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Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
—  Paul R. Halmos
To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive and the ability to visualize and guess.
—  Paul R. Halmos
Mathematics is not a deductive science -- that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.
—  Paul R. Halmos
The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me -- both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension.
—  Paul R. Halmos
... the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
—  Paul R. Halmos

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