20 quotes
My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer.
Journey Through Genius (1990)
In mathematics the art of asking questions is more valuable than solving problems.
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.
Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand!
Letter (1885)
Each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite.
A set is a Many that allows itself to be thought of as a One.
Infinity and the Mind (1995)
The totality of all alephs cannot be conceived as a determinate, well-defined, and also a finished set.
Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number.
Letter to Richard Dedekind (1899)
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.
Infinity and the Mind (1995)
The essence of mathematics lies entirely in its freedom.
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.
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.
However, he was born in Copenhagen, of Jewish parents, of the Portuguese Jewish community there.
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.
I entertain no doubts as to the truths of the transfinites, 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.
Modern Mathematicians (1995)
I believe that there is no part of matter which is not — I do not say divisible — but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures.
Out of the Mouths of Mathematicians : A Quotation Book for Philomaths (1993)
Infinity, in its first form (the improper-infinite) presents itself as a variable finite [veranderliches Endliches]; in the other form (which I call the proper infinite [Eigentlich-unendliche]) it appears as a thoroughly determinate [bestimmtes] infinite.
This view [of the infinite], which I consider to be the sole correct one, is held by only a few. While possibly I am the very first in history to take this position so explicitly, with all of its logical consequences, I know for sure that I shall not be the last!
Journey Through Genius (1990)
The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.
Mind Tools: The Five Levels of Mathematical Reality (1988)
What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending ladder of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers.
Understanding the Infinite (1994)