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It is not so much whether a theorem is useful that matters, but how elegant it is.
--
Chapter 15, Random Reflections on Mathematics and Science, p. 274

 

Stanislaw Ulam

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Elegance is necessarily unnatural, only achieveable at great expense. If you just do something, it won't be elegant, but if you do it and then see what might be more elegant, and do it again, you might, after an unknown number of iterations, get something that is very elegant.

 

Erik Naggum

 

We re-make nature by the act of discovery, in the poem or in the theorem. And the great poem and the deep theorem are new to every reader, and yet are his own experience, because he himself re-creates them.

 

Jacob Bronowski

 

We re-make nature by the act of discovery, in the poem or in the theorem. And the great poem and the deep theorem are new to every reader, and yet are his own experience, because he himself re-creates them.

 

Jacob Bronowski

 

It is an arithmetic, moreover, which cannot be denied even though we nearly all try to deny it. The arithmetic is simply this: Any positive rate of growth whatever eventually carries a human population to an unacceptable magnitude, no matter how small the rate of growth may be unless the rate of population growth can be reduced to zero before the population reaches an unacceptable magnitude. There is a famous theorem in economics, one which I call the dismal theorem, which states that if the only thing which can check the growth of population is starvation and misery, then the population will grow until it is sufficiently miserable and starving to check its growth. There is a second, even worse theorem which I call the utterly dismal theorem. It says that if the only thing which can check the growth of population is starvation and misery, then the ultimate result of any technological improvement is to enable a larger number of people to live in misery than before and hence to increase the total sum of human misery.

 

Kenneth Boulding

 

The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around.

 

Richard Hamming