Georg Cantor Quotes

There are infinite numbers between 0 and 1.

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.

The transfinite numbers are in a sense the new irrationalities [ ... they] stand or fall with the finite irrational numbers.

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.

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.

Something is infinite if, taking it quantity by quantity, we can always take something outside.

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

Some infinities are bigger than other infinities.

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

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.

Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number.

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

For mathematics, even to the logical forms in which it moves, is entirely dependent on the concept of natural number.

While we can prove that almost all numbers in the continuum are random, we cannot prove that any specific number is indeed random.

I tell you, with complex numbers you can do anything.

Renormalization is just a stop-gap procedure. There must be some fundamental change in our ideas, probably a change just as fundamental as the passage from Bohr's orbit theory to quantum mechanics. When you get a number turning out to be infinite which ought to be finite, you should admit that there is something wrong with your equations, and not hope that you can get a good theory just by doctoring up that number.

Topology and number theory are my faves.

If "Number rules the universe" as Pythagoras asserted, Number is merely our delegate to the throne, for we rule Number.

One may say that mathematics talks about the things which are of no concern to men. Mathematics has the inhuman quality of starlight - brilliant, sharp but cold ... thus we are clearest where knowledge matters least: in mathematics, especially number theory.

Number, place, and combination . . . the three intersecting but distinct spheres of thought to which all mathematical ideas admit of being referred.

Two infinities: the one that stretches to the beginning but never touches-when you halve and halve and halve, infinitely-and then the one that spreads out into the endless, endless future, the endless, endless, distance.The set of infinities that is itself infinite.

Number is the Word but is not utterance it is wave and light, though no one sees it it is rhythm and music, though no one hears it. Its variations are limitless and yet it is immutable. Each form of life is a particular reverberation of Number.

Has it been found that bodies of men act with more rectitude or greater disinterestedness than individuals? The contrary of this has been inferred by all accurate observers of the conduct of mankind; and the inference is founded upon obvious reasons. Regard to reputation has a less active influence, when the infamy of a bad action is to be divided among a number than when it is to fall singly upon one.

To make our position clearer, we may formulate it in another way. Let us call a proposition which records an actual or possible observation an experiential proposition. Then we may say that it is the mark of a genuine factual proposition, not that it should be equivalent to an experiential proposition, or any finite number of experiential propositions, but simply that some experiential propositions can be deduced from it in conjunction with certain other premises without being deducible from those other premises alone.

Number theorists are like lotus-eaters -- having once tasted of this food they can never give it up.