文档介绍：Number and Numbers 1
Alain Badiou
Number and Numbers
(Des Travaux/Seuil 1990)
Translation © 2005 by Robin Mackay
***DRAFT***
updated 12 September 2005
Number and Numbers 2
Number and Numbers 3
Contents
0. The Necessity of Thinking Number 5
1. GENEALOGIES: FREGE, DEDEKIND, PEANO, CANTOR 9
1. Greek Number and Modern Number 10
2. Frege 20
3. Additional note on a contemporary usage of Frege 29
4. Dedekind 38
5. Peano 55
6. Cantor: The "Well-Ordered" and the Ordinals 64
2. CONCEPTS: NATURAL MULTIPLICITIES 72
7. Transitive Multiplicities 73
8. Von Neumann Ordinals 80
9. ession and limit. The Infinite. 87
10. Recurrence, or Induction 98
11. The Whole Natural Numbers 109
Number and Numbers 4
Introduction
Number and Numbers 5
0. The Necessity of Thinking Number
. A paradox: we live in the era of the despotism of number, thought is submitted to
the law of denumerable multiplicities, and yet (or rather precisely in so far as this
default, this failure, is nothing but the obscure obverse of a submission without
concept) we have at our disposal no recent, active idea of what number is. The
question has been the subject of immense intellectual effort, but for the most part the
significant achievements of this labour belong to the beginning of the twentieth
century: they are those of Dedekind, Frege, Cantor, and Peano. The factual impact of
number brings with it only a silence of the concept. How can we understand today
Dedekind’s question, posed in his 1888 treatise, Was sind und was sollen die Zahlen1?
What purpose numbers serve, we know very well: they serve, strictly speaking, for
everything, they provide the norm for everything. But what they are we don’t know,
or we repeat what the great thinkers of the end of the eenth century – no doubt
anticipating the extent of their future domain – said they were.
. That number reigns, that the imperative must be: "count!" – who doubts this
today? And not in the sense of that maxim which,