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Revision of Mathematicians need to be metamathematicians from Sun, 14/12/2008 - 23:35

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If you want to prove a theorem, then one way of looking at your task is to regard it as a search, amongst the huge space of potential arguments, for one that will actually work. One can often considerably narrow down this formidable search by thinking hard about properties that a successful argument would have to have. In other words, it is a good idea to focus not just on the mathematical ideas associated with your hoped-for theorem, but also on the properties of different kinds of proofs. This very important principle is best illustrated with some examples.

What can a lower bound say about potential proofs of the upper bound? Brief summary ( There are certain kinds of arguments that always lead to certain kinds of bounds. Therefore, if you are trying to prove a bound, you can often be sure that certain techniques will not be sufficient. This makes the search for techniques that do work more efficient. )

When does reformulating a problem count as progress?

If you are getting stuck, then try to prove rigorously that your approach cannot work Brief summary ( Sometimes one gets stuck simply because one has run out of ideas. But sometimes it is for a different reason: one has ideas, but one gets bogged down in technical details and cannot get them to work, or cannot face trying. If that happens, it can be very helpful to try to formulate precisely what one's approach actually is. Sometimes it can then be made to work, and sometimes one can then prove rigorously that it cannot work. (The latter will happen if, for instance, one turns out to be using just a set of assumptions that does not in fact imply the conclusion one wishes to prove.) This article contains several examples of approaches to problems that are known not to work. )

Sharp results need sharp lemmas Brief summary ( If you are trying to prove a sharp result, then all the intermediate steps need to be sharp as well. This greatly restricts what they can be, and therefore makes searching for them easier. For this reason it can be a good idea to try to prove a sharp result even when all you actually need is a weak bound.)

Two proofs of the same theorem usually have deep similarities Brief summary ( If you are trying to find a new proof of a result, then even if your proposed argument is quite different from others that are known, there are certain features that it is likely to share. In particular, there seems to be a law that one might call conservation of work: at a deep level, any two sensible proofs of the same theorem need the same amount of effort. )

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General problem-solving tips

Comments

"Think about the converse"

"Think about the converse" should be added as a link in this article.