Logic sometimes breeds monsters. For half a century there has been springing up a host of weird functions, which seem to strive to have as little res… - Henri Poincaré

Logic sometimes breeds monsters. For half a century there has been springing up a host of weird functions, which seem to strive to have as little resemblance as possible to honest functions that are of some use. No more continuity, or else continuity but no derivatives, etc. More than this, from the point of view of logic, it is these strange functions that are the most general; those that are met without being looked for no longer appear as more than a particular case, and they have only quite a little corner left them. Formerly, when a new function was invented, it was in view of some practical end. Today they are invented on purpose to show our ancestors' reasoning at fault, and we shall never get anything more than that out of them.

La pensée ne doit jamais se soumettre, ni à un dogme, ni à un parti, ni à une passion, ni à un intérêt, ni à une idée préconçue, ni à quoi que ce soit, si ce n'est aux faits eux-mêmes, parce que, pour elle, se soumettre, ce serait cesser d'être.

John Stuart Mill understood the word existence in a material and empirical sense; he meant that in defining a circle we assert that there are round things in nature.
In this form his opinion is inadmissible. Mathematics is independent of the existence of material objects. In mathematics the word exist can only have one meaning ; it signifies exemption from contradiction.

Doubting everything and believing everything are two equally convenient solutions that guard us from having to think.