Analysis

Abstract

Teaching the techniques of differentiating and integrating and of developing functions in series, will not be discussed in the present chapter. Neither do I intend to relegate analysis into a little corner of a larger room, which under the name of “Topology” or “Mixed Structures” would fit better into a system of mathematics. Even less than in other fields will analysis be treated as a structure that ought to inspire deferential adoration, but rather as a tool which is badly needed by those who learn to handle it, and handled efficiently if need be. This requires other abilities than knowing elegant linguistic expressions that define what an open set is, or a limit, a differential quotient* or an integral; it is more important to have undergone these notions geometrically and numerically even if they cannot be fitted into a definition that is above reproach.

From his early childhood Chasles drank water only… A famous mathematician was pleased to remark: If Chasles would drink wine, he would possibly work on integral calculus.

Freshmen are still allowed to believe in irrational numbers, thankfully and with no criticism to view the complex domain as an indispensable tool, to naively enjoy limit processes, and to accept infinity as the multiplication table.

Should we discuss the indispensable fundamental concept in the introduction to higher analysis with due thoroughness… it seems to me that the well-known saying “a bad bargain is dear at a farthing” applies here too. I am convinced that the effort that is indispensable to learn mathematics, is not increased by the logical reinforcement of the method of proof.

J. Bertrand, Eloges acad., n.s. 1902, p. 40

E. Netto, Elementare Algebra, 1904, Vorwort

A. Pringsheim, Jahresb. D.M.V., 7 (1897-8), 142–143