A Museum for Mathematics
| The Garden of Archimedes
A Museum for Mathematics |
The arrangement of analysis operated by Cauchy exposed a series of problems connected to the concept of continuity of real numbers. These also emerged in the study of the convergence of Fourier's series and in that of discontinuity and derivability, in the definition of function itself and in the use of Dirichlet's principle. In his lectures and in some papers presented to the Academy of Berlin Weierstrass had already raised the issue of a rigorous definition of real numbers, which he considered to be a necessary step for his theory of analytic functions.
In 1872 Edward Heine (1821-1881) gave the first systematic presentation of the ideas of Weierstrass in publishing the article Die Elemente der Functionenlehre in the "Journal für die reine und angewandte Mathematik", founded by Crelle. An edition very widely read and studied in Italy on the subject is the Essay of introduction to the theory of analytic functions according to the principles of the prof. Weierstrass, which opens with the explanation of fundamental principles of arithmetic, the theory of integers and rationals and the theory of real numbers. The essay was published in 1880 in the "Giornale di Matematiche" by Salvatore Pincherle (1853-1936), a student of Betti and Dini in Pisa, who went to Berlin to attend Weierstrass' lectures.
In the year of the publication of the article by Heine, the article of an ex student of Weierstrass, Georg Ferdinand Cantor (1845-1918), appeared on the "Mathematische Annalen", Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen, that is "on the extension of a theorem of the theory of trigonometric series". Here, having to consider infinite sets of points in relation to the problem of the convergence of series, he first states an arithmetic theory of real numbers to be able to operate rigorously. Real numbers were defined by employing sequences of rational numbers under the condition known today as "of Cauchy". For such numbers, he defined the concept of equivalence and the usual arithmetic operations and then separated the set of points in different sets, depending on the derivative nth sets.
A similar plan is found in the Nouveau précis d'analyse infinitésimale, published in 1872 by Charles Meray (1835-1911) and anticipated in 1869 in one of the notes he released on the "Revue des Sociétés Savantes".
Of a slightly different nature is the other fundamental contribution, due to Richard Dedekind, that also appeared in 1872 Stetigkeit und irrationale Zahlen, that is "continuity and rational numbers" . Dedekind had been a student of Gauss in Gottinga where he had also attended the conferences Dirichlet gave and, while teaching elements of calculus in Zurich, he developed his considerations on a rigorous basis of the idea of continuum. Starting from the study of the properties of rational numbers, Dedekind states that "the essence of continuity" resides in what is known as the "axiom of Dedekind". Real numbers are, therefore, created, abandoning geometric intuition in favour of the arithmetic of rational numbers, through sections and then demonstrating the properties of system, defining the usual arithmetic operations and the concept of limit.