46
II - Convergence: Discrete variables
7r is the ratio between the length of a circumference and that of the diameter, presumes that we have a mathematically exact definition of "lengths",
and not just a cadastral or physical one; which is far from obvious. Since,
what is more, the modern algebraic notation had not then been developed,
everything had to be explained geometrically. The mathematicians of the
Renaissance and of the xvn th century inherited this point of view, too rigorous for the age; the invention of the differential and integral calculus about
1665-1700 made it disappear, banished it to a lower level, to the benefit of
the prodigiously effective quasimechanical methods of Calculus, whose lack
of rigour would certainly have scandalised the Greeks. One had to wait until
the second half of the XIX th century to clarify and simplify, by reversing the
procedure: the real numbers were then defined by procedures as rigorous as
those of arithmetic, and one was then able to define precisely the length of a
curve, the area of a surface, etc. In other words, geometry was no longer the
foundation of analysis, but the other way round: even though, of course, the
first still continues to provide indispensable intuitions.
We spoke above of "so-called" real numbers; why so-called? One might
insist that a magnitude to be measured, say the diagonal of a square, or
a circumference, is an eminently real object. But it all arises, in the final
analysis, from the desire to arrive at an absolute exactitude which certainly
exists in the minds of mathematicians, if not in physical reality, where, even
leaving aside the non-euclidean character of the universe, one never meets
"points", "lines", "squares" or "circumferences" in the mathematical sense
of the term. Apart perhaps from the whole numbers, mathematical objects,
starting from the numbers we call real, are only, at best, idealised models
of real objects. Add to this the essential fact that it is impossible to define
an irrational number 2 without the intervention of infinitely many elementary
arithmetical operations or of rational numbers, a situation which does not
arise in "reality" or "Nature", and even less, if this were possible, in the
experimental sciences.
True, physicists, engineers, etc. constantly use the number IT, with only
a bare mention, not bothering to reflect on its exact mathematical meaning.
the history and the construction of the rational numbers, real or complex, of
7r and other more advanced subjects, see H. D. Ebbinghaus et al., Numbers
(Springer, 1991), a book to be recommended from all points of view.
2 The real and complex numbers divide into two categories. First there are the
algebraic numbers which, by definition, satisfy algebraic equations with rational
coefficients: the rational numbers, and those obtained by extracting roots (for
example i,~, V's), the roots of the equation x 1848 - 3.14159x 1789 + 2.718 = 0,
etc. This set is a field included between Q and C. The other numbers are called
transcendental; 7r is one such. The set of algebraic numbers is countable, but not
the set of transcendental numbers, so not R. A simple procedure for constructing
transcendental numbers was discovered by Joseph Liouville in 1844: suppose that,
for n large, the nth decimal digit is 1= 0 if and only there exists apE N such that
n = 1.2.3 ... p = p!. See for example Christian Houzel, Analyse mathematique
(Belin, 1996), p. 64.
II - Convergence: Discrete variables
7r is the ratio between the length of a circumference and that of the diameter, presumes that we have a mathematically exact definition of "lengths",
and not just a cadastral or physical one; which is far from obvious. Since,
what is more, the modern algebraic notation had not then been developed,
everything had to be explained geometrically. The mathematicians of the
Renaissance and of the xvn th century inherited this point of view, too rigorous for the age; the invention of the differential and integral calculus about
1665-1700 made it disappear, banished it to a lower level, to the benefit of
the prodigiously effective quasimechanical methods of Calculus, whose lack
of rigour would certainly have scandalised the Greeks. One had to wait until
the second half of the XIX th century to clarify and simplify, by reversing the
procedure: the real numbers were then defined by procedures as rigorous as
those of arithmetic, and one was then able to define precisely the length of a
curve, the area of a surface, etc. In other words, geometry was no longer the
foundation of analysis, but the other way round: even though, of course, the
first still continues to provide indispensable intuitions.
We spoke above of "so-called" real numbers; why so-called? One might
insist that a magnitude to be measured, say the diagonal of a square, or
a circumference, is an eminently real object. But it all arises, in the final
analysis, from the desire to arrive at an absolute exactitude which certainly
exists in the minds of mathematicians, if not in physical reality, where, even
leaving aside the non-euclidean character of the universe, one never meets
"points", "lines", "squares" or "circumferences" in the mathematical sense
of the term. Apart perhaps from the whole numbers, mathematical objects,
starting from the numbers we call real, are only, at best, idealised models
of real objects. Add to this the essential fact that it is impossible to define
an irrational number 2 without the intervention of infinitely many elementary
arithmetical operations or of rational numbers, a situation which does not
arise in "reality" or "Nature", and even less, if this were possible, in the
experimental sciences.
True, physicists, engineers, etc. constantly use the number IT, with only
a bare mention, not bothering to reflect on its exact mathematical meaning.
the history and the construction of the rational numbers, real or complex, of
7r and other more advanced subjects, see H. D. Ebbinghaus et al., Numbers
(Springer, 1991), a book to be recommended from all points of view.
2 The real and complex numbers divide into two categories. First there are the
algebraic numbers which, by definition, satisfy algebraic equations with rational
coefficients: the rational numbers, and those obtained by extracting roots (for
example i,~, V's), the roots of the equation x 1848 - 3.14159x 1789 + 2.718 = 0,
etc. This set is a field included between Q and C. The other numbers are called
transcendental; 7r is one such. The set of algebraic numbers is countable, but not
the set of transcendental numbers, so not R. A simple procedure for constructing
transcendental numbers was discovered by Joseph Liouville in 1844: suppose that,
for n large, the nth decimal digit is 1= 0 if and only there exists apE N such that
n = 1.2.3 ... p = p!. See for example Christian Houzel, Analyse mathematique
(Belin, 1996), p. 64.