The apex of mathematical achievement occurs when two or more fields which were thought to be entirely unrelated turn out to be closely intertwined. Mathematicians have never decided whether they should feel excited or upset by such events.
[In mathematics] There are two kinds of mistakes. There are fatal mistakes that destroy a theory, but there are also contingent ones, which are useful in testing the stability of a theory.
The advice we give others is the advice that we ourselves need.
Running overtime is the one unforgivable error a lecturer can make. After fifty minutes (one microcentury as von Neumann used to say) everybody's attention will turn elsewhere.
Very little mathematics has direct applications - though fortunately most of it has plenty of indirect ones.
Stan Ulam was lazy, ... He talked too much ... He was self-centered ... . He had an overpowering personality.
Mathematicians - for what they do - are really poorly rewarded. And it's a very competitive field, almost as bad as being a concert pianist.
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