Every definition implies an axiom, since it asserts the existence of the object defined. The definition then… — Henri Poincaré
Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted.
Toute définition implique un axiome, puisqu'elle affirme l'existence de l'objet défini. La définition ne sera donc justifiée, au point de vue purement logique, que quand on aura démontré qu'elle n'entraîne pas de contradiction, ni dans les termes, ni avec les vérités antérieurement admises.
p. 131Source: Wikiquote · CC BY-SA 4.0
More from this book
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.
p. 139
It is because simplicity and vastness are both beautiful that we seek by preference simple facts and vast facts; that we take delight, now in following the giant courses of the stars, now in scrutinizing the microscope that prodigious smallness which is also a vastness, and now in seeking in geological ages the traces of a past that attracts us because of its remoteness.
p. 23
I think I have already said somewhere that mathematics is the art of giving the same name to different things.
p. 31
