Science is built up of facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.

Every good mathematician should also be a good chess player and vice versa.

The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.

Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose.

Doubt everything or believe everything: these are two equally convenient strategies. With either we dispense with the need for reflection.

Geometry is the art of correct reasoning from incorrectly drawn figures.

It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.

Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long the relations don't change. Matter is not important, only form interests them.

If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living

It is through science that we prove, but through intuition that we discover.

In the old days when people invented a new function they had something useful in mind.

Pure logic could never lead us to anything but tautologies; it can create nothing new; not from it alone can any science issue.

One geometry cannot be more true than another; it can only be more convenient.

All of mathematics is a tale about groups.

The mind uses its faculty for creativity only when experience forces it to do so.

All great progress takes place when two sciences come together, and when their resemblance proclaims itself, despite the apparent disparity of their substance.

Mathematical discoveries, small or great are never born of spontaneous generation.

It is far better to foresee even without certainty than not to foresee at all.

...the feeling of mathematical beauty, of the harmony of numbers and of forms, of geometric elegance. It is a genuinely aesthetic feeling, which all mathematicians know

Mathematicians do not study objects, but relations between objects.

The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances. I am far from despising this, but it has nothing to do with science. What I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp.

Mathematicians are born, not made.

It is by logic we prove. It is by intuition we discover.

There are no solved problems; there are only problems that are more or less solved.

Mathematics is the art of giving the same name to different things.

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