The manuscript for my lecture at the QUANTITY AND QUALITY, THE PROBLEM OF MEASUREMENT IN SCIENCE AND PHILOSOPHY conference at UC Davis, CA, April 5/6 2014, organized by Prof. Nathan Brown.

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CONTENTS

The first cycle: Homothesis as the Locus in Quo of the Universal’s Presence

1st Iteration (acquiring a space of possibility)

2nd Iteration (learning to speak a language in which no one is native)

3rd Iteration (setting the stage for thought to comprehend itself)

4th Iteration (Intelligence that is immanent and coextensive with the Universe)

5th Iteration (Inventing a scale of reproduction)

6th Iteration (the formula, a double-articulating application)

Second Cycle: Universal text that conserves the articulations of a generic voice

1st Iteration (marking all that is assumed to be constant with a cypher)

2nd Iteration (confluence of multiple geneses)

3rd Iteration (the residence of that which is genuinely migrational)

4th Iteration (universal genitality)

5th Iteration (mathematics is the circuit of cunning reason)

6th Iteration (the real as a black spectrum)

Coda – Realism of Ideal Entities: Conceiving, Giving Birth to, and Raising Ideas on the Stage of Abstraction

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“Thales, who reads in the traces of the body, deciphers, ultimately, only one secret, that of the impossibility to enter the Arcanum of the solid body in which knowledge resides, buried forever, and out of which wells up, as if from a ceaselessly springing source, the infinite history of analytical progress.”^[1]

In his text “What Thales saw at the foot of the Pyramids”, Michel Serres argues that the birth of pure geometry has never taken place,^[2] and that precisely because it has never taken place, it can keep to give room for things whose happening is highly unlikely. This shift in perspective, inverting the very subject matter so much mourned by people like Edmund Husserl, Martin Heidegger, Jacques Derrida, Bernard Stiegler – the loss (or rather mulitplication?) of eidetic intuition – owes everything from looking at the algebraic make up of the relation between geometry and theory, from (1) interpreting the triangle (epitome for measuring) as a geometrical atom, (2) from considering atoms as the characters of a universal script, and (3) from postulating that precision in measuring strictly depends upon idiosyncratic treatment of “bodies of pure regularity” as a dark spectrum.

My paper is organized in two cycles, each of which is made up of several iterations.

 

The first cycle:

Homothesis as the Locus in Quo of the Universal’s Presence

1st Iteration (acquiring a space of possibility)

In Serres’s text, we find ourselves in the desert with Thales, facing, in the pyramid, an impenetrable constellation. We might well recognize the pyramid’s outline as a triangle, but we know not how to measure it. We are taken to accompany Thales on an adventure that is pure concentration, a tour during which we reach, eventually, in a circuitous manner, what is straightforwardly and directly inaccessible—a space in which measuring the pyramid becomes possible. It is an adventure in archly reasoning, reasoning that is proceeded by an act of double duplication: on the one hand, we duplicate the situation in which we find ourselves, and on the other hand, simultaneously, we duplicate ourselves as we find ourselves comprehended in that situation. All that is left for us to do, if we follow Thales and Serres, is to give an account of how we proceed by aspiring to measure each repetitious step taken. The cunning that drives such reasoning never properly manifests itself, neither positively nor negatively. It establishes, through what I will call double duplication, a stage of abstraction that is capable of hosting a play of homothesisasthe dramatizing establishment of“homology between the crafted and the craftsman.”^[3] The cunning by which we are driven manifests in no other way but in tending to its own continuation. Tended by his own cunning, Thales’s double duplication introduces a time that might remain, by giving way to the unlikeliness of finding an accord in which it (measuring what is overpowering, colossal, and immense) acquires a space of possibility, exposed from elaborating the soundness of the presumed accord by computing auxiliary structures in all of which the same invariant quantity is at work. The postulation before Thales’s inner sight—a postulation in theory—of a module, from Latin modus, literally “a measure, extent, quantity, manner,” is enough to stage the invariant quantity at stake. This is what Serres tells us.

But how to find this quantity? All that there is to be contemplated, for finding an answer, so Serres tells us, is that Thales must find a unit of procedure, and that the quantity of this unit ought to be, if the procedure be feasible and valid, conserved by a structure. Thus, Thales must attempt to stage abstractly the very act of virtually en-familiarizing himself with what is colossal and immense. Thales knows that the interiority of the pyramid is inaccessible, that it would be an unworthy violation to force his access into it. Thus, Thales pays all due respect to that, and premises for his own symbolical double duplication that the interiority spaced out in it be inaccessible as well. He treats the size of his triangle purely structurally—without knowing, at first, anything about the structure, nor how he could possibly apply his triangle for measuring. We thus learn that Thales begins this elaboration by building a stock of experience—Serres calls it a résumé, from Latin resumere “take again, take up again, assume again.” Before Thales will be able to actually draw a circle, we learn, he has to actually go in circles. Many times. Learning to measure, even in theory, Serres tells us, is an operation of application. One has to “blossom into” the capability of doing it. Thus Thales keeps beginning, summing up what he finds along his iterations, and treats the sums he comes up with as a product of reciprocity, from reciprocus, “returning the same way, alternating.” Gradually, so we are told, he invents a scale of reproduction. How? All that we can say in this first iteration is that Thales measures the pyramid by postulating—on grounds no more “solid” than the immateriality of a desire—that it be possible, and by striving to elaborate the conditions for his own postulation.

2nd Iteration (learning to speak a language in which no one is native)

One idea Thales substantiates in the course of the elaboration of his postulate—the postulate being that the inaccessible pyramid is measurable—is that the pyramid incorporates the principle of homothesis. Homothesis is, as we learn from Serres elsewhere, “the same way of being there, of being placed.”^[4] The space of homothesis is a space of dislocation, deferral, and adjournment, “with or without rotation,” as he puts it.^[5] Things that are governed by this principle, things that are tributary to the space of homothesis, are things that can be considered as equally bounded. In short, they can be considered as things that are commensurate. But what can be the source that sheds light onto such a space for abstract intellection, and hence open it up to our intuitive sight? It is the sun that treats all things equally. Yet this equality, Serres warns us, cannot in any direct manner be found in the sun itself, as if it gave each thing its natural gloss immediately. Nevertheless, we are told, the sun facilitates that an abstract space may be engendered. The engendering of such an abstract space is, for Serres, the Greek miracle whose revelation eventually made possible what he calls the fabrication of a mathematical language, the sole language “capable of halting conflicts and which never needs translating.”^[6] The language spoken in such abstract space is the sole language in which there are no barbarians, because everyone speaks it as an immigrant, with no political obligations of conforming to the mother tongue spoken by natives.^[7]

3rd Iteration (setting the stage for thought to comprehend itself)

This language allows articulations on the stage of abstraction, and for Serres, its possible articulations open up and constitute the scene of writing. Within a space governed by the principle of homothesis, the scene of writing is constituted around homology. For Serres, it is the Greek understanding of logos that will allow alphabetic writing to think of the cosmos no longer in terms of genesis and progeny, but in terms of a logics that comprehends the cosmos within the universe. Homology, he tells us, is threefold: number, relation, and invariance. Arithmetics, geometry, and physics. This fantastic premise of one universal logos, Serres maintains, allows Thales to see in the pyramid a manifestation of the homothetical principle. On this assumption, Thales can postulate the invariance of form to complement the variations of quantity. Armed with such thinking, the colossality of the pyramid becomes less daunting, and this without the need to divest its constitutive secret, its inaccessible interiority (the origin of knowledge). The archly reasoning that supports such thinking is not the reasoning of an individual subject rising up against the principle that governs its own predication. In Thales circuitous thought, there is nothing revolutionary here whatsoever. The reasoning exerted in support of homology is an automatic reasoning, we are told, from autos “self” + matos “thinking, animated.” As Serres puts it, it is the reasoning that happens as the world exerts itself upon itself,^[8] a world that thrusts forth and pushes out of itself, in order to adjoin to itself what happens to it. Serres calls this the reasoning of how the gnomon counts, the reasoning that seeks to account for the objective ruler that sets the natural play of shadow and light in scene by collecting it with its own apparatus of capture (the casting of shadows). “Who knows? Who understands? Never did Antiquity ask these two questions,” Serres maintains.^[9] The gnomon allows to indicate time, but foremostly it is an observatory that does not, like modern telescopes, bundle what gathers into something specifically for the sight of an individual subject. In the events the gnomon is capable of staging, Thales (and anyone else) participates as nothing more than as a pointer, an index or cursor, since “standing upright we also cast shadows, or as seated scribes, stylus in hand, we too leave lines.”^[10] But aware of this precise circumstance, Thales now sets out to reason about how the gnomon stages, as an apparatus of capture, the play of shadow and light. In his double duplication, Thales literally tries to catch up with the course of what he himself (as a gnomon) indicates, and hence makes observable. It is by trying to catch up with his own significance within the situation that Thales eventually begins to substantiate the concept of similarity as an invariance—or, to make Serres’s point more clear, as an idea contemplated by the world in its own automatic reasoning. Even though Thales is trying to catch up with his own significance within the situation, the active center of knowing resides outside of Thales himself: “The world renders itself visible to itself, and regards this rendering of itself: here resides the meaning of the word theoria. To put it more clearly: a thing—the gnomon—intermits the world through stepping in, such that the world may read on its own surface the writing it leaves behind on itself. Recognition: a purse, or a fold.”^[11]

For Serres, the scene of writing is automatic too, as it is for Derrida as well. But unlike its characterization by Derrida, for Serres the scene of writing unfolds on the stage of abstraction, and this stage is a dramatic, not a mystical, space. But it too is a space that knows no individual poets or playwrights. The dramas it puts forth are authored by a collective subjectivity that spells out the reasoning of a world that exerts itself upon itself.

4th Iteration (Intelligence that is immanent and coextensive with the Universe)

Such a collective subjectivity depends upon an artificial memory. Serres finds such a memory in the canonical lists and tabular organization of practical problems—the preparation of how certain results to certain problems may be found more easily, based on how problems of a same kind have already been resolved whenever they have imposed themselves previously.^[12] The problems thereby treated are mainly economical problems; they revolve around how to count what is given—but not around how we might account for the manners in which we do count that which is given. The tables in which the treatment of these problems is organized must be ordered around a step-by-step procedure that will lead whoever follows it to the desired decision or solution. Such methodical, goal oriented procedures are what Serres calls algorithms.^[13] They spell out how to reach all intermediary steps as one attempts to multiply quantities, to divide them, to raise them to a different power than that in which it is given, to extract the roots of a quantity or to sum up or divide them. The overall framework of these operations, one might say, consists in finding ways of counting, as exhaustively as possible, the possibilities hosted in a quantity’s reciprocal value—these possibilities are the very substance of economic thought.^[14] The methods of how such tabular organization is gained, is strictly algorithmic. An algorithm is made up of techniques or operations of how to count—what we today summarize as the operations of arithmetics. Its procedures know three classes of numbers: the givens (data), the results, and the constants, which are the stepping stones from given to desired results.^[15] As long as possible manners of accounting for how what is given is counted by these tables, quantities lack a proper generality; they are always concrete and singular. Generality is not seen with regard to the things given, it applies to procedures only: an algorithm is an algorithm (and not an account of one’s experience, like a fable or a tale, for example) because it is a general rule that can be reproduced in its experiential value by anyone who follows its steps. Once a specific procedure is put in numerical form, one and the same algorithm can be applied arbitrarily to particular situations. Such algorithmic procedures usually end with the formulation: “Behold, one will do likewise for any fraction which occurs.”^[16]

Against this background we can understand Serres’s admiration for archly reasoning that has not the particular economical interest of a people at its core, but that fantasizes a reasoning proper to the world itself. The homological dramas that unfold in his homothetical space of abstraction, and that are expressed in the scenes of writing that accrue from it, are full of brilliance; yet the intelligence that shines in it is not that of an extraordinary priest, king, or an official expert. Archly reasoning differs from algorithmic reasoning mainly in that the manners of accounting in which that which counts expresses its power, are being treated wittily, and challengingly. The brilliance that shines in the archly reasoning of a world that exerts itself upon itself, by double duplication, is the brilliance of a world that collects and discretizes itself in a genuinely public language (that of mathematics). For Serres, “intelligence is immanent and, probably, coextensive with the Universe.”^[17] The world owns a huge stock in forms, he tells us, “there is a vast objective intelligence of which the artificial and the subjective constitute small subsets.”^[18] The new economy that corresponds to the archly reasoning of the world feeds from the cornucopia of ideas that the world might recognize as its own, while trying to keep track, in its reasoning, with who and what it actually is.

5th Iteration (Inventing a scale of reproduction)

So let us turn back to Thales, and how he gradually invents a scale of reproduction for measuring the colossal manifestation of the pyramid. Thales sees in the pyramid the eminence of a principle, we said, that of homothesis. But how can we learn to en-familiarize ourselves with the meaning of this? What we can learn from Serres is that homothesis abstracts from the tabulatory accounts that preserve and collect, in their algorithmic tables, all that the gnomon indicates. One way to put it is to say that Thales steps out of the apparatus of capture’s reign, and that he dares to multiply the very principle of its regime.

Let us recapitulate and see how Thales proceeds. Thales has no direct access to the object he wishes to measure, and sets out to establish the possibility of an indirect way, by double duplicating the situation and engendering the form of this double duplication as a reduced model. He proceeds to measure the pyramid by postulating that it be possible, and elaborating his own fantastic postulation before his inner sight, that is, in theory. He begins this elaboration by building a stock of experience—a résumé—or, as we might say now, by treating what appears to be a given as data to be organized in algorithmic tables. What appears as a given, he dares to think, is given by the gnomon and can count only as indexes to something that is not exhaustively given in what the gnomon collects. This something, he considers, must be of such a magnificent quantity that the form of reciprocity that hosts it also hosts the size of the pyramid as one of its possible variations. If one were to en-familiarize oneself with the dimensions of the monument, and hence be capable of measuring it, this magnificent quantity is what one would need to comprehend with more capacity. Thus, after having stepped out of the immediate reign of the gnomon’s apparatus, Thales gives way to a thrusting forth of his mind beyond what it is yet capable to encompass. He wants to learn. Following Serres in his account, we can remind ourselves that before Thales will know, and be able to draw his famous circle in order to measure the pyramid, he has to iterate and go in circles, on grounds no more solid than his desire that it be possible. He has to assume a result that seems, from all he can know, beyond reach—and it is on the premise of its assumption that he must try to find an algorithm that will guide his way to the result whose solvability he presumes against all odds. Thus Thales gradually builds up his résumé. He continuously sums up what he finds along his iterations, and attempts to treat the sums he comes up with as values proper to his hypothetical form of reciprocity of a quantity so magnificent that it hosts the invariant quantity that makes the pyramid comparable to the reduced models he is trying to build.

But from what stock of experience does he draw when attempting to build a model? Going around in his circles, Thales regards the pyramid as an objective ruler. He begins by regarding it, as is the common manner of thinking, Serres suggests, as a sundial. He expects the pyramid to speak about the sun, and to indicate the hours of measuring. He marks the outlines of its shadows as time goes by, and faces a growing number of varying outlines, the longer he goes on. As he continues his circles, he begins to consider that the outlined shadows (which build his stock of experience, his résumé) must all be variations commensurate with one another by that module of which he knows nothing more than that he must proceed according to its proportionality in his attempted act of double duplication. The way how Thales eventually succeeds in abstracting from the idea of the gnomon, explains Serres, is by changing the real setting of his exercise into a formal setting in theory: instead of bringing the pyramid to speak about the sun, he can now ask the sun to speak about the pyramid.^[19] This perspective, which is now a theoretical one, no longer based on experience alone, does not, as before, require that the magnificent quantity, whose form of reciprocity hosts the invariance he seeks, be real and actually given; it may remain a secret—like those secrets, inherent to materials and to tools, which forever inspire the development of a craftsman’s mastership.

Hence, we can imagine how Thales’s view gradually begins to change. He ceases to contemplate the variations he observes and registers, as he goes in circles, for the sake of finding in them a new “given,” from whose concrete shape he learns a general procedure. Yet with it, he cannot mechanically compute, as it was custom with the algorithmic way of thinking, what may count as constant and common throughout the transformations among all the outlined shadows. No, he begins to take the stance of the artistic craftsman—and he is well aware that what he attempts to craft must remain abstract. He sets out to craft a genuinely theoretical object, one that duplicates the objectivity of the ruler. Now, the variations begin to interest him because they must host, he thinks, the essence of an invariant quantity that, like a guest, can never appear in its familiarity as long as it is respected as a guest (and not subjected to the customs of one’s own home). Like a guest who is familiar and strange not due to willed disguise, but by lack of alphabetized commensurability, the invariant quantity must be treated in a space, and in a language, in terms of which the artistic craftsman too is an immigrant and a stranger. It cannot be the concretely objective space of collective memory that allows for the dramatic act of an inceptive conception, rather it must be an abstract space which is capable of staging the intuitive concreteness of collective memory. From now on, Thales strives to en-familiarize himself with the immenseness of the pyramid; he no longer hopes to succeed in subjecting it to an order that he would already be familiar with. He aspires to do so by expecting from that which changes ceaselessly (the shadows) that it be capable of speaking about what is stable in an abstract and non-concrete manner (the measured pyramid). He thinks about the setting in which he finds himself (at the foot of the pyramid, in the desert) as a formal setting, not as a real setting, and with this, Thales can find a trick to render—against all likeliness—the course of the sun permanent. He no longer participates in the dictate of the gnomon as a real ruler, where what it points to must belong to what is already given, but to what can be seen in what is given only by pointers to something whose magnitude is magnificent, and as such bound to remain immense, and barred from being directly experienced.

With this leap into theory, Thales no longer uses space to indicate time; he arrests time through generalizing one particular, and real, moment—that when our shadows and our bodies have the same length. As Serres puts it: “He homogenizes the singularity of each day in favor of a general case—one has to stop time in order to evoke geometry.”^[20] In other words, Thales must symbolize a world in which he could relate to a monument of such awesome greatness and vastness, from Latin colossus “a statue larger than life.” Like this, Thales can think with all the cunning and conquering reason he is capable of, and yet without being disrespectful to the secret at the center of the pyramids. Such is the symbolical nature of intellection, Serres seems to be saying, an intellectual nature that is not at odds with an ethics of mutual respect. We can see in the birth of mathematical theory the unlikeliness of beginning to converse abstractly.

6th Iteration (the formula, a double-articulating application)

Thales’s double-articulating application of the gnomon contemplates all possible variants of a triangle by inscribing them, theoretically, into a common compass: the course of the sun’s permanency. This is how Thales eventually succeeds in conserving, in his textual formula of right-angled triangles, a universal and formal concept of similarity. Its compass is conceived by a reasoning that is proper to the world as it exerts itself upon itself—the course of the sun as collected by a duplication of the gnomon. Thales’s theorem states, as a means of conservation, that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ∠ABC is a right angle.

For Serres, as we will see in a moment, recounting what Thales might have seen at the foot of the pyramid is inevitably a text about originality. Like Thales himself, Serres is not interested in revealing the signification of this origin by claiming to be familiar with it, but instead he wants to postulate, again like Thales, further theorems of universal value. Let’s see what some of Serres’s own postulations are, and how he sets out to elaborate on them.

 

Second Cycle

Universal text that conserves the articulations of a generic voice

1st Iteration (marking all that is assumed to be constant with a cypher)

First we must see what is the object of Serres’s own double duplication. Thales, we said, double duplicated the algorithmic mode of iteration and established a textual formula that conserves an infinite amount of variations. As Thales puts the algorithmic mode of iteration at stake in order to abstract and generalize from its naturalized custom, Serres puts Thales’s own archly reasoning in duplicating the scene at stake—in order to abstract and generalize from it. What happened in this “Thales moment” counts to Serres not so much as the origin of geometry (which is today’s customary association with this event), but as the inception of a stage for abstract thought. The inception of such a stage is necessary, he maintains, for developing proper alphabets of formal reasoning out of the formality of mathematical statements—alphabets that, like any alphabet, allow for expressing an infinity of articulations by a finite stock of elements. Thus, if Thales was capable of formulating his theorem by attending—theoretically—to the permanence of the sun’s course, Serres wants to reintroduce temporality and the vividness of real happenings into the formal settings established by Thales. If Thales questioned the principle of the gnomon by multiplying it, and thereby invented the space of theory (homothesis and homology, organized according to an abstract principle of similarity), Serres sets out to question the principle of theory by multiplying it, and to inventing an alphabetic view on the timeless space of formal theory. Such an alphabetic view is what to him counts as the birth of physics from the spirit of mathematics.^[21]

Serres’s account sets out to speak about how the abstractness of an architectonics of formal ideality had been fabricated. The proposal is simple. What Thales realized, according to Serres, is threefold: (1) the possibility of reduction: Thales creates a model that extracts from the given situation a skeleton reduced from all singular context, and that is in favor of a general case; (2) Thales affirmed the idea of a module: that throughout different sizes and scales, the quantities at stake must be commensurate; (3) Thales conceived of the model in a general, not in an iconic, representational manner: he invented a scale of reproduction.^[22]

These are the conditions that make the creation of a model possible, as an intellectual act of engendering. Yet, as conditions, they depend upon being bracketed and enciphered: Thales, trying to win the immense for a mutual encounter in a realm in which both are immigrants, all familiar constancy in terms of space, time, practice, perception must be questioned and marked with a cypher. Driven by his desire, Thales treats them as coefficients that must, in some way of which he knows he can never apprehend of it in an immediate manner, be at work within what he seeks. And indeed, once Thales comes to measure the pyramid, each condition will be raised in their powers: space will host something that does not exist, a general model; time is arrested and one of its moments is rendered perennial; practice comes to envelop not a necessity, but something that appears necessary (a theory); measuring does not depend upon tactile perception, but upon visual sense. Thales, in the account Serres gives of him, invented the stage of abstract conception by conquering, without disgrace, what is, in its dignity, impenetrable: the arcanum of the pyramid’s lasting and unviolated immenseness.

2nd Iteration (confluence of multiple geneses)

Serres’s own double duplication of the Thales situation constitutes, in turn, a model. What he sees while tracing the conquering movement of Thales’s act of intellection, lets him face something that appears to him as immeasurable as the pyramid must have appeared to Thales—let us call it the graceful desire by which he sees Thales moved. The desire that desires the arcanum. The desire for revelation of what must remain, if one does not want to violate it, concealed. So what does Serres do, in his account of Thales? He sees in the Thales situation a multiplication of originality in procedural, operative terms: algorithmic originality times gnomonic originality times formulaic originality times textual originality (the originality he adds to it when he reads Thales’s story as a story of origins).^[23] The multiplication of origins supports a multiplication of how we can account with givens by rooting them in enciphered constants, and by symbolically domesticating the growth of what can be yielded from these roots (the variables in all possible variation) if we carefully tend to their tabular organization. The careful tending of such graceful desire consists in treating formulaic statements as theoretical fabrics, which aspire to caress the integrity of the colossal through offering dramatizations of possible rapports, in which the terms of such statements feature as protagonists, as actors on stage in texts of proper originality. In the plurality of such dramatized theoretical fabrics, we can render the givens comparable as things that remain, essentially, elusive and come to the world from an outer space of universal intellection, as pointers to a magnitude with which we can en-familiarize ourselves only if we collect what marks it as indexical pointers to be integrated into a commensurate compass; stating what can be conserved into a formula depends upon abstract conception in a realm of theory, and this realm is, essentially, inexhaustible. More concretely, in his multiplication of originality, Serres faces an immense product, a result that integrates the streams that spring from all these different originalities, as the confluence of multiple geneses.^[24] The alphabetization of the theoretical space must attempt to draw balances from this immense product.^[25]

So how does Serres imagine that the Greeks were able to conceive of the abstract stage of geometry? Through a fourfold genesis, he suggests: (1) a practical genesis which consists in “producing a reduced model, coming up with the idea of a module, tracing back what is afar to what is near”; (2) sensorial genesis which consists in “organizing the visual representation of that which cannot be sensed immediately by touching” (3) a civic or epistemological genesis which consists in “departing from astronomy and inverting the problem of the sundial”; (4) a conceptual or aesthetic genesis which consists in “stopping time in order to measure space, swapping the functions of variability and invariance.”^[26]

3rd Iteration (the residence of that which is genuinely migrational)

From within this insubordinate happening of confluent streams, which Serres recounts while contemplating what Thales might have seen, Serres identifies three conditions that will firmly support to gracefully appropriate a sense of inner sight (theory) by building schemata in the form of optical diagrams.^[27] Optical diagrams contain the essence of theory, he holds, yet this essence is an act: that of transportation.^[28] Theory, by sending on travels whoever reasons in its terms, allows him or her to grow more familiar with what manifests itself as immense. Let us recapitulate Serres’s reasoning. The sense of sight, and that which is seen, premise the following givens: position and angle, a source of light, and an object that is viewed as either dark or light.^[29] The confluent streams are treated as processes of transportation, and the questions to be asked, Serres maintains, are questions of where that which is caught up in transport properly resides:

(1) “Where is the proper residence of position and angle? Anywhere. Where the source of light resides. Application, relation, measurement are possible because field markers are brought into constellation; one can see the sun and the peak of the pyramid in constellation, or one can see the peak of the tomb and the uttermost end of the shadow in constellation.”^[30]

(2) “Where is the proper residence of the object? Also the object must be transportable. And in fact it is transportable: either because of the shadow which it casts, or because of the model that emulates it.”^[31]

(3) “Where is the proper residence of the source of light? It varies, one only considers the sundial. It transports the object in the appearance of the shadow. It resides within the object, this, we will call the miracle.”^[32]

It is an enchanted world, the world in confluent streams of multiple geneses, and yet it is a world of objective reasoning. It is a world in which what testifies the immenseness of life and death can be encountered gracefully. Where a monument evokes a sense of tremendousness and seems to demand subordination, Thales shows us (via Serres) how we can en-familiarize ourselves with it by considering abstractly and carefully superordinate concepts, hypernyms,by dramatizing them. To conceive of the world abstractly is a form of conquering that never annexes what it desires to co-extend with. To conceive abstractly brings to work what one is familiar with from where one comes from in an altogether original manner, by treating what appears to be constant as cyphers that need to be rooted in symbolic domains yet unknown, to be engendered by no other way than by archly reasoning.

4th Iteration (universal genitality)

On the stage of abstraction, all that features in it is migrational, and hence welcomed as immigrant. It is the stage on which to conceive of things in their genericness, and in their universal genitality. It is a theorematical stage, and it enables the unfolding of plays in the scene of writing: plays that perform the measurement of originality in theory. Nothing in these plays is native to their plotlines; everything that features in them is on travel. With regard to such measurement, no one can possibly be at home when he or she dares to make statements about what happens in a scene of originality. Such measurement depends upon one’s own en-familiarization with what is awe-inspiring—on the sole condition that we can count, if only the ways of conduct are not without grace, on the colossal’s hospitality: “The theatre of measurement performs how a secret may be deciphered, how an alphabet may be deciphered, and how a drawing may be read.”^[33]

In Serres’s account of theory, mathematics is the key to history, not the other way around. A scene of originality cannot be witnessed, he insists.^[34] In it, something immense is posed at the discretion of a theory, and a theory is the dramatization of an arcanaum, a secret. Mathematics is archly reasoning that seeks to engender a circuit. Nothing more. It cannot be witnessed, it can only be actualized. If the essence of theory is transport, as Serres maintains,^[35] then theory is never about identifying with the revelation that takes place in abstract conceptions that are attributed to count as scenes of originality—like that of Thales and the inception of the theorem of angular measurement within a circle. It is not important whether Thales draws the circle around himself, or around a simple stick, as far as the statement of the scene in the form of a theorem is concerned. A theorem expresses a schema, an optical diagram, and the schema is a stable auxiliary construction that allows a thing to be transported. Such auxiliary constructions render all things mobile; they are vehicles.^[36] They facilitate within the reality of the universal the migrational activity of that about whose essence we can say nothing more than that it is immense, a crystallization between life and death, a being about which no one knows anything beyond what can be stated of it in the universal terms of mathematical agreement. As a thing stated like that, in its dramatized originality, one can tap into the circuit of activity that is organized in its statement. And this without, properly speaking, understanding it.

But one needs to understand the theorem. And this involves, ever again, to pay ones coordination of familiarity, the elements of one’s world, as a tribute to the spelling out of the theorem. That is why mathematics, to Serres, is the key to history. What can be told by theorematical statements are dramatizations of an immense content, and in that, they are not much different from how the schemes in mythical tales work: a schema is what remains invariant regardless of the number of times a story is told. But the schema is not the origin of this invariance, it is its vehicle.^[37] Every mythical tale is the dramatization of a given content. The relation between a schema, and the mobilization of an original thing that the schema affords, is essential for a tale to become tradable. Mathematics is a language, but one can speak in it only in the terms of a private, unpublished story. Because what it expresses cannot be witnessed; it can only be actualized. Knowing a theorem means to have lived up to encounter the arcanum it hosts with grace. It can only be talked about from afar, through anecdote, on the relation between two cyphers that are, ultimately, not to be deciphered:

“Thales’s geometry expresses, in the form of a legend, the relation between two blindnesses, that of the result of practice, and that of the subject of practice. It formulates and measures the problem yet without resolving it; it dramatizes the problem’s concept, yet without explaining it; it poses the question in admirable manner yet does not answer it; it recounts the relation between two cyphers, that of the mansion and that of the monument, yet it deciphers none of them.”^[38]

 

5th Iteration (mathematics is the circuit of cunning reason)

A theorem renders available certain techniques, because techniques envelop a theory. They are stable coatings that package the acts of archly reasoning in scenes of originality, in abstract conception. In order to take these practices and do something with them, in order to apply these techniques, one needs not know the theory that they envelop. But without knowing it, one doesn’t touch upon the question of originality. It cannot be separated from the pride of a craftsman who seeks to become masterful, in the sense of conquering his material without disgracing it. As Serres puts it:

What is the status of knowledge that is contained in a technique? A technique is always a practice that envelops a theory. The entire question—in our case that of originality—reduces here to a question of mode, the modality of this envelop. If mathematics springs one day from particular techniques, it is without doubt because of an explication of such implicit knowledge. And if the arcanum (the secret) plays a certain role in the tradition of craft, then certainly because its secret is a secret for every one, including the master. There is a transparent knowledge that resides hidden in the hands of a craftsman and his relation to stone. It resides hidden, it is locked in by a double bar; it remains in the dark. It lies in the dark shadow of the pyramid. This is the scene of knowing, it is here that the possible, the dreamt, conceptualized origin is staged and put in scene. The secret of the architect and the stonemason, a secret for himself, for Thales and for us, this secret is the scene of shadow plays. In the shadow of the pyramid, Thales finds himself within the implicitness of knowledge, which the sun is supposed to render explicit from behind, in the absence of us.^[39]

All things stated are artifacts, and artifacts conserve an implicit knowledge. Grasping how it is implied is the truly difficult thing, the impossible thing, because if one desires not to violate the secret, there will always be a remainder left. The circuit that can be established by archly reasoning cannot possibly exhaust its source. What reveals itself in scenes of originality, by abstract conception, is always impure. The universality of geometry resides in its application, and only there. In terms of purity, geometrical universality can never be born.^[40] In other words, it can never become physics, it can never be considered natural. Mathematics as language, on the other hand, allows us to consider all things natural. This is how Serres can claim that mathematics is the circuit of cunning reason, or archly staged scenes of conception. If originality is actualized in such scenes through theory, and if theory is transport and a theorem is a vehicle, then we can regard mathematical formulas as textual in a sense not unlike semiconductors are for electronics. This is indeed what Serres suggests:

Measuring, the direct or indirect field survey, is an operation related to application. In the sense, evidently, in which a metrics, a metretics, relies on an applied science. In the sense that in most cases, measuring constitutes an application in its essence [Wesen der Anwendung]. But most of all in the sense of touching. A unit of measure or leveling rod is being applied to a thing which is to be measured, it is being laid alongside it, it touches, and this as many times as necessary. A direct or indirect measuring is possible or impossible when such application is possible. Inaccessible is, hence, what I cannot touch, where I cannot lay the leveling rod, what I cannot apply my measuring unit to. In such cases, so people say, we must go from practice to theory, we must come up with an artfulness and devise a replacement for those sequences that are inaccessible to my body, the pyramid, the sun, the ship at the horizon, the riverbank at the other side. Mathematics were, so considered, the quasi-electric circuit [Stromkreis] of these cunnings.^[41]

6th Iteration (the real as a black spectrum)

However, to see in mathematics the quasi-electric circuit of cunning reason would be to underestimate the scope of practical activities. Because the established circuit is a bridge, archly, between tactility and sight. To theorize means to organize sight according to the quasi-tactility of a conceptual body that lives in the scenes that unfold on the stage of abstraction. Measuring puts two things in mutual relation, and a relation presumes a transport—of the levering rod, of the angle, of the things applied when measuring. There is an inexplicable intimacy between knowing and the problem such knowing lays out theoretically. Homothesis constitutes the stage of abstraction, and the homology—the variable equivalence—that can be expressed by the statements of homothesis belongs to the reality between product and producer. What is formulaically set up as equivalent is an invitation to read into what the formula states; it is not a question of addressing and answering. Reading mathematically means to stage a scene that supports trading the secret of the manifest body through scenes that are accessible only to an intellectual sense of sight. The anecdotes in which the origin of a theorem can be told imply a schema that lives forth in the dramatizations it supports. The schema, the optical diagram, can be traded only in written form. It keeps what is enveloped by practices through not explicating it. In proceeding like this, the schema demarcates something real, something stable and lasting that belongs to the manifest body one seeks to measure: its arcanum, its secret. And it demarcates this secret by treating it as an invariance that can only be conceived abstractly, by attributing it a measure, as a manner of how to proceed. The stage of abstraction is the theater of measuring—what is being measured, by dramatization, is the real as a black spectrum.^[42] From the point of view of the craftsman who seeks to understand more about the origins implied in his material, the material’s original reality resides in the shadow cast by the sun. It is the shadow that bursts with spectral information: “Knowledge of things resides in the essential darkness of manifest bodies, in their compactness behind its faces.”^[43] Knowledge about the real is natural not despite but only because it is conceived and born abstractly. It is impure because it was conceived within the happenings of confluent streams of geneses, whose pool of possibilities cannot possibly be exhausted. It is from the essential darkness of things that can be rendered apparent on the stage of abstraction, in the plays that unfold in the scene of writing, where knowledge of real things lies buried, Serres maintains. From its source springs the infinite history of analytical progress: “The body which can never be exhaustively described from analyzing its bounding surfaces retains in the safe depth of the bounding surfaces’ shadows a dark kernel.”^[44]

Remembering the stage of abstraction that supports real knowledge allows us to see the purity of mathematics instead of an ideality of representations. The purity of mathematics is constituted by nothing more and nothing less than the presumption that there be contained, within manifest bodies, ever more that can be explicated in theory. To see ideality in the geometrical forms, as Plato did, instead of assuming purity in the mathematical theorems, means to dislocate homothetics and homology into the eternity of the one moment that Thales arrested when he wished that time—the epitome of change—might speak about the solidity of the thing he faces. It means that geometry is conceived yet cannot be born. It means to postulate that there be no reality to desiring conquest, that technics be either divine fate (Prometheus, Pandora, etc.) or the stigma of decadence. It holds that revelation be apocalyptic, purifying, in that it clears the spectrums of recognition into the whiteness of virginity. This white spectrality, which supposedly allows us to recognize the identity of things as they ideally are, behind their disturbed appearance in actual existence, constitutes the idea of pure intuition. By insisting on the essential darkness of things, Serres may well sound like a worried prophet; yet it would be the prophecy of a worldly nature and a natural sexuality that is driven by the desire to conquest and master what is never intended as possession:

[But] when the moment has come and this postulation of the purity of geometrical form, inherited from the Platonic legacy, will die because nothing can be supported by intuition, when the theatre of representation has closed its doors, then we will see secrets, shadows and implication explode anew in the world beneath abstract forms, and before the eyes of surprised mathematicians—explosions which have been prefiguring long before these deaths. The line, the plane, the volume, their distances and regions will once more be viewed as chaotic, dense, compact […] entities, full of dark and secret angles. The simple and pure forms are not that simple nor that pure; they are no longer things of which we have, in our theoretical insight, exhaustive knowledge, things that are assumedly transparent without any remainder. Instead they constitute an infinitely entangled, objective-theoretical unknown, tremendous virtual noemata like the stones and the objects of the world, like our masonry and our artifacts. Form bears beneath its form transfinite nuclei of knowledge, with regard to which we must worry that history in its totality will not be sufficient for exhausting them, nuclei of knowledge which are profoundly inaccessible and which pose themselves as problems. Mathematical realism wins back in weight and re-adopts that compactness which had dissolved beneath the Platonic sun. Pure or abstract idealities will cast shadows once more, they are themselves full of shadows, they are turning black again like the pyramid. Mathematics unfolds, despite its maximal abstractness and the genuine purity which is proper to it, within the framework of a lexicon which results, partially, from technology.^[45]

Technology manifests, as implicit ideality, that whose theorems are mobilized in the representations of its variables and coefficients, representations which are dramatized in myth and transported through language. Technology is bursting with implicit knowledge. Every technology is a text that hosts an account given about a scene of originality, of abstract conception. And this, following Serres, is no embrace of mysticism.

Coda – Realism of Ideal Entities: Conceiving, Giving Birth to, and Raising Ideas on the Stage of Abstraction

“Language faces a truly boundless realm, that of the thinkable. It must make an infinite use of a finite stock of means, and it can achieve this through the identity of the power that engenders thought as one and the same with that which engenders language. Language is not to be treated as a once engendered something that is now dead. It is not an oeuvre (ergon), but itself activity (energeia).”^[46]

The nature of the universal, according to the perspective we owe to Serres’s reading of Thales, can be separated neither from concrete sensible reality nor from the conceptual reality of that which is only intelligible. The nature of the universal is real, virtual, and distributed equally much throughout the intelligible as the sensible. The presence of what belongs to no thing in particular insists as the noisy confusion between the two spheres, and is hosted in nature’s comprehensive and bursting quickness of all that grows and decays.

The core question that preoccupies me is in what kind of world we would find ourselves if we began to consider that through information technology, universal algebra is de facto constitutive for nearly all domains in how we organize our living environments today. Two things seem crucial: (1) we would have to assume that what we can calculate is not the necessary but the possible; and (2) theory must provide a basis for decision rather than relieving thought from the demand of “transcendental deliberation”^[47] – what many like to call by the name of “corralationalism”. If we regard mathematics (algebra) as a language, we must assume that ideas are essentially problematical and dependent upon clarification. As a consequence, reasonable thought alone does not liberate us from the responsibility of power and the associated challenging task of dealing with the questions of values.

Text written in algebraic symbols is not the scene of writing that hosts life-in-general; rather we might see in it the body of universal genitality. This body is the residence of the mathematical principle, which is host to all things generic and pre-specific. It governs magnitude, multitude, and value—symbolically. It is the master of all things that are most unlikely to ever happen or turn real. Universal genitality, incorporated in the principle of mathematics, is capable of performing incredible acts—like giving multitude an extension in time that is subjected to the fullness of space (Aristotelian ontology); or magnitude an extension in the fullness of time without having one in space (Dynamics); or it can give multitude an extension in an abundant plenty of space, together with a distributed-yet-collected extension in time (probability amplitudes in quantum physics). Universal text is the body of an infinitely wealthy principle, its content is arithmetic and its form is restlessly generous; and yet it cannot give without demanding: it demands mastership in logics and in geometry by those who desire to receive what it has to give. Universal text as the natural residence of intellectuality is also the collective body to think in. It is genealogical without originarily determined pureness; it is corporeal and yet arcane; it is natural in the sense of being sexed and gendered, yet impredicatively so: universal text is universal genitality.

The architectonics of formal ideality is neither constructed from ultimate elements nor does it grow according to ultimate morphological body plans, rather, we might say perhaps, it takes shape through blossoming. It cannot be decided whether the character of the principle that is master in this residence (mathematics), is a One or a Many. Rather, it is, symbolically, both at once: it collects and comprehends confluxes from many geneses. This principle, which masters the natural residence of collective intellectuality, demands of its subjects nothing more than reasoning in a manner that proceeds archly as well as utterly precise, such that it may provide auxiliary structures of symbolical stages for abstract thought to conceive and engender objective ideas. And objective ideas are present everywhere: Even the mathematical and formal descriptions of things chemical, physical, or biological, are capable of manifold representation. Matter that is informed can be assumed to exist in universal and original form as little or as much as this can be assumed of language itself. This inverses the legendary confusion of speaking in many tongues, which is said to have come from Babylon. While the Babylonian confusion usually exhibits that we have many names for the same thing, the informability of matter inverses the situation: we now have many things for the same name.

Hence, what I would like to suggest is a realist approach to the universal, which would consider it not as a space that gives room and passively hosts the extension to all things, insofar as they are pure and do not contradict each other. In a realist understanding, the form of comprehension that is proper to the universal is communicational, and its nature is vivid and of infinite capacity. Dialectics is virtualized by mathematics, if we conceive of it as a language that alloys atomism, alphabet, and idiosyncrasy. Unlike a notion of space that hosts the extension of things, which is supposed to be only giving without ever demanding anything, the communicational nature of the universal must be considered as being equally giving as it is demanding: it gives everything that can be the object of intellection, and it demands to be received, spelt out, interpreted, formulated, and integrated into the architectonics of its formal ideality. It is a consequence of such communicational nature that nothing that corresponds to it—nothing that can be called universal—can ever be owned. But at the same time, it is not actually real unless it is being conquered and appropriated, intellectually. All communicational reasoning in the terms of universal text is archly reasoning; it is not reflective or projective reasoning. The nature of the universal is self-engendering; it does not, properly speaking, ever cease to take place or actually happen as long as its demand finds response and respect. We may think of it perhaps as an intermitting point, a moment that resides in its own lasting, or as a circle that desires to comprehend everything that it encompasses.

 

^[1] Ibid., 232.

^[2] “If one were to understand by the birth of geometry the rise of absolute purity out of the grand ocean of these shadows, then we might as well, a few years after geometry’s death, say that it had never actually been born.” This is Serres’s answer to Husserl’s mourning in the end of his article. Ibid., 238–39.

^[3] Ibid., 226.

^[4] Michel Serres, “Gnomon: The Beginnings of Geometry in Greece,” in A History of Scientific Thought, 88–89.

^[5] Ibid., 88.

^[6] Ibid., 77.

^[7] Ibid. “All the cultural hegemonies of the world are impotent against this community and against the universality of this teaching.”

^[8] From ex-, “out,” + serere, “attach, join.”

^[9] Serres, “Gnomon,” 80.

^[10] Ibid.

^[11] Ibid. My translation here deviates from the proposed one, which suggests the following: “The world lends itself to be seen by the world that sees it: that is the meaning of the word theory. Even better: a thing—the gnomon—intervenes in the world so that it might read on itself the writing it traces on itself. A pocket or fold of knowledge” (86).

^[12] Ibid., 86–87.

^[13] Ibid., 86. See footnote 10, p. 725: “Algorithm: contrary to appearances, the word does not come from Greek but from Arabic and means a finite set of elementary operations for a computational procedure or the resolution of a problem”.

^[14] In these descriptions, I follow mainly the account given by James Ritter in his articles “n –1800”, 17–43, as well as “Measure for Measure,” 44–72.

^[15] Ritter, “Measure for Measure,” 62.

^[16] Ritter cites from the Rhind Papyrus, ibid., 69.

^[17] Ibid., 96.

^[18] Ibid.

^[19] Serres, “Was Thales am Fusse der Pyramiden gesehen hat,” 218.

^[20] Ibid., 219.

^[21] See Michel Serres, The Birth of Physics, trans. Jack Hawkes (CITY: Clinamen Press, 2000 [1977]).

^[22] Serres, “Was Thales am Fusse der Pyramiden gesehen hat,” 214.

^[23] Ibid., 219 As Serres literally puts it: “a multiplication of genetic procedures” and “the origin of geometry is a conflux of geneses.”

^[24] Ibid., 219–20.

^[25] In “Gnomon,” Serres writes: “And so it does not appear that the Ancients sought or thought of elements absolutely first or last: there are elements everywhere, in local tables” (112). He explains: “The term Elements, which translates into Latin and our modern languages the title used by Euclid and probably Hippocratus of Chios before him, originates from the letters L, M, N, in the same way as the alphabet spells the first Greek letters: alpha, beta, and the sol-fa sings the notes: sol, fa. The authentic title Stoicheia does indeed mean letters, understood as elements of the syllable or of the world” (111). And, further: “Again, what is an element? This mark, this line, the dash, the hyphen, in general the note, as these words were used by Leibniz. And in the plural, a series of these notes, a series generally grouped in a table or a chart of points and lines, in well-ordered lines and columns. As far as I know, the Elements of geometry also consisted of points and lines that we have to learn to draw. Today, as in the past, everywhere we see similar tables: the letters of the different alphabets, numbers in all bases, axioms, simple bodies, the planets, markings in the sky, forces and corpuscles, the functions of truth, amino acids […] Our memory preserves them so easily that they themselves constitute a memory in the triple sense of history—hence the commentaries—of automation and of algorithms” (113).

 

^[27] Ibid., 221.

^[28] Ibid.

^[29] Ibid., 219–20.

^[30] Ibid., 220.

^[31] Ibid.

^[32] Ibid.

^[33] Ibid., 229.

^[34] Ibid., 221.

 

^[36] Ibid., 221.

^[37] Ibid.

^[38] Ibid., 225.

^[39] Ibid., 223.

^[40] Ibid., 238–39.

^[41] Ibid., 215; my insertions.

^[42] Ibid., 232.

^[43] Ibid., 232.

^[44] Ibid.

^[45] Ibid., 237.

^[46] Wilhelm von Humboldt, Über die Verschiedenheit des menschlichen Sprachbaus, 1836, § 13; my own translation. The original German: “Denn sie steht ganz eigentlich einem unendlichen und wahrhaft grenzenlosen Gebiete, dem Inbegriff alles Denkbaren gegenüber. Sie muss daher von endlichen Mitteln einen unendlichen Gebrauch machen, und vermag dies durch die Identität der Gedanken- und Sprache erzeugenden Kraft. Man muss die Sprache nicht sowohl wie ein totes Erzeugtes, sondern weit mehr wie eine Erzeugung ansehen. Sie selbst ist kein Werk (Ergon), sondern eine Tätigkeit (Energeia).”

^[47] Andrew Brook and Jennifer McRobert, “Kant’s Attack on the Amphiboly of the Concepts of Reflection,” Theory of Knowledge (August 1998), https://www.bu.edu/wcp/Papers/TKno/TKnoBroo.htm.