50
II - Convergence: Discrete variables
- the set of real numbers, contained in P(Q), - possessing these properties
(Chap. I, nO 4).
This is a construction of a "modern mathematical" type, very much less
intuitive than writing a nonterminating sequence of digits onto a strip of
paper of infinite length and keeping on moving always further right along
the line in the eternally frustrated hope of arriving one day at the aim of
your quest, the Grail: the last digit to the right. It at least has the advantage
of being mathematically correct, which could be the reason why Richard
Dedekind (1831-1916), a through and through algebraist (theory of fields of
algebraic numbers) who would not confuse mathematics and physics, invented
it in 1858 and published it in 1872; one might consider this the birth date of
modernity in mathematics, which consists of constructing all mathematical
objects with the aid of logic and set theory and establishing their properties
starting from there.
This construction strips the real numbers, for example 'If, of all their mystery: reasoning about 'If becomes, in this perspective, reasoning about all the
rational numbers < 'If. All the calculators, and in particular the physicists
and engineers, have always done this, since they replace the nonterminating
decimal expansions of 'If by, for example, its 25 first digits. But the genius
of Dedekind's idea was to see that instead of privileging one or other more
or less arbitrary or artificial procedure of approximating a real number by
rational numbers, it was simpler to identify it with the totality of its rational
approximations from below, i.e. to a subset of the set Q of rational numbers,
and to use this to define the algebraic operations and inequalities in JR..
For the reader's convenience we recall briefly the definition of complex
numbers which we shall use constantly, a much easier enterprise than defining
the real numbers.
It is often believed that they were invented to provide roots for the
quadratic equations ax 2 + bx + c = 0 when b 2 - 4ac < O. This is not so:
the Italians of the XVI th century invented them because having found miraculous formulae for solving third degree equations, they discovered that these
formulae, although sometimes featuring square roots of negative numbers ..
thus apparently "impossible" - nevertheless provided a real root 6 when one
substituted the formula in the equation calculating d la Bertrand Russell,
i.e. not knowing of what one speaks. They were thus led to introduce new
"numbers" of the form a + bA, where a and b are usual numbers, and to
calculate mechanically with them, bearing in mind the "fact" that the square
of A is equal to -1. Later, Euler introduced the convention of denoting
this strange number by the letter i; it took a long time before everyone, and
particularly the "users", adopted it.
6 An equation of odd degree with real coefficients always possesses at least one
real root.
II - Convergence: Discrete variables
- the set of real numbers, contained in P(Q), - possessing these properties
(Chap. I, nO 4).
This is a construction of a "modern mathematical" type, very much less
intuitive than writing a nonterminating sequence of digits onto a strip of
paper of infinite length and keeping on moving always further right along
the line in the eternally frustrated hope of arriving one day at the aim of
your quest, the Grail: the last digit to the right. It at least has the advantage
of being mathematically correct, which could be the reason why Richard
Dedekind (1831-1916), a through and through algebraist (theory of fields of
algebraic numbers) who would not confuse mathematics and physics, invented
it in 1858 and published it in 1872; one might consider this the birth date of
modernity in mathematics, which consists of constructing all mathematical
objects with the aid of logic and set theory and establishing their properties
starting from there.
This construction strips the real numbers, for example 'If, of all their mystery: reasoning about 'If becomes, in this perspective, reasoning about all the
rational numbers < 'If. All the calculators, and in particular the physicists
and engineers, have always done this, since they replace the nonterminating
decimal expansions of 'If by, for example, its 25 first digits. But the genius
of Dedekind's idea was to see that instead of privileging one or other more
or less arbitrary or artificial procedure of approximating a real number by
rational numbers, it was simpler to identify it with the totality of its rational
approximations from below, i.e. to a subset of the set Q of rational numbers,
and to use this to define the algebraic operations and inequalities in JR..
For the reader's convenience we recall briefly the definition of complex
numbers which we shall use constantly, a much easier enterprise than defining
the real numbers.
It is often believed that they were invented to provide roots for the
quadratic equations ax 2 + bx + c = 0 when b 2 - 4ac < O. This is not so:
the Italians of the XVI th century invented them because having found miraculous formulae for solving third degree equations, they discovered that these
formulae, although sometimes featuring square roots of negative numbers ..
thus apparently "impossible" - nevertheless provided a real root 6 when one
substituted the formula in the equation calculating d la Bertrand Russell,
i.e. not knowing of what one speaks. They were thus led to introduce new
"numbers" of the form a + bA, where a and b are usual numbers, and to
calculate mechanically with them, bearing in mind the "fact" that the square
of A is equal to -1. Later, Euler introduced the convention of denoting
this strange number by the letter i; it took a long time before everyone, and
particularly the "users", adopted it.
6 An equation of odd degree with real coefficients always possesses at least one
real root.