Someone asked me if I was ever sorry I had chosen mathematics. I said, “I didn’t choose! Mathematics is an addiction with me!”
If we are to teach mathematics at all, real success is not possible unless we know that the subject is beautiful as well as useful. Mere utility of the moment without any feeling of beauty becomes a hopeless bit of drudgery, a condition which leads to stagnation.
…What would mathematics have amounted to without the imagination of its devotees—its giants and their followers? There never was a discovery made without the urge of imagination—of imagination which broke the roadway through the forest in order that cold logic might follow.
The desire to explore thus marks out the mathematician. This is one of the forces making for the growth of mathematics. The mathematician enjoys what he already knows; he is eager for more knowledge.
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful. The ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.
And here is a bonus—an article by Sawyer about Hardy:
Denise Gaskins' Let's Play Math 
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