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Mathematical discoveries, small or great are never born of spontaneous generation.
– Henri Poincare
Thought is only a flash between two long nights, but this flash is everything.
It has adopted the geometry most advantageous to the species or, in other words, the most convenient.
A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations.
The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law.
The mind uses its faculty for creativity only when experience forces it to do so.
Mathematicians do not study objects, but relations between objects.
It is through science that we prove, but through intuition that we discover.
Just as houses are made of stones, so is science made of facts.
Need we add that mathematicians themselves are not infallible?
Point set topology is a disease from which the human race will soon recover.
If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment.
Mathematics is the art of giving the same name to different things.
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.
In the old days when people invented a new function they had something useful in mind.
Mathematicians are born, not made.
Hypotheses are what we lack the least.
To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection.
Geometry is not true, it is advantageous.
It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.
It is far better to foresee even without certainty than not to foresee at all.
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.
How is an error possible in mathematics?
Science is facts.
If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.
Ideas rose in clouds; I felt them collide until pairs interlocked, so to speak, making a stable combination.