Georg Cantor > Quotes

“My beautiful proof lies all in ruins.”
― Georg Cantor

“The essence of mathematics is in its freedom.”
― Georg Cantor

“In Mathematics the art of proposing a question must be held of higher value than solving it.”
― Georg Cantor

“Some infinites are bigger than other infinites.”
― John Green via Augustus Waters

“In der Mathematik ist die Kunst Fragen zu stellen wertvoller als Probleme zu lösen.”
― Georg Cantor

“Je le voie, mais je ne le crois pas.”
― Georg Cantor

“In particular, in introducing new numbers, mathematics is only obliged to give definitions of them, by which such a definiteness and, circumstances permitting, such a relation to the older numbers are conferred upon them that in given cases they can definitely be distinguished from one another. As soon as a number satisfies all these conditions, it can and must be regarded as existent and real in mathematics. Here I perceive the reason why one has to regard the rational, irrational, and complex numbers as being just thoroughly existent as the finite positive integers.”
― Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers

“what wonderful power there is in the real numbers, since one is in a position to determine uniquely, with a single coordinate, the elements of an n-dimensional continuous space.”
― Georg Cantor

“A false conclusion once arrived at and widely accepted is not easily dislodged and the less it is understood the more tenaciously it is held.”
― Georg Cantor

“One other thing to keep in mind, though, is that Cantor's transfinite math will end up totally undercutting Aristotelian objections like the above (b) to Dedekind's proof, since Cantor's theory will constitute direct evidence that actually-infinite sets can be understood and manipulated, truly handled by the human intellect, just as velocity and acceleration are handled by calculus. So one thing to appreciate up front is that, however abstract infinite systems are, after Cantor they are most definitely not abstract in the nonreal/unreal way that unicorns are.”
― Georg Cantor

“The idea of considering the infinitely large not only in the form of the unlimitedly increasing magnitude and in the closely related form of convergent infinite series...but to also fix it mathematically by numbers in the definite form of the completed infinite was logically forced upon me, almost against my will since it was contrary to traditions which I had come to cherish in the course of many years of scientific effort and investigations.”
― Georg Cantor

“The transfinite numbers themselves are in a certain sense new irrationals, and in fact I think the best way to define the finite irrational numbers is entirely similar; I might even say in principle it is the same as my method for introducing transfinite numbers. One can absolutely assert: the transfinite numbers stand or fall with the finite irrational numbers; they are alike in their most intrinsic nature; for the former like these latter are definite, delineated forms or modifications of the actual infinite.”
― Georg Cantor