Mathematicians stand on each other's shoulders.

We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.

I have had my results for a long time: but I do not yet know how I am to arrive at them.

To praise it would amount to praising myself. For the entire content of the work... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years.

God does arithmetic.

To such idle talk it might further be added: that whenever a certain exclusive occupation is coupled with specific shortcomings, it is likewise almost certainly divorced from certain other shortcomings.

I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible.

It may be true, that men, who are mere mathematicians, have certain specific shortcomings, but that is not the fault of mathematics, for it is equally true of every other exclusive occupation.

When a philosopher says something that is true then it is trivial. When he says something that is not trivial then it is false.

Life stands before me like an eternal spring with new and brilliant clothes.

You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.

The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic.

Mathematicians stand on each other's shoulders.

I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect.

Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.

The enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it.

When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again.

It is not knowledge but the act of learning not possession but the act of getting there which grants the greatest enjoyment.

Mathematics is concerned only with the enumeration and comparison of relations.

Mathematics is the queen of science.

As I say, there was this movement to try to bring philosophers and mathematicians together into an organization where they would talk to each other. An organization wasn't effective unless you had a journal. That's about all I know.

Mathematicians are born, not made.

As far as I know, only a small minority of mathematicians, even of those with Platonist views, accept the idea that there may be mathematical facts which are true but unknowable.

Therefore I would not have it unknown to Your Holiness, the the only thing which induced me to look for another way of reckoning the movements of the heavenly bodies was that I knew that mathematicians by no means agree in their investigation thereof.

Mathematicians may flatter themselves that they possess new ideas which mere human language is as yet unable to express.

It may be true, that men, who are mere mathematicians, have certain specific shortcomings, but that is not the fault of mathematics, for it is equally true of every other exclusive occupation.

Mathematicians aren't satisfied because they know there are no solutions up to four million or four billion, they really want to know that there are no solutions up to infinity.

The greatest mathematicians, as Archimedes, Newton, and Gauss, always united theory and applications in equal measure.

Pure mathematicians just love to try unsolved problems - they love a challenge.

The greatest problem for mathematicians now is probably the Riemann Hypothesis.

Mathematics is written for mathematicians.

First rate mathematicians choose first rate people, but second rate mathematicians choose third rate people.

Abel has left mathematicians enough to keep them busy for 500 years.

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.

Mathematicians are like managers - they want improvement without change.

The truth has never been of any real value to any human being - it is a symbol for mathematicians and philosophers to pursue. In human relations kindness and lies are worth a thousand truths.

For example, the philosophers who were interested in logic were probably rather logical for mathematicians. But the ASL got us together, so we could talk to each other and publish in the same journal.

The only thing that might have annoyed some mathematicians was the presumption of assuming that maybe the axiom of choice could fail, and that we should look into contrary assumptions.

Programming is one of the most difficult branches of applied mathematics; the poorer mathematicians had better remain pure mathematicians.

Need we add that mathematicians themselves are not infallible?

Most were beginning to feel they had learned enough to last for the rest of their lives. They remained mathematicians, but largely went their own way.

Mathematicians stand on each other's shoulders.

The way in which mathematicians and physicists and historians talk is quite different, and what a physicist means by physical intuition and what a mathematician means by beauty or elegance are things worth thinking about.

It would be better for the true physics if there were no mathematicians on earth.

I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future.

Physicists are more like avant-garde composers, willing to bend traditional rules... Mathematicians are more like classical composers.

Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.

All musicians are subconsciously mathematicians.