Mathematical Anamneses, from Hermes I, by Michel Serres

A translation (February, 2019) by Randolph Burks of Les Anamnèses mathématiques from Michel Serres, Hermès I: La communication, (Paris: Éditions de Minuit, 1969), pp 78-112.

MATHEMATICAL ANAMNESES

By Michel Serres The philosophy of science has, since August Comte, at least in France, readily set up its project as a philosophy of the history of science. Whether this history be conceived as dialectical or stadial evolution, as rational or psycho-sociological genesis, as the terrain for an erudite or psychoanalytic archaeology, the great patronymics of the discipline are immediately recognized: from Duhem and Brunschvicg all the way to Bachelard. Inside the angle formed by these philosophies of history and formal logic (finally rediscovered), a certain synchronic epistemology risks being found to be slumbering. Whether it is so de facto or by full right is a question.

Besides an excess of pretension, there is no doubt a hidden paradox in again taking the history of a science that is nothing other than the excellence of the logos as the object of a discourse. In fact, the scandal doesn't merely reside in the place where words would expect it, but rather here: how is it possible (and so, by what condition is it possible?) to affect a mathematical truth with an index of historicity; how can invariants such as rigor and purity vary? If mathematics is a well-formed language, transforming this language is in appearance useless or contradictory. Yet everyone goes around repeating that a scientific truth has no value except in reference to the global system that contains it and makes it possible: an allegation that makes the most sense in the universe of mathematical discourse.

Here, quickly said, truth is only a certain relation that a sentence or a word maintains with its language, that a systematic atom maintains with its family, in short, that a system maintains with itself. All paradox ceases when the history is no longer seen as the sequence of manifestations of a pure logos, but rather as sequences of (meta)morphoses of a logos referring to itself – mathematics being the science of this self-reference, rigor being the science of this application. In the avenue of mirrors Lautréamont talks about, the light rays' course, continuous or interrupted, remains to be followed. This open avenue is the very history of mathematics, the history of a language whose words strictly respond to each other, of a language endlessly translated into new but homologous languages, the history of systems that are self-referential, therefore closed, referring to other systems that are therefore open, but referring to new systems similarly mathematical, therefore closed..., the history of forms taking on meaning in a system, forms therefore involuted, but sometimes and as though all of a sudden taking on a different meaning from the native one, going beyond their internal self-reference and therefore evolving outside the system, like a pathological outgrowth, towards a new internal systematic reference, like a lost ray in search of its mirror..., the history of truths always in quest of a closed universe that would seal them over themselves, that would give them possibility and existence, until the demand for rigor would render the internal application untenable and break open the door to bring about a reference that would be broader and better closed over itself... Hence the implacable dynamism steering of itself towards the universal in act, totally open and totally closed, which is the end, always remote, of their history.1 1

Hence the wanted condition: on the one hand, the historical truth of idealism is, quickly said, physics; as soon as a

Serres: Mathematical Anamneses We can stop along this path for a local examination; we can try to travel it for a global synopsis. In the first case, it's fair game to choose a system and to see how it reduces historical questions; in the second one, it is good strategy to create models to explain the succession of forms and systems that are victorious over the confusion of languages. A character awaits us at the crossroads of this always the same and always different path: the Meno's little slave boy.

My best experience in these matters is that of a failure, which I can better confess than teach. It concerns Leibniz's philosophy, which I will use as a paradigm or prototype for a beginning of analysis.

And, first of all, it's a question of a good example, since his work is indistinctly a systematic philosophy, a scientific encyclopedia2 and an erudite's doxographical accumulation. Having posited this, I believe I can put forward that the metaphysics of Pre-Established Harmony is a supple, indefinite and complicated system of translations of theses into each other, theses sometimes (often) having a scientific character, it being understood that the word “science� is being given the broadest sense, that is, the encyclopedic sense. Better or more profoundly, it's a question of a network of correspondences ensuring the universal possibility of every translation of every thematic into any other and conversely.

If one gives up recounting, relating or repeating, the problem of explication here is immediately complicated, redoubled. Should one for example translate some thesis into the mathematical language characteristic of the author, it is brought to demonstration, something Leibniz announced and desired; although possible and sometimes realizable, this explication technique is not sufficient: for it errs on the side of lateralization or scenography, like moreover every other explication referring to a regional science, dynamics, to give other examples, or the theory of law. This is explication via reduction of a system to a local region; this is reducing to a single language the very theory of every possible passage of any language to another. Consequently, and paradoxically, demonstrating isn't explicating; on the contrary, it's implicating; it's implicating the theory of translation into a positive language; it's enveloping the very theory of explication, of the explication of what a content of knowledge envelops that's implicit; now, for Leibniz, the implicit of a region is precisely the totality of the system. In other words, the Leibnizian system explicates itself by endlessly applying itself onto itself. For example, the theory of the point of view is quite easily translated into the geometrical and perspectival language of conical sections, a translation that allows us to bring a thesis, let's say a philosophical one, to demonstration; but conversely the theory of conics envelops the idea of harmony, the problem of error (theory of shadows), the principle of continuity, the questions concerning infinity, the existence of an invariant in a series of metamorphoses, the establishment of a classification of natural beings, etc., and of course the question of the point of view, of perception and of expression in general. The first technique, a demonstrative one, is an implication; the second one is a development, that is to say, an

2

science attains maturity and folds reflexively over itself, it suddenly expresses its philosophical truth; and therefore, contemporary physics stages the ego as the condition of possibility of its own constitution as science. It becomes aware of an I which had never stopped being present to it in its historico-philosophical context. On the contrary, for mathematics: having attained maturity, it expresses the truth that, from its Hellenic dawn, has never stopped being its own, namely, the most radical bracketing possible of the subject. In other words, the condition of possibility for the application to the world resides in the transcendental field in subjecto, whereas the condition of possibility for the application onto itself resides in the very field in which it is carried out. We mean by this that mathematics is a transcendental field as such, and quasi objective.

A scientific encyclopedia time hasn't, as though by a paradox, progressively consecrated to obsolescence, but rather has, through continuing recurrence, rediscovered and reestablished its living presence.

2

Serres: Mathematical Anamneses explication: the system is at once to be explicated and explicates (explicandum et ultimum explicans).3 Whence it happens that Leibniz's philosophy is written in a universal language endlessly translatable into every positive language of the country of the Encyclopedia, or better, whence it happens that it is constructed as a multilingual dictionary with several entrances. In other words, pluralism is not merely ontological and substantial, it is, let's say, structural. Couturat, Russell and others had tried to write the grammar of this language as syntax and morphology; the semantics of the system remained to be established, that is to say, the dictionary in question remained to be constituted. Even though it may be complicated or almost infinite, this task was possible.

The difficulty I alluded to just now presents itself right here: for as long as one remains in an ideal system, and in particular in a system that includes the Combinatory Art as an element, the architectonic disposition of things only requires a simple patience and an ordinary scientific technology. As soon as one has a piano, it's easy, at least in principle, to draw from it as many harmonic and melodic sequences as you like. Leibniz never described his system as halted, ideally frozen or congealed. On the contrary, he had the sharpest awareness of epistemological becoming, of heritage and the encyclopedic tradition, of scientific prospection in general: we are living, he said, in a certain infancy of the world, in quodam mundi infantia; further, he readily sacrificed the rigor of concepts in favor of their capacity to succeed, in favor of their fertility, their effectiveness for “gaining ground”, according to his express words. In short, he put the Ars inveniendi above the Method of Certainty.4 What is more, Discovery for him wasn't tied to a tabula rasa erased of precursors, but on the contrary to a methodic accumulation of tradition and to its reactivation: finding the gold veins in the sterile rocks.

Consequently, the Structural Dictionary is not merely an ideal and formal architectonic; it is not of the synchronic type; it takes into account the diachronies of all the languages it mobilizes. Hence we find present there the history of the sciences, the history of languages – in the regional sense of philology –, the history of institutions – politics, diplomacy, law –, the history of religions, ethnology (Novissima Sinica) and mythology, the natural history of living things and the archaeology of the earth (Protogaea), or geology of deep sedimentary layers. The Structural Dictionary is no longer merely the basic instrument of thematic correspondences in general or of semantic inter-expressions; it is also an etymological, genetic and prospective dictionary. The atoms of meaning there are both formal and in formation.

Etymological, this goes without saying since:  every evolution flows back to a pre-establishment which is translated regionally as preformation, pre-existence, predestination, predemonstration, predetermination, etc.;  since in every discipline invention and project are based on the search for primitive elements, translated regionally into primitive numbers, a primitive active force, primitive ideas, primitive words, a primitive or prehistoric language, an alphabet of human thought, etc.;  since all in all the word originatio (De Rerum Originatione Radicali) never meant anything but etymology, the word radicalis designating the roots of words.

3

4

Hence explication via mathematics is valid and insufficient: we must explicate mathematically why mathematics is only one explication among other possible ones.

Or rather, it was Leibniz who discovered (or rediscovered) the idea that rigor has heuristic power, that rigor leads to invention.

3

Serres: Mathematical Anamneses Genetic, this goes without saying once again since any element whatsoever envelops its past, its present and its future, in the form of an original divine inscript: true of the monad, this theme is translated, invariantly, for the activity of knowledge in the understanding (omniscient passive memory – continuous activity of rediscovery), for the evolution of vital germs and of the organism (involution – metamorphosis), for the historical adventure of the individual (Caesar, Alexander, Sextus), for the supernatural destination of the sinful soul, etc., but also for the very content of our knowledge (the theory of progress of the Enlightenment).

In short, the formal atomic element that is translatable everywhere in the system is also a summary of history, enveloping its radical origin, the law of its evolutionary sequence and the horizon of its finality.

At any moment of its serial development, it is in principle possible to read on it, as on an erased palimpsest, its forgotten origin, which is the key to its end in the kingdom of ends. From which you can draw all the models of history you might wish, models to which I shall return elsewhere: linear series unfolding to infinity, in monodromic style, circularity, recurrence, spiral models, decline, static immobility and so on. Since time is only an order, all orders are conceivable.5 The universal possibility of translating themes is joined by the universal possibility of making them vary in themselves so as to account for their formation. So history is taken up again by formal thought, which gives it the range of its meanings, that is to say, the totality of conceivable meanings. As seen by the system, history has every meaning; as seen by history, the system has, at the same time, one meaning and an infinity.

This is how things are, quickly said, with a system whose value as an example in these questions resides in the high perfection of its architecture and in the exceptional and totalizing welcome it reserves for history. It is the science of science, the history of science, but also the science of history and the history of history. In a certain sense, its value as a paradigm is not so different from the value that can be given to Euclid's or Bourbaki's Elements, which can also be considered as quasi-perfect ideal architectures, but equally as summaries of history: encapsulations of heritage, a synchronic crosssection of the idealities united by the system – cross-sections in permanent danger of obsolescence or already obsolete –, an opening of meaning for mathematicians to come.

Hence the difficulty, which is due to the way we have to choose to apprehend these exemplary systems, of apprehending them historically. How for example are we to date a mathematical concept in Leibniz? Or even Bourbaki? It has at least three ages: the age of its appearance in the mathematical tradition, the age of its reactivation in the system that gives it a new meaning, the recurrent age of its power of fertility, which we can be the judges of now. From the point of view of ordinary – chronological – history, it's the first one that counts; from the point of view of truth in the synchrony of the system, it's the second; from the point of view of the complete diachrony of mathematics, it's obviously the third. Hence there are at least three historical meanings for any ideality: its birth meaning, thenceforth sedimented, naturalized; the set of its meanings at each reactivation, which takes charge of this set again for a new value by naturalizing the preceding reactivations; its recurrent meaning for the retrograde judgment of the most recent restructuration of the mathematical edifice. The last meaning is uniquely its scientific truth.

So the norms of historical faithfulness explode: if I approach Leibniz's mathematics, for example, equipped with the recurrent judgment of contemporary algebra, I confer its truth to it; in other words I 5

The reduction of time to an order permits the application of combinatorics to history.

4

Serres: Mathematical Anamneses filter its teleology, but I am unfaithful to a certain history, let's say the history of ideas as the catalogue of the results of the day. Moreover, the teleological truth I confer to it is limited to my current reference: by covering over the past meaning I also risk covering over an inconceivable meaning to come. I interest the current scientist by giving him a precursor, but I'm not a historian in the accepted sense. On the contrary, if I approach Leibniz's mathematics equipped only with synchronic references, I am certainly a faithful historian, but I ignore the essential, which is the finalized truth of mathematics: faithful to sedimented history, unfaithful to science as history, unfaithful to the truth, which is nothing other than teleology. Hence this principle of uncertainty or indeterminism, so tricky to reduce: if I speak true regarding Leibniz's meaning, I don't necessarily speak “true� in every case; if I speak true, I don't necessarily speak true regarding Leibniz's meaning in every case. I'm forced to collide head-on with either the mathematician, for whom the historical concept is burdened with sediment, or the historian, for whom the true concept is sometimes nothing but a fossil. In brief: either I know the position of the concept and don't know its speed, its particular motion, which is its veracity, or I know its speed and don't know its position. This indeterminism has its limit in the question of error, which the historian is forced to reactivate as a situatable truth, which the scientist is forced to cover over and forget. As historians we interest ourselves in the dross of Galileo; scientists interest themselves in the genius intuitions of Messier which had no meaning in his era. Historical truth can become dross; dross can be reactivated as truth. Hence the limit: if I speak true regarding Galileo's meaning, I might possibly speak false; if I speak true, I might possibly speak false concerning Galileo's meaning. In return, this indeterminism defines the history of science, not as a continuous tradition, but as an always interrupted, discontinuous framework.

It is possible that all this is due to the exceptional situation of the history of the sciences and, as we have recently learned, of the sciences themselves, as the contact site for historicity and ideality or, to speak generally, for two modes of beings that meet two completely different norms. The principle of indetermination is the first step in the exploration of this contact site, an exploration achieved, as we have just suggested, by bringing two normative references closer to this site, one of which would stem from historicity and the other from ideality, but in giving this latter an original historicity. In fact, there is tangency because science itself is a history. The principle therefore was indeed a principle for the history of science. Can we turn this principle around by exploring this contact site from the point of view of science itself? Let's consider a system of idealities at a given moment (for example, Bourbaki's Elements in 1966) so that each concept it mobilizes is cut at the same instant of reactivation.6 The contact here is well established; I make a synchronic cross-section in the ideal system for the actual time of history and for the ideal time of reactivation. Let's consider this cross-section as historians in the usual sense (not for recurrent diachrony, something impossible,7 but for sedimented diachrony): then a quite remarkable principle of indetermination presents itself; for I date the idealities of fibered, foliated, scattered, chaotic, compact spaces from the 1940s; from 1955, the ideality of categories; the ideality of sets from the 19th century; the ideality of functions from the 18th; that of integration from the 17th; that of the diagonal from the 5th century BC; that of addition from the first millennium and so on.

The temporality proper to the system is homogeneous; the temporality proper to the atoms of the system, if they are only taken as sediments that haven't been reactivated by the restructuration in question, this temporality is indeterminate; it is rent, chaotic, and, seen from outside, aleatory.

The contact site for the historicity proper to the sciences, as a system of idealities, and for history in the standard sense is therefore such that it is in one sense subject to contradiction and in the other 6 7

Translator's note: cut=coupĂŠ [the past tense of couper, to cut]; cross-section=coupe.

Something which ought to do away with the preceding indeterminism since the entire question would be present.

5

Serres: Mathematical Anamneses indeterminate. The situation is quite exceptional, but it is paradoxical. These paradoxes, and those that are to follow, form the deep reason for the disinterest shown by the scientist towards the history of science8 as a catalogue of successive results or as an evolution of ideas; for he inhabits a system whose catalogue is no longer anything but fossilized fallout; he lives an original teleology whose historical evolution is a definitively exhausted possible insofar as it is an instituted chain that only an institutor is to revitalize so as to transmit it as the tradition of a world which, as in the Meno, would be forgotten without him. In other words, our history of science (that of Montucla, of Mortiz Cantor, etc.) is a history of professors of science which aims to ensure the transmission of a communication that the scientist as inventor instead aims to reassess; it's a history we are seeking to make connected, to make continuous, by filling in its cuts, while the inventor-scientist chops it up and makes it discontinuous: we are trying to prevent communication from being broken, whereas inventive activity breaks it of itself.

The union of inventors of a proof doesn't have the same language as the union of the transmitters of the proof. This is verified in our time in a very acute historical experiment. The traditional history of science projects the always recommencing disruptions of anterior orders, the always new combinations of reversible sequences, onto an invariable and irreversible linearity (onto a tradition).

In the final analysis, three types of history would exist: 1) the history of science conceived as the accumulative totalization of the tradition, as the gathering of the totality of documents, whose ideal would be the absence of loss, both in collection and in communication along the ordinary diachrony. This would be the connected history of the professors, one and totalizing; 2) recurrent history, built upon the latest truth, that is to say, the truth. Only the most recent of 8

While I'm dealing with the question of this disinterest, I'd like to note that it is perhaps less interesting to ask oneself the question “which history of science interests or doesn't interest the scientist?”, than the question “which scientist is interested or not interested in the history of science?” So we could at least roughly distinguish two types of inventors: a) The inventor who continues the path in the naturalized sequence of the preceding results: this person only needs to reactivate the temporality proper to the chain he is working on, from the particular axiomatic origin to the successive theorems. b) The inventor who promotes a global restructuration of the system and who needs to reactivate the entirety of the tradition. This person is generally a historian and needs an enormous doxographical culture; even if he writes a historically false history, he writes a teleologically true history (e.g. Leibniz, Chasles, Bourbaki).

Consequently, there are as many histories of science (all different) as there are globalizing scientific inventions. In other words, for each restructuration of the system there corresponds a different type of totalization of the tradition, a different type of teleology taken up again by a recurrent judgment. The same relation exists from one history to another then as from the new science to the preceding one, that is, the precise relation of history to prehistory, to an era for which the new language hadn't yet been invented or written. Euclid's geometry is for us as prehistoric as Egyptian surveying is in relation to the Greek miracle.

It therefore seems inexact to speak with Kant of an “attempt from which the path to be taken onward could no longer be missed” (Preface to the 2nd edition of the Critique of Pure Reason [Bxi]) or, with Husserl, of affirming that “geometry was born one day and from then on remained present as an ancient tradition” (Origine de la Géométrie, Derrida, p. 176).

Or else this tradition is present as the erased layer of a palimpsest in the same way as the forgotten languages preceding Thales. There are as many origins of Geometry requiring the effort of birth that was achieved by the mythic Thales as there are of the histories I spoke about: Desargues, Galois, Cantor and Hilbert took up radical newnesses in linguistics, writing and the promotion of idealities.

The origin is infinitely withdrawn to the left; it is infinitely moved forward to the right, as end and τέλος [telos], but it is also and above all in some points or other, situated points, of ordinary history. At the limit, it would be in every point (if in mathematics the inventor is always of the second type): there would then be as many histories as one might like.

This situation is again paradoxical, but it explains, I believe, many things. It rediscovers the Parian marble dating system, by a fixed-mobile reference starting from the present.

6

Serres: Mathematical Anamneses these uchronia via selection would be selected. This is the history that every restructuring invention of the system trails behind it. There are a plurality of them, and their principle attribute is to be filtering.

All things considered, the set of these histories is presented as a succession of filters placed on other filters. As history, the system here is different from totalization. It is more selective than accumulative;9 3) the history that science itself is as original movement, as the indefinite formation of a system.

Of course, the diversity of these types of history corresponds to diverse conceptions of temporality.

Hence the depth of Leibniz's solution: reducing time to an order and considering the preceding indeterminations as the possibility of a range of solutions. Systematic space restores all possible chronic lines.

It is necessary then to take the question by the other end: instead of going from the description of a system to the diverse possibilities of historical projections, going from historical descriptions to the possibility of projecting them in a system. Let's try then to recount a single and totalizing history; after that let's try to put successive filters on it, sub-engendered by different systems.

Plato asks: where is the square, where is the diagonal? Not on the soil, not on the sand I'm writing it on. It's a form in the heaven of forms. We are no longer asking the question: where; we are asking the question: when. At which moment, during which epoch did the Meno's diagonal occur as the pure form envisioned by Plato? What does this question mean, this question that leads us to substitute the changing heaven of cosmogony for the eternal motionless heaven of cosmology?

And so, there once was the square of Pythagoras, a mythical blazon bearing around its neck the diagonals of the pons asinorum. Then came the square of the crisis and its irrational diagonal, shipwreck in the absurd. Euclid conceived the square again in a coherent universe. There were the squares of Archimedes, those of quadratures, and the imaginative square of those who dreamed to cover over the circle with it. In ordering the plane with axes of reference, Descartes paved it with a network of parallelograms which, very quickly, transformed into a paving of squares. At the same time Arnault, Pascal and others had at their disposal arithmetic squares, magical squares, magico-magical squares, soon satanic ones. The old logical square of minor logic reappeared with Leibniz, who divided concepts according to this form, endlessly iterated, a new model of dichotomy. Soon, algebra would know the squared determinants whose diagonals are sometimes remarkable; it would manipulate matrices, sometimes square. The calculus of probabilities can no longer do without Latin squares. A day came when the diagonal became again, in geometry, what it ought never to have ceased to be, a vector. The already old combinatorial topology called the archaic square a Jordan curve, homeomorphic to a circle, to an ellipse, to every closed curve. Cantor's methods ended up attributing the cardinality of the continuum to the set of the square's points through equinumerosity with the set of points on the segment (0, 1). At the same time, diagonalization became a classic method in algebraic geometry, in algebraic topology, even in set theory. And now, the diagonal and the square are schemas in the sense of the new algebra, or graphs in the sense of graph theory.

Upon inspection, the historical variation is far from being complete, but for the moment this variation suffices to let us see the quasi chaotic becoming of an ideal form; so chaotic, moreover, that no mathematician would accept seeing a history there; to his eyes, that is, to the gaze of truth, it is never or almost never a question of the same form; or if one does accept, roughly, seeing the same form there, it 9

The opposition between these two types of history, between accumulation and selection, naturally accounts for the indetermination indicated above. It's the opposition of necessary loss and the absence of loss given as an ideal.

7

Serres: Mathematical Anamneses is never the medium of the same thought.10 The Platonic firmament is the seat of a becoming for which the new problem is to know what its modality is. Can one imagine a model, or models of the evolution of a pure ideality? Is this monograph possible? Is it the monograph of the same graph?

The evolution complicates to the point of being chaotic. An atom of form has neither the same situation, nor the same weight, nor the same meaning in any system punctuating the diachrony. Each synchronic cross-section sets up such a redistribution, implements such a restructuration, that the form being followed is, here, a first-principle element, there, an abandoned bit of dross, there again, an archaic sediment taken up again, reintegrated, reactivated by the generalization. Is it a question then of the same form, or of a form that's always other? In general, is the historicity of science continuous or discontinuous? In both cases, what is its meaning?

We are familiar with the story of the Meno, the recreation by an ignorant of a demonstrative sequence, which is said to be an anamnesis. Thanks to the geometric chain, communication is reestablished with a forgotten world. Beyond the anecdote's autochthonous signification in Platonism, is there a way take it seriously within the context of our questions? It's because it brings several types of temporality into play: a tear, first, in the tradition, then a re-established continuity; a recurrence, first, a return, then a teleology re-established in such a way that the teacher and the ignorant are together in a new quasi circular temporality, endlessly repeatable.

This Platonic situation is an ordinary mathematical situation. Let's reread for example chapter V of Le Rationalisme appliqué: Gaston Bachelard takes the Pythagorean theorem up again in the contemporary language of group theory. This is once more the situation of the Meno, strengthened by the similarity of the problems. Bachelard's text gives existence again to a geometric imagery forgotten by the theory of structures; it explains the historical situation of a lost geometry by means of a new priority; it unearths a covered-over origin thanks to a recovered origin; it rediscovers an archaic world as a marginal consequence, as a trivial technological model of the new world.11 But the interest of this text redoubles if one becomes aware of the reversal of the Meno's situation: here Socrates is the ignorant, that is to say, the traditionalist, he who knows Pythagoras and knows nothing but him, he who has “forgotten” group theory. But the scientist, he who knows structures, has precisely “forgotten” the Pythagorean theorem; and he asks the ignorant to forget the traditional idealities as quickly as possible.

Bachelard addresses the Socrates-ignorant: “If some Sphinx one day posed this riddle to you, don't get lost in the twists and turns, in the dark jumble of diagonals, instead,” citing Peer Gynt, “make a detour”; it would have been better to cite Galois: “jump with both feet over calculations,” jump with both feet over the Platonic graph, forget the world we were formerly supposed to remember, tear up traditional continuity, and this forgetfulness will lead you to a more distant and more deeply buried origin, to a new world still erased by forgetfulness. Hence again several types of temporality: the discontinuity of inventive temporality is deeper than the continuity of the tradition; the idea of group is anterior or posterior to the geometrical theorem; we don't know which is first, teleology or the judgment of recurrence. Reactivating a sediment is designating it to be such: this is what the Pythagorean theorem was, this is what it ought to have been, this is what it ought never to have stopped being, this is what it is, this is why it is no longer anything: the resurfaced Pythagoras explains why he is dead, why he is no longer present except as a shadow. From the Meno to the Dialogue of the Dead.

It is easy to multiply examples: easy to explain how Schwartz's theory of distributions is a Dialogue of the Dead with Leibniz and his insane idea of real order derivation, dead yesterday, living today, 10

11

Or better, of the same system. The form is an atom for a system. Hence it changes meanings with the system that contains it and makes it possible: a word changes meanings as soon as language changes.

Translator's note: rediscovers=retrouve; covered-over=recouverte; recovered=recouvrée.

8

Serres: Mathematical Anamneses unlike Pythagoras; to explain how Hilbert enters into communication with Euclid, but how current mathematics presents the Euclid-Hilbert diachrony as archaic and rends this connection again. This is still the adventure of the diagonal: at the dawn of geometry, the triangle was considered to be the figura simplex of space, after the segment and the angle. Hence the traditional richness of its analysis, via ultra-elementary triangulations, by means of bisectors, perpendicular heights, medians, right bisectors...

Everyone knows how the Timaeus triangulates the cosmic elements. Hence again the Pythagorean theorem, among the first expressions of the metric, of metric space. Supposing that Euclid and his predecessors had considered the triangle to be half of a square, or better, half of a parallelogram: they would have immediately been led to the vector, that is to say, to the structure of space as vector space.

Here we are again at the origin, and we're taking up a good piece of history again: the point, the segment, the angle, then the open triangle (three segments with a common vertex, part of the parallelogram) – and not the closed trilateral we call the triangle –, from which vector addition is drawn, by components and resultant, which in return causes the in its turn first-principle idea of the vector to rebound onto the idea of the segment and the idea of the null vector onto the point, and so on: the structure of vector space is little by little unveiled in a primary simplicity and over the course of a spiral temporality passing through the origin many times, and which I can only bring to light again through the reverse direction of my recurrent judgment. The questions of norm and scalar product coming after these elements, the Pythagorean theorem is pushed far back along the chain as a trivial application. Hence the historical judgment: by proceeding in this way we could have saved more than twenty centuries of hesitations and superficial analysis of space. Everything happens as though we were covering over the ordinary tradition in order to situate ourselves upstream from the Greek origin.

The metric diagonal was historically lived as a drama of the irrational and the death of pure thought: it is lived by us as what could have been the first step of a higher rationality than Euclid's, so much higher that the former pure is impure, mixed, poorly analyzed. So the drama changes camps: it's Euclidean rationality that becomes transhistorical drama, it's the Greek miracle that becomes bad luck or infelix culpa; the idea of vector space forces me to forget, to reduce an entire diachrony, an entire history, which is no longer for clear thought anything but the drama of a blindness. The idea of vector space causes me to jump with both feet over the axiomatics of Euclid-Hilbert, in whatever sense I may take it.

Here again recurrence rends, disconnects traditional communication, which I would no longer be able to follow except as a sedimented cultural layer. The history of this science is no longer anything but the history of a certain mode of nescience, of a certain modality of non-knowledge, of a certain type of impurity. The reversal of teleology is patent, however far recurrence may flow back: the diagonal was suicide and shipwreck; it ought to have been a birth, a resurgence, the rebirth of a higher and more profound geometry, its very origin, through the liminary scissiparity of the metric and the vectorial.

This example is ordinary; it expresses the habitual situation of mathematics as living movement.

Let's begin the same history again: let's no longer base the recurrent judgment on group structures or vector space, but rather on topological structures. We are led back to the origins: not to the logical or historical origin, but to the fundamental conditions for the constitution of spatial idealities. Hence idealities, designated by Ideen... I to be morphological, are discovered at the base of geometry, not in an intentional style or in the archaic terrain of pre-geometry, but in an already thematized course, but in geometry itself. This is because mathematical thought already knew how to invade, during the very era when Husserl was writing, the idealities of round, notched, etc. – Jordan curves, Riemann surfaces, spheres equipped with cross-caps, etc. – before consenting to being given the pseudo-original instruments of the Pythagorean metric; this is because it knew how to erase the historical confusion of pure mathematics and the metric, that equivocity constitutive of the tradition that led philosophers to 9

Serres: Mathematical Anamneses believe they had freed themselves from the mathematon as soon as they had succeeded in freeing themselves from the metric. Through this retro-analysis, geometric thought discovers a new purity that owes nothing to measurement, anterior to measurement, and again suspends twenty centuries of equivocal tradition, perceiving them to be impure and confused, technological and applied, in brief non-mathematical, recovering them as absent and missed (contrary to Kant's and Husserl's terminology). It again reverses our vision of the origin by turning the miracle into a scandal. How did the tradition manage to take root right in the middle of the trunk, in an arbitrary site – miraculous because arbitrary? It's a miracle, that is to say, chance and accident, that the Greeks were able to jump onto a moving train at the moment when everything had already been decided, when the concepts had been overdetermined thousands of times – not a miracle of ultra-elementary purity, but a miracle of having designated a complex and mixed ore as being pure. Topological regression necessitates forgetting the tradition and remembering a spatial constitution covered over by the Greek miracle, covered over by the equivocality of the Greek miracle; this regression suspends traditional language as ambiguous and practices liminary dissociation of non-metric purity and measurement. Once again, the entire history of this science is nothing but the history of an impurity, that is to say, of a certain type of non-mathematicity.12 So mathematics is situated in a trans-historical dialogue – in the direct and (or) opposite direction to the Meno –, in a continuing dialogue with an ignorant traditionalist scientist, that is to say, with the historian of his own science, with a doxographer of what goes beyond the doxa, in order to forget some knowledge and remember a preliminary non-known, in order to make a decisive choice between reactivations and coverings-over. Likewise, it is doesn't matter that Pascal reinvented Euclid, as is recounted – which is a historian's myth at least twice over –, but it does matter that he re-engendered geometry starting from deeper priorities, which were Apollonian and were to become Desarguian: hence the choice between several forgettings and several rememberings. Thus all won ground illuminates or occults the history of science, in semi-aleatory rhythms: the current invention invents precursors, or sediments confusions. It isn't surprising that traditional history is indeterministic since it is post-ordered to an unforeseeable teleology. Even more, it is post-ordered to the indetermination I pointed out above: for the complexity of the system, which is the true reference of the judgment of recurrence, makes it difficult to distinguish between the traditions and origins it is vital to cover over and the origins and tradition it is urgent to remember. I would like to designate this difficulty as the living focus of mathematical historicity in general, as the site where connections are tied, where impure adherences, destined to become sedimented, are cut, in brief, the luminous point of invention.13 Here the mathematician never ceases suspending the tradition and returning to the origin (logical and constituting at the same time), or covering the origin over and reactivating the tradition, never ceases cutting and (or) connecting durations intersected in every conceivable manner. Here, the inventive mathematician is master of time and history; he invents the time of his science and, by that very fact, 12

13

This bracketing of the tradition of course includes non-Euclidean geometries as the final achievement of the metric in general.

Mathematical invention is what remains of a bet of the imagination and the counter-examples created against it. It is the residue of conjecture and critique, of dream and error. This description is not psychologistic: modal logics analyze this state of affairs marvelously. Necessity there is given as one time possibility and two times negation, what cannot not be.

If one displays logical definition in genesis, it remains to establish the possible and to destroy the counter-examples that destroy the possible. Conjecture remains one time and critique two times. Imagination plays, of course, the first possible.

It is interesting to see Leibniz for example invent an art of inventing inside the core of a metaphysics supported by modal logic, that is to say, inside the logico-metaphysical square: possible, impossible, necessary, contingent. There is therefore a genesis of necessity, which is the art of inventing in and through the rigorous.

10

Serres: Mathematical Anamneses the time of the history we're trying to take up again after him. Like Leibniz's god, he reads on a formalin formation ideality the occulted past, the active present and the possibles; he applies teleology onto recurrence in the focal point I was speaking of; in a system that's a network whose every element is a knot of anachronic diachronies, he is free to cut or retie: from the Dialogue of the Dead to the kingdom of the Fates. The taking charge of mathematicity, the responsibility assumed for purity as living becoming, imply an original, exceptional and free attitude with regard to historicity. Not only is every promotion of a form reform of temporality or constitution of a new mode of history, but above all the ahistorical character of pure form causes it to evolve in a time that is the projection of every imaginable modality of temporality. Ahistoricity is discovered not as the absence of time but as the fusion of every possible time: times that are unpredictable, determined and overdetermined, irreversible and reversible, finalized and recurrent, connected and always rent, referred to one, two, a thousand origins, times that are dead, forgotten, taken up again, accelerated in a lightning fast manner and so on.14 That fact that there is a history of ahistorical idealities cannot be understood unless a pan-historicity is conceived, a complex temporality, finely fibered or layered. In a certain way, the outline thematized by Husserl in the Krisis is enveloped by mathematics, as a particular or simplified case: mathematics is necessarily always in crisis and always in the process of resolving it. I will no doubt have to return to this point.

We must now jump with both feet over the examples and try to reconstitute, starting from the simple, the complex and intersected entanglement of the diverse modes of temporality they present. I cannot devote myself to this examination, whose naivetÊ will have to be pardoned, except by the method of models. Likewise, faced with the spatio-temporal complexity of our information about the world – that world the Greeks precisely considered to be eternal –, the cosmologist attempts to create models that account for the maximum of phenomena.

We have up to now encountered four basic concepts: the historicity proper to the sciences (mathematics) could be connected and (or) discontinuous; it could be read (with reservations regarding the question: who reads it in these ways?) in the direct direction of teleology or in the opposite direction of recurrence. There would be, as a first approximation, four types of elementary models: direct and recurrent connected ones, and direct and recurrent non-connected ones. What states of affairs do these models account for?

CONNECTED

NON-CONNECTED

DIRECT

Connected Direct

Non-Connected Direct

RECURRENT

Connected Recurrent

Non-Connected Recurrent

1. The direct connected models are both the traditional models and those of the tradition.

They have an interest in expressing the following well enough: a) the temporality of deduction or of rigorous linked sequence, in the style of Descartes.

14

Including the possibility of rewriting the Uchronia of mathematics a number of times: a conversation about the plurality of forgotten worlds.

11

Serres: Mathematical Anamneses It's impossible to skip a link along this uninterrupted linear process; however you start, “this path can no longer be missed.” The speed of propagation along this chain is variable, and can be lightning-fast, as can be seen with reasoning by recurrence. But this is not the form of this temporality that directly interests us here. b) the form of educational communication, of the perfect transmission of information.

The term “mathematics” here takes on its first sense of μανθάνειν [manthanein], to learn, to have learned. This is because mathematics provides the first example of an almost perfect communication, of information that's univocal at transmission and reception. This is so true that it isn't forbidden to think that its very origin lies in a dialogue in which two interlocutors fight together against the powers of noise, that mathematics is established from the moment the victory remains in their hands. So it is natural for Platonism to present a philosophy of the pure mathematon and a dialectic at the same time, this latter term taken in Benoit Mandelbrot's sense. I have attempted to show this above15 by defining the role of a third man, or of a third person jamming the dialogue, whose exclusion the entire Platonic effort aims at practicing. The dematerialization described by Mugler would reduce then to this exclusion, which would be a condition for pure thought, in transcendental intersubjectivity. Let no one ignorant of geometry enter here. With this posited, mathematics is easily defined as the world of communication maximally purged of noise and, consequently, as the world of traditionality subject to the minimum of loss: the communication route is essentially connected everywhere and without interruption, the limit case, exceptional and no doubt paradoxical, of historicity in the ordinary sense.

Therefore this continuous path drawn by the model can no longer be missed because it is essential that the information be preserved in its significative totality, because it is impossible for the communication to suffer interference or rupture except by falling into non-mathematicity. In other words, mathematics is entirely transmitted or not at all. Recollection in the Meno is a reconnection or a complete taking on by the inheritor, by the one being taught, of a tradition that's not open to misinterpretation, equivocations or gaps. Conversely, a common conception of history that would have this connected model for support is an illusion of pure reason, stemming from the exceptional or limit form of mathematical traditionality. c) It follows from this that the model lastly expresses a form of continuous historicity, polarized in an irreversible way by an end and forever abandoning its origin: the act of birth or constitution starting from prehistoric archaisms would be a point of no return.

Naturally, the progressive extension of the mathematical field, the continued purification of its concepts, the always strengthening power of its methods, the forward movement towards a mathematics conceived as horizon give us to think an evolving form that's connected, punctuated with stages or steps, to speak like Brunschvicg, or better with crises, to speak like the set theorists from the beginning of the 20th century. These stages or crises would only be global reorganizations of a knowledge transmitted without any loss, therefore incessantly accumulated. The path, once again, could no longer be missed because it is accumulative, because each stage, as a remarkable accumulation point, would only be a reorganization of an overly dispersed aggregate, a systematization of scattered elements. The path inflects because mathematization is no longer made to focus on the 15

Translator's note: “above” meaning earlier in the book from which the present essay is taken, a chapter translated as “The Platonic Dialogue” in Hermes: Literature, Science, Philosophy, eds. Bell and Harari (Baltimore: Johns Hopkins, 1982).

12

Serres: Mathematical Anamneses atoms but on the distributive totality of the disciplines. Each point of inflection is a point of inflation and of reconstruction. Thus Euclid, Leibniz, Cauchy, etc., recuperate the totality of history in a totalizing system – condensation and consistency. A good mathematical system, that is to say, a universal system, would be given as a synchronic cross-section at a moment of inflection of the diachrony. Bachelard saw this situation well: “It's at the moment a concept changes meaning that it has the most meaning.”16 The truth of these historical fragments of meaning is in a certain manner given by philosophy itself: Plato and the irrationals, Descartes and algebraic geometry, Leibniz and infinitesimal calculus..., Husserl and the crisis of foundations.

The starting model becomes more refined: it is no longer linear, rather it schematizes a diachrony by stages, intervals or diastemata, united by moments of system, of global reorganization. Any synchronic cross-section in the intervals reveals the preceding system as well as new layers that aren't part of it and aren't integrable into it. It's an endlessly to be reconstructed Tower of Babel, which it is urgent to reconstruct as soon as the new promotions can no longer use the same language among themselves or with the preceding system. It is necessary then to reunify by means of a system, which then is only a dictionary created for a new perfect communication. Working on a common systematic base, Gergonne, Cauchy, Abel, Galois, Cantor, etc., go beyond it, creating a confusion of languages such that one could think for a moment that mathematics was dead and such that one is led to reconstruct a new base that gathers the common etymology of their languages, that therefore causes mathematicity to be reborn, and so on, all the way to the reunification by Grothendieck, etc. Thus Plato, Leibniz and their contemporaries created languages, new universal characteristics. At the beginning of our century we found ourselves in a Leibnizian situation.

2. Recurrent Connected Models. This analysis tends to show that mathematics wasn't once and forever in the situation of origin. The construction of a new language for a new perfect communication, the constitution of new idealities, the taking on of the totality of the edifice lead the scientist, at the time of great systematic enterprises, to take the whole of the path traversed up again. That's why the judgment of recurrence is not only historical practice but above all epistemological practice.

Questioning backwards, questioning the foundations, and the refined analysis of original elementary idealities perceived retroactively as layered, stratified ideas, as complex particular cases of idealities that are even more original still, are ordinary attitudes of the mathematician. We have seen above the triple return to pseudo-elementary or pseudo-primitive Euclidean spatial forms. We would never get to the end of repeating how many times the question has returned about the real line, about zero, about whole numbers, about equality, about the diagonal and the circle, how many times the answer to this question was found to be an ideality actually founding the ideality being questioned, not only through its axiomatically defined structure, but in its very constitution; – the example of straight line R about which we have long wondered whether it had a natural topology or whether it had been endowed with certain topologies.

Everything happens as though it was necessary to combine the direct movement of teleology and the inverse movement of recurrence into a circular, or better, spiral diagram, as though the development of a theory only drew its effectiveness from the endless iteration of passages through the origin, itself reconsidered by means of the methodical weapons forged in the course of the extension. There would be here a kind of feedback model where the development feeds back through the source and the source through the development. We find again, if not Antaeus, who only regains his strength by putting his 16

Translator's note: this could equally be read as: it's at the moment a concept changes direction that it has the most meaning.

13

Serres: Mathematical Anamneses foot back on the Earth, at least the anecdote of the Meno; and the latter at least three times: through the combining of direct progress and anamnesis; through the mathematical example that we discover is essential since only mathematics furnishes the path of a lightning-fast and unequivocal communication with the origin, a communication no other historical experience can give any idea of; lastly, through the endlessly possible iteration of the process: as Leibniz indicates, it would be possible to practice on a slave from the forgotten world the anamnesis of a world twice forgotten, et ita porro. The origin of mathematics is laid bare at each great moment of reconstitution (historically, this is visible from the outside) and by each reconstitution (the movement is perceptible from the inside).

I repeat, recurrence is not first a historiographical movement; it's not enough to say that every leap forward demands rewriting the uchronia of what preceded, demands rectifying the entire perspective upstream in terms of “what ought to have been thought”. It's not enough to say that the history of mathematics has a dating scale similar to that of Parian marble. Recurrence is first a movement proper to mathematical temporality as such, in so far as it presents itself as continuing systematic restructuration. The properly historical recurrence is only the second consequence of this internal and original movement. Bourbaki's Elements of the History of Mathematics is the mirror-image portrait of the Elements of Mathematics, the projection in a diachrony of what in fact happens in the system, the displaying in a historical genesis of the systematic deduction. Such promotions – that of infinitesimal calculus, of group theory, set theory, category theory – reverberate globally in the entire edifice and propagate in a lightning-fast manner down to its original bases, entirely as though the last thing constituted called into question the whole of the constitution. And once again, it's not only a question of logico-axiomatic conditions, it is also a question of conditions of constitution: at the dawn of infinitesimal calculus what was questioned was not merely the true or the false and the rigor of the linked sequence, it was mathematicity in its entirety and even more its foundation on a world; what was in question was the Earth and the fixed stars. This recurrent movement, propagating vertically in the system starting from these promotions, shows that there is a contemporary archaeology of decisive advances; better yet, it shows that a given advance is only decisive when it lays bare originary archaisms at the very moment it is promoted. There is simultaneity of the teleological acceleration and the archaeological recurrence. Hence the originality of mathematical temporality, which in the same moment makes its way towards its τέλος [telos] and its beginning. It follows from this practically that if I want to study the historical, or logical, or gnoseological, or transcendental question of the origin of mathematics, I can question Thales or Pythagoras in myth, Desargues or Descartes in history, Bourbaki, Serre or Grothendieck in the living present. Any origin is the origin itself.17 Better yet, this study shows structures common to each of them, structures that answer the question.

Hence the following simple model: I observe that the first schema is no different from a cone – a model which hasn't been new since Bergson or Einstein –, that each synchronic or systematic crosssection is a cross-section of this cone, as Desargues would say, that in the interval between these crosssections all the appropriate geodesics are drawn, traced in a helix on its nappe. This model's interest lies in the fact that the geodesics advance indifferently from the front to the rear or from the rear to the front: which combines teleology and recurrence. What's more, the whole of the figure is projected into two new schemas, depending on the point of view. What can be said about them is the progressive development of the theory, its closure and the conjunction of extension and of the endlessly iterated passage through the origin. The second point of view is perhaps the more interesting inasmuch as it shows that a continuing archaeological deepening corresponds to every development: we have seen for 17

Hence the question: is Thales and Pythagoras' mythic origin truly (historically) the first? Nothing could be less certain.

14

Serres: Mathematical Anamneses example how the new geometry had founded Euclid's spatial idealities via idealities that were constitutively deeper, group structure, vector space, topological manifold. We can moreover wonder whether the schema has to be read in progression or regression, so much does the precise analysis of conditions suffice to immediately broaden the field. This is because the axiomatic method scarcely leaves certain origins anymore.

The origin of mathematics then is present in the entire course of its history; it is a percurrent origin.

The return to originary conditions is historical (recurrence), logical (axiomatic), transcendental (constitution).18 3. Non-Connected Models. The preceding models don't take an essential phenomenon into account.

The teleological movement is a movement towards the elementary specifications of mathematics in general, conceived as horizon: towards rigor, purity, analytical refinement, etc. So every systematicsynchronic cross-section is more mathematical than the preceding one; at the limit, this latter is nonmathematical for recurrent judgment, which is a judgment of truth: it is impure, confused, hardly rigorous – confused insofar as it confounds into a single structure structures to be dissociated. So the recurrent judgment becomes judgment of application. For us, Thales' geometry is a master mason's metric; Desargues' is from an expert in cutting stones, in squinches and stairs; Cartesian geometry is that of an engineer; Monge's is from an architect at his wash drawing (it was said to be descriptive); the geometries said to be non-Euclidean are metrics of the physicist; the mathematics of Lorentz and Einstein is a mathematics applied to the cosmic or electronic world. Mathematicians sometimes say, as a joke, that they are geographies – a term that has meaning for us philosophers. This signifies that it's a question of mathematics that have been sedimented, reduced to technology by the movement of purification; it is to be remarked that they become artifacts all the more the older the sedimentation.19 In this sense, they are forgotten: we find the Meno once again and a kind of necessary covering over, the cutting, the discontinuity of mathematical time. Hence the history of the diagonal and the square, which I related above, is a false and unfaithful history, denuded of sense for the mathematician: it's a flatprojected catalogue in which it is impossible to see the superposition of the lamina of meaning, the stratification of the layers of different ages, the exasperated topography of forgotten worlds.20 This history should be read as a complex surface, made up of “chimneys” of strong acceleration, “cols” of stoppage of a rising, zones of stationary values, tears and so on, like the surfaces conceived by Euler or Riemann.21 This is because a given system doesn't recuperate all the ancient sediments; it doesn't presentify the whole of the tradition: on the contrary, it makes a choice, a selection in its recurrent movement; it lets concepts fossilize as technological gangue. In the preceding model, there are absent geodesics, ruptures of connection, definitively cut adherences: the system functions like a filter; the 18

19

20 21

As we shall see, the model of science we can introduce approaches the model that science creates of the world. Here, it's not the imperishable heaven that's in question, but rather the incorruptibility of atoms. Infinitely hard and indivisible, they evade history, the erosion of use. We now know that they can be broken, but above all that they are regenerable in the event of a return to initial conditions. As a result, the model of a “first creation”, relatively stable from Epicurus to Newton, can only be abandoned in favor of a model whose originary condition is a current event, percurrent, taking “place” everywhere and at every moment of “time.”

The question should be asked: is the technological origin of mathematics an illusion of recurrence or a discovery by recurrence?

Translator's note: “exasperated” is probably being used in its etymological sense of “made rough.”

It would even be interesting to take nonorientable surfaces as models insofar as we need to evoke a historicity developing indifferently in two directions, a historicity sometimes connected and sometimes rent. The most elementary topology offers this, as everyone knows, in superabundance.

15

Serres: Mathematical Anamneses teleological filtering of purity, of rigor, etc., eliminates fossils. The river is all the more transparent for discharging finer and finer alluvia. As soon as Euclidean space gets plucked into topological space, metric space, vector space, a group of displacements, etc., all that remains of it is the trihedron of walls and ceiling that protects me in my house. I know of no technique so luminous for designating archaisms as this filtering for purity achieved by the very movement of mathematics. In every point along its course it is easy to find evidence of the origin carried up to there and abandoned through the contemporary filtering, evidence of prehistory: the situation is the same as in astronomy, in which you can receive information from worlds that no longer exist.

This designates two distinct archaeologies: the one proper to mathematical movement as such, which never ceases reactivating its origins and deepening its foundations through the iteration of its internal recurrence, the one that extricates originary idealities, which weren't mathematical and become so, the one that little by little historicizes prehistory and gives a language to what was deprived of it: thus topology invades and thematizes said morphology. On the other hand, the one that consists in reading prehistory in abandoned concepts that were mathematical and no longer are, in reading dead prehistory in fossils carried along by history and abandoned by it. The first one is the archaeology intrinsic to science, the second one is extrinsic; it reconstructs the lost genesis of a lost ideality: and this is the case with Euclidean space. The first one is regressive and progressive at the same time because it follows the double movement of teleology and recurrence; the second one can only be regressive: that's why it has the power to uncover prior soils, while being struck with impotence to explain the effective foundation, that is to say, to go back over itself by following the progressive movement; this movement is forbidden to it, this path is cut off since the ideality it deals with is no longer mathematical. As, on the contrary, the first one combines the two movements, it is easy to define mathematics itself as an autochthonous technique of archaeological research, as has already been said, although unwittingly, in the Meno. In a certain way there is a continuing solution to the old problem of the origin of mathematics, and that solution is endlessly readable inside the mathematical process: I mean by this that a cultural formation is only accessible as pre-mathematical in and through the autochthonous process of mathematics. When topological graph theory mathematized knots, labyrinths and paths, then and only then was I able to understand that the weaver was a pre-mathematical technician more ancient than the surveyor, the way the taut plumb line is only a metric modality of the same cord bent or knotted in a thousand ways; then and only then was I able to understand Gordium and Minos to be premathematical mythic schemas that were more deeply buried than the myths of builders. No other archaeological technique would have had the power to lead me below traditional surveying. Whence it happens that the shaky square drawn in the sand, the hesitant and anexact graphe22 that Plato refused to see, is of a sensible status and purely mathematical at the same time. Whence it happens that the world of the shaky graphe is the world forgotten by Plato himself, anterior to intelligible metric, and which twenty-five centuries after him we wind up remembering. Whence it happens that the mathematization of the anexact causes me to discover all graphism in general to be a pre-mathematical manipulation of topological manifolds in general. Mathematization leads me to the pre-mathematical. The problem of the origin of mathematics is a problem that's endlessly resolved and posed again by mathematicity in general, conceived as recurrent and teleological temporality. It's in studying the dynamics of rivers that I understand the processes of sedimentation and the existence of forgotten meanders. I go directly from 22

Translator's note: graphe=graphe, which normally means graph, but which Serres seems to be using in this passage mostly in its etymological sense of writing or drawing. Hence I write it as “graphe” to indicate a non-standard usage.

“Graphism” two sentences below and hereafter translates graphisme, which normally refers to the way a language is represented by written signs.

16

Serres: Mathematical Anamneses the square in the sand to the topological manifold, abandoning the Euclidean meander: a lightning-fast short-circuit with a little slave boy who's the son of the earth. And once again, the situation is the same as in astronomy, in which I can endlessly expect from the future information issuing from the most distant worlds.

Leibniz and later Engels, among others, uttered the fear that the accumulation of knowledge leads as inevitably to barbary as its very absence. Science would collapse beneath its own proliferation. This amounts to believing that the progressive advance in what we know is a recommencing recuperation of the distributive totality of previous knowledge: an accumulative process of an encyclopedia that would snowball on itself; this amounts to having faith in the connected models of history. As far as mathematics is concerned, it is clear that things don't happen like this:23 it filters its heritage rather than taking it up in its entirety; or better, it takes it up by filtering it. By this very fact, mathematics shortens by augmenting, diminishes by accumulating. Three volumes of calculation on Harmony by Mersenne are rendered useless by some given theorem on the arithmetic triangle; a page from the De Arte Combinatoria does away with diverse techniques of Lull's type; some structure or other at once takes up an entire gallery of models. So the history of mathematics is a history of the theory of theory: the science of science endlessly substitutes for science itself, as though synthesis succeeded dispersal so as to annihilate it with a stoke of the pen, as though one attained the possibility to say an entire Sisyphean labor in one word. So recurrent judgment discovers a science of repetition, the iteration here and there of a word one didn't know how to say and which, as soon as it is said, stops the adventure. It's in this sense that Descartes said of Desargues that he had set out “the metaphysics of geometry”, that Leibniz reproached the scientists of his day for “always rolling the same rock”, that Galois recommended to “jump with both feet over calculations”, that Bachelard counseled not to wander in the dark jumble of graphism. It's to say that a great invention is annulation, doing away with a field of knowledge, just as much as it is promotion of knowledge: with its open sesame, it closes an entire domain, which is hardly understood after this invention except as the underworld where the daughters of Danaus strive.

Through such invention, it happens that the entire corpus is put into short-circuit and remains in history as a forgotten loop: taken up however by the short-circuit, but forgotten as far as the loop's entire circumference is concerned. Progress becomes possible again through doing away with certain repetitions, and recurrent judgment indicates stagnations. The history of science then seems a series of putting into short-circuit, a sequence of putting out of circuit. Hence the lightning-fast communication with the origin at the very moment when invention brings the ὲποχή [epokhē] of its heritage. Hence the points of rupture, of stoppage and of resumption for a non-connected model.

Hence the ruptures of connection and the always missed path: on the one hand, I possess traditional information stemming from vanished worlds; on the other, I discover new information stemming from worlds foreign to the tradition, worlds come to me by the shortest path. Mathematics is archaeology, but archaeology by the shortest path, by continuously abandoning traditional meanders. This situation defines the extreme boundaries of the filter: what the present leaves and finds, what archaeology finds again and abandons, the entirety of the same movement of birth and rebirth, and of death with no return. That said, we must examine the filter inside these boundaries. Let there be then two synchronic cross-sections: mathematical language A is anterior to language B in the ordinary diachrony. It is almost always possible to translate A into B; conversely, it is rare to be able to translate B into A. For 23

Of course, the Leibnizian fear must still haunt us if it is true – and it is true – that the scientific city is now constituted, to turn Auguste Comte's words around, by more living people than dead.

17

Serres: Mathematical Anamneses example, Euclidean space can be translated into topological, metric or vector language: it is a model of such-and-such structures; conversely, in the Euclidean repertoire, there is no term that corresponds to “topological manifold”... Insofar as the path of recurrence is only considered to be the reverse of diachrony, this path is cut off – most often; communication is cut off because the intersection of the two repertoires can be empty.24 And since the path is punctuated with points of no return, the futility of a regressive archaeology that would confine itself to reversing history, that would not take the original movement of the science into account, can be measured. On the contrary, designating deeper layers, this movement reinterprets in return surpassed idealities or better, again defines a system of translations. Each synchronic cross-section includes its conditions of translatability. The judgment of recurrence doesn't go from topological space to Euclidean space; it goes from the topological presuppositions of Euclidean space to the global reinterpretation of Euclid's corpus. The new language is at the same time anterior and posterior to the preceding one; it makes it explode, it cuts it up, filters it, eliminates the impure, retains from it only the gold of mathematicity. Each restructuration is a kind of earthquake that abruptly uncovers archaic layers and buries recent sediments. If I am in lightningfast communication with the origin, it isn't through the traditional historical channel, it's through the effort of foundation of mathematics itself. My regression doesn't follow the path of tradition, endlessly out of circuit, but rather the vertical path of mathematical deepening: it's starting from this that I reinterpret the historical tradition.

We have observed that Leibniz's system was open to a self-explication via application of itself onto itself. We have just evoked the possibilities of translating one mathematical language into another so that the development of that science can be envisioned as a series of failures and successes in such an undertaking of translation: the strongest success would naturally be defined here as the establishment of a language common to a plurality of dialects, dialects previously differentiated and henceforth referring to a mother-language, the most recent example being contemporary mathematical language, predominantly algebra. Invention then would be a successful application of a region onto one or several others and, at the limit, a self-application of the system onto itself.

On the contrary, mathematics would be in a state of crisis when an application of this nature fails.

This leads to the reciprocal idea that every mathematical system – like the one by Leibniz – is, taken globally, an ars inveniendi, endlessly. Their history is a trans-lation, repeated at every moment, a history of discoveries or coverings-over.

That said, let's return to that Greek miracle, which is manifestly no longer anything but a word that escaped from Renan during a moment of elation. Aren't we in the presence, here as elsewhere, that is to say, as in every origin moment, of an application of a certain mathematical language onto another, of a certain graphic process onto another? The geometrical forms – square, triangle, circle, tetrahedron... –, whose metric “perfection” we now know is not a necessary condition for mathematicity, these forms were known and practiced well before Thales, as the decorative arts and technologies – pottery, basketry, transport, construction – of the preceding civilizations, from Egypt to Sumer, superabundantly show. No monument can give us information about the gnoseological attitude of their contemporaries regarding these forms. But what we are certain of is that the Greeks began speaking about them, taking them as the objects of their discourse, that they invented a logos suited to their analysis (to a certain type of analysis), that they began to translate them into a sayable and universally communicable language, that they began to decode them, to decipher them, that they moved from a spatial schematism 24

This is made worse the moment the reasoning is iterated: this is because the intersection isn't transitive. The fact that the intersection of repertoires A and B is non-empty, the same as the intersection of repertoires B and C, doesn't imply that the intersection of repertoires A and C is non-empty.

18

Serres: Mathematical Anamneses involuted over itself, immobile and communicable in the secrecy of the skilled hand, to a language that designated part of its meaning; in other words, that they substituted, for an ideographic writing of the geometric forms, a signaletic writing, by letters and signs, which applied best to the former: rigor being the rigor of this application, of this translation. The Greek miracle is this miracle, so ordinary in mathematics, which consists in recognizing an ideography in a form, a meaning or several in a symbol, in knowing how to translate them into a signaletic and communicable graphism, so that both languages, both writings might be in the most exact relation. In doing so, a correspondence is invented between a symbolic schematism and analytic characters – as, in early arithmetic, between things and numbers, that is to say, the letters of the alphabet. But since analysis by characters doesn't, in the end, manage to exhaust the schema's compact meaning, since conversely writing by signs reveals absurd secrets which geometrical ideography doesn't directly exhibit – a counter-proof soon inflicted on the Pythagorean consciousness by the crisis of the irrationals –, the translational correspondence failed as soon as it succeeded: it became urgent to pursue perfect applications towards the ever pushed-back horizon. The Greek miracle no longer designates the origin of geometry, but a point of departure for the history of a certain mathematics: it opens the historicity of science. The idea – here regional – of translating a schema into characters inaugurates an endless series of applications of the same type, sown with failures and triumphs, opens the ideograms up from their prehistoric (and not ahistoric or transhistoric, as a certain Platonism tries to teach) immobility, disenchants them from the closure of their meaning, from the communication by invariant transport they were subject to in art and technology; from now on, history is open, in which characters are going to be able to convert, across thousands of languages, the meanings enchained in the square schema. As soon as Socrates gives the ignorant the possibility to talk about it, the latter remembers his mute prehistory as a forgotten world: anamnesis is the memory, across a communicable language, of what is only structured as a schema25 and which was transmitted in prehistory as a hieratic, invariant, inaudible, manual symbol. Thus the origin of history begins again with each translation into a new language: the institution, for example, of characters that have the power to translate, to decipher knots, paths, interlacings or labyrinths, that free meaning from schemas transmitted from hand to hand by weavers, decorators, scribes and helmsman in the prehistory of the logos. Of course, the reverse application is just as familiar to the mathematician when he envelops in a schema a plurality of meaning stemming from characters. The Greek miracle is the miracle of the history of science, without the geometer philosophers having had any consciousness of it other than a mythic one: the forgotten world there is only an image of heaven, whereas heaven is only the myth of prehistory. The history of mathematics is the history of miracles of the same nature.

It therefore seems essential to rectify the connected models – models which would remain valid in the exceptional cases where there would still be a common repertoire. So it would be necessary to read the final projection as a series of geological cross-sections, the final one of which is always deeper, giving the preceding ones to be understood, but precisely thereby designating their lack of interest, their superficial and problematic character, their prehistoric and pre-mathematic nature. From which a significant result comes: if there is no continuity between the properly mathematical cross-sections since each one places the preceding one in short-circuit, how much less continuity is there between cultural formations as such and the formations that are differentiated from the first ones by the fact that 25

This structuration as a schema of prehistory in general – scientific in particular – or of the non-conscious unconsciousness of its knowledge or of its logos – of its science, in particular – explains, in turning around fashionable aphorisms, many contemporary works of the interpretative or archaeological type (Leroy-Gourhan).

19

Serres: Mathematical Anamneses they carry the truth along with them?26 And once again it's starting from the second ones that we are obliged to reinterpret the first ones. Up to the present, I have seen no hope of freeing the chthonic foundations of mathematics except by following the autochthonous movement of mathematics itself, since the act of putting the science out of circuit is rigorously carried out within the very interior of its historicity. There is a research of a transcendental type that is proper to mathematical historicity, or even better, mathematical historicity is also transcendental. Mathematics as a systematic organon that's formal and in formation is an objective and intersubjective field. Mathematics is both a formal ontology and a transcendental logic.

This incessant putting out of circuit accounts in depth for the principle of indeterminism indicated above: either one returns through cultural formations and never encounters science as original and veracious movement, or one returns through science itself and ceaselessly reinterprets the cultural formations by always pushing the cultural as such farther back into the process of digging deeper. By endlessly making its way towards mathematicity, mathematics (and science in general) makes its way backwards towards another τέλος, that of the prehistory of prehistories.

In a certain way science tends to do away with the traditional characteristics of the model of time: its directional and irreversible character, the arrow and fletching of its vector,27 its continuous character, its forgettings and its mnemonic accumulation; through its iterated choice between a lightning-fast communication and a putting out of circuit, it sometimes plays Socrates's game and sometimes that of the little slave boy. In a word, it is master of a new time; it invents a new time by constituting it historically from the scattered elements of the former model's bursting. It's no longer a question of time or eternity, of any tangency between time and what's outside time, it's a question of the constitution of a historicity that reconstructs at leisure its former characteristics: this is why I spoke about panchronism and uchronia, or about non-orientability.

Is it possible to determine a principle of choice among the envisioned models? Let's observe, in the first place, that the process of sedimentation proper to the course of mathematics doesn't leave concrete enough formations behind it so that only a knowledge other than mathematics would be able to rescue their meaning: the Meno's inaugural lesson of anamnesis. Nevertheless, sedimentation continues through concretion of the abstract. In these concretions, a certain mathematicity is preserved and remains invariant so that historicity retains its point of reference inside the organon in general, so that historical experience – partly integrative – remains mathematical experience and sometimes reciprocally. In other words, the history of mathematics presents an original mode of sedimentation of the clear – doubtless not of the distinct – and of the true. The true remains invariant across the diachronic transformations; what changes is the concept of truth. Mathematical truth, index sui et falsi, the automatic essence of this true remains stable – and stable because automatic –, and mathematics is stable or better, mathematicity; what varies, let's dare the word, is the philosophy of mathematics, that is to say, the mode of being of this true: but, since this philosophy is autochthonous, mathematics again transforms. It is therefore the invariance of the true in an always connected diachrony and the variance 26

27

In the Krisis (Part 3, paragraph 31), Husserl speaks of “mental formations of a certain type called theoretical”: a theoretical stratum would be a singular species of the genus formation. This assumes that the movement of science has endlessly stretched the link of theoretical thought and the lived, but hasn't broken it. The entire question is there: has it broken it or not?

The most recent models of physics attempt to explain, via symmetries, cases of returning to initial conditions, as though the first creation took place at each moment of time.

20

Serres: Mathematical Anamneses of the modality of the true in an always broken course.

Look at the field of dead histories: Greek geometry, Enlightenment analysis, modern mathematics, which has stopped being contemporary. Dead and not false: what is this death of the true that never changes into error? A singular death of surrectibles, clarity become dark or cold with an inextinguishable light, the sedimented concepts never stop being clear and remain truthful: from the depths of the ages, the so-called Pythagoras always talks rigor and would not be able to either mislead us nor be mistaken in a language that would still be audible; and recently, the so-called Bourbaki put forward the everlasting true. But clear concepts are sedimented, semi-concrete, involuted in a gangue that only new truth can dissolve in order to reveal the new true of the old one. The only admissible historicity then is the historicity of systems (and, in the intervals, that of their constitution): the mode of being of the true lies precisely in its relation to the system. This is what lives and dies, as fertile opening, then as stifling gangue. Broken historicity is that of alethologies, connected historicity is that of an alethic.

It seems that I find at this crossroads the most ancient and recommencing of the philosophical traditions, according to which the most rigorous of the paradigms of theoretical thought resides in the contemplation of the heavens. Everything happens as though the models that philosophy can create of science were isomorphic to the models that science creates of the world. I don't emphasize this heritage that, from the Ionians to the Timaeus or De Caelo, from Ptolemy to Bruno, from Tycho-Brahe to Pascal and Leibniz, from Copernicus to Newton and Kant, from Laplace to Comte, from William Thompson to Nietzsche, all the way up to Husserl's great text on the motionlessness of the Earth, has never faltered. How are things today with this philosopheme? It has been preserved invariant, across the variations of the theory of the world and of the theory of theory.

First of all, we have brought history into the domain of the ideal model at the same time as into the model of the Universe. Even though the objects in the sky seemed to our precursors to be as stable and pure as the idealities of theoretical thought, we now know that rigor and purity are evolving, in the same way that stars are born, grow old and die in their novas, leaving remnants populating the trash can-universes. Theory is a history; purity has its own time, the way cosmology now has its cosmogony: not rest, motion, regulated movements, but origin, evolution and disappearance. This is an astrophysical revolution that brings rigor to variance without variation of rigor, the way in the past the Copernican Revolution had changed the references for thought.

Next and above all, I observe the sky the way I observe the system of knowledge. Here and now, visible or Hertzian waves are giving me incoherent or aleatory pieces of information to read regarding my time, regarding the time of my history. One informs me about a recent event, another about an event that's earlier by several millennia, another lastly about an event that has no meaning on the historical scale. It's no longer eternity I discover here, but the infinite confusion of chronological trails; I simultaneously hold, I reactivate in one same thought two, three, n anachronic elements. This sky of today, constituted presently before my eyes, this pure thought whose history I want to write, these two systems, of the universe and of knowledge, put me simultaneously in lightning-fast communication with events whose dates are dispersed in every imaginable way. And yet, if I want to understand, I must understand the site of contact between my living present and this theoretical-concrete spectacle that rends, confuses and complicates the temporal sequences in an almost random way, the site of contact between a chroneme and an enormous number of anachronemes, between a time and a distributive panchrony. We know that, regarding this indetermination, a certain number of models of the world have 21

Serres: Mathematical Anamneses been proposed. No doubt there are as many models of the history of science: Leibniz had perceived this and had precisely had the relativist theory of time as order. So there is a relativist revolution to be carried out here as well in what concerns our view of the theoretical universe. If Bachelard had analyzed the essential complexity of science as such – instead of making use of this concept as a descriptive attribute –, he would have necessarily arrived at the idea of Astro-Physical Revolution.

Counter to Husserl, I will readily say that it's not the earth that's the originary ground from which theoretical thought is drawn; it's not the earth that gives meaning to movement and rest, since these concepts are already superficial and residual; it's the totality of the universe as both evolving and anachronic that gives its meaning to time and the absence of time, to time and the multitude of times.

The world as always anachronic and uchronic (achronic and panchronic) is once again philosophy's paradigm, the model that founds the possibility of the site of tangency between our time and the absence of time, between our time and the totality of possible times.

Kant described a history of science that was a history of purity and found the Copernican Revolution in that history to be an event to be repeated for the henceforth rigorous metaphysics; Husserl describes a history empty of science, at the risk of confusing a pre-scientific layer and a scientific layer, an error carried out in his theory of morphological idealities, and discovers the earth as the originary fixed point and transcendental reference. We must now write the history that science as such is, that is to say, scrambled and complicated temporalities in a single-totalizing temporality and, to this end, practice a Revolution without eponym. This is the return to the world itself, that is to say, to the New World.

And again we are at the origin, in quodam mundi infantia. As in every moment of history, we have to take up a new knowledge, discover a new world whose inchoative history no longer takes our culture back to history but to prehistory. Everything happens as though we were in the closest proximity to archaic layers to be forgotten and to new idealities to be understood. Under the guise of contemporary mathematicians, astronomers, astrophysicists and biochemists, Thales is again among us in order to invite us to new translations, to new anamneses.

April-May 1966

22