2
I - Sets and Functions
always shaky, even though the results were themselves correct. This way of
doing mathematics reached its apogee at the beginning of the XIXth century with Joseph Fourier and his trigonometric series, without which a large
part of present day mathematics and physics would not have been possible.
Fourier's "proofs" are not only invalid; they are also unmeaningful, resting
as they do, even if not explicitly, on equations as absurd as
1 - 3 + 5 - 7 + . . .
0,
1 3 - 3 3 + 53 - 7 3 + . . .
0,
1 5 - 3 5 + 55 - 7 5 + . . .
0
etc. His results, here again, are nevertheless correct; his first formula is the
series for the "square wave" as used by all electricians. His general theorems
on periodic functions were soon correctly proved by quite other methods
by more serious mathematicians, Abel and above all Dirichlet, who were the
first, with Cauchy, to introduce rigour and precision into analysis in the 1820s;
nevertheless Fourier had seen all the simple and fundamental results and had
invented a method which, right up to our days, has been unceasingly exploited
in situations far more general and difficult than he could have known.
One now enters progressively into a new age, when, particularly in Germany, while waiting for the French and Italians at the end of the XIX th
century, mathematicians systematically put to the test all the concepts -
limits, convergence, irrational numbers, continuity, differentiation, integration, etc. - which involve, implicitly and more often explicitly, an infinite
number of operations, and so were not defined in a perfectly clear and unambiguous way; they challenged all the "geometrically obvious" statements
which were incorrectly proved and even sometimes false if taken literally, as
was finally understood. Sometimes apparently monstrous creatures arrived
on the scene, notably in connection with Fourier series, which continued to
have surprises in store, and still, in 1997, pose very difficult problems: discontinuous functions which jump brusquely and at all rational values of the
variable, continuous curves not admitting even a single tangent, functions
which one did not know how to integrate, not because the "formula" was
not known, but because they eluded all known definitions of an integral,
continuous trajectories which passed through all the points of a square, etc .
. On the other hand, there is a branch of mathematics which has always
escaped these crises because the infinite plays no role there: arithmetic, or
algebra, and particularly the classical theory of numbers and of algebraic
equations, particularly the study of "algebraic" numbers, that is, roots of
polynomial equations with integer coefficients, to which one attempts to generalise such classical results as decomposition into a product of prime factors.
Carl Friedrich Gauss, who did many other things as well as mathematics, considered arithmetic so understood to be the "Queen of the Sciences"; he himself
was the King of Arithmetic between 1800 and 1830 .,. The study of these
numbers presents enormous methodological difficulties - it took decades of
I - Sets and Functions
always shaky, even though the results were themselves correct. This way of
doing mathematics reached its apogee at the beginning of the XIXth century with Joseph Fourier and his trigonometric series, without which a large
part of present day mathematics and physics would not have been possible.
Fourier's "proofs" are not only invalid; they are also unmeaningful, resting
as they do, even if not explicitly, on equations as absurd as
1 - 3 + 5 - 7 + . . .
0,
1 3 - 3 3 + 53 - 7 3 + . . .
0,
1 5 - 3 5 + 55 - 7 5 + . . .
0
etc. His results, here again, are nevertheless correct; his first formula is the
series for the "square wave" as used by all electricians. His general theorems
on periodic functions were soon correctly proved by quite other methods
by more serious mathematicians, Abel and above all Dirichlet, who were the
first, with Cauchy, to introduce rigour and precision into analysis in the 1820s;
nevertheless Fourier had seen all the simple and fundamental results and had
invented a method which, right up to our days, has been unceasingly exploited
in situations far more general and difficult than he could have known.
One now enters progressively into a new age, when, particularly in Germany, while waiting for the French and Italians at the end of the XIX th
century, mathematicians systematically put to the test all the concepts -
limits, convergence, irrational numbers, continuity, differentiation, integration, etc. - which involve, implicitly and more often explicitly, an infinite
number of operations, and so were not defined in a perfectly clear and unambiguous way; they challenged all the "geometrically obvious" statements
which were incorrectly proved and even sometimes false if taken literally, as
was finally understood. Sometimes apparently monstrous creatures arrived
on the scene, notably in connection with Fourier series, which continued to
have surprises in store, and still, in 1997, pose very difficult problems: discontinuous functions which jump brusquely and at all rational values of the
variable, continuous curves not admitting even a single tangent, functions
which one did not know how to integrate, not because the "formula" was
not known, but because they eluded all known definitions of an integral,
continuous trajectories which passed through all the points of a square, etc .
. On the other hand, there is a branch of mathematics which has always
escaped these crises because the infinite plays no role there: arithmetic, or
algebra, and particularly the classical theory of numbers and of algebraic
equations, particularly the study of "algebraic" numbers, that is, roots of
polynomial equations with integer coefficients, to which one attempts to generalise such classical results as decomposition into a product of prime factors.
Carl Friedrich Gauss, who did many other things as well as mathematics, considered arithmetic so understood to be the "Queen of the Sciences"; he himself
was the King of Arithmetic between 1800 and 1830 .,. The study of these
numbers presents enormous methodological difficulties - it took decades of