文章与日志

数学是否需要一个基础？这个问题在19世纪变得尖锐和热烈起来。
Felix Klein在1893年8、9月间，非常感性地描述了当时的三类数学家，他们对于数学的基础抱有相当不同的态度和观念。

Among mathematicians in general, three main categories may be distinguished; and perhaps the names logicians, formalists, and intuitionists may serve to characterize them, (1) The word logician is here used, of course, without reference to the mathematical logic of Boole, Peirce, etc. ; it is only intended to indicate that the main strength of the men belonging to this class lies in their logical and critical power, in their ability to give strict definitions, and to derive rigid deductions therefrom. The great and wholesome influence exerted in Germany by Weierstrass in this direction is well known. (2) The formalists among the mathematicians excel mainly in the skilful formal treatment of a given question, in devising for it an "algorithm." Gordan, or let us say Cayley and Sylvester, must be ranged in this group. (3) To the intuitionists, finally, belong those who lay particular stress on geometrical intuition (Anschauung), not in pure geometry only, but in all branches of mathematics. What Benjamin Peirce has called " geometrizing a mathematical question" seems to express the same idea. Lord Kelvin and von Staudt may be mentioned as types of this category. 1

1. 1. Felix Klein,Lectures on Mathematics