The Birth of Physics - Michel Serres

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The Birth

of Physics

MICHEL SERRES

Translated by

Jack Hawkes

Edited, introduced and annotated by

David Webb

CLiNAMEN PRESS

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Introduction and annotation © David Webb 2000 Clinamen Press Limited

Enterprise House Whitworth Street West Manchester M1 5WG www.clinamen.net

Published in French by Les Editions de Minuit as La naissance de fa physique dans Ie texte de Lucrece

© Les Editions de Minuit, 1977 7, rue Bernard-Palissy, 75006, Paris

All rights reserved. No part of this edition may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or othetwise) without the written permission of the publishers. A catalogue record for this book is available

from the British Library

ISBN 1 903083 04 4 (hardback) ISBN I 903083 03 6 (paperback)

7 9 8 6 4 2

Typeset in Adobe Garamond by Koinonia, Manchester

Printed and bound in the UK by Redwood, Trowbridge, Wiltshire

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CONTENTS

Introduction-vii Editions of Lucretius-xxi

Translator's note-xxii PROTOCOL

First model: of declination in a fluid environment page 3

Turbulence 6

MATHEMATICS

An analysis of the hydraulic model 9

Archimedes' work 13

Archimedes, or the concept of deviation 19

RETURN TO THE MODEL

Turba, turbo 27

Slope and extrema 31

Flows and paths 49

EXPERIENCES The Meteora

Experimentation: magnetism

CONDITIONS Epistemological conditions

Observation and simulacra IOI

Cultural conditions

Violence and contract: science and religion I07

APPLICATION: GENESIS OF THE TEXT 135

Atoms, letters, cyphers 139

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Coding

Fall and rhythm

H I STORY Antiquity, modernity

MORALITY The soul and the descent to the underworld The garden and the local

Index-I93 Vi 147 151 165 172

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INTRODUCTION

This extraordinary and passionate book is devoted to the renewal of atomism in contemporary philosophy and science. Neither orthodox classical commentary, nor traditional history or philosophy of science, the work traces a relation to the text of Lucretius that is at once exemplary in its scholarship and sensitive to the aspirations of atomism itself. In doing so, it both revises our conception of ancient atomism and challenges our understanding of the paradigms that underlie much recent and contemporary science. More specifically, it presents a philosophical basis for a distinct approach to movement and relation, to time, to language and to the question of physical law. Running through all of this are the themes of form, abstraction, the emergence of sense and the need to reach a new and less violent relation to nature. Not least, the book is also an eloquent tribute to the precision and intensity of Lucretius' own text.

Provocatively, Serres names as the birthplace of physics the text of an author, Lucretius, who is roundly ignored by most histories of science. It is thought that Titus Lucretius Carus was born a Roman citizen in the first century BC and he wrote his only known work, De rerum natura, as an exposition of the atomist philosophy of Epicurus (341 BC-C.270 BC). Philosophically, the book is taken today as primarily a point of departure for looking back towards ancient atomism. AB a scientific text, it is not taken seriously at all.' The atomism Lucretius espoused, along with that of his precursors Leucippus, Democritus and Epicurus, is treated as a curiosity and a cul-de-sac having little or nothing to do with its modern scientific counterpart. Against this prevailing view, the claims made here by Michel Serres on behalf the contemporary relevance of Lucretius may seem fanciful. However, Lucretius' modest profile in the history of science and philosophy is no accident, Serres argues, for his neglect is of a piece with the enthusiasm for classical rationalism that has dominated our intellectual tradition from Plato through Descartes to Newton and beyond. In spite of the recurrence of corpuscular philosophy in the 17th

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and 18th centuries, there is something in the ancient doctrine of atomism that has always been anathema to this classical tradition. Yet as the underlying ideals of classical and modern rationalism no longer have the force they once did, the time is ripe for Serres to reveal that there has always been a secret current of atomism running through European philosophy and science. Strangely, while atomism has been suppressed by our philosophical and scientific tradition, it has continued at the same time to inform that tradition from within: not just as an excluded negative, but as a positively contributory factor, establishing such recur­ rent themes as fluidiry, inclined planes, the fall, equilibrium and disequi­ librium. To recover a little of the riches of ancient atomism therefore means neither relating it to its accomplishment in modern science, nor turning against modern science and back into the past. Indeed, the renewal of atomism in Serres' work speaks of fracture, convolution and of unexpected kinships. It complicates relations: historical and discursive. And just as the scope of the ancient doctrine, and Lucretian physics in particular, extended beyond what we regard today as the province of the natural sciences, so the atomism Serres advances here bears directly on epistemology, the philosophy of language, and morality. Above all, this new translation of The Birth of Physics is a timely reminder of the important contribution Serres has made to the debate on the relation between philosophy and science.

Atomism ancient and modern

It is generally said that modern atomic theory was animated by the desire to break down inert matter into its smallest constitutive elements in order that the laws governing their motion may be determined and that these laws might in turn underpin and ultimately explain higher­ order phenomena. Its determination of the smallest element of matter was in pursuit of a reductive science. In this way, modern physics set itself apart from ancient atomism by the provision of a mathematical basis that ancient atomism apparently lacked. By contrast to the law­ governed behaviour of matter in early modern physics, the ancient doctrine of atoms combining as they fall through the void seemed primitive and unsystematic, an eccentric yet lucky intuition. While this story of modern science is broadly speaking true, it veils a less straight­ forward series of developments.

Modern atomism grew up alongside the dynamics formalised in Newton's laws of motion. The relationship between the two was neither

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Introduction

straightforward nor always free from antagonism. Newton's laws described the movement of matter in a perfectly efficient system that would remain stable as long it were not disturbed by some external force and where all change was in principle reversible. His introduction of a universal force of gravitation depended on the idea of action at a distance and thereby dealt a serious blow to the principle of mechanism that underlay the corpuscular conception of matter. The triumph of Newton's thesis seemed to cut the last thread connecting modern science to ancient atomism. However, this view was soon to be complicated by the later developments of thermodynamics, statistical mechanics, quantum theory and non-linear dynamics, where the ideals of reversi­ bility and closed systems were thrown into question. It is here in the elements of uncertainty and openness characteristic of such theories that ancient atomism re-asserts itself in modern physics. Serres' work accentuates this counterpoint to the classical ideal not just by referring to the renewed importance of certain general ideas drawn from atomism, but by bringing the richness of Lucretian physics back to life, and above all by correcting the widespread perception that it lacked a mathematical language. All the components required for the mathe­ matical expression of atomist physics - a geometry of spirals and revolution, an infinitesimal calculus, a hydrostatics and more - are to be found, Serres reminds us, in the works of Archimedes.

But there is more, something quite specific, that sets the body of Archimedes' work apart from the mathematical principles and ideals that have informed philosophical thought from Pythagoras, Plato and Aristotle, to Descartes, Leibniz, Kant and down to Husserl and much contemporary thinking. The geometry underlying so much of the metaphysical tradition is concerned with the construction of figures and the measure of their elements; with sides and lengths. Serres, by contrast, embarks on a 'history of the angle'. Incidental to the principal interests of geometry, the angle resists the push toward quantification and remains 'a shape, a corner, like a quality'. The problem, Serres remarks, is that in conceding to the imperative to measure, geometry has conflated the study of form with its relation to number. As a result, it has confused rigour with exactitude, and crucially ceased to attend to the qualities of form, direction and inclination that do not lend themselves to measure. In short, it developed essentially as an applied mathematics. 2

Compound this with the tendency towards uniformity arising from the need for reliable units of measurement and a bias against the irregular and the non-linear is installed from the very beginning. By contrast,

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Archimedes is interested in the minimal angle of deviation, the departure of the tangent from the curve, an element that is indiscernable to the eye and which by definition escapes measurement. It is through such imperceptible departures from an ideal path that the forms of things really emerge. The mathematical treatment of such form is concerned less with the pursuit of simplicity and purity than with the attempt to express the singular, and must therefore remain open to the complexity, impurity and ultimately the concreteness of existing things. Every form, writes Serres, 'is draped in an infinity of adherences', 'differential robes' that slide over them (BP 130:I03) . The differential geometry he finds in Archimedes is less an abstraction than 'a phenomenology of the caress, voluptuous knowledge' (BP 134:I07) . In this way Archimedian mathematics is at once closer to the concreteness of naturally occurring forms and yet less able to derive universal principles by which they may be mastered. Perceiving this as a simple weakness, history has regarded Archimedes as a brilliant yet unsystematic innovator. However, the alliance between atomism and Archimedian mathematics enriches them both, giving Lucretian physics a mathe­ matical language and the work of Archimedes a new coherence. 'It no longer forms a library, a tub ric of results and methods among others; they are an encyclopedia, a monument that testifies to a world' (BP 34:24).

The world to which it testifies, the world described by Lucretius, is a place of turbulent flows, of chaos and the emergence of order by what classical metaphysics has taught us to call chance, but which ancient atomism also knew as necessity.3 Everything begins with atoms falling through the void. Were this flow to remain laminar, the trajectory of each atom continuing parallel to that of every other, there would be no collisions and thus no combinations from which worlds are formed. Such a state is what Serres calls the 'first chaos', the absence of order in perfect order, the absence of all relation.4 But Lucretian physics is not governed by a principle of inertia and bodies do not have to await the action of an external force before undergoing a deviation from their path. Deviation occurs spontaneously, with no cause and to no end. This declination - the central concept in Lucretian atomism - is the clinamen.

The clinamen is the minimal solid angle of deviation from a laminar flow required to create turbulence, which is the condition for atoms to meet and combine and is therefore also (paradoxically) a stimulus for the emergence of order. From the chaotic turba, the confused tumult of atoms, arises the turbo, the spiral, vortex or spinning cone. Already one can begin to see that what separates ancient atomism from its modern

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Introduction

counterpart is the conception of movement and relation that accompanies it. In fact, while the material atom is a necessary and fundamental component in Lucretian physics, it is not the focus of its conceptual invention and endeavour. The ancient doctrine is an atomism of angle more than of matter. Its building block is not essentially the irreducible solid unit, but the imperceptible angle of deviation from a path that peels off into a proliferation of new forms, the fluctuation around a point of equilibrium. The treatment of motion in modern physics was inspired by the revolution of the planets, that is, by solid bodies on stable paths. As Serres tells us, for Lucretius 'the subjects of physics are mass, fluids and heat' (BP I2:5) . Everything flows, and if we are not to consider atomist physics absurd and archaic, we must, he writes, give up the general framework of solid mechanics altogether. From the Lucretian perspective, fluidity is not a particular and rare case of the general condition of solids, but rather the model from which all physics begins. 5 Solid bodies are just exceptionally slow moving fluids. Stable order exists not through resistance to change, but through the temporary maintenance of structured change. Form itself is never static and local order, which from within may give the appearance of stability, is a minimally open system that will in time return to the global flow from which it arises. It is this fluidity that is constant, what Lucretius described as the fall of atoms, and not the brief and contingent pockets of order that appear within it. Homeostasis is a local exception to global homeorrhesis.6

Inceptions: against finality

In spite of the many examples of complex flows in the world around us, the idea of movement without finality can still strike us as strange. As reluctant or recovering Aristotelians, we may continue to be swayed by the idea that order is prefigured in a potentiality that already existed in advance of its emergence, and that only what is potential can become actual, can happen. This formulation was Aristotle's response to the perceived paradox of becoming: what is cannot come from what is not, but if what is comes from what is, no change has occurred. Seeing that the paradoxical nature of this problem lay in the monolithic deter­ mination of Being as One, he allowed for a multiplicity of possible significations of Being, and in particular for the division between potential and actual Being. What exists as actual could then come to be from what exists as potential. In this way, Aristotle resolved the paradox,

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while preserving a commitment to the original unity of Being, re­ inscribed now in the structure of change itself, whose end is in every case written into its beginning.

With the idea of original multiplicity (everything that is may be unitary, but there is a countless number of such units), atomism intro­ duced at once a new theory of movement and an alternative doctrine of contingency. Where Aristotle could only regard chance as an inter­ ruption in a pre-determined causal sequence, atomism allowed for con­ tingency as the outcome of an indeterminate beginning. In Lucretius, this fundamental indeterminacy is articulated via the clinamen, which cannot therefore be interpreted as a cause of any kind. To be a cause -even as accidental or chance - is to be isolable. The clinamen, however, is by definition concealed beneath the lowest possible threshold of measurement. Its angle of deviation is indiscernable. In the same way, as an event that occurs over a time span shorter than can be detected, it eludes any attempt to identify it as having taken place at a given time.? Indeed, given the continual variation of form, even in relatively stable systems, there is no reason to suppose that it is a rare event at all. When Serres follows Lucretius in saying that the clinamen occurs at a time and place that are 'indefinite' [incerto tempore, incertisque locis ], he does not only mean that it can occur without warning, but rather that its occur­ rence cannot be localised at ali.8 The difference is important. The clinamen should not be treated as an occasional 'chance' event - that is, as a 'rogue' cause that does not obey the law - but rather as the theoretical expression of an irreducible complexity in the order of events.

It is ultimately in the void that the aleatory character of Lucretian physics resides. Without it, local conditions would in every case be continuous with the universal order (an hypothesis at play in both Laplace's dream of perfectly determinate systems and in Leibniz's monadology). Conversely, it is as a consequence of the ontological inter­ ruption by the void that local states are not continuous with global states. But if the void is the physical condition of locality vis-a.-vis the global, the dynamic condition is the clinamen itself, which from the outset interrupts the chain of cause and effect linking global or universal states to the behaviour of the local environment. The clinamen is the dynamic condition of locality. And of course this works both ways, also undermining the possibility of generalization on the basis of local observations.

If, through the clinamen, the origin of every event or system is always multiple, then every attempt to reduce change to a unilinear process

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Introduction

must necessarily fail.9 There is, therefore, no universal history, no unilinear development and thereby no single frame of reference within which all events may be encompassed (this is encapsulated in the idea of fluidity as a general phenomenon. fu Serres, writes, even the vessel that contains a given fluid is porous and an open system, it 'is itself a flow, although thicker and more complex' (BP 87=69)). There cannot even be a reliable rule of translation by which we can navigate our way from one frame or region to another. For example, we can, Serres writes, identifY several orders or rhythms of time. Events do not unfold uniformly. Indeed, time itself is described as the 'fluctuation of turbulences' (BP II5:9I) that open the dimension of time as a pocket, or pockets, of local and short­ lived order within the laminar flow. For there to be a navigable history that was anything more than an abstraction, we would have to integrate the times inherent within this multiple and variable turblence.

Without the vantage point required to survey the whole, thinking and writing chart their own paths as they find their way along, pro­ ducing local cartographies that reflect the specificity of environments, both physical and epistemological. Metaphysical thought long ago realized that human intuition is imperfect, its grasp of the whole sequential and partial when compared to God's total and instantaneous - that is, timeless - understanding. Yet it has laboured on trying to compensate for this weakness. The ideal of epistemological certainty, the persistent tug towards unity and universality, the residual distrust of the indeterminate, the episodic, idiom and noise all testifY to the extent to which, as Nietzsche predicted, we have continued to live in the shadow of God. One of the attractions of atomism for Serres is that it was never in thrall to such ideals. Its expression of human transcendence within and through nature is not strained by a negotiation with the residual language and habits of a philosophy antagonistic to such an idea.

Let us note briefly that when Serres speaks of the event of the clinamen as an instantaneous declination that falls beneath the threshold of possible measurement, this is not just the expression of a failure in the accuracy of our instruments. Lucretius has not been simply undermined by technological progress. For the breakdown in the determinable order of cause and effect has repercussions beyond the strictly scientific question of our ability to ascribe a serial order to natural events. For the introduction of an irreducible ambiguity into the relation between the determining and determined elements opens not just on to the problem of time, but also onto that of sensibility, in the guise of our affection by things, and even on to the relation between the sensible and the

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ligible, in the guise of our conceptualisation of what is given through sensibility. In short, the renewal of atomism offers important new perspectives on problems that have been at the vety heart of philosophy for a long time, and most especially modern philosophy since Kant.

Language and perception

Compared with much post-Kantian philosophy, there is a danger that Lucretian atomism may look naIve. In fact, in a certain sense it is naIve, and deliberately so. It is true, for example, that it recognises no division between reality and appearance and does not engage in a 'hermeneutics of suspicion'. The problem is less this naivety itself than the possibility that it may be mistaken for foolishness or regarded precisely as a

dangerous lack of sophistication. The issue here is that of the emergence of meaning: how do we negotiate the passage between the appearance of a thing and our speaking of it. What is the relation between nature and the language in which it is expressed?

The question leads us back from a consideration of our own language to that attributed to nature itself 'Atoms, we know, are letters or are like letters' (BP 175:141) [BP - The Birth of Physics]. Finite in variety, their

possible combinations are nonetheless infinite: letters link together into words and texts, just as atoms combine to form bodies. 'The analogy of behaviour is perfectly apt. It is a metaphor and it is not' (BP 175:141). In this way, Serres proposes an intimate relation between nature and language, such that the combinations in the one are mirrored in the other. 'Language is born with things and by the same process' (BP 153:123). Accordingly, 'Things appear bearing their language' (BP 153:123). This is not only a thought that bypasses the whole of post­ Kantian and phenomenological philosophy in its concern with how things are given and the relation between such givenness and the language in which it is ultimately expressed, it also reaches much further back into more ancient speculations on the curse that accompanies the blessing of language. When the Chorus in Sophocle's Antigone speaks of humankind as the strangest beings on earth, it seems to acknowledge the

alienation from nature that the power of language brings with it.IO For Serres, however, atomism provides a theory of language that does not imply this rupture and which leaves human beings at home in the world. A few pages after the lines quoted above, he writes: 'That atoms are letters, that connected bodies are sentences is certainly not a metaphor; were it not so, there would be no existence' (BP 185:150)." Yet iflanguage

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Introduction

is not metaphorical, a difficulty arises that seems to threaten the materialism Serres is proposing. It will be helpful to explore it briefly. Serres describes materialism as the reference of all accounts of 'feder­ ation' back to nature itself, as opposed to an idealist tendency to discover in nature some pre-determined conception of order that it had secretely placed there in advance (BP 150:121) . Yet this aspiration towards a naive empiricism is not obviously compatible with Serres' own insistence that wherever one looks, one finds the same model of movement, order and relation - that of turbulent flows and the clinamen. If nature (in all its manifestations), language, economics and morality are all isomorphic, we are bound to ask after the status of the model on which the accounts of all phenomena are based. Conversely, if the rule is variation, declination and the strict locality of order, might we not expect to find departures from the model? Should we not be suspicious of the very globality, not to say universality, of the dynamics Serres discovers in all phenomena?" The answer to this question points in two directions.

First, it is significant that in discussing the isomorphism between discourses, Serres explicitly rejects any priority of one instance over the other. Economic law is not to be derived from physical law, or vice versa: 'They are the same, that's all.' (BP 68:53). The perfection of the analogy between nature does not entail the endless reiteration of the same. In fact, paradoxically, it is precisely because the model applies everywhere that turbulence is possible not only as a feature of the phenomena described by a particular discourse, but also between discourses, in and through translation and communication. In this way, the isomorphism that seems to elevate a certain configuration above the fray in fact facilitates the declination that interrupts the perceived invariability in the instantiations of the model. Isomorphism thereby allows distortion at the local level. Given how easily materialism is associated with reductionism (a consequence of the way that materialism has been approached from within the classical paradigm), it is important to see that it is precisely by virtue of its material basis in atomism that Serres' epistemology is in fact non-reductionist.

However, and this is the second point, there is still a model said to be repeatedly instantiated in the various isomorphic discourses; namely, the mathematical model of turbulent flows. To this extent, it seems that Serres may be proposing a simple universal theory that commits him to an 'analogical unity' grounded in the mathematical model. This would indeed be a problem, were the mathematical model to be an ideal. As we have already seen, Serres' objection to geometry arises from its emphasis

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on metrics.13 The preoccupation with measurement has obscured the exploration of form itself. It is the 'applied' character that geometry takes on by virtue of its fascination with measurement that orients it towards idealism: the form it determines will be the abstracted truth of the physical thing, of nature. By contrast, for Serres it is the precisely the purity of pure mathematics that frees it from such a commitment to measurement and thereby also from any pretensions to speak the truth of nature. As pure, it is at once closed in on itself and, insofar as it is not local, universal. Above all, it is not derived from, or otherwise developed with regard to, the physical world.

The occurrence of the model therefore does not amount to the reiteration of an ideal essence. More importantly still, neither physical law nor economic law, nor any of the other isomorphic discourses can be directly read off from the mathematical model. This is encapsulated in Serres' own statement of condition of generality that keeps this from being the case: 'The theory of flows and paths is general, but ceaselessly in deviation from the general' (BP 121:96). The model therefore cannot dictate the exact and necessary form of each of its instantiations. As Serres remarks, '[declination] interrupts the universality of the laws. It opens the closed system. It places the physical laws under the rule of exception' (BP 97:77).14

The question of law in nature

We have seen already that form is not something that pre-exists the contingent formation of things, or survives their dissolution. We have also seen that for atomism there can be no reliable map by which to navigate berween the local and the global. As a result, the longstanding model of physical law as universal must be untenable. This classical conception of laws of nature as universal formulae with which nature must comply has a theological provenance. Nature is subordinate to its creator, and by extension to anyone who can divine its eternal laws. In addition, it is, for Lucretius and Serres alike, literally Martial Law: the law of Mars, the god of war. Following Lucretius, Serres calls this configuration of law the fledera foti.

All this is called into question by the atheist Lucretius, and by atomism in general. As there is no entirely stable order, no continuity between local and general, no pure form hidden beneath the flesh of things, there can be no universal laws. There is in nature simply nothing in relation to which a universal law could obtain. As in the case of

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Introduction

geometry, the universality of law is won at the price of ignoring the real complexity of open turbulent systems. In contrast to the fledera fati, Serres traces the introduction of the fledera natura -a pact, an alliance,

and not a law. Serres is careful to argue that the idea of such a pact does not reflect the projection onto nature of a political convention or arrangement. On the contrary, atomism itself deals with combinations, accomodations and temporary couplings, all of which are destined to be undone in the end. It is then our own political conventions that have modelled themselves on nature, albeit often unwittingly. For Serres, this discovery draws us into a relation with nature that in principle acknowledges the fragile condition of the regularities it describes; as opposed to calling us to uphold a law that stands over and above nature as the condition of its mastery, and which at the same time attributes to the natural order a stability and resilience that excuses our violence towards it.

Philosophy, science and mathematics

For a long time, the official story has been that whatever modest dialogue has existed between philosophy and science has been confined to particular areas of shared interest: epistemology, methodology, and to a lesser degree the philosophies of mind and language. Beyond that, relations were clouded by a mutual mistrust. Science, it was said, could not think; and once philosophy went beyond the bounds defining its incipient naturalisation, ultimately the bounds of empiricism, then it lost itself in useless metaphysical speculation. What this account would not acknowledge was the possibility that philosophy and science might communicate about more fundamental matters: ontology, the nature of space, of time, relation, number, limits. Yet if established channels of communication were indeed in poor repair, many outstanding figures in both fields succeeded in opening up important dialogues outside the generally accepted borders of their fields. In addition to Serres himself, Poincare, Weyl, Einstein, Bergson, Heisenberg, Husser!, Bachelard, Heidegger, Schrodinger, Thorn, Deleuze and Prigogine are all exem­ plary in this respect, and there are others.

If this dialogue is gaining momentum, there may be at least two contributory factors. First, as we have mentioned already, what distin­ guishes pure mathematics for Serres is its isolation both from the natural world and from the subjective ego.I5 It was precisely this, however, that

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made it less interesting to the philosophical tradition that passed via Descartes and Kant which has been preoccupied with the relation between these two domains. Where mathematics has featured in that tradition, it has done so in the form of what Serres regards as the applied discipline of geometry, thereby introducing metrics into the original structure of our experience. This situation is now changing. AB post­ Kantian and phenomenological philosophy in particular have addressed the structure and genesis of experience, they have begun to explore the limits of truth, sensibility and givenness that have characterised the philosophy of the subject. AB the subject-object framework is increas­ ingly left behind by philosophy, the possibilities for thinking themes such as space, time, relation, limits, continuity and discontinuity offered by pure mathematics will become important. Second, from the side of mathematics and science, the emergence of topology from the shadow of geometry has strengthened the resources available to pure mathematics, while the development of non-linear dynamics and chaos theory has given new impetus to the critique of finality and classical ontology that was already underway in philosophy.

Once again, central to all this is the work of Michel Serres. One of his most important contributions in this respect has been to break down the exclusive and proprietorial relation that the natural sciences, and physics in particular, have had to mathematics, yet without compromising the integrity of mathematics itself. Serres' work, far from encouraging non­ scientific disciplines to 'borrow' from mathematical and scientific discourse, has demonstrated that mathematics was never the exclusive property of what we now call the natural sciences. In addition, as interest in original multiplicity, non-linearity, contingency, local order, fragmented times and spaces, continuity and discontinuiry, fluidity and non-violence continues to grow, we are indebted to Serres above all for renewing our relation to atomism, and for reminding us that Lucretius is still our contemporary.

Staffordshire University David Webb

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Introduction

NOTES

I Literature on Lucretius is mostly made up of philological scholarship that provides exhaustive analysis of the text in the relation to ancient atomism and other ancient and classical literature. Pre-eminent amongst these are the following editions with commentaries: H. A.]. Munro, Vols 1-2 (Cambridge, 1864): C. Giussani, Vols 1-4 (Ermanno Loescher, Turin, 1896-8): C. Bailey, Vols I-III (Oxford University Press, 1947). One full length study worth particular mention for its attempt to develop a more philosophical inter­ pretation of the text is M. Bollack, La Raison de Lucrece, (Editions de Minuit, Paris, 1978). Another book that has sought to integrate the poetic and philosophical aspects of Lucretius' work is M. Gale, Myth and Poetry in Lucretius

(Cambridge University Press, Cambridge, 1994), which also contains an extensive bibliography. The most accessible translations at present are: Lucretius, On the Nature of the Universe trans. R. E. Latham (Penguin,London) and the translation by C Bailey (updated) included in ]. Gaskin, The Epicurean Philsophers (Everyman, London, 1995).

2 Referring to the idea, expressed by Husserl, that the figure of the salinon could not be mathematical because it was inexact, Serres writes: 'a superb and ancient confusion of the pure and the metrical, to which all of geometty from its origins to topology, from the Greeks to Riemann, bears witness rJait justice] in its histoty and its work. It is rigorous, anexact. And not precise, exact or inexact. Only a metric is exact' [BP 29:19]. It is only with the inception of topology that the dominant link to measurement is broken.

3 For Aristotle, only what has a final cause is a candidate for necessity. The winds, for example, though always caused in the sense that there is an antecedant to every event, do not conform to such an order. For Democritus, necessity denoted this order of events, whether we can ascribe a reason to it or not. C£ Aristotle, Physics II iv-vi: C. Bailey, The Greek Atomists and Epicurus (Oxford University Press, oxford, 1928), pp. 139-43.

4 'We say of this flow that is laminar. Everything happens as if each separable lamella in the flow acts without regard for any other. Hence there is only one question: how, in this flow, does turbulence happen? Or: how does a laminar flow become turbulent?' (BP 103:82). The further question Serres addresses via Lucretius is how order can emerge from either state; that is, how order exists as a suspension between the two extreme states of chaos.

'But what distinguishes [Lucretian physics] from modern mechanics, which will adopt the same methodology, is that shape is not metrical, and movement is not that of a solid. The form is described qualitatively, the flow is that of a liquid, a current.' (BP 124:99).

6 The term 'homeorrhesis' denotes stable flow. As an example of the contrast that Serres intends, he discusses the well known Heraclitean fragment according to which one cannot step into the same river twice (B91a,b) and concludes that while the flow (and on a larger scale, the water cycle) is stable, the banks are continually worn away by the current: if anything, the reverse is true, and one cannot sit on the same bank twice to watch the river (BP 189:153).

7 This interpretation has been advanced and explored by G. Deleuze, Logic of

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8 The phrase incerto tempore incertisque locis (Lucretius, II, 218-19) 'does not signify the nullity of place or time, and thereby the passage to the soul outside of sensible quality, but simply aleatory scattering.' (BP 140:112).

9 Deleuze makes this point too in the text cited above. C£ also Foucault's affirm­ ation of an irreducible multiplicity of causes, for example in his interest in

Entsteheung (emergence) as opposed to Ursprung (origin) in his essay 'Nietzsche, Genealogy, History', P. Rabinow (ed), The Foucault Reader (Penguin, London, 1984) pp. 76-100.

10 'Greek wisdom arrives at one of its most important points here. Where man is in the world, of the world, in matter and of matter. He is not a stranger, but a friend, a familiar, a companion and an equal' (BP 162:131).

II 'That atoms are letters is not an arbitrary theory or a decision or a metaphor. It is a necessity of what Lucretius and his predecessors called nature' (BP 182:147] . 12 C£ Shoshana Felman: 'De la nature des choses ou de l' ecart a l' equilibre' in

Critique January 1979 Vol. XXXV, No. 380, pp. 3-15.

13 'Mathematics and Philosophy: What Thales Saw . . . ' M. Serres, Hermes: Liter­ ature, Science and Philosophy, ed. J. V. Harari and D. F. Bell (London: Johns Hopkins Press, 1982). 'Ce que Thales a vu au pied des pyramids' from M. Serres,

Hermes III: La traduction (Paris, Minuit, 1974). The Origin of Geometry', 'Origin de la geometrie V' - originally appeared in Diacritics 8, no. I (spring 1978).

14 Finally, the separation and relation between the model and its instantiations in various phenomena and discourses cannot itself be regarded as metaphorical, since deviation, the ecart is itself 'the primary space in which every metaphor finds its place and time. The clinamen is transport in general' (BP 185=150).

15 C£ M Serres, Hermes I La communication (Editions de Minuit, Paris, 1969), p. 72.

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EDITIONS OF LUCRETIUS

This text has used a number of editions, in Latin and English, of De rerum natura. Where a specific translation has been used in this book, a reference has been given. All other translations have been prepared specifically for this edition by Fiona Forsyth and Clinamen Press.

Michel Serres used the Alfred Ernout French/Latin edition in the preparation of the original text.

Latin texts:

Bailey, C. (ed.) Lucretius, De rerum natura Vols I-III (Oxford University Press, 1947)

Costa, C. D. N. (ed.) Lucretius, De rerum natura Vol. V (Oxford, Clarendon, 1984)

Giussani, C. (ed.) Lucretius, De rerum natura Vols 1-4 (Ermano Loescher, Turin, 1896-98)

Munro, H. A. J. (ed.) Lucretius, De rerum natura Vols 1-2 (Cambridge, 1864)

Translations:

C. Bailey in: Gaskin, John (ed.) The Epicurean Philosophers (London, Everyman 1995)

Humphries, Rolfe, Lucretius, The Way Things Are, De rerum natura of Titus Lucretius Carus (Bloomington, Indiana University Press, 1969) Latham, Ronald Lucretius, On the Nature of the Universe (London,

Penguin 1994) Parallel Texts:

Latin/English - Rouse, W. H. D. (trans. & ed.) Lucretius, De rerum natura (Loeb Classical Library, Harvard 1975)

Latin/French - Ernout, Alfred (trans. & ed.) de Natura rerum Vols 1-2 (Paris, Les Belles Lettres, 1972)

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In a story entitled 'Averroes's Search' Borges describes that situation which is the secret nightmare of every translator. The great twelfth­ century Islamic philosopher, responsible for the preservation of much of what Greek philosophy remains to us, is busily at work translating Aristotle's Poetics from a version which, tantalizingly, still contains both the section on tragedy which we have and the section on comedy which subsequently perished. Averroes is pleased with his work, but there are still two little details which trouble him: he is unable to find an adequate translation for two words, 'comedy' and 'tragedy', two words which are, as we know, absolutely those most critical to his text. Furthermore, we recognize what Averroes cannot: not only doesn't he know what these words mean, he is incapable of understanding them, since he belongs to a culture to which the very concept of drama is foreign.

Translating Michel Serres' Birth of Physics reminded me of Borges and Averroes, since two of Serres' most important words in this work really do not go into English very well. The first is tourbillon, which can mean a vortex, an eddy, a whirlwind, a waterspout, a whirlpool, a whirling or rotational movement, and has associations both with the Old Testament and Cartesian philosophy. The second is eeart, (from the popular Latin *exquartare, to quarter), which can mean a deviation, a spatial or temporal separation, a difference, a discrepancy, a gap, or (with overtones of the clinamen) a swerve.

In both cases I have opted for consistency,: I have translated tourbillon as 'vortex' and icart as 'deviation' throughour. In this way I hoped to make the reader aware of the importance of these two concept to Serres' argument, and of the structure of that argument; but some of the richness of his text has thereby inevitably been lost.

Jack Hawkes

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PROTOCO L

First model: declination in a fluid medium

Everyone knows, everyone concedes that atomic physics is an ancient doctrine but a contemporary discovery. It is a scientific matter, the science of Perrin, Bohr or Heisenberg; the ancient doctrine is only 'philosophy,' or even poetry. Like history in general, the history of the sciences has a pre-history. Just as there is no mathematics before the Greek miracle, that ofThales or Pythagoras, so there is no physics before the blessed classical age, before what has been called, roughly since Kant and the Enlightenment, the Galilean revolution. During this pre­ history, 'philosophy' slumbered. We recognise, I believe, ideologies, religious or otherwise, by their use of the calendar as a dramatic device: before or after the birth of Christ, before or after the foundation of Rome or the first year of the Republic, before or after the establishment of the positivist doctrine, before or after the Galilean revolution. Nothing will ever again be as it was. Here is the metaphysical age; there is the positivist age.

From Cicero to Marx and beyond, down to us, the declination of atoms has been treated as a weakness of the atomic theory. The clinamen is an absurdity. A logical absurdity, since it is introduced without justification, the cause of itself before being the cause of all things; a geometrical absurdity, in that the definition that Lucretius gives is incomprehensible and confused; a mechanical absurdity, since it is contrary to the principle of inertia, and would result in perpetual motion; an absurdity of physics in general, since experimentation cannot possibly reveal its existence. No-one has ever seen a heavy body swerve suddenly from its path as it falls. Therefore we are not concerned with science. The clinamen, consequently, finds a haven in subjectivity, moving from the world to the soul, from physics to metaphysics, from the theory of inert bodies in free fall to the theory of the free movements of living beings. It is the last secret of the decision of the subject, its inclination. Lucretius' text itself points in this direction, speaking soon enough of the will as torn from destiny, and of horses that throw

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themselves from their stalls at the races. Modern materialists are most unhappy with this rupture in determinism and its reinterpretation as the

idealism of a free subject. The entire discussion of indeterminism will subsequently reproduce the classical debate on the subject of the clinamen in the domain of the sciences.

On the other hand, the absurdity of such a principle is another proof, and a decisive one this time, of the prehistoric status of Greco-Roman physics. This was not a science of the world, but an impure mixture of metaphysics, political philosophy and musings on individual freedom, projected onto the things themselves. Hence the crude finding of criticism: there was no atomist physics in Antiquity. What is more, no applied sciences in general; and the clinamen on which it is based is just an immaterial property of the subject. We must read Lucretius' De rerum natura as humanists or philologians, and not as a treatise on physics. Let us go back to Book II, where declination is introduced. It is characterised primarily by two phrases. Paulum, tantum quod momen mutatum dicere possis (II, 219-20): atoms, in free fall in space, deviate from their straight trajectory 'a little, just so much that you can call it a change of movement'. Their deviation is as small as possible, and the alteration in their movement is as small as description allows. Lucretius repeats and redefines this deviation a little further: nec plus quam minimum (II, 244), 'no more than the minimum'. classical editions remark on a rhetorical device in these lines. The thing is so absurd and so far from our experience that the physicist minimises it, as if to hide it. Now, anyone who has ever read any Latin texts on mathematics, and more specifically on differential calculus will recognise here two canonic definitions of the potential infinitely small and the actual infinitely small. This is not an anachronism; the relationship of atomism to the first attempts at infinitesimal calculus is well known. From the outset, Democritus seems to have simultaneously produced a mathematical method of exhaustion and the physical hypothesis of indivisibles. We can see here one of the earliest formulations of what will be called a differential. The clinamen is thus a differential, and properly, a fluxion.

On the subject of fluxions, let us examine the atomic cataract in which this infinitely small angular deviation is produced. In the lines which precede it, Lucretius shows that the movement of bodies cannot take place from low to high. The examples he cites are instructive; to explain the movement of fire, he uses liquid models: the flow of blood, the red gush which spurts, the fluidity of water, umor aquae (II, 197). In the same way, just prior to the long passage on the clinamen, he shows us

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Protocol

the lightning path obliquely crossing the rainfall, and shows it nunc hinc, nunc illinc (II, 214), now on this side, now on that. And the same rain is taken up again in the definition of declination, imbris uti guttae, like the drops of rain. There we have it.

The absurdity of all this to critique, and perhaps the whole question, arises from what has always been considered the original fall of atoms in the global framework of the mechanics of solids. All the more so since the emergence of the inaugural Galilean moment within this discipline. For us, mechanics is first and foremost that of solids. It's clear cut. The mechanics of fluids is or was only a special case, which the most important texts, that of Lagrange for example, only take up in the final pages and as an afterthought. But now we must reverse the perspective.

Modern science is born, or has its renaissance, in the works of Torri celli , Benedetti, Da Vinci, those of the Accademia del Cimento, which concern fluids as much if not more than solids. All of the Latin world is unified on this subject: Vitruvius expressly devotes a book of his treatise on architecture, the eighth, to the flow of water, and Frontinus writes an entire book on the aqueducts of Rome. A century before Lucretius, the works of Archimedes had raised hydrostatics to a state of perfection equal if not superior to that of ordinary statics. And both in his own time and before him the works and achievements of the Greek hydraulic engineers were remarkable.

Consequently, if it is absurd that a small solid mass might at some moment deviate from the orbit of its fall, let us examine whether the same may be said where the primary atomic cataract is like a stream, like a flux, like the flow of a liquid. Lucretius says elsewhere that the subjects of physics are mass, fluids and heat. And since for him everything flows, nothing is truly of an invincible solidity, except for atoms.

In the primary cataract, atoms are not touching. When encounters and connections occur, bodies are characterised according to their resistance. The hardest, like diamond, stone, iron or bronze, owe their solidity to the fact that their atoms are tangled, branching, knotted into a tightly-packed fabric. As we move towards the fluids and gases, the atoms are rounder and smoother rather than hooked, of course, but in particular they are less tangled among themselves .. We could even say that when the fabric is completely unravelled we are in the presence of a very subtle flow, in any case one that is not globally solid.

So there is flow; we will call it a laminar flow. This means that however small may be the lamina cut from the flow, the movement of each is strictly parallel to the movement of the others. This model is

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faithful to the description in De rerum natura. These lamellae are its elements; they are solids but the cataract is fluid. Now a laminar flow is ideal and in effect theoretical. In the real world it is very rare that all the local flows remain parallel. They always become more or less turbulent. The question to be raised, which we ask here, is the following: how do vortices form? How does turbulence appear in a laminar flow? Parallel flow is taken up in the first place as a simple model. Perhaps originary, I don't know, but in any case much less complicated or tangled than a swirling flow. Now the question that we raise and which we are in the process of resolving, through many experiments and localised theories, is exactly Lucretius' question. I will formulate it again: the fall of atoms is an ideal laminar cataract, what are the conditions under which it enters into concrete experience, that of vortical flow?

Turbulence

Now this vortex, tourbillon -OlV1], dine, Otvo�, dinos -is none other

than the primitive form of the construction of things, of nature in general, according to Epicurus and Democritus. I The world is first of all

this open movement, composed of rotation and translation. The latter is given by the flow and the fall, the laminar cascade. Question: how does rotation appear? Answer: the clinamen is the smallest imaginable condition for the original formation of turbulence. In the De finibus, Cicero wrote that atomorum turbulentit concursio. Atoms meet in and by turbulence.

Let us return to the text: just as a lightning bolt creates its oblique flight across the parallel lines of rain nunc hinc, nunc illinc, here and there, so declination appears in laminar flow as the minimum angle in the inception of turbulence, incerto tempore, incertisque locis (at an indefinite time and place, II, 2I8-19). A fresh argument with which traditional science may accuse Lucretius of ignorance and caprice. This has nothing to do with science, since the incident is indefinite in time and indefinite in place, and in any case undetermined. The argument says nothing about the model nor the description, but a great deal about its own ideal of science. For it to carry weight, knowledge should have nothing to say about chance distribution. What Lucretius says, however, remains true - that is, faithful to the phenomenon: turbulence appears stochastically in laminar flow. Why? I don't know why. How? By chance, with respect to space and time. And, once again, what is the clinamen? It is the minimum angle of formation of a vortex, appearing by chance in a laminar flow.

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Protocol

The only line of Lucretius that everyone knows by heart is the very famous Suave mari magno, generally translated as a rhapsody to selfish serenity.2 It opens Book II, where declination is introduced. Now, our cultural memory only recalls the first part. The passage continues, turbantibus aequors ventis.! Here are vortices in a fluid medium, water and wind, presented as a heading, at the beginning of the world. A repetition of the Democritean dine.

A first model may already be constructed. A working hypothesis and experimental protocol. To understand the atomistic undertaking and not consider it absurd and archaic, we must give up the general frame­ work of solid mechanics. It is that of our modern world, its very tech­ nicality and its speculation. Perhaps the Mediterranean world needed water more than tools, perhaps it was more preoccupied with rain, storms, rivers. It built reservoirs, aqueducts. Hydraulics was important to it. What is hard to understand here isn't the local occurrence of declination, but its inscription in the framework of another mechanics, another science than that of fluids. For Lucretian physics is entirely immersed in it.

Who can fail to see that a flow does not remain parallel for long, who can fail to see that a laminar flow is merely ideal and theoretical? Turbulence soon appears. In relation to theory, the appearance of con­ crete experience is contemporaneous with that of vortices. Declination is their beginning. Nothing is absurd here, everything is exact, precise, and even necessary.

We must therefore outline a sheaf of parallels. Then at some point in the flow or cataract, mark a small angle, and from this, a spiral. Within this movement the atoms, separate until now, meet: atomorum turbulenta concursio.4 But the text is still more precise: it refers to a mathemathics, a differential calculation, to the ideal of a great number, to a whole corpus implicit in the model. We need then to look for a man, one who wrote and conceived this corpus.

The work of physics begins. Here is the protocol. Here are the experi­ ments, the complete models, the awaited mathematisation, and the innumerable applications.

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The French term tourbillon that Serres adopts to denote the turbulent yet stable structure that forms in fluid flow has been translated throughout as 'vortex'. This has the. virtue of being familiar and precise in its signification of a 'three­ dimensional spiral' or 'cyclonic' structure. It is also neutral with regard to the medium in which it forms, generally water or air. Its drawback is that one loses the important etymological link between the French tourbillon and 'turbulence', 'turbination' and, ultimately, the Latin turba and turbo. It is helpful to bear these links in mind in reading the text that follows.

2 II, 1. 'How sweet it is when on the open sea . . . '

' . . . the winds are troubling the waters.'

4 Cicero, De finibus, I, 20. Cf. also Academia, I, 6.

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M AT H E M AT I CS

An analysis of the hydraulic model

History of the angle. When the classics try to describe will, freedom, or uncertainty, they often appeal to the image of a pendulum or balance. The infinitesimal angle of the beam, the smallest change in the balance of the pans, here is decision, determination, sometimes anguish, unrest. This is not' declination, says Leibniz, it is inclination. These simple machines are models. And poor models, because they are static. They were theorised theory, at the time, in terms of equilibrium, their machines were stators. Statues. And their psychology was a mechanics; or rather, the image of a statics. You forget about geometry and think you're talking about the subject. But in fact you're only talking about machines. This forgetting will last for a long time, enough at least so that at the beginning of the nineteenth century the angle in the atom is nothing but the freedom of the subject. Reality grows faint, like a dream of the soul. We must therefore go back to the Greeks.

Their classical method is the measurement of segments. Hence their sections and their polytomies. Their primary figure, the triangle, is in fact a trilateral; primary in the possible construction of figures on a plane, and thus primary in the world, as we see in the Timaeus. We have to wait a considerable time for the measurement of angles to be added to the measurement of other elements, sides or other things, that is, for the formulation of trigonometry. The angle remains a shape, a corner, like a quality, it resists efforts at quantification. Its trisection remains, for example, a very difficult problem. It is acute, pointed, obtuse, notice­ able. Less easily abstracted than a length or a segment, by which I mean that it is less easily related to number. Perhaps more to movement; this is why figures must be superposed, thus transported, with regards to measurement; it is precisely because they are angled.

N ow the first possible angle that we may construct or perceive, or the smallest that may be formed, so that nothing can be inserted between the two lines which open, is that which lies between a curve and its tangent. In the language of geometry it may be called nee plus quam

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minimum; or in the language of mechanics paulum, tantum quod momen mutatum dicere possis.' In other words, the angle appears at the same time as the curvature. Between two straight lines or two line segments, this minimal angle makes no sense. If we are calculating with shapes or rectilinear solids we only need, in general, ordinary mathematics. If, on the contrary, we square or cube curved elements, we must at least switch to a differential proto-calculus. And thus to Democritus. He left two lost books on irrational lines and solids, and it is reasonable to suppose, as do Heiberg and Tannery, that the theory of irrational numbers served him as a springboard to atomic interpretation.2 In both cases, it is a question of divisibility and indivisibility. In both cases, the last division recedes beyond our reach. This is not all: we know, from a reference in Plutarch and by a section of Archimedes' Method, that Democritus provided solutions for the volume of a cone or a cylinder, or for that of their sections, and doubtless more generally for that of a solid of revolution. Heiberg and Philipps on think, correctly, that he achieved this by integration. This presupposes a differential division, and so once again an atomist interpretation. Democritus is the Pythagoras on the side of things, of the irrational and of the differentiable. It is inevitable that the first integrator should take things to be formed of a crowd of subliminal atoms. Not yet of an infinite 'sum' of infinitely small things, but of a very great number of subdivided things. In this way, one crosses the

threshold of perception at the same time as that of operation.

This is still not all, this is nothing compared to the fact that the man of the philosophical pentathlon - the gold medal conferred on the Abderitan by Diogenes Laertius - indeed left a treatise, lost like all the others, concerning contact between the circle and the sphere. In which he discussed the angle of tangency, opposing Protagoras' view that the straight line touches the circle at more than one point. We do not know the detail of the polemic, but we know that it concerned osculation and the elements of what we would call differential geometry. What happens in the closest proximity of the curve to its tangent? What happens in the case of the smallest angle possible? And for the sake of symmetry, in the case of contact between two circles? For tangency and contingency? In passing, it is interesting to read the classics on this matter: when they write about mathematics, they speak of the angle of contingency; when they discourse on metaphysics, they write of contingency for what exists without necessity. Physics is indeed an affair of angles. End of demonstration: what we can re-establish of this sleeping pentathlon is coherent with the physics that has come down to us. Not only did the

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Mathematics

atom have to be born by way of the treatment of curved elements, in the irrational and differential, or by way of the indefinitely divisible, stopped short by arbitrarily determined limit, but also and especially by way of this minimal angle, this atom of angle, this first angle, whose idea long appeared so monstrous to modern scholarship, and which, nonetheless, is more logical or more obvious than that of the atom. This is because the angle of contingency may not be subdivided: it is demonstrably minimal. It is null, but without the lines which form it overlaying one another. It is more atomic, so to speak, than the atom. As

a result of the first infinitesimal calculus, there can be no atomism without curved elements; and no curve without a tangent, no curve without a minimal angle; thus, no atomism without declination. No atomism without the full schema of an inflected path. Cogitur flecti (II, 283) . The clinamen, like the spiral, is present and possible, from the beginning, in the geometry of the first atomist. I am not saying that Democritus himself immediately made a physics of it; we have no record of this. Except for the turbulence which, in Diogenes Laertius, is said to be the universal cause. I am just saying that his mathematics, at least

what remains of it, gives all these outlines a geometric coherence, a systematicity. What is called rigour. Atomist physics has never forgotten geometry; witness Lucretius and his definitions: nec plus quam mini­ mum, and so on. It is the commentators who have forgotten it. As, later, they forgot the angle, in their pathetic dissertation on contingency and subjective freedom. And, as far as I know, in the first of Euclid's Definitions, the angle is in fact called clisis, XAWU;, bending or inclination. What we need for the model proposed - that is, the link between atom, angle, curves - is thus certainly to be found in Democritus. We will never know how this writer, who wrote a treatise on liquids and a debate on the clepsydra, was able to realise this model in what we would today call fluid mechanics. We will never know either, parenthetically, if his three books on the plague and pestilential illnesses furnished inform­ ation for the closing lines of the De rerum natura. But there are too many vortices in Epicurus' Letter to Pythocles, for the resources of hydraulics not to have occurred at one time or another to one author or the other.

I am looking for a man, I wrote above, as I finished drawing up the

protocol. I am looking for a man, an organon.

So here, once again, is the model. First a sheaf of parallels, where a laminar flow slips by. At some point, that is to say by chance, a deviation, a very small angle is produced. A vortex forms at once from this point on. I will break down the model, I will divide it into elements.

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1. A large atomic population.

2. A tangent to a curve, an angle of contingency.

3. A solid angle, a cone.

4. A curved vortical line. 5. Infinitely small elements. 6. Balance and deviations. 7. Flows, a fluid medium.

To mathematize the model successfully, I therefore need: 1. A mathematical or arithmetic theory of elements. 2. A geometrical theory of the tangent.

3. A geometry of forms of revolution. 4. A theory of spirals.

5. An infinitesimal calculus. 6. A mechanics of equilibrium. 7. A hydrostatics.

Now, as if miraculously, this list of requisites corresponds exactly to a

very well-known catalogue of works. Suppose a mathematician had written: 1. A book entitled the The Sand-Reckoner.

2. A theory of tangency to the spiral.

3. A treatise On Conoids and Spheroids, and On the Sphere and Cylinder.

4. A book On Spirals.

5. Treatises on Measurement of the Circle and Quadrature of the Parabola.

6. A book On Plane Equilibriums 7. A treatise On Floating Bodies.

He would fulfill the requisite conditions. This man is Archimedes. Born barely twenty years before the death of Epicurus, murdered about a century before Lucretius' work. I have found the corpus. The whole corpus and nothing but the corpus.

I am in a position, then, to demonstrate several propositions. First, to

show the general unity of Archimedes' entire work. The list of what remains to us will no longer be just a rubric, a catalogue, it will designate a global system. A system which describes, in a refined mathematics, the physical model of the Epicurean world. Next, and conversely as it were,

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Mathematics

to show that atomist physics is not non-mathematical, as was believed, but that on the contrary it is analogically given mathematical expression in the Archimedian model. From which follows, in general, that the Greeks did not conceive of mathematical physics in the same way as we have done since the Renaissance. We mix experiments with equations. And we accompany the protocol, step by step, with formalism and metrics. Without this continual proximity, no experimentation, nor law. The Greeks would, I believe, have been strongly repulsed by this mixture. They did not have, as we do, a unitary mathematical physics. Theirs was double. They produced rigorous formal systems and dissertations upon nature, like two separate linguistic families, like two disjunct wholes. And, since they are often signed with completely different proper names, no one dares to think that they are structurally isomorphic. We would need a local and subtle blend of the two and we have only scattered monuments. Hence the strange idea, common to the history of the sciences, that there could have been no mathematical physics in Greece. There was, but we have to see it. And to see it by way of an example, we might carefully link Epicurus to Archimedes. Or Lucretius and his theory to the work of the Syracusan.

Archimedes' work

Silius Italicus: 'He knew the cause of the movement of the waves on the sea, what law the ocean follows in the ebb and flow of its waves. '3 He had repulsed the ordered Roman armies, from the heights of the city's ramparts to the shores of the sea. A sophisticated genius, serene, in high places fortified by the science of the wise. I like to see his great shadow there at the beginning of the Book I1.4

What is, first of all, the subject of the The Sand-Reckoner? Technically, it concerns numeration, series, the theory of increase. The first discovery of large numbers. Now, ancient atomism, like all atomic theories in general, implies the manipulation of very large populations, since the elements are subliminal. In any case, one must apprehend the per­ ceptible, and the world, in fairly compact assemblages. That said, what good is it to think of filling the sphere of the fixed stars with grains of sand gathered, or fitted, together in larger and larger balls, if we don't to some extent want, by means of an arithmetic, to make a certain model of the world rational? At least to make it possible?

This was the reasoning followed by the calculator Leibniz in the Baroque period, encouraged by his own conception of monads and by

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the animalcules discovered with the microscope.5 Archimedes, like Leibniz after or Democritus before him, is a geometer of the infini­ tesimal. In the end he arived at indivisibles, in the manner of Cavalieri, like Leibniz with the monad and Democritus with the atom. Or Giordano Bruno with his unities, or Pascal with some mite. And so he had no choice but to refer to the grain. To any grain, taken in general, a grain of sand, for example. And to dream for a moment of constituting the universe by these simple means, as every circle, every sphere and every spheroid had taught him in geometry. Thus his scales of order. Thus also this model which has remained canonical. Each time it is reconstructed in history, it is redesigned by a worker of the infinitesimal who is at the same time, from a certain point of view, an atomist. Bruno, who quotes Lucretius by name, Leibniz, and a few others, holds together what was separated, but analogous, in Archimedean mathematics and Epicurean physics. Hence the sphere of the fixed stars is filled with a sea of sand. Relatively filled, to be sure, since voids appear, lacunae at the tangencies or the contacts of the grains among themselves. A first model, naive, minimal.

The Sand-Reckoner achieves results and forges methods: the theory of ordered intervals, what we may call an approximate arithmetic cubature, what we may call Archimedes' axiom. But this harvest, brilliant as it is, may hide the essential point. The Sand-Reckoner builds a world and places all these means at the service of a model. So powerful that history will take it seriously, though as false, and will reiterate it each time the new calculus encounters arithmetic. Now this schema, there is no getting round it, is atomistic. In the final assessment, the universe is filled with grains and their lacunae, that is to say atoms and voids. Of course, here, things are homogeneous, the model is set up as closed, static, without movement, almost geometric; but we should not forget that Archimedes always thinks of this in terms of maxima and minima. There may exist at least the ten to the power sixty three of these grains. Elsewhere, in The Cattle Problem, even more, perhaps: bulls and cows, different colours. The model is naive, the model is the limit. This is the infinite in Gauss' sense: mathematically finite and physically infinite. Finally, the strategy of progressive orders clearly suggests that one may as well never stop. This clarifies the discussions of the finite and infinite; either by the formation of large numbers, or by the notation of ordered progressions. Epicurus' reflection on the limitless whole in the forty-first paragraph of the Letter to Herodotus, elucidated by Lucretius at the end of Book I by the example of the archer who shoots an arrow beyond the boundaries of

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References

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