In mathematics, the art of asking questions is more valuable than solving problems.
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.
Great innovation only happens when people aren’t afraid to do things differently.
To ask the right question is harder than to answer it.
A set is a Many that allows itself to be thought of as a One.
The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.
Mathematics is entirely free in its development, and its concepts are only linked by the necessity of being consistent, and are co-ordinated with concepts introduced previously by means of precise definitions.
The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great.
There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated.
Don’t always blindly follow guidance and step-by-step instructions; you might run into something interesting.
My beautiful proof lies all in ruins.
The essence of mathematics lies precisely in its freedom.
I entertain no doubts as to the truths of the tranfinites, which I recognized with God’s help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science.
Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.
Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number.
I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.
Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand!
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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.