Number and Numbers

Alain Badiou

• Publisher: Polity
• Year: 2018
• Print ISBN: 9780745638799, 9780745638782
• eBook ISBN: 9781509534067
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy & Ethics in Science

The political regime of global capitalism reduces the world to an endless network of numbers within numbers, but how many of us really understand what numbers are? Without such an understanding, how can we challenge the regime of number?

In Number and Numbers Alain Badiou offers an philosophically penetrating account with a powerful political subtext of the attempts that have been made over the last century to define the special status of number. Badiou argues that number cannot be defined by the multiform calculative uses to which numbers are put, nor is it exhausted by the various species described by number theory. Drawing on the mathematical theory of surreal numbers, he develops a unified theory of Number as a particular form of being, an infinite expanse to which our access remains limited. This understanding of Number as being harbours important philosophical truths about the structure of the world in which we live.

In Badiou's view, only by rigorously thinking through Number can philosophy offer us some hope of breaking through the dense and apparently impenetrable capitalist fabric of numerical relations. For this will finally allow us to point to that which cannot be numbered: the possibility of an event that would deliver us from our unthinking subordination of number.

1
Genealogies: Frege, Dedekind, Peano, Cantor

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 trait² 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:

1. The logicist perspective must be abandoned for reasons of internal consistency: it cannot satisfy the requirements of thought, and especially of philosophical thought.
2. 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.
3. 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, includi...

Table of contents

• Cover
• Table of Contents
• Title page
• Copyright page
• Translator's Preface
• 0: Number Must Be Thought
• 1: Genealogies: Frege, Dedekind, Peano, Cantor
• 2: Concepts: Natural Multiplicities
• 3: Ontology of Number: Definition, Order, Cuts, Types
• 4: Operational Dimensions
• Index
• End User License Agreement