4
I - Sets and Functions
mathematical object, and of reasoning about such objects; the appearance of
the "actual infinity" in mathematics, a concept which for centuries has set
cohorts of metaphysicians and theologians cogitating, and often deliriously
- Cantor knows and quotes them - and which therefore at the beginning
provoked violent opposition from among the mathematicians 2 . Cantor and
Dedekind, with whom he cooperated, were at first interested in more and
more complicated sets of real numbers - here again it was the theory of
trigonometric series which furnished Cantor with his initial motivation and
examples, though the fields of algebraic numbers and Dedekind's "ideals" are
also sets of numbers, even if much simpler 3 , - then to sets which, in the eyes
of many mathematicians of the period, were a matter for metaphysics rather
than normal mathematics, a criticism which, at the beginning, was legitimated by the logical paradoxes which arose on applying modes of reasoning
that rely on an abusive extension of ordinary language to them. One should
note that before attempting the assault on "transfinite numbers" which, a
century later, are rarely used in practice, Cantor introduced extraordinarily useful concepts into the study of sets of numbers or points - open and
closed sets, "accumulation points" etc. - which do not seem to pose logical
problems 4 , which many mathematicians then revealed in their most orthodox
research, and which would complete the clarification of analysis undertaken
by Weierstrass, of whom, moreover, Cantor had been a student.
Then a period opens which sees the birth of mathematical logic, of the
theory of "abstract" sets, and first attempts to axiomatise all of mathematics, starting with the theory of integers, i.e. arithmetic. Founded by Gottlob
Frege and pursued by Giuseppe Peano, Bertrand Russell and Alfred North
Whitehead, Ernest Zermelo, David Hilbert, etc., to mention only authors
known or famous before 1914, these new theories were intended to codify
on the one hand the rules of construction of mathematical arguments (elementary logical operations, use of "variables", the formulation and calculus
of "propositions", etc.), and on the other hand the rules of construction of
mathematical objects (numbers, functions, sets, etc.), and finally to isolate
2 The opposition came not so much from the fact that the bizarre sets that Cantor
constructed contained an infinite number of elements: no one objected to considering a line, a plane, a curve, as an infinite collection of points, nor to considering,
for example, the intersection of a plane and a surface. The major objection arose
from the fact that Cantor sometimes constructed sets by an infinite number of
intermediate constructions, with arbitrary choices at each stage, inexplicit, and
even impossible to describe explicitly. Present day mathematicians do this every
day, but it was not so around 1870-1880.
3 In the ring of rational integers Z (Le. of arbitrary sign) an ideal is a set I possessing the property that ux + vy E I for every x, y E I and u, v E Z. Such an
ideal is the set of multiples of some integer.
4 But fifty years it later would be discovered that one of the conjectures which
Cantor and others endeavoured in vain to prove (the Continuum Hypothesis)
is in fact unprovable and one can, at will, accept or reject it, as with Euclid's
Parallel Postulate.
I - Sets and Functions
mathematical object, and of reasoning about such objects; the appearance of
the "actual infinity" in mathematics, a concept which for centuries has set
cohorts of metaphysicians and theologians cogitating, and often deliriously
- Cantor knows and quotes them - and which therefore at the beginning
provoked violent opposition from among the mathematicians 2 . Cantor and
Dedekind, with whom he cooperated, were at first interested in more and
more complicated sets of real numbers - here again it was the theory of
trigonometric series which furnished Cantor with his initial motivation and
examples, though the fields of algebraic numbers and Dedekind's "ideals" are
also sets of numbers, even if much simpler 3 , - then to sets which, in the eyes
of many mathematicians of the period, were a matter for metaphysics rather
than normal mathematics, a criticism which, at the beginning, was legitimated by the logical paradoxes which arose on applying modes of reasoning
that rely on an abusive extension of ordinary language to them. One should
note that before attempting the assault on "transfinite numbers" which, a
century later, are rarely used in practice, Cantor introduced extraordinarily useful concepts into the study of sets of numbers or points - open and
closed sets, "accumulation points" etc. - which do not seem to pose logical
problems 4 , which many mathematicians then revealed in their most orthodox
research, and which would complete the clarification of analysis undertaken
by Weierstrass, of whom, moreover, Cantor had been a student.
Then a period opens which sees the birth of mathematical logic, of the
theory of "abstract" sets, and first attempts to axiomatise all of mathematics, starting with the theory of integers, i.e. arithmetic. Founded by Gottlob
Frege and pursued by Giuseppe Peano, Bertrand Russell and Alfred North
Whitehead, Ernest Zermelo, David Hilbert, etc., to mention only authors
known or famous before 1914, these new theories were intended to codify
on the one hand the rules of construction of mathematical arguments (elementary logical operations, use of "variables", the formulation and calculus
of "propositions", etc.), and on the other hand the rules of construction of
mathematical objects (numbers, functions, sets, etc.), and finally to isolate
2 The opposition came not so much from the fact that the bizarre sets that Cantor
constructed contained an infinite number of elements: no one objected to considering a line, a plane, a curve, as an infinite collection of points, nor to considering,
for example, the intersection of a plane and a surface. The major objection arose
from the fact that Cantor sometimes constructed sets by an infinite number of
intermediate constructions, with arbitrary choices at each stage, inexplicit, and
even impossible to describe explicitly. Present day mathematicians do this every
day, but it was not so around 1870-1880.
3 In the ring of rational integers Z (Le. of arbitrary sign) an ideal is a set I possessing the property that ux + vy E I for every x, y E I and u, v E Z. Such an
ideal is the set of multiples of some integer.
4 But fifty years it later would be discovered that one of the conjectures which
Cantor and others endeavoured in vain to prove (the Continuum Hypothesis)
is in fact unprovable and one can, at will, accept or reject it, as with Euclid's
Parallel Postulate.