Mathematics is the art of giving the same name to different things.
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.
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.
Point set topology is a disease from which the human race will soon recover.
In the old days when people invented a new function they had something useful in mind.
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.
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.
It is far better to foresee even without certainty than not to foresee at all.
Geometry is not true, it is advantageous.
If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.
How is an error possible in mathematics?
Mathematicians are born, not made.
It is through science that we prove, but through intuition that we discover.
To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection.
Mathematicians do not study objects, but relations between objects.
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