Translator's Preface

Alain Badiou's Number and Numbers, first published two years after his Being and Event, is far from being the specialist work its title might suggest. In fact, it recapitulates and deepens Being and Event's explosion of the pretexts upon which the ‘philosophy of mathematics’ is reduced to a theoretical ghetto; and their kinship to those reactionary modes of thought that systematically obscure the most pressing questions for contemporary philosophy. Neither does Number and Numbers balk at suggesting that even the greatest efforts on the part of number-theorists themselves have fallen short of the properly radical import of the question of number. Badiou's astonishing analyses in the historical section of the book uncover the inextricable bond between philosophical assumptions and mathematical approaches to the problem in these supposedly ‘merely technical’ works. The aim of Number and Numbers, then, is certainly not to mould the unwilling reader into a calculating machine, or a ‘philosopher of mathematics’: its exhortation is that we (mathematicians, philosophers, subjects under Capital) systematically think number out of the technical, procedural containment of which its quotidian tyranny, and the abysmal fear it strikes into the heart of the non-mathematician, are but symptoms. Symptoms, needless to say, whose expression within the situation of philosophy is a pronounced distaste for number-as-philosopheme – whence its recognisable absence in much ‘continental philosophy’, except where it is pilloried as the very nemesis of the ontological vocation. So if the ‘return of the numerical repressed’ proposed here will, by definition, excite a symptomatic resistance, for Badiou it alone can clear the way for the proper task of philosophy; as a working-through of the mathematical ontology presented in Being and Event, Number and Numbers is a thorough conceptual apprenticeship preparatory to the thinking of the event.

For the great thinkers of number-theory at the end of the nineteenth century, the way to an ontological understanding of number was obscured by calculatory and operational aspects. Today, according to Badiou, the political domination of number under capitalism demands that the project be taken up anew: only if contemporary philosophy rigorously thinks through number can it hope to cut through the apparently dense and impenetrable capitalist fabric of numerical relations, to think the event that can ‘subtract’ the subject from that ‘ontic’ skein without recourse to an anti-mathematical romanticism.

Whilst this doubtless demands ‘one more effort’ on the part of the non-mathematician, it would be a peevish student of philosophy who, understanding the stakes and contemplating the conceptual vista opened up, saw this as an unreasonable demand – especially when Badiou offers to those lacking in mathematical knowledge the rare privilege of taking a meticulously navigated conceptual shortcut to the heart of the matter.

Badiou's remarkable book comprises a number of different works – a radical philosophical treatise, a contribution to number-theory, a document in the history of mathematics, a congenial textbook and a subtle and subversive exercise in political theory – whose intricate interdependencies defy any order of priority. The translator's task is to reproduce, with a foreign tongue, that unique voice that can compel us to ‘count as one’ these disparate figures. In negotiating this challenge, I have sought to prioritise clarity over adherence to any rigid scheme of translation, except where mathematical terminology demands consistent usage, or where an orthodoxy is clearly already in force within extant translations of Badiou's work. In the latter case, my references have been Oliver Feltham's landmark translation of Being and Event,^¹ with which I have sought to harmonise key terms, Peter Hallward's invaluable A Subject to Truth,^² and Ray Brassier and Alberto Toscano's collection of Badiou's Theoretical Writings.^³ Apart from these, in translating chapters 2 and 3 I referred closely to Sam Gillespie and Justin Clemens' translation in UMBR(a), Science and Truth (2000). Finally, whilst seeking also to maintain continuity with long-standing English translations of number-theoretical works, some classics in their own right, occasionally the rigour of Badiou's thinking has demanded a re-evaluation of their chosen translations for key terms.^⁴ Translators also find themselves obliged to arbitrate between a fidelity to Badiou's in many ways admirable indifference to the pedantic apparatus of scholarly citation, and the temptation to pin down the allusions and quotations distributed throughout his work. Badiou's selection of texts is so discerning, however, that it is hardly a chore to return to them. Having thus had frequent recourse to the texts touched on in Number and Numbers (particularly in the first, historical part), I have seen no reason not to add citations where appropriate.

One presumes that those self-conscious styles of philosophical writing that necessitate laboured circumlocutions or terminological preciosity on the part of a translator would for Badiou fall under the sign of ‘modern sophistry’, taken to task herein, as elsewhere in his work. Nevertheless, the aspiration to universal conceptual transparency does not preclude consideration of Badiou as stylist: firstly, as Oliver Feltham has remarked, Badiou's sentences utilise subject/verb order in a characteristic way, and I have retained his tensile syntax whenever doing so does not jeopardise comprehension in translation. Perhaps just as importantly, Badiou does not achieve the deft and good-humoured development of such extremely rich and complex conceptual structures as are found in Number and Numbers without a generous and searching labour on behalf of the reader, not to mention a talent for suspense. Although the later sections of Number and Numbers may seem daunting, I hope to have reproduced Badiou's confident, meticulous, but never stuffy mode of exposition so as to ease the way as much as possible. In fact, in contrast to his own occasionally chilly edicts, I would venture to suggest that here, ‘in his element’, Badiou allows himself a certain enthusiasm. One certainly does not accompany him on this odyssey without also developing a taste for the ‘bitter joy’ of Number.

This translation slowly came to fruition on the basis of a somewhat impulsive decision; it may not have survived to completion without the enthusiasm and aid of an internationally dispersed group of friends and acquaintances, actual and virtual, with whom I shared the work in progress. I would like to extend my thanks to those who helped by pointing out errors and offering advice on the evolving manuscript: Anindya Bhattacharyya, Ray Brassier, Michael Carr, Howard Caygill, Thomas Duzer, Zachary L. Fraser, Peter Hallward, Armelle Menard Seymour, Reza Negarestani, Robin Newton, Nina Power, Manuela Tecusan, Alberto Toscano, Keith Tilford, David Sneek, and Damian Veal. My thanks also to Alain Badiou for his generous help and encouragement, and to the Institution and Staff of the Bodleian, Taylor Institution, and Radcliffe Science Libraries in Oxford. Part of my work on the translation was undertaken whilst in receipt of a studentship from the Centre for Research in Modern European Philosophy at Middlesex University, London.

My greatest debt of gratitude is to Ruth, without whose love and understanding my battles with incomprehension could not even be staged; and to Donald, a great inspiration, for whom the infinite joys of number still lie ahead.

Robin Mackay

Notes

0
Number Must Be Thought

0.1. A paradox: we live in the era of number's despotism; thought yields to the law of denumerable multiplicities; and yet (unless perhaps this very default, this failing, is only the obscure obverse of a conceptless submission) we have at our disposal no recent, active idea of what number is. An immense effort has been made on this point, but its labours were essentially over by the beginning of the twentieth century: they are those of Dedekind, Frege, Cantor, and Peano. The factual impact of number only escorts a silence of the concept. How can we grasp today the question posed by Dedekind in his 1888 treatise, Was sind und was sollen die Zahlen?^¹ We know very well what numbers are for: they serve, strictly speaking, for everything, they provide a norm for All. But we still don't know what they are, or else we repeat what the great thinkers of the late nineteenth century – anticipating, no doubt, the extent of their future jurisdiction – said they were.

0.2. That number must rule, that the imperative must be: ‘count!’ – who doubts this today? And not in the sense of that maxim which, as Dedekind knew, demands the use of the original Greek when retraced: ἀεὶ ὁ ἄνθϱωπος ἀϱιθμητίζει^² – because it prescribes, for thought, its singular condition in the matheme. For, under the current empire of number, it is not a question of thought, but of realities.

0.3. Firstly, number governs our conception of the political, with the currency – consensual, though it enfeebles every politics of the thinkable – of suffrage, of opinion polls, of the majority. Every ‘political’ convocation, whether general or local, in polling-booth or parliament, municipal or international, is settled with a count. And every opinion is gauged by the incessant enumeration of the faithful (even if such an enumeration makes of every fidelity an infidelity). What counts – in the sense of what is valued – is that which is counted. Conversely, everything that can be numbered must be valued. ‘Political Science’ refines numbers into sub-numbers, compares sequences of numbers, its only object being shifts in voting patterns – that is, changes, usually minute, in the tabulation of numbers. Political ‘thought’ is a numerical exegesis.

0.4. Number governs the quasi-totality of the ‘human sciences’ (although this euphemism can barely disguise the fact that what is called ‘science’ here is a technical apparatus whose pragmatic basis is governmental). Statistics invades the entire domain of these disciplines. The bureaucratisation of knowledges is above all an infinite excrescence of numbering.

At the beginning of the twentieth century, sociology unveiled its proper dignity – its audacity, even – in the will to submit the figure of communitarian bonds to number. It sought to extend to the social body and to representation the Galilean processes of literalisation and mathematisation. But ultimately it succumbed to an anarchic development of this enterprise. It is now replete with pitiful enumerations that serve only to validate the obvious or to establish parliamentary opportunities.

History has drawn massively upon statistical technique and is – even, in fact above all, under the auspices of academic Marxism – becoming a diachronic sociology. It has lost that which alone had characterised it, since the Greek and Latin historians, as a discipline of thought: its conscious subordination to the real of politics. Even when it passes through the different phases of reaction to number – economism, sociologism – it does so only to fall into their simple inverse: biography, historicising psychologism.

And medicine itself, apart from its pure and simple reduction to its scientific Other (molecular biology), is a disorderly accumulation of empirical facts, a huge web of blindly tested numerical correlations.

These are ‘sciences’ of men made into numbers, to the saturation point of all possible correspondences between these numbers and other numbers, whatever they might be.

0.5. Number governs cultural representations. Of course, there is television, viewing figures, advertising. But that's not the most important thing. It is in its very essence that the cultural fabric is woven by number alone. A ‘cultural fact’ is a numerical fact. And, conversely, whatever produces number can be culturally located; that which has no number will have no name either. Art, which deals with number only in so far as there is a thinking of number, is a culturally unpronounceable word.

0.6. Obviously, number governs the economy; and there, without a doubt, we find what Louis Althusser would have called the ‘determination in the last instance’ of its supremacy. The ideology of modern parliamentary societies, if they have one, is not humanism, law, or the subject. It is number, the countable, countability. Every citizen is expected to be cognisant of foreign trade figures, of the flexibility of the exchange rate, of fluctuations in stock prices. These figures are presented as the real to which other figures refer: governmental figures, votes and opinion polls. Our so-called ‘situation’ is the intersection of economic numericality and the numericality of opinion. France (or any other nation) can only be represented on the balance-sheet of an import–export business. The only image of a country is this inextricable heap of numbers in which, we are told, its power is vested, and which, we hope, is deemed worthy by those who record its mood.

0.7. Number informs our souls. What is it to exist, if not to give a favourable account of oneself? In America, one starts by saying how much one earns, an identification that is at least honest. Our old country is more cunning. But still, you don't have to look far to discover numerical topics that everyone can identify with. No one can present themselves as an individual without stating in what way they count, for whom or for what they are really counted. Our soul has the cold transparency of the figures in which it is resolved.

0.8. Marx: ‘the icy water of egotistical calculation’.^³ And how! To the point where the Ego of egoism is but a numerical web, so that the ‘egotistical calculation’ becomes the cipher of a cipher.

0.9. But we don't know what a number is, so we don't know what we are.

0.10. Must we stop with Frege, Dedekind, Cantor or Peano? Hasn't anything happened in the thinking of number? Is there only the exorbitant extent of its social and subjective reign? And what sort of innocent culpability can be attributed to these thinkers? To what extent does their idea of number prefigure this anarchic reign? Did they think number, or the future of generalised numericality? Isn't another idea of number necessary, in order for us to turn thought back against the despotism of number, in order that the Subject might be subtracted from it? And has mathematics simply stood by silently during the comprehensive social integration of number, over which it formerly had monopoly? This is what I wish to examine.

Notes

1
Greek Number and Modern Number

1.1. The Greek thinkers of number related it back to the One, which, as we can still see in Euclid's Elements,^¹ was considered not to be a number. From the supra-numeric being of the One, unity is derived. And a number is a collection of units, an addition. Underlying this conception is a problematic that stretches from the Eleatics through to the Neoplatonists: that of the procession of the Multiple from the One. Number is the schema of this procession.

1.2. The modern collapse of the Greek thinking of number proceeds from three fundamental causes.

The first is the irruption of the problem of the infinite – ineluctable from the moment when, with differential calculus, we deal with the reality of series of numbers which, although we may consider their limit, cannot be assigned any terminus. How can the limit of such a series be thought as number through the sole concept of a collection of units? A series tends towards a limit: it is not the addition of its terms or its units. It cannot be thought as a procession of the One.

The second cause is that, if the entire edifice of number is supported by the being of the One, which is itself beyond being, it is impossible to introduce, without some radical subversion, that other principle – that ontological stopping point of number – which is zero, or the void. It could be, certainly – and Neoplatonist speculation appeals to such a thesis – that the ineffable and archi-transcendent character of the One can be marked by zero. But then the problem comes back to numerical one: how to number unity, if the One that supports it is void? This problem is so complex that, as we shall see, it remains today the key to a modern thinking of number.

The third reason, and the most contemporary one, is the pure and simple dislocation of the idea of a being of the One. We find ourselves under the jurisdiction of an epoch that obliges us to hold that being is essentially multiple. Consequently, number cannot proceed from the supposition of a transcendent being of the One.

1.3. The modern thinking of number thus found itself compelled to forge a mathematics subtracted from this supposition. In so doing, it took three different paths:

Frege's approach, and that of Russell (which we will call, for brevity, the logicist approach), seeks to ‘extract’ number from a pure consideration of the laws of thought itself. Number, according to this point of view, is a universal trait^² of the concept, deducible from absolutely original principles (principles without which thought in general would be impossible).

Peano's and Hilbert's approach (let's call this the formalist approach) construes the numerical field as an operational field, on the basis of certain singular axioms. This time, number occupies no particular position as regards the laws of thought. It is a system of rule-governed operations, specified in Peano's axioms by way of a translucid notational practice, entirely transparent to the material gaze. The space of numerical signs is simply the most ‘originary’ of mathematics proper (preceded only by purely logical calculations). We might say that here the concept of number is entirely mathematised, in the sense that it is conceived as existing only in the course of its usage: the essence of number is calculation.

The approach of Dedekind and Cantor, and then of Zermelo, von Neumann and Gödel (which we shall call the set-theoretical or ‘platonising’ approach) determines number as a particular case of the hierarchy of sets. The fulcrum, absolutely antecedent to all construction, is the empty set; and ‘at the other end’, so to speak, nothing prevents the examination of infinite numbers. The concept of number is thus referred back to an ontology of the pure multiple, whose great Ideas are the classical axioms of set theory. In this context, ‘being a number’ is a particular predicate, the decision to consider as such certain classes of sets (the ordinals, or the cardinals, or the elements of the continuum, etc.) with certain distinctive properties. The essence of number is that it is a pure multiple endowed with certain properties relating to its internal order. Number is, before being made available for calculation (operations will be defined ‘on’ sets of pre-existing numbers). Here we are dealing with an ontologisation of number.

1.4. My own approach will be as follows:

The logicist perspective must be abandoned for reasons of internal consistency: it cannot satisfy the requirements of thought, and especially of philosophical thought.
The axiomatic, or operational, thesis is the thesis most ‘prone’ to the ideological socialisation of number: it circumscribes the question of a thinking of number as such within an ultimately technical project.
The set-theoretical thesis is the strongest. Even so, we must draw far more radical consequences than those that have prevailed up to the present. This book tries to follow the thread of these consequences.

1.5. Whence my plan: To examine the theses of Frege, Dedekind and Peano. To establish myself within the set-theoretical conception. To radicalise it. To demonstrate (a most important point) that in the framework of this radicalisation we will rediscover also (but not only) ‘our’ familiar numbers: whole numbers, rational numbers, real numbers, all, finally, thought outside of ordinary operational manipulations, as subspecies of a unique concept of number, itself statutorily inscribed within the ontology of the pure multiple.

1.6. Mathematics has already proposed this reinterpretation, as might be expected, but only in a recessive corner of itself, blind to the essence of its own thought: the theory of surreal numbers, invented at the beginning of the 1970s by J. H. Conway (On Numbers and Games, 1976),^³ taken up firstly by D. E. Knuth (Surreal Numbers, 1974),^⁴ and then by Harry Gonshor in his canonical book (An Introduction to the Theory of Surreal Numbers, 1986).^⁵ Any interest we might have in the technical details of this theory will be here strictly subordinated to the matter in hand: establishing a thinking of number that, by fixing the latter's status as a form of the thinking of Being, can free us from it sufficiently for an event, always trans-numeric, to summon us, whether this event be political, artistic, scientific or amorous. Limiting the glory of number to the important, but not exclusive, glory of Being, and thereby demonstrating that what proceeds from an event in terms of truth-fidelity can never be, has never been, counted.

1.7. None of the modern thinkers of number (I understand by this, I repeat, those who, between Bolzano and Gödel, tried to pin down the idea of number at the juncture of philosophy and the logico-mathematical) have been able to offer a unified concept of number. Customarily we speak of ‘number’ with respect to natural whole numbers,^⁶ ‘relative’ (positive and negative) whole numbers, rational numbers (the ‘fractions’), real numbers (those that number the linear continuum) and, finally, complex numbers and quaternions. We also speak of number in a more directly set-theoretical sense when designating types of well-orderedness (the ordinals) and pure quantities of any multiple whatsoever, including infinite quantities (the cardinals). We might expect that a concept of number would subsume all of these cases, or at least the more ‘classical’ among them, that is to say, the whole natural numbers (the most obvious schema of discrete ‘stepwise’ enumeration) and the real numbers (the schema of the continuum). But this is not at all the case.

1.8. The Greeks clearly reserved the concept of number for whole numbers, which was quite in keeping with their conception of the composition of number on the basis of the One, since only natural whole numbers can be represented as collections of units. To treat of the continuum, they used geometrical denominations, such as the relations between sizes or measurements. So their powerful conception was marked through and through by that division of mathematical disciplines on the basis of whether they treat of one or the other of what were held by the Greeks to be the two possible types of object: numbers (from which arithmetic proceeds) and figures (from which, geometry). This division refers, it seems to me, to the two orientations whose unity is dialectically effectuated by effective, or materialist, thought: the algebraic orientation, which works by composing, connecting, combining elements; and the topological orientation, which works by perceiving proximities, contours and approximations, and whose point of departure is not elementary belongings but inclusion, the part, the subset.^⁷ This division is still well-founded. Within the discipline of mathematics itself, the two major divisions of Bourbaki's great treatise, once the general ontological framework of set theory is set out, deal with ‘algebraic structures’ and ‘topological structures’.^⁸ And the validity of this arrangement subtends all dialectical thought.

1.9. It is nevertheless clear that, ever since the seventeenth century, it has no longer been possible to place any sufficiently sophisticated mathematical concept exclusively on one side of the opposition arithmetic/geometry. The triple challenge of the infinite, of zero and of the termination of the idea of the One disperses the idea of number, shreds it into a refined dialectic of geometry and arithmetic, of the topological and the algebraic. Cartesian analytic geometry radically subverts the distinction from the very outset, and what we know today as ‘number-theory’ had to call on the most complex resources of ‘geometry’, in the extremely broad sense in which the latter has been understood in recent decades. Moderns therefore can no longer accept the concept of number as the object whose provenance is foundational (the idea of the One) and whose domain is prescribed (arithmetic). ‘Number’ is said in many senses. But which of these senses constitutes a concept, allowing something singular to be proposed to thought under this name?

1.10. The response to this question, in the work of the thinkers I have mentioned, is altogether ambiguous and exhibits no unanimity whatsoever. Dedekind, for example, can legitimately be named as the first one to have, with the notion of the cut, convincingly ‘generated’ the real numbers from the rationals.^⁹ But when he poses the question: ‘What are numbers?’ he responds with a general theory of ordinals which certainly, as a particular case, might found the status of whole numbers, but which cannot be applied directly to real numbers.^¹⁰ In which case, what gives us the right to say that real numbers are ‘numbers’? Similarly, in The Foundations of Arithmetic^¹¹ Frege offers a penetrating critique of all previous definitions (including the Greek definition of number as a ‘set of units’)^¹² and proposes a concept of ‘cardinal number’ that in effect subsumes – on the basis of certain arguable premises, to which I shall later return – cardinals in the set-theoretical sense, of which natural whole numbers represent the finite case. But at the same time he excludes ordinals, to say nothing of rational numbers, real numbers or complex numbers. To use one of his favourite expressions, such numbers do not ‘fall under the [Fregean] concept’ of number. Finally, it is clear that Peano's axiomatic defines whole numbers and them alone, as a rule-governed operational domain. Real numbers can certainly be defined directly with a special axiomatic (that of a complete, totally ordered Archimedean field). But, if the essence of ‘number’ resides in the specificity of the statements constituting these axiomatics, then, given that these statements are entirely dissimilar in the case of the axiomatic of whole numbers and of that of real numbers, it would seem that, in respect of their concept, whole numbers and real numbers have nothing in common.

1.11. It is as if, challenged to propose a concept of number that can endure the modern ordeal of the defection of the One, our thinkers reserve the concept for one of its ‘incarnations’ (ordinal, cardinal, whole, real …), without being able to account for the fact that the idea and the word ‘number’ are used for all of these cases. More particularly, they prove incapable of defining any unified approach, any common ground, for discrete numeration (whole numbers), continuous numeration (real numbers) and ‘general’, or set-theoretical, numeration (ordinals and cardinals). And yet it was precisely the problem of the continuum, the dialectic of the discrete and the continuous, which, saturating and subverting the ancient opposition between arithmetic and geometry, compelled the moderns to rethink the idea of number. In this sense their work, admirable as it is in so many ways, is a failure.

1.12. The anarchy thus engendered (and I cannot take this anarchy to be innocent of the unthinking despotism of number) is so much the greater in so far as the methods put to work in each case are totally disparate:

Natural whole numbers can be determined either by means of a special axiomatic, at whose heart is the principle of recurrence (Peano), or by means of a particular (finite) case of a theory of ordinals, in which the principle of recurrence becomes a theorem (Dedekind).
To engender negative numbers, algebraic manipulations must be introduced that do not bear on the ‘being’ of number, but on its operational arrangement, on structures (symmetricisation of addition).
These manipulations can be repeated to obtain rational numbers (symmetricisation of multiplication).
Only a fundamental rupture, marked this time by a shift towards the topological, can found the passage to real numbers (consideration of infinite subsets of the set of rationals, cuts or Cauchy sequences).
We return to algebra to construct the field of complex numbers (algebraic closure of the Real Field, adjunction of the ‘ideal’ element i = $c1-math-0001$ , or direct operational axiomatisation on pairs of real numbers).
Ordinals are introduced through the consideration of types of order (Cantor), or through the use of the concept of transitivity (von Neumann).
The cardinals are treated through a totally different procedure, that of biunivocal correspondence.^¹³

1.13. This arsenal of procedures was historically deployed according to overlapping lines which passed from the Greeks, the Arab algebraists and those of Renaissance Italy, through all the founders of modern analysis, down to the ‘structuralists’ of modern algebra and the set-theoretical creations of Dedekind and Cantor. How are we to extract from it a clear and univocal idea of number, whether we think it as a type of being or as an operational concept? All that the thinkers of number have been able to do is to demonstrate the intellectual procedures that lead us to each species of ‘number’. But, in doing so, they left number as such in the shadow of its name. They remained distant from that ‘unique number which cannot be any other’,^¹⁴ whose stellar insurrection Mallarmé proposed.

1.14. The question, then, is as follows: is there a concept of number capable of subsuming, under a single type of being answering to a uniform procedure, at least natural whole numbers, rational numbers, real numbers and ordinal numbers, whether finite or infinite? And does it even make sense to speak of a number without at once specifying which singular, irreducible apparatus it belongs to? The answer is yes. This is precisely what is made possible by the marginal theory, which I propose to make philosophically central, of ‘surreal numbers’.

This theory offers us the true contemporary concept of number, and in doing so it overcomes the impasse of the thinking of number in its modern-classical form, that of Dedekind, Frege and Cantor. On its basis, and as the result of a long labour of thought, we can prevail over the blind despotism of the numerical unthought.

1.15. We must speak not of a single age of the modern thinking of number, but of what one might call, taking up an expression Natacha Michel applies to literature, the ‘first modernity’ of the thinking of number.^¹⁵ The names of this first modernity are not those of Proust and Joyce, but those of Bolzano, Frege, Cantor, Dedekind and Peano. I am attempting the passage to a second modernity.

1.16. I have said that the three challenges to which a modern doctrine of number must address itself are those of the infinite, of zero and of the absence of any grounding by the One. If we compare Frege and Dedekind – so close on so many points – on this matter, we immediately note that the order in which they arrange their responses to these challenges differs in an essential respect:

On the infinite Dedekind, with admirable profundity, begins with the infinite, which he determines with a celebrated positive property: ‘A system S is said to be infinite when it is similar to a proper part of itself.’^¹⁶ And he undertakes immediately to ‘prove’ that such an infinite system exists. The finite will be determined only subsequently, and it will be the finite that is the negation of the infinite (in which regard Dedekind's numerical dialectic has something of the Hegelian about it).^¹⁷ Frege, on the other hand, begins with the finite, with natural whole numbers, of which the infinite will be the ‘prolongation’ or the recollection in the concept.^¹⁸

On zero Dedekind abhors the void and its mark, and says so quite explicitly: ‘[W]e intend here for certain reasons wholly to exclude the empty system which contains no elements at all.’^¹⁹ Whereas Frege makes the statement ‘zero is a number’^²⁰ the rock of his whole edifice.

On the One There is no trace of any privileging of the One in Frege (precisely because he starts audaciously with zero). So one – rather than the One – comes only in second place, as that which falls under the concept ‘identical to zero’ (the one and only object that falls under the concept being zero itself, we are entitled to say that the extension of this concept is one). Dedekind, on the other hand, retains the idea that we should ‘begin’ with one: ‘the base-element 1 is called the base-number of the number-series N’.^²¹ And, correlatively, Dedekind falls back without hesitation on the idea of an absolute All^²² of thought, an idea that could not appear as such in Frege's formalism: ‘My own realm of thoughts, i.e. the totality S of all things, which can be objects of my thought, is infinite.’^²³ Thus we see that, in retaining the rights of the One, the All is supposed, because the All is that which, necessarily, proceeds from the One, once the One is.

1.17. These divergences of order are no mere technical matter. They relate, for each of these thinkers, to the respective centre of gravity of their conception of number and – as we shall see – to the simultaneous stopping point and founding point of their thought: the infinite and existence for Dedekind, zero and the concept for Frege.

1.18. The passage to a second modernity of the thinking of number obliges thought to return to zero, to the infinite and to the One. A total dissipation of the One, an ontological decision as to the being of the void and that which marks it, a lavishing without measure of infinities: such are the parameters of such a passage. Unbinding from the One delivers us to the unicity of the void and to the dissemination of the infinite.

Copyright page

Translator's Preface

Notes

0
Number Must Be Thought

Notes

1
Genealogies: Frege, Dedekind, Peano, Cantor

1
Greek Number and Modern Number

Notes