*I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail.* (Abraham Maslow, “Psychology of Science”)

You will find, when listening to talks or reading papers, that there will be problems which interest you which were solved using an unfamiliar tool, but seem out of reach of your own personal “bag of tricks”. When this happens, you should try to see whether your own tools can in fact accomplish a similar task, but you should also try to work out what made the other tool so effective – for instance, to locate the simplest model case in which that tool does something non-trivial.

Once you have a good comparison of the strengths and weaknesses of the new tool in relation to the old, you will be prepared to recall it whenever a situation comes up in the future in which the tool would be useful; given enough practice, you will then be able to add that tool permanently to your repertoire. Thus it is worth investing some time in learning about other tools, even if they are outside your field. One way to do this is to read survey articles in other fields aimed at a general audience (one good source for this are the quadrennial ICM proceedings).

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15 June, 2008 at 3:02 pm

这等牛人也在wordpress上写blog！ « Just For Fun[...] to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a [...]

15 June, 2008 at 4:20 pm

Jonathan Vos PostThis may be a dumb question, or cluster of questions, coming from a former adjunct professor in two different subjects, but how many “tricks” are there?

Does the answer depend on what we know and how we came to know it, as a state variable of our mathematical knowledge or as a history variable?

Does it depend on our evolved animal terrestrial barins? That is, how many tricks might extraterrestrials have?

Is there a hierarchy of tricks?

I know that, on another thread, you’ve said that many great mathematicians have few tricks, and gave Hilbert and Feynman examples.

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