A History of Mathematics From Mesopotamia to Modernity

256 A History ofMathematics

2

1

0

• 1 –0.5 0 0.5 1
  x

1.5

–1

–2

0

0

Fig. 8Graph ofy^2 =x^3 −x.

Fig. 9A torus (again); this is the shape of a (complex) elliptic ‘curve’ in the complex projective plane; two real dimensions, one

complex dimension.

Appendix A. From Bourbaki, ‘Algebra’, Introduction

HOW TO USE THIS TREATISE

1. The treatise takes mathematics at its beginning, and gives complete proofs. Consequently its
   reading presupposes, in principle, no mathematical knowledge, but only a certain habit of
   mathematical reasoning, and a certain ability to abstract.
   Nonetheless, the treatise is particularly aimed at readers who have at least a good knowledge
   of the subjects taught, in France, in courses of ‘mathématiques générales’ (abroad, in the first or
   first two years of university), and, if possible, a knowledge of the essentials of the differential
   and integral calculus.


2. The first part of the treatise is devoted to the fundamental structures of analysis (on the
   meaning of the word ‘structure’ see book I, chapter 5); in each of the books into which this
   part is divided, we study one of these structures, or several structures which are closely related
   (book I,Theory of Sets; book II,Algebra; book III,General Topology; books to follow:Integration,
   combinatorial topology, differentials and integrals of differentials, etc.)...

The method of exposition followed in the first part is axiomatic and abstract; it proceeds on the

whole from the general to the particular. The choice of this method was imposed by the principal

aim of this first part, which was to provide solid foundations for all the rest of the treatise, and even

for the whole of modern mathematics.

Appendix B. Turing on computable numbers

We have said that the computable numbers are those whose decimals are calculable by finite means.
This requires rather more explicit definition. No real attempt will be made to justify the definitions