The definition of number-equality through the concept of reciprocal one-to-one correlation

pp. 101-115

in: Edmund Husserl, Philosophy of arithmetic, Berlin, Springer, 2003

Abstract

Ever since Euclid's "Elements" attained the status of the model of scientific exposition, mathematicians have followed the principle of not regarding mathematical concepts as fully legitimized until they are well-distinguished by means of rigorous definitions. But this principle, undoubtedly quite useful, has not infrequently and without justification been carried too far. In over zealousness for a presumed rigor, attempts were also made to define concepts that, because of their elemental character, are neither capable of definition nor in need of it. Of this sort are the so-called "definitions" of equality and difference with respect to number whose refutation will now engage us. And they have indeed a special claim on our interest precisely because they have led to a class of definitions of the number concepts themselves. These definitions, baseless and scientifically useless, have nevertheless, in virtue of a certain formal character, found favor among mathematicians and among the philosophers influenced by them.