Generosity: No doubt, but at times excessive and delusive

One of the ways in which the artificial languages of mathematics are ‘‘generous'', that is, in which they assist the advance of thought, is through their establishment of advanced operatory structures that permit an even further advance of intuition. However, this generosity may be delusive, suggest ideas which in the longer run turn out to be untenable. The paper analyses two cases of ‘‘honest generosity'', namely a ‘‘proof'' of the sign rule ‘‘less times less makes plus'' from the 1340s and a result in partition theory obtained by Euler by means of rash manipulations of infinite series and products, one perilous case-Cantor's introduction of transfinite numbers from 1895-and (in modern terms) a failed attempt to extend the semi-group of algebraic powers into a complete group, also from c. 1340