely the sought-for singularity. The roads for Daulis and Thebes meet at Megas; they form the road that rises toward Delphi through the valley. At Megas, the bifurcation. This is a very good point of departure, since in a diagram the example is trivial. There a road passes between two high rocks, as in a crevice or a narrow defile. Crossroads: cross, passage of a road across a ribbon that divides space, passing over a crack. Bridge : connection through the disconnected. To the left, ignorance, blindness, or the unconscious-the unknown and the unsaid. To the right, knowledge, the conscious, the sacred, Delphi and the signifier, the word. Oedipus is driven back from the narrow defile by Laius' team of horses, insulted by Polyphontes. The fact that the murder of the father takes place at this cross, this interrupted, joined edge, this limit or fault, is a catastrophe. Thus the circumstance is the murder and the law is traced upon the ground. To cross the broken threshold of the word. The essential thing is indeed the bifurcation. As soon as the father is involved, once again we come back to the law, a bifurcation traced on the family tree: father, mother, son, here again on the graph is inscribed the triviality of the narrative. To the left the one, to the right the other, and incest, we have already seen, is still another connection upon the disconnected. The text turns inside out like a glove and shows its function: the establishment of separations between spaces and their difficult junction. One can say that Oedipus kills Laius at this place, and miss the place, and thus repress the place of the repressed; or one can say instead that this place is such that Oedipus kills his father there, that it is a point so catastrophic and so confined that he must kill father and mother to go past it. To be the son or to place oneself at the crossroads: two bifurcations and two catastrophes that the myth joins together by its very word. Furthermore, the fact that the son's name is Oedipus repeats the same law. How can one move about in space when one's feet are afflicted? Now, to prevent him from journeying, his parents hang him feet up in the air. Oedipus regains his feet, he sets out for Delphi, the myth regains its feet.

The Sphinx, then, and the same law is repeated. This watchdog of Thebes dies as the result of a solution and lives as a result of solutions of continuity. She keeps close watch on the closed road where Thebans no longer pass, where they are devoured, in pieces. She is a chimera, half­ lion and half-woman; half four-legged, also, and half two-legged, and perhaps partly bird. She is a body sewn back together, badly sewn: two parts related by dichotomy, joined in the form of a Chl crowned by wings; she is a crossroads, with wings that protrude for one who no longer needs feet. The Sphinx is a bifurcation, and conversely. And the crossroads is a chimera. Thus everything is repeated, enigma and knowl­ edge, on the road to Thebes and the road to Delphi, catastrophe and passage, tear and connection. Oedipus is indeed the last descendant of the Sparto}, of disseminated spaces, of catastrophic separation, of the continuous that must be recovered. Everything is repeated once again when Jocasta recognizes her son by the scar on his feet, a scar in which the lips of a crevice connect. Now, Sophocles gives another version of the recognition scene, and his translation is faithful-Oedipus recognizes himself as a murderer at the moment in the narrative when Jocasta, the mother, mentions the crossroads, the chi. It is not I but Sophocles and the son and the mother together who draw the law out of the discourse.

From the beginning of the world protrayed in Plato's Timaeus, after reference to the chara, matrix and mother, in which we recognize a topological space, the Same and the Other, separated, are rejoined by the Demiurge in the figure of a chi. This figure is formed by the inclination of the ecliptic on the equator; the world is a chimera. The space of the world is described as requiring artful connection.

Now, then, at a certain beginning of a certain story, on the family tree containing ordered paths structured by some ordered relation, incest describes a loop that turns back upon itself toward a previous crossroads and strongly reconnects the spatial complex.

From this results the general and simple idea that mythical spaces are chimerical. This is a theorem containing a literal tautology but which uncovers a complicated state. Parts as separate as the Same and the Other are to be joined. Oedipus' itinerary crosses spatial accidents, bifurcations, catastrophes, and loops. Oedipus' discourse (discours) is identical to this itinerary (parcours). It poses chi's on cracks, crossroads between spatial varieties that do not have common boundaries. This in turn presupposes that before it, in other words, before discourse, there existed a multi­ plicity of unrelated spaces : chaos.

It would be necessary to demonstrate the generality of the hypothesis. The theme of the Odysseus cycle is not space, this discrete unit redis­ covered indefinitely or by repetitions along its discursive sequence. The plurality of disjointed spaces, all different, is the primal chaos, the condi­ tion of the series that assembles them. Ulysses' journey, like that of Oedipus, is an itinerary. And it is a discourse, the prefix of which I can now understand. It is not at all the discourse (discours) of an itinerary (parcours), but, radically, the itinerary (parcours) of a discourse (discours), the course, cursus, route, path that passes through the original disjunction, the bridge laid down across crevices. And the separation is of an im­ pregnable rigor.(All the spaces encountered are perfectly defined, without waver or blur. It'ls impossible to connect them among themselves. They cannot be composed to form a single homogeneous space. They combine such categories as open and closed, exterior and interior, boundary and limit, vicinity and adherence, and so forth, all concepts characteristic of the numerous spaces of topology. Hence comes everything one might desire in the text: inaccessible islands, and countries from which one cannot escape; the beach upon which the catastrophe casts you; the breaking of the waves; the shores from which one is hurled as one ap­ proaches. The intrusion of a wooden horse into the heart of the enclosed eitadel, where the warriors are at the same time inside the city but outside it by being inside the closed compartment that is inside the closed citadel. The exit of a ram, this bridge, out of an enclosed cavern in which a dan­ gerous fire burns; ram, horse anew, with the difference that the space of touch full of voids is more important here than optical or visual space.9 Hence the blindness of the Cyclops, in order to demonstrate that a closed system is not the same for the clairvoyant and for one who is reduced to his sense of touch.

Likewise, the attractive passage by the Sirens' shore where a vicinity, an adherence, is skirted, open for the deaf and closed for every listener. Original spaces proliferate on the map of the journey, perfectly disseminated, or literally sporadic, each one rigorously de­ termined. The global wandering, the mythical adventure, is, in the end, only the general joining of these spaces, as if the object or target of dis­ course were only to connect, or as if the junction, the relation, constituted the route by which the first discourse passes. Mythos, first logos; transport, first relation; junction, condition of transport. Thus we have Penelope at the theoretical position: the queen who weaves and unweaves, the originally feminine figure who, become male, will be Plato's Royal Weao;er. As Descartes says in Rule X, a tapestry intermingles threads with infinitely varied nuances.!o Infinitely: the rational and the irra­ tional. Descartes says this of a barbarous mathematics. Here we are once again. Barbarous or feminine, the logos is present, but still at the level of the hands. They connect. Penelope is the author, the signatory of the dis­ course; she traces its graph, she draws its itinerary. She makes and undoes this cloth that mimes the progress and delays of the navigator, of Ulysses on board his ship, the shuttle that weaves and interweaves fibers separated by the void , spatial varieties bordered by crevice s . She is the em­broideress, the lace-maker, by wells and bridges, of this continuous flux interrupted by catastrophes that is called discourse. In the palace of Ithaca, Ulysses, finally in the arms of the queen, finds the finished theory of his own mythos. The heroine of La Debacle, on the contrary, finds along her catastrophic route the weaver whose loom has just burned.

No one leaves here and no one enters-except those who speak geometry, the discourse that has com­ munication as its goal. Myth attempts to transform a chaos of separate spatial varieties into a space of communication, to re-link ecological clefts or to link them for the first time: from the mute animal to the proto-speaker. At the theoretical position in universo is she who conditions and who prepares the work of the weaver herself, she who produces and who gives the thread: Ariadne.

This can be general. All of Greece about which I am speaking is Dichotomy, Polytomy: Zeno's paradox, the Platonic classificatory trees, the division of segments by relations and proportions in the Euclidean manner, logos and analogy, the sharing of riches on Aristotle's scale, to each his part, his destined part. . . . This unitary discourse through dis­ tinctions and partitions, this discourse of the beginnings of mathematics, miraculously established, flows back toward its Pythagorean origin where the speakable, namely, the rational, is the split whole that we call fractions, the set of numbers that are the very things themselves. Here the method, road, path, track propose and set forth medians: the middle term between two terms. The completion of an interval is a problem that has not varied from the dawn of time up to Cantor. Let this bridge be lost and the endless nature of the path or the inaccessibility of the opposite shore be discovered, and we have the crisis, the shipwreck of Hippasus of Meta­ pontum, he who can no longer cross the sea. No one can speak any longer, and we have the irrational or the unspeakable-the incommu­ nicable, to be very precise. In fact, we have the return to the state of things before the establishment of rational discourse, the time when spaces were poorly joined, when transport and itinerary were only myth. The sect is dissolved when faced with the infinitely divisible-until the atomism of Democrites. Hippasus is shipwrecked like Ulysses, both of Metapontum (meta-pons, "metabridge"). Pythagoreanism had turned its back upon barbarous topology; it founders once again in myth with the discovery of the topology of real numbers. It had established a space of mediations, of communication, and it dies from losing it.

All this was rational, discursive, and speakable, all this was mathematical and logical, but in the closest vicinity to the sources, to the possibility of speaking to one another. The Greek cities were dispersed, reciprocally closed insu­larities, islands separated like the Sporades, in which every man worthy of the name, in other words, measure of all things, was inside, while on the exterior of this political space animals, barbarians with growling languages, circulated in a chaotic multiplicity of sociopolitical spaces: the world before its formation, the practical world before the emergence of scientific knowledge. This logos was first myth, in order to succeed in creating at least one koine.12 All the principles of the Greek cities go beyond this arm of the sea, before Troy, in order to found a language of communication-that the gods first make possible. The gods are en­ countered as the same-here, everywhere-because in their other space they enjoy a single space. It is essentilotl that one no longer know where Dionysus was born, where Oedipus and Theseus died. Anywhere : this is far preferable. Thus, in this discourse, chaos begins again: scattered members, the diasparagmos, the bones of Mother Earth, the first family of Spartol, dissemination in space, or, rather, dissemination of mor­ phologies themselves. Whereupon the first problem: to find the single space or the set of operators by which these spatial varieties in impracti­ cal, inconceivable vicinity will be joined together. To open the route, way, track, path in this incoherent chaos, this tattered cloud, whose dichotomic thicket is reformulated in the common space of transport when it is reconstructed. To find the relation, the logos of analogy, the chain of mediations, the common measure, the asses' bridge; to find the equilibrium or the clinamen.
The clinamen is the minimum angle to the laminar flow that initiates a turbulence".
Second answers, second words, where measure and correct measure presuppose a homogeneous space which is posited as reference and which is the answer to the first question asked: the unitary space of possible transports or of always possible transfers.

And thus one must find first, find conditionally, a word, a logos, that has already worked to connect the crevices which run across the spatial chaos of disconnected varieties. One must find the Weaver, the proto-worker of space, the prosopopeia of topology and nodes, the Weaver who works locally to join two worlds that are separated, according to the autochton's myth, by a sudden stoppage, the metastrophic caesura amassing deaths and shipwrecks: the catastrophe. He works, according to Plato, in a discourse where rational dichotomy and the myth of the two space-times, common measure and the Weaver, all converge. He untangles, interlaces, twists, assembles, passes above and below, rejoins the rational, the irra­ tional, namely, the speakable and the unspeakable, communication and the incommunicable. He is a worker of the single space, the space of measure and transport, the Euclidean space of every possible displace­ ment without change of state, royally substituted one fine day in place of the proliferating multiplicities of unlinked morphologies.

In order to practice dichotomy and its connected paths, one must know that its clefts follow and overlap the ancient mythical narrative in which worlds are torn asunder by a catastrophe-and only the Weaver knows how to link them again or can reunite them. Then and only then geometry is born and myth falls silent. Then the logos or relation unfolds, the chains and networks on the smooth space of transport, which itself alone replaces the discourse (discours) of itineraries (parcours). Linked homogeneity erases catastrophes, and congruent identity forgets difficult homeomorphisms. Reason, as the saying goes, has triumphed over myth. No, it is Euclidean space that has repressed a barbarous topology, it is transport and displace­ ment without obstacles that have suddenly taken the place of the journey, the ancient journey from islands to catastrophes, from passage to fault, from bridge to well, from relay to labyrinth. Myth is effaced in its original function, and the new space is universal, as is reason or the ratio that it sustains, only because within it there are no more encounters. As Plato says, one can walk there on two or four legs, follow the diagonal, freely choose the longest or shortest road, route, ode, or period, and so on, as much as one wishes. The earth is measured (geo-metry) by means of just measure (the King). The multiplicity, the dangerous flock of chaotic morphologies, is subdued. Thus the Statesman is written.

Hence the two great vicissitudes of the nineteenth century. Beneath the apparent unity of Euclidean space, mathematics, turning back toward its origins, rediscovers the teeming multiplicity of diverse and original spaces-and topology emerges as a science. We have not finished nor shall we ever again finish dealing with spaces. At the same moment, in an aged Europe asleep beneath the mantle of reason and measure, mythology reappears as an authentic discourse. The coupling of these rediscoveries becomes clear : Euler's bridge and the vessels' bridge across the Hellespont during the storm, Listing's or Maxwell's complex and the Cretan maze. Let us not forget that Leibniz, proto-inventor of the new science, said in time and against his time that one should listen to old wives' tales." [Hermes]

The Law of Contradiction.

Serres wrote:

"The ca­cophonous speaker and the auditor, exchange their reciprocal roles in dialogue, where the source becomes reception, and the reception source (according to a given rhythm). They exchange roles sufficiently often for us to view them as struggling together against a common enemy. To hold a dialogue is to suppose a third man and to seek to exclude him. a successful communication is the exclusion of the third man. The most profound dialectical problem is not the problem of the Other, who is only a variety -or a variation-of the Same, it is the problem of the third man. We might call this third man the demon, the prosopopeia of noise.

For example, the Metaphysical Meditations can be ex­ plained according to these principles: the Meditations seek out the other with whom one must join in order to expel the third man. Dialectic makes the two interlocutors play on the same side; they do battle together to produce a truth on which they can agree, that is, to produce a successful communication. In a certain sense, they struggle together against interference, against the demon, against the third man. Obviously, this battle is not always successful. In the aporetic dialogues, victory rests with the powers of noise; in the other dialogues, the battle is fierce - attesting to the power of the third man. Serenity returns little by little when the exorcism is definitively(?) obtained.

The subject of abstract mathematics is the "we" of an ideal republic which is the city of communication maximally purged of noiselO (which, parenthetically, shows why Plato and Leibniz were not idealists). In general, to formalize is to carry out a process by which one passes from concrete modes of thinking to one or several abstract forms. It means to eliminate noise as well, in an optimal manner. It means to become aware of the fact that mathematics is the kingdom that admits only the absolutely unavoidable noise, the kingdom of quasi-perfect communication, the manthdnein, the kingdom of the excluded third man, in which the demon is almost definitively exorcised. If there were no mathematics, it would be necessary to renew the exorcism.

With Plato, on the contrary, the discussion is full and complete : it makes the recognitoin of the abstract form and the problem of the success of the dialogue coexist. When I say "bed," I am not speaking of such and such a be?, mine, yours, this one or that one; I am evoking the idea of the bed. When I draw a square and a diagonal in the sand, I do not in any way want to speak of this-wavering, irregular, and inexact graph; I evoke by it the ideal form of the diagonal and of the square. I eliminate the empirical, I dematerialize reasoning. By doing this, I make a science possible, both for rigor and for truth, but also for the universal, for the Universal in itself. By doing this I eliminate that which hides form-cacography, inter­ference, and noise-and I create the possibility of a science in the Uni­ versalfor us. Mathematical form is both a Universal in itself and a Uni­ versal for us: and therefore thefirst effort to make communication in a dialogue successful is isomorphic to the efort to render aform independent ofits empirical realizations. These realizations are the third man of the form, its interference and its noise, and it is precisely because they intervene ceaselessly that the first dialogues are aporetic. The dialectical method of the dialogue has its origins in the same regions as mathematical method, which, moreover, is also said to be dialectical.

To exclude the empirical is to exclude differentiation, the plurality of others that mask the same. It is the first movement ofmathematization, of formalization. there would have to be a different word for every circle, for every symbol, for every tree, an d for every pigeon; and a different word for yesterday, today, and to­ morrow; and a different word according to whether he who perceives it is you or I, according to whether one of the two of us is angry, is jaundiced, and so on ad infinitum. At the extreme limits of empIrICISm, meaning is totally plunged into noise, the space of communication is granular,ll dialogue is condemned to cacophony: the transmission of communication is chronic transformation. Thus, the empirical is strictly essential and accidental noise. The first "third man" to exclude is the em­ piricist, along with his empirical domain. And this demon is the strongest demon, since one has only to open one's eyes and ears to see that he is master of the world.l2 Consequently, in order for dialogue to be possible, one must close one's eyes and cover one's ears to the song and the beauty of the sirens. In a single blow, we eliminate hearing and noise, yision and failed drawing; in a single blow, we conceive the form and we under­ stand each other. And therefore, once again, the Greek miracle, that of mathematics, must be born at the same time-historical time, logical time, and reflexive time - as a philosophy of dialogue and by dialogue.

In Platonism, the link between a dialectical method-in the sense of communication - and a progressive working diagram of abstract idealities in the manner of geometry is not an accident in the history of ideas, nor just an episode in the willful decisions of the philosopher: it is inscribed in the nature of things. To isolate an ideal form is to render it independent of the empirical domain and of noise. Noise is the empirical portion of the message just as the empirical domain is the noise of form. In this sense, the minor Socratic dialogues are pre-mathematical in the same way as is the measurement of a wheat field in the Nile valley.


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When two speakers have a dialogue or a dispute, the channel that connects them must be drawn by a diagram with four poles, a complete square equipped with its two diagonals. However loud or irreconcilable their quarrel, however calm or tranquil their agreement, they are linked, in fact, twice: they need, first of all, a certain intersection of their repertoires, without which they would remain strangers; they then band together against the noise which blocks the communication channel. These, two conditions are necessary to the dia­ logue, though not sufficient. Consequently, the two speakers have a common interest in excluding a third man and including a fourth, both of whom are prosopopoeias of the powers of noise or of the instance of intersection'! Now this schema functions in exactly this manner in Plato's Dialogues, as can easily be shown, through the play of people and their naming, their resemblances and differences, their mimetic preoccupations and the dynamics of their violence. Now then, and above all, the mathe­ matical sites, from the Meno through the Timaeus, by way of the Statesman and others, are all reducible geometrically to this diagram. Whence the origin appears, we pass from one language to another, the language said to be natural presupposes a dialectical schema, and this schema, drawn or written in the sand, as such, is the first of the geometric idealities. Mathematics presents itself as a successful dialogue or a communication which rigorously dominates its repertoire and is maximally purged of noise. Of course, it is not that simple. The irrational and the unspeakable lie in the details; listening always requires collating; there is always a leftover or a residue, indefinitely. But then, the schema remains open, and history possible. The philosophy of Plato, in its presentation and its models, is therefore inaugural, or better yet, it seizes the inaugural moment.

To be retained from this first attempt at an explanation are the expul­ sions and the purge. Why the parricide of old father Parmenides, who had to formulate, for the first time, the principle of contradiction? To be noted here again is how two speakers, irreconcilable adversaries, find themselves forced to turn together against the same third man for the dialogue to remain possible, for the elementary link of human relationships to be possible, for geometry to become possible. Be quiet, don't make any noise, put your head back in the sand, go away or die. Strange diagonal which was thought to be so pure, and which is agonal and which remains an agony.

The second attempt contemplates Thales at the foot of the Pyramids, in the light of the sun. It involves several geneses, one of which is ritual.2 But I had not taken into account the fact that the Pyramids are also tombs, that beneath the theorem of Thales, a corpse was buried, hidden. The space in which the geometer intervenes is the space of similarities: he is there, evident, next to three tombs of the same form and of another dimension - the tombs are imitating one another. And it is the pure space of geometry, that of the group of similarities which appeared with Thales. The result is that the theorem and its immersion in Egyptian legend says, without saying it, that there lies beneath the mimetic operator, constructed concretely and represented theoretically, a hidden royal corpse.

The third attempt consists in noting the double writing of geometry.Using figures, schemas, and diagrams. Using letters, words, and sentences of the system, organized by their own semantics and syntax. Leibniz had already observed this double system of writing, consecrated by Descartes and by the Pythagoreans, a double system which represents itself and expresses itself one by the other. He sometimes liked, as did many others, to privilege the intuition, clairvoyant or blind, required by the first [diagrams] over the deductions produced by the second [words].

There were even, as usual, two schools at odds over the question. One held the Greeks to be the teachers of geometry ; the other, the Egyptian priests. This dispute caused them to lose sight of the es­ sential: that the Egyptians wrote in ideograms and the Greeks used an alphabet. Communication between the two cultures can be thought of in terms of the relation between these two scriptive systems (signa/Cliques). Now, this relation is precisely the same as the one in geometry which separates and unites figures and diagrams on the one hand, algebraic writing on the other. Are the square, the triangle, the circle, and the other figures all that remains of hieroglyphics in Greece? As far as I know, they are ideograms. Whence the solution: the historical relation of Greece to Egypt is thinkable in terms of the relation of an alphabet to a set of ideograms, and since geometry could not exist without writing, mathematics being written rather than spoken, this relation is brought back into geometry as an operation using a double system of writing. There we have an easy passage between the natural language and the new language, a passage which can be carried out on the multiple con­ dition that we take into consideration two different languages, two dif­ ferent writing systems and their common ties. And this resolves in turn the historical question: the brutal stoppage of geometry in Egypt, its freezing, its crystallization jnto fixed ideograms, and the irrepressible development, in Greece as well as in our culture, of the new language, that inexhaustible discourse of mathematics and rigor which is the very history of that culture . The inaugural relation of the geometric ideogram to the alphabet, words, and sentences opens onto a limitless path.

This third solution blots out a portion of the texts. The old Egyptian priest, in the Timaeus, compares the knowledge of the Greeks when they were children to the time-worn science of his own culture. He evokes, in order to compare them, floods, fires, celestial fire, catastrophes. Absent from the solution are the priest, history, either mythical or real, in space and time, the violence of the elements which hides the origin and which, as the Timaeus clearly says, always hides that origin. Except, precisely, from the priest, who knows the secret of this violence. The sun of Ra is replaced by Phaethon, and mystical contemplation by the catastrophe of deviation.

We must start over - go back to those parallel lines that never meet. On the one hand, histories, legends, and doxographies, composed in natural language. On the other, a whole corpus, written in mathematical signs and symbols by geometers, by arithmeticians. We are therefore not concerned with merely linking two sets of texts; we must try to glut: two languages back together again. The question always arose in the space of the relation between experience and the abstract, the senses and purity.

Can you imagine (that there exists) a Rosetta Stone with some legends written on one side, with a theorem written on the other side? Here no language is unknown or undecipherable, no side of the stone causes problems; what is in question is the edge common to the two sides, their common border; what is in question is the stone itself.

What separates the Greeks from their possible predecessors, Egyptians or Babylonians, is the establishment of a proof. Now, the first proof we know of is the apagogic proof on the irrationality of V2 [square root of 2].

And so, legends, once again. Euclid's Elements, Book X, first scholium. It was a Pythagorean who proved, for the first time, the so-called irra­tionality [of numbers]. Perhaps his name was Hippasus of Metapontum. Perhaps the sect had sworn an oath to divulge nothing. Well, Hippasus of Metapontum spoke. Perhaps he was expelled. In any case, it seems certain that he died in a shipwreck. The anonymous scholiast continues: "The authors of this legend wanted to speak through allegory. Everything that is irrational and deprived of form must remain hidden, that is what They were trying to say. That if any soul wishes to penetrate this secret region and leave it open, then it will be engulfed in the sea of becoming, it will drown in its restless currents."

Legends and allegories and, now, history. For we read a significant event on three levels. We read it in the scholia, commentaries, narratives. We read it in philosophical texts. We read it in the theorems of geometry. The event is the crisis, the famous crisis of irrational numbers. Owing to this crisis, mathematics, at a point exceedingly close to its origin, came very close to dying. In the aftermath of this crisis, Platonism had to be recast. The crisis touched the logos. If logos means proportion, measured relation, the irrational or alogon is the impossibility of measuring. If logos means discourse, the alogon prohibits speaking. Thus exactitude crumbles, reason is mute.

Hippasus of Metapontum, or another, dies of this crisis, that is the legend and its allegorical cover in the scholium of the Elements. Parmenides, the father, dies of this crisis-this is the philosophical sacrifice perpetrated by Plato. But, once again, history : Plato portrays Theaetetus dying upon returning from the the battle of Corinth (369), Theaetetus, the founder, precisely, of the theory of irrational numbers as it is re­ capitulated in Book X of Euclid. The crisis read three times renders the reading of a triple death: the legendary death of Hippasus, the philo­ sophical parricide of Parmenides, the historical death of Theaetetus. One crisis, three texts, one victim, three narratives. Now, on the other side of the stone, on the other face and in another language, we have the crisis and the possible death of mathematics in itself.

Given then a proof to explicate as one would a text. And, first of all, the proof, doubtless the oldest in history, the one which Aristotle will call reduction to the absurd. Given a square whose side AB = b, whose diagonal AC = a.


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We wish to measure AC in terms of AB. If this is possible, it is because the two lengths are mutually commensurable. We can then write A C/AB = a/b. It is assumed that a/b is reduced to its simplest form, so that the integers a and b are mutually prime. Now, by the Pythagorean theorem: a2 = 2b2• Therefore a2 is even, therefore a is even. And if a and b are mutually prime, b is an odd number.

If a is even, we may posit: a = 2c. Consequently, a2 = 4c2. Consequently 2b2 = 4c2, that is, b2 = 2c2• Thus, b is an even number.

The situation is intolerable, the number b is at the same time even and odd, which, of course, is impossible. Therefore it is impossible to measure the diagonal in terms of the side. They are mutually incommensurable. I repeat, if logos is the proportional, here a/b or 1/V2, the alogon is the incommensurable. If logos is discourse or speech, you can no longer say anything about the diagonal and V2 is irrational. It is impossible to decide whether b is even or odd.

1) What does it mean for two lengths to be mutually commensurable? It means that they have common aliquot parts. There exists, or one could make, a ruler, divided into units, in relation to which these two lengths may, in turn, be divided into parts. In other words, they are other when they are alone together, face to face, but they are same, or just about, in relation to a third term, the unit of measurement taken as reference. The situation is interesting, and it is well known : two irreducibly diferent entities are reduced to similarity through an exteriorpoint ofview. It is fortunate (or necessary) here that the term measure has, traditionally, at least two meanings, the geometric or metrological one and the meaning of non-disproportion, of serenity, of nonviolence, of peace. These two meanings derive from a similar situa­ tion, an identical operation. Socrates objects to the violent crisis of Callicles with the famous remark: you are ignorant of geometry. The Royal Weaver of the Statesman is the bearer of a supreme science : superior metrology.

2) What does it mean for two numbers to be mutually prime? It means that they are radically different, that they have no common factor besides one. We thereby ascertain the first situation, their total otherness, unless we take the unit of measurement into account.

3) What is the Pythagorean theorem? It is the fundamental theorem of measurement in the space of similarities. For it is invariant by variation of the coefficients of the squares, by variation of the forms constructed on the hypotenuse and the two sides of the triangle. And the space of similarities is that space where things can be of the same form and of another size. It is the space of models and of imitations. The theorem of Pythagoras founds measurement on the representative space of imitation.  

4) What, now, is evenness? And what is oddness? The English terms reduce to a word the long Greek discourses: even means equal, united, flat, same; odd means bizarre, un­ matched, extra, left over, unequal, in short, other. To characterize a number by the absurdity that it is at the same time even and odd is to say that it is at the same time same and other.

Conceptually, the apagogic theorem or proof does nothing but play variations on the notion of same and other, using measurement and com­ mensurability, using the fact of two numbers being· mutually prime, using the Pythagorean theorem, using evenness and oddness.

It is a rigorous proof, and the first in history, based on mimesis. It says something very simple : supposing mimesis, it is reducible to the absurd. Thus the crisis of irrational numbers overturns Pythagorean arithmetic and early Platonism.

Hippasus revealed this, he dies of it-end of the first act.

It must be said today that this was said more than two millennia ago. Why go on playing a game that has been decided? For it is as plain as a thousand suns that if the diagonal or v'2 are incommensurable or ir­ rational, they can still be constructed on the square, that the mode of their geometric existence is not different from that of the side. Even the young slave of the Meno, who is ignorant, will know how, will be able, to construct it. In the same way, children know how to spin tops which the Republic analyzes as being stable and mobile at the same time. How is it then that reason can take facts that the most ignorant children know how to establish and construct, and can demonstate them to be irrational? There must be a reason for this irrationality itself.

In other words, we are demonstrating the absurdity of the irrational. We reduce it to the contradictory or to the undecidable. Yet, it exists; we cannot do anything about it. The top spins, even if we demonstrate that, for impregnable reasons, it is, undecidably, both mobile and fixed. That's the way it is. Therefore, all of the theory which precedes and founds the proof must be reviewed, transformed. It is not reason that governs, it is the obstacle. What becomes absurd is not what we have proven to be absurd, it is the theory on which the proof depends. ["An apagogic proof is one that proceeds by disproving the proposition which contradicts the one to be established, in other words, that proceeds by reductio ad absurdum. -Ed.] Here we have the very ordinary movement of science: once it reaches a dead-end of this kind, it immediately transforms its presuppositions.

Translation : mimesis is reducible to contradiction or to the undecidable. Yet it exists; we cannot do anything about it. It spins. It works, as they say. That's the way it is. It can always be shown that we can neither speak nor walk, or that Achilles will never catch up with the tortoise. Yet, we do speak, we do walk, the fleet-footed Achilles does pass the tortoise. That's the way it is. Therefore, all of the theory which precedes must be trans­ formed. What becomes absurd is not what we have proven to be absurd, it is the theory as a whole on which the proof depends.

Whence the (hi)story which follows. Theodorus continues along the legendary path of Hippasus. He multiplies the proofs of irrationality. He goes up to 07. There are a lot of these absurdities, there are as many of them as you want. We even know that there are many more of them than there are of rational relations. Whereupon Theaetetus takes up the archaic Pythagoreanism again and gives a general theory which grounds, in a new reason, the facts of irrationality. Book X of the Elements can now be written. The crisis ends, mathematics recovers an order, Theaetetus dies, here ends this story, a technical one in the language of the system, a historical one in the everyday language that relates the battle of Corinth. Plato recasts his philosophy, father Parmenides is sacrificed during the parricide on the altar of the principle of contradiction; for surely the Same must be Other, after a fashion. Thus, Royalty is founded. The Royal Weaver combines in an ordered web rational proportions and the irra­ tiona:Is; gone is the crisis of the reversal, gone is the technology of the dichotomy, founded on the square, on the iteration of the diagonal. Society, finally, is in order. This dialogue is fatally entitled, not Geometry, but the Statesman.

The Rosetta Stone is constructed. Suppose it is to be read on all of its sides. In the language of legend, in that of history, that of mathematics,that of philosophy. The message that it delivers passes from language to language. The crisis is at stake. This crisis is sacrificial. A series of deaths accompanies its translations into the languages considered. Following these sacrifices, order reappears: in mathematics, in philosophy, in history, in political society. The schema of Rene Girard allows us not only to show the isomorphism of these languages, but also, and especially, their link, how they fit together.6 For it is not enough to narrate, the operators of this movement must be made to appear. Now these operators, all constructed on the pair Same-Other, are seen, deployed in their rigor, throughout the very first geometric proof. Just as the square equipped with its diagonal appeared as the thematized object of the complete intersubjective relation, formation of the ideality as such, so the rigorous proof appears as such, manipulating all the operators of mimesis, namely, the internal dynamics of the schema proposed by Girard. The origin of geometry is immersed in sacrifical history and the two parallel lines are henceforth in connection. Legend, myth, history, philosophy, and pure science have common borders over which a unitary schema builds bridges.

Metapontum and geometer, he was the Pontifex, the Royal Weaver. His violent death in the storm, the death of Theaetetus in the violence of combat, the death of father Parmenides, all these deaths are murders. The irrational is mimetic. The stone which we have read was the stone of the altar at Delos. And geometry begins in violence and in the sacred." [Hermes]

The Birth of Geometry.

Serres wrote:

"What Thales Saw. . .

Hieronymus informs us that he [ThalesJ measured the height ofthe pyramIds by the shadow they cast, taking the observation at the hour when the length of our shadow equals our height.

- Diogenes Laertius

The height of a pyramid is related to the length of its shadow just as the height ofany vertica4 measurable object is related to the length ofits shadow at the same time of day.

- Plutarch


As Auguste Comte says: "In light of previous experience we must acknowledge the impossibility of determining, by direct measurement, most of the heights and distances we should like to know. It is this general fact which makes the science of mathematics necessary. For in renouncing the hope, in almost every case, of measuring great heights or distances directly, the human mind has had to attempt to determine them indirectly, and it is thus that philosophers were led to invent mathematics. Geometry is a ruse; it takes a detour, an indirect route, to reach that which lies outside immediate experience. In this case the ruse is the model : the construction of the summary, the skeleton of a pyramid in reduced form but of equivalent proportions. In fact, Thales has dis­ covered nothing but the possibility of reduction, the idea of a module, the notion of model. The pyramid itself is inaccessible; he invents a scale, a type of ladder.  

To measure the inaccessible consists in mimicking it within the realm of the accessible.

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 OIl. top of the object, it touches it. And this is done as often as is 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. Som.e 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.

This amounts to underestimating the importance of practical activities. For in the final analysis the path in question consists in forsaking the sense of touch for that of sight, measurement by "placing" for measure­ ment by sighting. Here, to theorize is to see, a fact which the Greek language makes clear. Vision is tactile without contact. Descartes knew this, just as he understood better than most what measurement is. The inaccessible is at times accessible' to vision. Can one measure visually the distance to the sun, to the moon, to a ship, to the apex of a pyramid? This is the whole story of Thales-, who discovered nothing but the precise virtues of the human gaze, just as, somewhat later, Berkeley organized in

an erudite manner a spectacle of light beneath his microscope, a rigorous organon of optical representation. Since he cannot use his ruler, he sets up lines of sight or, rather, he lets light project them for him. As far as I know, even for accessible objects, vision alone is my guarantee that the ruler has been placed accurately on the thing. To measure is to align ; the eye is the best witness of an accurate covering-over. Thales invents the notion of model, of module, but he also brings the visible to the tangible. To measure is, supposedly, to relate. True, but the relation implies a transporting: of the ruler, of the point of view, of the things lined up, and so on. In the realm of the accessible, the transporting is always possible: i n the realm o f the inaccessible, vision must take care o f displacements : hence the angle of sight, hence the cast shadow. Measurement, the problem of relation ; sight, the cast shadow ; in any case, the essential element is the transporting.

Let us return to the schema, that of Diogenes or Plutarch. It deals with things in motion or at rest. The constant factor is the pyramid, immobile for ten centuries beneath the Egyptian sky. The apparent movement of the sun, the length and the position of the shadow are all variable. Everyday experience tells us that the latter depend on the former. Hence the initial idea : the clock. The pyramid is a gnomon and the line of its shadow tells time. Measurement and the gauging of the shadow's varia­ tions mark the rhythm of the sun's course. The gnomon, stable and arbitrary, is only an intermediary object; the variances echo one another.

The goal is either civil or astronomical. With a sundial, the measurement of space only measures time. The sundial, whose origin is lost somewhere at the dawn of time,S will disappear during the quarrel of the Ancients and the Moderns, a quarrel particularly acerbic in regard to clocks. Hence, in Diogenes and Plutarch, the remains of what was once the problem of time: to wait for the moment of equality between an object's shadow and its height, or to observe the two shadows at the same time of day; to keep the sun and its daily course in mind,  

Thales's idea (for we must give it a name) consists simply in turning the process around, that is, in considering and then resolving the reverse problem of the gnomon. Instead of letting the pyramid speak of the sun, or the constant determine the scale of the variable, he asks the sun to speak of the pyramid; that is, he asks the object in motion to provide a constant flow of information about the object at rest. This ruse is much more clever than the one we described earlier: the constant is no longer what gauges the regular intervals of the variable ; on the contrary, Thales gauges, within the variable realm, the stable unknown of the constant. Or rather, with the gnomon, whoever measured space also measured time. By inverting the terms, Thales stops time in order to measure space. He stops the course of the sun at the precise instant of isoceles triangles; he homogenizes the day to obtain the general case. And so do Joshua and Copernicus. Hence it hecomes necessary to freeze time in order to conceive ofgeometry Once the gnomon has disappeared, Thales enters into the eternity of the mathematical figure. Plato will follow him. This is the old Bergsonian conclusion.

How did geometry come to the Greeks? A practical genesis: build a reduced model, have a notion of the module, bring the distant to the immediate. A sensorial genesis: organize a visual representation of that which defies physical contact. A civil or epistemological genesis: take astronomy as a starting point, reverse the question of the gnomon. A genesis that is either conceptual or esthetic: erase time in order to measure and master space. Exchange the functions of the variable and the invariable. The origin of geometry is a confluence of geneses. We must follow the other affluents.

We have said that the essential element is the transporting. For even if measurement can be exact or precise, only the relation is rigorous: the reference of a giant schema to a reduced model. These initial geneses are acts of transporting: reduction, the transition from touching to seeing, and vice versa, the reversal of the gnomonic function, the exchange of the stable and the variant, the substitution of space for time.

in case I wish to recall Thales's theorem, the story of the pyramid can serve as a mnemonic device. In a culture with an oral tradition, story takes the place of schema, and theater equals intuition. The diagram of the theorem can only be transmitted in written form, but, in an oral culture, drama is the vehicular form of knowledge. Myth then, the mythical tale, is less a legend of origin than the very form of transmission; it does not bear witness to the emergence of science so much as it communicates an element of science. Here mathematics is the key to history, not the contrary. The schema is the invariant of the tale instead of the tale being the origin of the schema. To know, then, and, in this case, to know Thales's theorem is to remember the Egyptian tale. To teach the theorem is to tell the pseudo-myth of origin. We know that all mythical tales are merely the dramatization of a given content. Only the mathematical decipherment of the text can demonstrate the relationship of the implied schema to the mobilization which turns it into a trans­missible tale.

What Thales or anybody else perceives cannot be anything other than objects of the same form but of different dimensions. The perception of three pyramids is developed within the space of similarities, and this space is constituted by choice in this place: each pyramid is different and yet the same, like the triangles in Thales's theorem. Hence the story is perfectly faithful to the concept, and similar to the idea of similarity. It is more a question here of tech­ nology than of perception: similarity is the secret to the triple edifice, the secret to its construction. The pure knowledge implicit within the design of the pyramids is certainly homothetic.

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 in the Platonic mode, in terms of implicit and explicit, of knowledge and practical operations : on the one hand, the sun of knowl­ edge and of sameness; on the other, the shadow of opinion, of empiricism, of objects.

These first two readings reveal convincingly the implicit knowledge that a fabricated object hides within itself.

The thing which is dif­ ficult and ultimately inextricable, which we attempt indefinitely to render explicit without being able to explain it completely, and which is thus forever clouded over, is the modality, the "how" of this implication, which, in an actual application, is clearer. The articulatory mode of luminous knowledge and blind practice is blinder in implication, more luminous in application. The origin of knowledge acquired through everyday practice is on the side of shadow; the origin of a practice ac­quired through knowledge is on the side of light.

In the initial technical activity, knowledge is in shadow, and we are also in the dark as acting beings, trying to situate theory in light.

The technique of measurement which is still a ruse of application, or, as Auguste Comte says, an indirect path, repeats the implication but does not explain it. Thales extracts a technique from a technique, and from a practice he gets another practice. The homology of repetition eventually designates the homothesis, but in each instance within the gangue of the applied. The theory expressed by shadows remains in shadow.

There is no longer any original miracle: different techniques give rise to other ones and perpetuate themselves in repetition; measurement and architecture see the theorem differently, that is all. And we remain in the immense shadow of the secret. For, again, one cannot conceive of the origin of technique except as the origin of man himself, faber as soon as he emerges, or rather, emerging because he is faber. Technique is the origin of man, his perpetuation and his repetition. Hence Thales repeats his very origin, and our own : his mathematics, his metrics of geometry, repeats in another way (and as simply as possible) and designates in another way the modality of our technical relationship to objects, the homology of the fabricator to the fabricated. His mathematics takes its place in the open chain of those utterances and designations, but it does not provide the key to the cipher; it does not excavate the secret articula­ tion of knowledge and practice in which the essential element of a possible origin is located. His mathematics is the relation between two shadows, two secrets, two forms and two ciphers, relation or logos, relationship and utterance to be transmitted, utterance which transmits a relationship. As is commonly said, it measures the problem, takes its dimensions, poses it, weighs it, demonstrates it, relates it, but never resolves it. The logos of shadows is still the shadow of the logos.

Still, what Thales's mathematics recounts, at its very inception, is the de-centering of the subject of clear thought with regard to the body that casts its shadow: the subject is the sun, placed beyond the object, on the other side of the shadow. This was also Copernicus' lesson. What this mathematics articulates is the Platonic decision that a geometry of mea­ surement is but a propaedeutic. What it announces, for the first time, is a philosophy of representation, dominating both the pure diagram and its dramatization beneath the torches of the solstice.

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.

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 informa­ tion 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 per­ spective (an architecture), it is a physics, an optics: the shadow is a black specter.

The theater of measurement demonstrates the decoding of a secret, the decipherment of a writing, the reading of a drawing. The sand on which the sun leaves its trace is the screen, the wall at the back of the cave. Here is the scene of representation established for Western thought for the next millennium, the historically stable form of contemplation from the summit of the pyramids. Thales's story is perhaps the instauration of the moment of representation, taken up ad infinitum by philosophers, but also and above all by geometers, from Descartes and his representational plane to Desargues and his point of view, from Monge and his descriptive diagram to Gergonne and his legislative transfers:8 the first word of a perspectival geometry, of an architectural optics of volumes, of an in­ tuitive mathematics immersed in a global organon of representation, the first instance of the Ptolemaic model of knowledge. But from Thales's time to the present day we have forgotten that the shadow was cast, transported by some supporting device, that it itself transported certain information.

The most important question - which messenger transports (and how?) which message?-was covered over for centuries by the f)linding scenography of the shadow-light opposition.

Thales's story is not unanalogous to that of Desargues: the size of the stones, a perspectival geometry, the theory of shadows.9 Nor, after all, to Plato's stories: the sun of the same, the other and empirical object, the cast shadow of the shaded surface , similarity, the cave of representation . Is it a tale of origin ? Yes, and in several ways : the origin of a technology, of an optics, of a philosophy of representation. Of a geometry? Per­ haps-if geo-metry is that triangulation which Plato scorned for being pre-mathematical . It is a mnemonic recipe, friend of the cultural memory because of its forceful dramatization and mythification under the sun of Ra, easy to transmit within a homogeneous cultural setting,IO the ruse of applied mathematics, of an architect and of an expert builder. Even Descartes, followed by Desargues and Monge, remains in the domain of applied geometry as well as that of representation; they perpetuate an engineering geometry that is metric and descriptive. They exhibit the archaic forms of pre-mathematics that run through history.

Like Thales, they impede the formation of pure mathematics. And the latter will emerge as soon as this geometry dies-very recently. And HusserI will write The Origin of Geometry as the bell tolls its disappearance, as if an immense historical cycle had finally come to an end. Thales's story tells something like the story of the birth of a geometry, the measured division of the earth and the differences in shadow and light written on the earth by solid figures and the sun; it does not tell of the birth of mathematics. As proof, let us cite Plato, who, in order to bring about this miracle, requires something else : the essential reality of idealities. Question : how can the pyramid be born as an ideal form?

Every­ thing transpires as if Plato had relegated Thales's story to the depths of his cave. The flat, even wall is always bright: on it the volume casts a shadow; light creates a shaded area. My knowledge is limited to these two shadows; it is only a shadow of knowledge. But there is a third shadow of which the two others provide only an image, or a projection,

and which is the secret buried deep within the volume. Now it is probable that true knowledge of the things of this world lies in the solid's essential shadow, in its opaque and black density, locked forever behind the multiple doors of its edges, besieged only by practice and theory. A wedge can sunder the stones, geometry can divide or duplicate cubes, and the story, indefinitely, will begin again; the solid, whose surfaces cannot be exhausted by analysis, always conserves a kernel of shadow hidden in the shade of its edges. Thales, while readi!1g and noting the volume's traces, deciphers no secret except that of the impossibility of penetrating the volume's arcana, in which knowledge has been entombed forever, and from which the infinite history of analytical progress bursts forth as if from a spring. In this case, his history tells the conclusion of a story, that of the confrontation with solid objects, that of the attack on compact volumes, comprehended like theoretical, objective, unconscious elements, like theoretical, objective, indefinite unknowns. In this case the thing exists qua thing, like an unknown and a correlate, like a secret involuted into thousands and thousands of replicas. Two decisions: either I recognize the object by its shadow, which gives rise to geometry or, better yet, to the idealism of representation, or I allow for a kernel of shadow within the object. In the latter case, theory and practice develop this secret infinitely in a perpetually open history, the history of science, which admits that the solid always envelops something that can be rendered explicit. In Plato, the idealism of representation appears re­ pressed in the depths of his cave, and realism is assumed. However, the story begun in the Nile delta will soon be completed by a sudden and incredibly audacious coup d etat: the radical negation of interior shadows. The Sun of Thales and Ra, the sun whose rays are shut out for an im­ peccable definition,ll is reduced to the meager fire of the prisoners of representation. Thales's theorem, schema of this story, is in the cave's shadow. Outside, the new sun gives off a transcendent light which pierces things and transmits an all-seeing vision. This is how the marvelous miracle is accomplished : the transparency of volumes, the metaphorical naming of the realism of idealities. From the cave to the world outside, the scenography turns into an ichnography : the shadow of solids played on the plane of representation and defined them by boundaries and partitions; now light goes through them and banishes the interior shadow. In place of a planar triangulation of geo-metry there is now a stereometry of empty forms in the epiphany of diaphanousness. 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 explicitations 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. And, as if to close the circle in all rigor and for the coherence of global history, the Timaeus will constitute the world by means of these five bodies : the first, the simplest, the tetrahedron in fact, will be fire. Plato has the pure pyramid come into existence beneath the fires of the sun, and from this tetrahedron he has fire born again: a double miracle that fulfills the scriptures, the Egyptian legend, and the initiation of intuition by posi­ tioning the source of light within the polyhedron. When the pyramid is itself fire (did its name influence its legend?), the sun passes through it.

The entire myth of origin, even that of The Republic, is thus immersed in a vision of fire and dramatizes a solar rite. The new Thales can no longer see any shadow beneath the furnace that pure form and the solar hearth constitute: original conjunction of mathematical stereometry and the mythical element, blinding atmosphere of the first philosophies of intui­ tion. The kernel of knowledge is continually enveloped by myth, and the myth is ceaselessly generated within the theater of representation. The­ory, vision; light, fire. We have here a new genesis with four branches where two tributaries are mixed: science and the history of religions. From astronomy to solar mythology.

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 the diachrony and the genealogy of forms.

The realism of transparent idealities is still immersed in a philosophy of representa­ tion. Of course, ichnography is substituted for scenography, but the former is a trans-representation from a divine point of view. To go beyond Thales's scene, the shadowless theater is still a theater. The inevi­ table realism is still an idealism: the geometric form clearly expresses this difficulty. 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 ge­ ometry, 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 mathematician s - ex­ plosions 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. Pure and simple forms are neither that simple nor that pure; they are no longer complete, theoretical knowns, things seen and known without residue, but rather theoretical, objective unknowns infinitely folded into one another, enormous virtualities of noemes, like the stones and the objects of the world, like our stone con­ structions and our wrought objects. Form hides beneath its form trans­-infinite kernels of knowledge which, one might fear, history will never exhaust; these highly inaccessible instances become our new tasks. Mathematical realism is weighed down and takes on the old density that Plato's sun had dissolved. Present-day mathematics, although maximally abstract and pure, is de­veloping in a lexicon that derives in part from technology. It is a new way of listening once more to Thales's old Egyptian legend.

What Thales saw at the base of the pyramids (the sun, the homo­ thetic 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 gnomons, 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." [Hermes]

Dynamics from Leibniz to Lucretius: The Abstraction of Physics

lIya Prigogine and Isabelle Stengers wrote:

"The Systeme de Leibniz begins by positing the major problematic of Serres's work: what is at stake in the hypothesis of the great classical rationalist who supposes that the passage from local to global is always possible? As we shall see, the question of the integrable world authorizes dreams of determinism. What we shall call here the classical style in physics is expressed in Laplace's dream of a world made of determinist and calculable trajectories. Laplace's demon observes the instantaneous state of the world and integrates its trajectory. He thus has access to both the past and the future in the minutest detail. This dream of omniscience translates Leibniz's baroque monadology using the unidimensional plati­ tude characteristic of the nineteenth century.

Nine years later, in La Naissance de fa physique dans fe texte de Lucrece, Serres takes up, fulfills, and modifies the project sketched out in Le Systeme de Leibniz: the confrontation of the rationalism of differential and integral calculus (Leibniz) with the rationalism of ancient atomism.

We observe that, as a good Epicurean, Lucretius answers "no" to the following question: "Is the passage from local to global always possible?"

This distinction between the pure and the impure crippled Leibniz's reputation as a physicist. Though his role in mathematics is recognized, in physics he figures more often than not as the inopportune and obstinate adversary of Newton, the person whose ambition to create communicative paths between physics and metaphysics led down the road to perdition.

Thus Leibniz's rigor was judged severely. It is commonly agreed that, though he was the creator of the term "dynamics," he nevertheless "missed" the mathematical physics created by Newton at that very mo­ ment. This is explained by the fact that for him philosophical rigor came before the needs of an inductive and necessarily approximate science. He refused to give up the idea of the rational nature of the real, measured not by the yardstick of man, who observes and generalizes, but by that of God, who, calculating, created the world. Thus Leibniz was unequivocally a "pre-Newtonian." This is a condemnation, moreover, that is sufficiently justified by his rejection of the principles of inertia and of interaction at a distance-in short, of Newtonian physics. In the face of this condemna­ tion, we can make three remarks.

In the first place, one might well ask, solely based on the facts, whether it is not the history of physics that has "missed" Leibniz. The discovery of his role and influence will undoubtedly offer a few surprises when the history of Continental physics is better known: Bernoulli, Euler, and D'Alembert were neither Newtonians nor pre-Newtonians. Second, the role played by God in Leibnizian physics does not allow the opposition of this physics to Newton's in the same way that metaphysical speculation might be opposed to positive scientific praxis. Think of the controversies between Leibniz and Clarke, Newton's proxy, or of Newton's own con­ siderations on the production of forces of attraction by the active prin­ ciples that show the action of God on the world :  these will suffice here as examples. Certainly Laplace (the "second Newton") was not wrong to say to Napoleon that his system had no need for the hypothesis of God: this remark merely expressed the fact that God, in the role assigned to him by Newton, did not resist the progress of dynamics. Last, it cannot be denied that, if not since Newton, then at least since Laplace, we have accepted the systems of interactions at a distance as part of our physical world.s But one must distinguish between such a conception and the de­ velopment of "mathematical physics," that is to say, the creation of the formalism that is today called dynamics. We have said, and we wish to show, that the language of dynamics has, in a certain sense, changed from a Newtonian to a Leibnizian one. The world of trajectories determined by forces can henceforth be thought of as being identical to the Leibnizian system of the world in which every point locally expresses the global law.

To introduce this thesis we propose some little-known satiric verses of James Clerk Maxwell, which are doubly interesting because they celebrate both what we call the "Leibnizian" transformation of dynamics and some­ one who was among the first to explore the possibilities and powers of a role that was then new within the scientific community, that of "textbook writer " :

But see! Tait writes in lucid symbols clear One small equation;

And Force becomes of Energy a mere Space-variation.

Force, then, is Force, but Mark you ! not a thing, Only a Vector;

Thy barbed arrows now have lost their sting, Impotent spectre !

Thy reign, O Force! is over. Now no more Heed we thine action ;

Repulsion leaves us where we were before, So does attraction.

Both Action and Reaction now are gone. Just ere they vanished,

Stress j oined their hands in peace , and made them one ; Then they were banished.

The universe is free from pole to pole, Free from all forces.

Rejoice! ye stars-like blessed gods ye roll On in your courses.

The stars and, following their example, all physical bodies travel through the universe like free and self-determined gods, each following its own law. Newtonian physics posits a body assumed to be isolated, endowed with a rectilinear and uniform inertial movement, and calculates the modifications of this movement as determined by the action of forces. For Leibniz, the forces are not "given" and are in no way the real causes of the modification of a movement but rather are local properties within a dynamic system: at every point, they characterize a momentary state belonging to a series regulated by a law.!

In the same way, since Lagrangell and especially since Hamilton, mathematical physics has abandoned Newtonian representation. Instead of calculating the action of each force on each point, it first of all proposes the system in its canonic form, and constructs a function (the Hamiltonian in particular, a representation of energy - see Maxwell's versesfthat de­ fines the global state of the system. From this function, the set of "forces" acting on each point at every moment can be derived. Forces are no longer responsible for accelerations; rather, they are deducible from the structure of the dynamic system defined by the Hamiltonian; they are the effects of the global law of evolution which the Hamiltonian ex­presses.

What we would discover is deducible from the fact that the dynamic problem is henceforth posited in the same way that Leibniz posited it: movement is produced within a full world, an interdependent world in which nothing can happen that has not been made possible by the state of the set of bodies according to a harmony that determines and checks at every moment the unfolding of the different movements. What Leibniz thought of as a preestablished harmony translated at every instant by the conservation of energy the physics of Lagrange put to work through the study of movement as the succession of states of equilibrium, disrupted and reestablished at every instant, and Hamiltonian formalism transformed into an a priori syntax of the formal language in which every dynamic problem can be posited. During the eighteenth century, in fact, physicists succeeded in inscribing Newtonian physics as a special a. posteriori case within the a priori conceptual framework of Leibnizian physics.

We know that for Leibniz the physics of aggregates of bodies affecting each other had only an imaginary character. It is a dream (but a coherent one) to attribute the variations of force at each point to external factors. That is to say, in reality no longer from a physical point of view but from a metaphysical one, the world is composed of unextended substantial elements-monads-each of which displays a predetermined internal law:

"The monad automatically deciphers, both in itself and for itself, a universe that is at once its closed interior, its own account of it, and the extensive entirety of its exteriority. . . . The monad is full to an inaccessible extent of attachments that are sufficient for representing  a full and compact nature like itself: the inherence of all the numbers belonging to deciphering. Impressed and expressive but never im­ pressionable."

It is God alone in the Leibnizian system who can know monads as such; it is he who, at the origin, calculated each one's individual law so that the monads express each other, so that each one, through its internal law, translates every change that has occurred in another-that is to say, so that the universe described by physics is imaginary but not illusory.

But the jump from the imaginary point of view of the monad dreaming itself and dreaming of things affected from the outside to the point of view of God-from the physics of aggregates to monadology-has now received a purely physical translation: every integrable system, every system whose equations of movement can be integrated, allows for a monadic repre­ sentation.

Let us first of all define what we mean by an integrable system. A problem put in the canonic language of dynamics is presented in the form of a set of differential equations that describes the following situation for every point: at every instant, a set of forces derived from a function of the global state (such as the Hamiltonian, the sum of kinetic and potential energies) modifies the state of the system. The.refore this function as well is modified : from it, a momentlater, a new set of forces will be derived . To resolve a dynamic problem is, ideally, to integrate these differential equations and to obtain the set of trajectories taken by the points of the system.

It is evident that the complexity of the equations to be integrated varies according to the more or less judicious choice of the canonic variables that describe the system. That is why dynamic physics as it was formulated in the nineteenth century is a theory of transformations among canonic languages, among points of view on a system, each de­ scribing this system in terms of a different set of canonic variables and thereby placing it within a different space defined each time by these variables. More precisely, it is a theory that allows for the choice of the best point of view so that the system can be integrated and the trajectories calculated.

But what integration could be easier than that of the movement of an isolated body, with no interaction with the rest of the world? No ex­ ternal perturbation determines a change in velocity, which thus remains constant, while the position is a linear function of time. All the energy is kinetic; the value of "potential energy" is zero.

The optimal point of view on a system, the best choice of variables, is therefore the one that cancels out the potential energy redefined in terms of these variables. And dynamic theory tells us that every integrable system can be represented in this way-can be redefined as a set of "units" evolving in a pseudo-inertial movement, without any interaction among the "units." Each "monadic unit" is no longer determined in each of its movements by interactions with the aggregate ; each deploys its own law for itself, alone in a system which it reflects intrinsically, because its very definition supposes and translates this system in every detail. There is full passage between the local and the global.

Michel Serres has shown that Leibniz did not "speak of" science, did not "speak about" it from an external position; he "spoke" science, and did so even when he "spoke" metaphysics. Thus, speaking the language of dynamics in a philosophical manner-moreover, a language which was based on his work in physics-Leibniz arrived at a conclusion which was only to be found by dynamics in its most abstract state. This is not an anachronism; Leibniz is not even a "precursor"; he introduces no new fact or concept. He simply undertakes-with the rigor for which he is criticized - an exploration of that internal coherence of the physical and mathematical language of his age which he contributed to creating. The point of view he attributes to God is a privileged point of view whose existence is affirmed by physics as soon as the system can be exactly integrated.

But-and this is how the difference between classical dynamics and Leibnizian metaphysics is now defined-we know that the class of in­ tegrable systems is extremely restricted (the theorem of Liouville). More­ over, our formalized science (we shall soon speak of this) is no longer limited to classical dynamics. The world described by the science of irreversible processes, the world described by quantum mechanics, is a world in which interactions play an ever more important role. The Leibnizian exploration of classical style therefore allows us to identify precisely what is now at stake in science: the description of a world of processes, the definition of entities that participate in the becoming of the world.

We have not shown that the Leibnizian system could be entirely reduced to a theorem oj dynamics, nor that the plurality of isomorphic languages could be reduced to one: the language of dynamics is only one model among others in the Leibnizian system. We have simply shown the pertinence of this model. In fact, for the past two centuries, there have been physicists who affirmed that the whole world could be described as if it were an integrable dynamic system : this is what we have called the Laplacean dream.

Let us examine the properties of the dynamic world based on.the model of integrable systems, which is also' the world whose legality Lucretius's clinamen will undermine. It is a world of determinist and re­ versible trajectories whose definition presupposes two disparate kinds of information: knowledge of the law of evolution which syntax allows one to formulate a priori from the definition of the forces of interaction and the binds inside the system, and knowledge of the description of any state of the system. From this point on, "everything is given." The law will lay out the trajectory taken both toward the past and toward the future. The law is generality itself: it defines the limits of all the possible evolutions of the system and defines them as equivalent to each other, each reflecting the arbitrary particularity of an initial condition.

The property of reversibility is given in a very simple manner: the law of dynamics is such that the operation of instantaneous inversion v -> -v of velocities at each point of the system is equivalent to the operation of inversion of the direction of the flow of time t -> -t. For any dynamic evolution it is thus possible to define an initial state (in fact, the one prepared by this operation of inversion of velocities) such that the system undergoes inverse evolution, "moving in reverse."

This property is an excellent illustration of the arbitrary and de­ terminist character of dynamic evolution. Generally, for any given state, the law permits the calculation of the initial condition needed for the system so that it ends up "spontaneously" at a specified moment at that state. In this world of automata, both arbitrary and inflexible, to know is in fact to dominate; the most extravagant evolutions are deployed in­ differently, translating the extravagance of an initial state. Among these extravagant evolutions, "moving in reverse" has the force of a symbol. Everyone knows the absurd impression provoked by movies shown in reverse: burning matches which become reconstituted; flowers which become buds; a wave of water in a swimming pool which projects a diver up onto the springboard. Dynamics describes and postulates this absurd world.

The notion of reversible and determinist trajectories does not belong exclusively to classical dynamics. It is found in relativity and in quantum mechanics; the evolution of the wave-function as defined by Schrodinger's equation also echoes the syntax of dynamics.

Rather unexpectedly, it is in quantum mechanics that the monadic character of every integrable system has been most evident. In Bohr's model of the atom each orbit is characterized by a well-determined energy level in which electrons are in steady, eternal, and invariable movement. The steady state of orbital electrons is the typical example of the monadic state. The orbits are defined as being without interaction with each other or with the world ; it is as though they were isolated, alone in the world. This monadic description was absorbed in the modern formulation of quantum mechanics by means of Schrodinger's equation : this description becomes a privileged representation such that quantum evolution is reduced to the evolution of a set of isolated steady states without inter­ actions which remain identical to themselves for an indefinite time.

One might object that we have not spoken of the second half of Bohr's model: electrons can jump from one orbit to another, emitting or ab­ sorbing a photon of energy corresponding to the difference of energy between initial and final levels. For that reason we can know that these levels exist; the electron without interaction is unknowable.

It is here that quantum mechanics decisively parts company from dynamics; quantum formalism does not define the determinist and re­ versible description as being complete. It associates a second type of evolution with it, one that is irreversible and discontinuous, the reduction of the wave-function that corresponds, for example, to the jump from one orbit to another with the recorded emission or absorption of a photon, or corresponds to any other interaction with an instrument of measure­ ment, after which one can deterministically attribute a numerical value to one of the quantum parameters. Thus irreversibility of the measure­ ment is necessary to the definition of the quantum phenomenon. Starting from quantum mechanics, the physicist knows that there are interactiom in the physical world which cannot be eliminated by a dynamic transfor­ mation. The process that ends with the amplification and recording of a quantum phenomenon at the macroscopic level is one such interaction, as is the world of unstable sub-quantum particles.

Quantum mechanics thus presents a reversal of perspective relative to classical style. It is no longer a question of looking for simplicity at the level of elementary behavior. Dynamic simplicity, as reflected by the possibility of a completely monadic representation, belongs, in fact, to the macroscopic world, to the world on our scale. Our physics is a science created by macroscopic beings, created with conceptual tools and instru­ ments that belong to the macroscopic world. It is from that position, when we question the world of quanta, that we must choose what will allow us to express matters in terms of measurable, reproducible, and communicable properties. We can no longer allow ourselves, as far as the physical world is concerned, the privileged point of view which, when pushed to its limit, we once could have identified as that of God.

Wte can se how unfortunate was the widespread assumption that quantum mechanics "discovered" that the process of measurement disturbs the system nwasured. uncontrollably modifying; the values of certain parameters in order to ascertain the value of others. Such an assumption in fact implies that only an arbitrary positivistic prohibition prevents us from speaking; of "hidden variables," that is to say, prevents us from affirming; that the system in question is, at every moment, defined by the set of physical parameters, even if all of them cannot be known simultaneously.

The actual situation is entirely different. The real discovery of quantum mechanics, as it is expressed by the inseparable character of reversible evolution and irreversible reduction, is not that tht, process of measurement disturbs, but rather that it participates in the defini­tion of, the measured parameter, so that this parameter cannot be attributed to the quantum system "in itself" and one cannot speak of "hidden variables." As Niels Bohr repeatedly said, quantum mechanics discovered the necessity of choice, choosing; what question to ask, in other words, choosing; both the instrumental fram('work of the qu('stion and on(' of the complementary descriptions articulated among; themselves by formalism but irreducible to a single description.

If dynamics is above all a science of the macroscopic world, the fol­ lowing question immediately comes to mind : in the natural world, where irreversibility seems to be the rule, what is the status of the reversible

descriptions of dynamics? Two solutions have often been proposed, and the ground common to both is denial of the existence of the problem. According to the first, reversible description is only an idealized and partial model that must be abandoned or "completed" adequately as soon as it is no longer valid. According to the other, irreversibility is only an illusion ; we are dream-like automata "swimming" in one direction in the sea of an eternal and legal world.

Neither of these solutions, too simple and especially too sterile, has ever really taken hold. The first amounts to the acceptance of a strict separation of disciplines and to the idea that a style is never more than an inflated paradigm. The second, in contrast, brings us back to a pseudo­ Leibnizian style, telling us that the irreversible world is only a well­ fabricated illusion determined by our subjectivity, and that objective reality is reversible, legal, and determinist.

But a style is not abandoned in the name of prudence and plausibility. Like the Laplacean dream, repeatedly declared defunct, it regroups and deploys, changes arenas, rises again under different theoretical guises. On the other hand, our style is no longer the classical style : from Laplace to Du Bois-Reymond, the nineteenth century, which made dynamics the basis of a conception of the world, was also the period in which a new history and culture arose, and thus a new style of science.

"What is the Industrial Revolution? A revolution operating on matter. It takes place at the very sources of dynamics, at the origins of force. One takes force as it is or one produces it. Descartes and Newton, crowned by Lagrange, chose the first alternative: force is there, given by the biotope, the wind, the sea, and gravity. It is beyond our control except insofar as men and horses are subject to it, but it is not under our dominion when it is a question of heavy bodies, of air, and of water. With it one produces motion, work, by using tools. . . . Then a sudden change is imposed on the raw elements: fire replaces air and water in order to transform the earth . . . . Fire finishes off the horses, strikes them down. The source, the origin of force is in this flash of lightning, this ignition. Its energy exceeds form; it trans­ forms. Geometry disintegrates, lines are erased; matter, ablaze, ex­ plodes; the former color-soft, light, golden-is now dashed with bright hues. The horses, now dead, pass over the ship's bridge in a cloud of horsepower."

From mechanics to thermodynamics, changes of style and society have occurred. And, in parallel fashion, dynamics developed and reached a point of formal perfection. For the last century we have been faced with an original scientific problem, that of the articulation of styles and of reversible and irreversible time. We no longer live in Leibniz's time: to speak the language of dynamics, and conclude from it that we are dreamt monads, is for us not to speak "science" but to speak "a science," not only against the style of the age and against eventual personal beliefs, but also against another science.

Michel Serres has often cited Leon Brillouin's response to the claims of dynamics: a dynamic description is only determinist if the description of its initial state is completely accurate - and accuracy is expensive. All the energy in the world could not pay the energetic debt of a completely determinist description on a global scale. In this form, the argument is perfectly correct: no description actually produced will be perfectly de­ terminist. However, determinism stands out as a limit, perhaps an inac­ cessible one in practice but one which nevertheless defines the series of increasing precision: style resists the argument of plausibility.

However, dynamics has discovered today that as soon as the dynamic system to be described is no longer completely simple, the determinist description cannot be realized, even if one dismisses questions of cost or of plausibility. In other words, we now know that there are dynamic systems of different sorts. There are the rare ones in which determinism exists as a limit-state, costly but conceivable, in which extrapolation is possible between the approximate description of any observer and the infinitely precise one of which Leibniz's God is capable. And there are systems in which the idea of determinist prediction conflicts with the laws of dynamics and in which the idea of determining the initial con­ ditions becomes unthinkable. In certain cases, the passage between local, dynamic descriptions and global vision is impossible.

Since the end of the nineteenth century, dynamics has had a new history, born of two necessities: one, coming from astronomy, the need to define exactly the trajectories that determine the interactions among more than two bodies, and the other, the need to derive from dynamics the description of irreversible evolution, typically defining the increase

in entropy tending toward a maximum. The first led to non-integrable systems, the second, to research on complicated dynamic systems which would not a priori exclude complicated evolutions (ergodic systems, mixing systems, etc.). The history of dynamics has therefore been marked by the coexistence of the two styles; the fruit of this coexistence, the formulation of a broadened dynamic theory in which an "operator" of entropy can be defined, does not belong to history but rather to current research in physics.

In fact, dynamics here encounters the mixture : the systems which, since Poincare, have been known not to be exactly integrable and the systems which are studied by statistical mechanics cloud the view of the observer, even the demon. The science of analysis and of separation must hence­ forth, as Serres says, becalm itself, feminize itself, erase itself, with observation disappearing in favor of relation : "The world as it is is not the product of my representation; my knowledge, on the contrary, is a product of the world in the process of becoming. Things themselves choose, exclude, meet, and give rise to one another."

The systems which aim for the theorem of impossibility of which we are speaking are called unstable systems. In order to understand what an unstable dynamic system is, let us describe a stable system. A rigid pendulum can have two kinds of movements that are qualitatively dis­ tinct : oscillation and rotation. The behavior of a pendulum is predictable and depends on the initial conditions. There is only one case of uncer­ tainty; it occurs when the initial acceleration is such that the pendulum attains the vertical with zero velocity. In that case, a perturbation "as small as one wishes" will be sufficient to determine which side it will fall to and thus what kind of movement it will adopt. The pendulum is thus the type of system for which, with the exception of these individual rare cases of uncertainty, an approximate description is sufficient to avoid any unexpected evolution and for which a determinist description is the limit. An unstable system, on the other hand, is a system in which the initial conditions determining various qualitatively distinct behaviors are not clearly separated but are, on the contrary, as close as one might wish. We are all familiar with this sort of intimate mixture - it is described by number theory: every rational number is surrounded by irrationals, and every irrational by rationals. Similarly, whatever the neighborhood defined for an initial state, one always finds at least one other state giving rise to a qualitatively different behavior, just as oscillation and rotation are qualitatively different.

Under these conditions, in order to predict deterministically the type of behavior the system will adopt, one would need infinite precision. It is of no use to increase the level of precision or even to make it tend toward infinity; uncertainty always remains complete-it does not diminish as precision increases. That means that divine knowledge is no longer implied in human knowledge as its limit, as that toward which one might tend with increasing precision ; it is something other, separated by a gap.

For the third time in the twentieth century, physics finds itself defined by the fact that we cannot observe and measure with positively infinite precision, no more than we can communicate faster than the speed of light or measure with instruments that are not macroscopic. Just as the demonstrations of impossibility in relativity and in quantum mechanics are tightly linked to the opening of a new conceptual field, the impos­ sibility of conquering the indeterminacy essential to unstable dynamic systems is not an epistemological discovery which only concerns the re­ lation of our knowledge to the world; rather, it offers a new method of positing problems of physics: the possibility of positing the problem of irreversibility within dynamics. It is not a question of recognizing that we are incapable of calculating such trajectories; rather, it is a question of realizing that the trajectory is not an adequate physical concept for these systems. Henceforth the field of dynamics will appear larger: systems described in terms of trajectories with their determinist and reversible properties are only a particular class within that field.

Where monadic physics ends and trajectories become unstable, the world of the irreversible begins, the open world in which, through fluctu­ ations and bifurcations, things are born, grow, and die. The instability of trajectories, their irreducibie and essential indeterminacy, have as a global result the heavy, macroscopic irreversibility of the self-organizing processes that make up nature.

To speak of unstable trajectories is to use the language of classical dynamics to introduce non-classical physics; it is to use the determinist and reversible model to construct the description of the irreversible; it is to invoke monads to describe interactions that cannot be reabsorbed in the monadic interior. In short, it is to repeat Lucretius's procedure, "starting" with the inflexible and legal order and then introducing dis­ turbance and indeterminacy. Things are born where the law is not suf­ ficient to exclude disturbance or to prevent the dynamic monads from interacting.

"Without the declination, there are only the laws of fate, that is to say, the chains of order. The new is born of the old; the new is only the repetition of the old. But the angle interrupts the stoic chain, breaks thefoederafati, the endless series of causes and reasons. It disturbs, in fact, the laws of nature. And from it, the arrival or life, of everything that breathes ; and the leaping of horses."

The model of falling atoms, parallel flows in an infinite void, eternally identical to itself, constitutes an exact image of the monadic evolutions of classical dynamics, parallel as well, without interactions, in a reversible, that is to say, in an indifferent, world. The fall is nothing but the uni­ versal without a memory whose every instant is the integral repetition of the preceding instant. Classical dynamics was the mathematico-physical effectuation of this ordered world, directed by a law.

But the parallel flow is only one of the models of the primitive base from which things are born. The second is that of the cloud, of stochastic chaos, closer to reality, Serres says. Here, atoms go in all directions and collide randomly; it is an immense, tumultuous population, a "disor­ ganized, fluctuating, brownian mass composed of dissimilarities and op­ positions." Here as well one must note the exactness. This is a descrip­ tion of another physical situation, a purely macroscopic one but one which also is integrally subject to a law: statistical equilibrium within a population in which all the processes and their opposites are produced simultaneously and compensate for each other. At this macroscopic level, just as at the microscopic level of the description of atomic trajectories, the apparent absurdity of the clinamen is repeated. Statistical disorder should not produce a difference, any more than a disturbance should occur in the established trajectory, for disorder is the state in which all differences are abolished in the indifferent, senseless tumult in which they all coexist.

Yet non-classical science has taught us that trajectories can become unstable and that stochastic chaos can become creative. In certain cir­ cumstances, evolution bifurcates, the homogeneous disorder is no longer stable, and a new order of organized functioning is established, with amplified fluctuation. For example, laminar flow in parallel sheets "spontaneously" becomes turbulent; that is to say, we now can calculate it. In this realm of the bifurcation, in which turba becomes turbo, rather a strange tumult reigns, the complete opposite of indifferent disorder. Creative chaos is illegality itself, for its description dissolves the distinc­ tion between the macroscopic state and the microscopic fluctuation; correlations can appear among distant events; local deviations echo through­ out the system - the matrix-state in which fluctuations are amplified and from which things are born.

And thus Serres is correct: the question is reversed. It is no longer necessary to ask where the clinamen comes from or how one might justify the disturbing of laws. All laminar flows can become unstable past a certain threshold of velocity, and that was known just as the productive nature of organized forms, of bifurcating evolution, of what we call dissipative structures, was known. One must ask how an abstraction of this knowledge could have been made to describe the world in order, subject to a universal law. We already know one answer given by Serres. Classical science is a science of engineers who knew, of course, that their flows were never perfectly laminar, but who made the theory of laminar flow perfectly controllable and directable, the only flow for which know­ ing is controlling.

"The technological model is in place. It is a physics of water mains. Our physics was first of all a physics of fountain-builders, of well­ diggers, or of builders of aqueducts. . . . Hence this physical world where the drainpipe is essential and where the clinamen seems to be freedom because it is precisely the turbulence that refuses enforced flow. Incomprehensible by scientific theory, incomprehensible to the hydraulic engineer."

The history of science particularly repeats itself when it is a question of mastery or control. We could rewrite the same text with irreversible replacing clinamen and the builder of thermic machines replacing the well-digger, with matter transformed by fire and heat replacing flowing water. This text would tell of the birth of the thermodynamics of equi­librium, the classical science from which will later be born the study of irreversible processes." [Excerpted in Serres, Hermes]

Serres is a Girardian.

His book 'Rome' is an attack on the blond beast, and the barbarous founding at the heart of the Empire.

He like Girard sees the scapegoat victim, as the centre around which a mob organizes itself into a Form.
Violence as the foundation.
Romus kills Remus.
Patrocles dies in stead of Achilles.
The King is annually felled like the sun-king, and so is twin to the scapegoat, the pharmakos.
Violence, as Girard noted, grows acute with the growing erasure of difference - twins, twin figures, and eventually the mob where one is just as well as any other.

The scapegoat is a ritual substitute, and therefore a proxy.
He likens this to the Joker card in a deck that by itself having no value, is capable of taking any value, capable of being anything and everything - from king to clown to victim…
It dons many masks.
The place-holder 'Joker' changes the series of relation.
The law of circulation makes possible a body - the king's body, a sovereign body.

From him, we may appreciate the word 'Vicar', which helps differentiate I.E. notions of the Druid or the Priest.

Vicar:

From Old French vicaire "deputy, second in command," also in the ecclesiastical sense (12c.), from Latin vicarius "a substitute, deputy, proxy," noun use of adjective vicarius "substituted, delegated," from vicis "change, interchange, succession; a place, position". The original notion is of "earthly representative of God or Christ;" but also used in sense of "person acting as parish priest in place of a real parson" (early 14c.)."

Vicarious:

"1630s, "taking the place of another," from Latin vicarius "that supplies a place; substituted, delegated," from vicis "a change, exchange, interchange; succession, alternation, substitution," from PIE root *weik- (4) "to bend, wind" (source also of Sanskrit visti "changing, changeable;" Old English wician "to give way, yield," wice "wych elm;" Old Norse vikja "to bend, turn;" Swedish viker "willow twig, wand;" German wechsel "change")."


The neutral symbol that can take on any signifier is a proxy, a ritual substitute, the king-And-the-scapegoat…
It circulates like a phantom taking on forms.
Substituting for the absolute absence, the Vicar who himself acts as a "proxy of God", Constructs and Founds the sacred around the ritual substitute, the scapegoat, the pharmakos.

In the sociological sense, this means, against Hobbes, the social is a formation not between man and man, but between the collective and the coveted/dispelled object.

Serres' view is as Xt. as Girard's, where everyone is constituted and mutually implicated through every other.
It de-centers identity - which is what postmodernism is.

Still, there is much to appreciate in all of Serres' books.