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.

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

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.

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

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.

So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the initimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality.

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

...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

Intuition is more important to discovery than logic.

Guessing before proving! Need I remind you that it is so that all important discoveries have been made?

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

Mathematics is the art of giving the same name to different things. [As opposed to the quotation: Poetry is the art of giving different names to the same thing].

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

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

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

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

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

Mathematicians do not study objects, but relations between objects.

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

All of mathematics is a tale about groups.

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

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

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