Georg Cantor Quotes

The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly disimilar to, and I might even say in priciple the same as, my method described above of introducing trasfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite.

A googolplex is precisely as far from infinity as is the number 1... no matter what number you have in mind, infinity is larger still.

I realise 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.

Nature never uses prime numbers. But mathematicians do.

There are infinite numbers between 0 and 1.

Transcendental [numbers], They transcend the power of algebraic methods.

If equations are trains threading the landscape of numbers, then no train stops at pi.

God made integers, all else is the work of man.

Some infinities are bigger than other infinities.

It is almost as hard to define mathematics as it is to define economics, and one is tempted to fall back on the famous old definition attributed to Jacob Viner, "Economics is what economists do," and say that mathematics is what mathematicians do. A large part of mathematics deals with the formal relations of quantities or numbers.

I can show you that the art of calculation has to do with odd and even numbers in their numerical relations to themselves and to each other.

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 lader 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.

Primes are the atoms of the arithmetic - the hydrogen and oxygen of the world of numbers

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.

One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery--and more totally useless--than the perfect numbers.

You have this world of mathematics, which is very real and which contains all kinds of wonderful stuff. And then we also have the world of nature, which is real, too.

Infinities are not for casual exploration. You could fall and keep falling. You could get lost.

Mathematics is the queen of science, and arithmetic the queen of mathematics.

It is strange that we know so little about the properties of numbers. They are our handiwork, yet they baffle us; we can fathom only a few of their intricacies. Having defined their attributes and prescribed their behavior, we are hard pressed to perceive the implications of our formulas.

We are the creatures of imagination, passion, and self-will, more than of reason or even of self-interest. Even in the common transactions and daily intercourse of life, we are governed by whim, caprice, prejudice, or accident. The falling of a teacup puts us out of temper for the day; and a quarrel that commenced about the pattern of a gown may end only with our lives.

Tell me then, does love make one a fool or do only fools fall in love?

A prime number is one (which is) measured by a unit alone.

I count Maxwell and Einstein, Eddington and Dirac, among "real" mathematicians. The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are at present at any rate, almost as "useless" as the theory of numbers.

Imaginary numbers are a fine and wonderful refuge of the divine spirit almost an amphibian between being and non-being.

Only Numbers. Pure math. You have to accustom yourself to thinking that way.