One of the ways in which the artificial languages of mathematics are “generous”, that is, in which they assists the advance of thought, is through its 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, 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.

Gewöhnlich glaubt der Mensch, wenn er nur Worte hört es müsse sich dabei wohl auch was denken lassen Goethe, Faust I, 2565-2566

He gives the kids free samples because he knows full well that today’s young innocent faces will be tomorrow’s clientele Tom Lehrer, “The Old Dope Peddler”