I - Sets and Functions
3
efforts by several mathematicians of the first rank before Dedekind discovered
after 1870 what has become the guiding thread up to our days, the theory of
ideals -, but the results obtained, though still partial and limited in scope,
and the methods used, left no doubt: the logic of proof was unassailable.
One knew exactly what one was speaking of when proving a theorem, even
though, to be sure, not all of the very complex mechanism which governs
the properties of algebraic numbers had yet been discovered. The problems
of arithmetic call for the detective's art; those posed by the fundamentals of
analysis are rather a matter for philosophical reflection.
The main part of these developments in arithmetic was due, here again,
to the Germans influenced by Gauss; he himself de facto founded a dynasty
which ruled the subject for most of a century. It is hardly surprising that, in
these circumstances, other Germans, sometimes they themselves (Dedekind),
little by little elaborated a program which was named to arithmetise analysis,
as Felix Klein put it in 1895: in other words, to substitute irrebuttable proofs
in place of the hazy or wrong arguments of the XVIIth and XVIII th centuries,
to substitute completely clear concepts for the vague intuitions founded on
deceptive geometric images - no one will ever draw the graph of a continuous
nondifferentiable function -, in sum, as much as possible to "replace computation by concepts" as, much later, Bourbaki would write in the Preface to
his Elements of Mathematics. As we shall see at the beginning of the next
chapter, one does not have to go very far to understand the need for this
arithmetisation: what is the number 1T" ?
There are two fundamental aspects here. In the first place, to elaborate
a systematic technique for manipulating approximations to allow one to give
perfectly precise definitions of such concepts as convergence, continuity, differentiability etc. This started, still rather vaguely, with Cauchy in about
1820, and was completed about 1870-1880 with Weierstrass and his usage of
c and 8, Epsilontik as our German colleagues term it; vast, more or less abstract generalisations emerged in the XXth century, where the c and 8 would
be replaced by the concept of "neighbourhood", but the ideas remained essentially his own, and his technique remained necessary and often sufficient
in the immense majority of branches of analysis. As I shall use them from
start to finish of this book (though my typist's habit is to use rand r' where
Weierstrass and all contemporary mathematicians use c and 8), it is needless
to say more here than, in essence, it consists of demonstrating equalities by
replacing them by more and more precise inequalities: one shows that a = b
by proving that la - bl < IlIon for every integer n. This is the fundamental
difference between analysis and arithmetic or algebra.
The other aspect, which spread less easily because it constituted a major
upset to the modes of thought of mathematicians, was the invention of Set
Theory by Georg Cantor (1845-1918) between about 1870 and 1890: to a first
approximation this consists of conceptualising that the totality of mathematical objects possessing a given property forms, in itself, a new quite distinct
3
efforts by several mathematicians of the first rank before Dedekind discovered
after 1870 what has become the guiding thread up to our days, the theory of
ideals -, but the results obtained, though still partial and limited in scope,
and the methods used, left no doubt: the logic of proof was unassailable.
One knew exactly what one was speaking of when proving a theorem, even
though, to be sure, not all of the very complex mechanism which governs
the properties of algebraic numbers had yet been discovered. The problems
of arithmetic call for the detective's art; those posed by the fundamentals of
analysis are rather a matter for philosophical reflection.
The main part of these developments in arithmetic was due, here again,
to the Germans influenced by Gauss; he himself de facto founded a dynasty
which ruled the subject for most of a century. It is hardly surprising that, in
these circumstances, other Germans, sometimes they themselves (Dedekind),
little by little elaborated a program which was named to arithmetise analysis,
as Felix Klein put it in 1895: in other words, to substitute irrebuttable proofs
in place of the hazy or wrong arguments of the XVIIth and XVIII th centuries,
to substitute completely clear concepts for the vague intuitions founded on
deceptive geometric images - no one will ever draw the graph of a continuous
nondifferentiable function -, in sum, as much as possible to "replace computation by concepts" as, much later, Bourbaki would write in the Preface to
his Elements of Mathematics. As we shall see at the beginning of the next
chapter, one does not have to go very far to understand the need for this
arithmetisation: what is the number 1T" ?
There are two fundamental aspects here. In the first place, to elaborate
a systematic technique for manipulating approximations to allow one to give
perfectly precise definitions of such concepts as convergence, continuity, differentiability etc. This started, still rather vaguely, with Cauchy in about
1820, and was completed about 1870-1880 with Weierstrass and his usage of
c and 8, Epsilontik as our German colleagues term it; vast, more or less abstract generalisations emerged in the XXth century, where the c and 8 would
be replaced by the concept of "neighbourhood", but the ideas remained essentially his own, and his technique remained necessary and often sufficient
in the immense majority of branches of analysis. As I shall use them from
start to finish of this book (though my typist's habit is to use rand r' where
Weierstrass and all contemporary mathematicians use c and 8), it is needless
to say more here than, in essence, it consists of demonstrating equalities by
replacing them by more and more precise inequalities: one shows that a = b
by proving that la - bl < IlIon for every integer n. This is the fundamental
difference between analysis and arithmetic or algebra.
The other aspect, which spread less easily because it constituted a major
upset to the modes of thought of mathematicians, was the invention of Set
Theory by Georg Cantor (1845-1918) between about 1870 and 1890: to a first
approximation this consists of conceptualising that the totality of mathematical objects possessing a given property forms, in itself, a new quite distinct