I - Sets and Functions
§1. Set Theory - §2. Logicians' Logic
It is generally understood that mathematicians concern themselves with objects or concepts about which they establish theorems by applying logical,
preferably irrebuttable, arguments. This last detail, which remained theoretical for a long time, apart from in the theory of numbers and in the elementary
geometry inherited from the Greeks, has always scared the majority of people,
accustomed as they are to produce and hear daily, even in exalted intellectual
activities, tens of arguments each more contestable than the other: life in s0ciety would be impossible if everyone had to provide incontrovertible proofs
of his assertions and to express himself clearly and without ambiguity!.
Up to the end of the XvrI th century the objects of mathematics were
numbers, geometrical figures, equations or functions more or less directly
arising from everyday life, astronomy or mechanics: the triangles and conic
sections of the Greeks, the whole numbers, rationals, or irrationals like v'2,
the simplest algebraic equations which one sometimes tried, like Fermat, to
solve in terms of integers, the trigonometric functions which had been extensively developed by the Greek and Arab astronomers before the Westerners,
Napier's logarithms, Galileo's parabolas, the velocity of a moving body and
the calculus of tangents to a curve, which, in the second half of the xvn th
century, led to the concept of the derivative, the computation of the area
bounded by a curve - the circle or the parabola in the case of Archimedes -
which in the same period led to the integral calculus etc. Although still very
primitive until about 1600, a short while later mathematics exploded thanks
to the creation of the infinitesimal calculus by Fermat, Descartes, Huyghens,
Wallis, Cavalieri, and, above all, between about 1665 and 1720, by Newton, Leibniz and the Bernoullis. This opened an epoch, when, by the new
methods, they solved an amazing number of problems without worrying too
greatly about the validity of their proofs; the then Prince of Mathematicians
was Leonard Euler (1707-1783), a man who "calculated as he breathed", and
was so inventive that he not only discovered innumerable formulae which are
still useful, but - and this is far more difficult - also provided proofs, almost
1 See various comments on this point in Chap. II, nO 7.
§1. Set Theory - §2. Logicians' Logic
It is generally understood that mathematicians concern themselves with objects or concepts about which they establish theorems by applying logical,
preferably irrebuttable, arguments. This last detail, which remained theoretical for a long time, apart from in the theory of numbers and in the elementary
geometry inherited from the Greeks, has always scared the majority of people,
accustomed as they are to produce and hear daily, even in exalted intellectual
activities, tens of arguments each more contestable than the other: life in s0ciety would be impossible if everyone had to provide incontrovertible proofs
of his assertions and to express himself clearly and without ambiguity!.
Up to the end of the XvrI th century the objects of mathematics were
numbers, geometrical figures, equations or functions more or less directly
arising from everyday life, astronomy or mechanics: the triangles and conic
sections of the Greeks, the whole numbers, rationals, or irrationals like v'2,
the simplest algebraic equations which one sometimes tried, like Fermat, to
solve in terms of integers, the trigonometric functions which had been extensively developed by the Greek and Arab astronomers before the Westerners,
Napier's logarithms, Galileo's parabolas, the velocity of a moving body and
the calculus of tangents to a curve, which, in the second half of the xvn th
century, led to the concept of the derivative, the computation of the area
bounded by a curve - the circle or the parabola in the case of Archimedes -
which in the same period led to the integral calculus etc. Although still very
primitive until about 1600, a short while later mathematics exploded thanks
to the creation of the infinitesimal calculus by Fermat, Descartes, Huyghens,
Wallis, Cavalieri, and, above all, between about 1665 and 1720, by Newton, Leibniz and the Bernoullis. This opened an epoch, when, by the new
methods, they solved an amazing number of problems without worrying too
greatly about the validity of their proofs; the then Prince of Mathematicians
was Leonard Euler (1707-1783), a man who "calculated as he breathed", and
was so inventive that he not only discovered innumerable formulae which are
still useful, but - and this is far more difficult - also provided proofs, almost
1 See various comments on this point in Chap. II, nO 7.