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Friday, September 5, 2008

The key to understanding proofs

is to realize that the exposition of a proof is distinct from its derivation. The (published) exposition is a very condensed description of the what and why. The (usually unpublished) derivation is about the how (the proof was derived). Reading carefully between the lines of the exposition one can often discern a faint outline of the creator's thought process but that's about it.

A (weak) analogy is having the binary code of a program that does something wonderful.Without the source code(the step by step derivation of the proof), one has to use disassemblers (digging into the exposition)to figure out how it works. Having the source code would be better in some contexts, but most proofs are "binary only".

Having said that, nothing stops me from starting from the givens and trying to prove the conclusion myself, which is a worthwhile exercise in itself. ("worthwhile" as in "what does not kill me makes me stronger" ;-) )

3 comments:

John said...

I acknowledge that there is not enough room in the journal to include all of the details of the derivation of the proof, but I am personally irritated by the way a week's worth of sleepless nights, possibly sobbing into a notebook is condensed into "It can be shown..."

On the one hand this tradition can keep mathematicians sharp as they work through the problems. Onn the other they are not doing a service for journal readers like me who have very minimal graduate training.

Ravi said...

"sobbing into a notebook" rings true! :-) :-(

Sam said...

I am reminded of Richard Feynman's response when asked about his method for solving problems:
1. Write down the problem
2. Think really hard
3. Write down the answer

Sounds easy enough, right? ;)