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.
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.
It is with logic that one proves; it is with intuition that one invents.
Science is built up with 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.
There are no solved problems; there are only problems that are more or less solved.
Doubt everything or believe everything: these are two equally convenient strategies. With either we dispense with the need for reflection.
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.
Often when works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, long or short, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind.
It is far better to foresee even without certainty than not to foresee at all.
Mathematicians are born, not made.
Experiment is the sole source of truth. It alone can teach us something new; it alone can give us certainty.
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?
Chance ... must be something more than the name we give to our ignorance.
If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living
Thought must never submit, neither to a dogma, nor to a party, nor to a passion, nor to an interest, nor to a preconceived idea, nor to whatever it may be, save to the facts themselves, because, for thought, submission would mean ceasing to be.
Every good mathematician should also be a good chess player and vice versa.
Mathematicians do not study objects, but the relations between objects.
Pure logic could never lead us to anything but tautologies; it can create nothing new; not from it alone can any science issue.
A small error in the former will produce an enormous error in the latter.
It is not order only, but unexpected order, that has value.
One geometry cannot be more true than another; it can only be more convenient.
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].
Mathematical discoveries, small or great are never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.
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