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Thread: Proofs and Proving

  1. #1 Proofs and Proving 
    Forum Freshman Wilhelm's Avatar
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    How does one go about to prove something in mathematics? What are the basic components in proving something and the usual methods mathematicians have taken in the past to prove things such as:

    Differentiation Rules
    Mean Value Theorem
    The Central Limit Law (Statistics)
    The Law of Large Numbers (Statistics)


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  3. #2  
    Forum Ph.D. william's Avatar
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    This sounds like the perfect job for Guitarist and River_Rat to explain....

    But...
    In a simplistic view, we start with definitions and maybe throw in a few axioms, then start building theorems, and from all those, we build even more theorems....

    E.g., for the differentiation rules, we start with the definition of a limit and then a derivative. Then, we apply that to specific functions, and so on.

    A better explanation will have to wait until RR or guitarist joins in....

    (Or is your question about specific types of proofs - induction, contradiction, etc.?)

    Cheers,
    william


    "... the polhode rolls without slipping on the herpolhode lying in the invariable plane."
    ~Footnote in Goldstein's Mechanics, 3rd ed. p. 202
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  4. #3  
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    Ha! Willaim flatters me. I get proofs by hitting the problem with a big stick until it tells me the answer itself!

    If you think I'm joking, I'm not. If I am asked to prove an assertion, I first look carefully at the given data, try to remember all the relevant definitions, think about all the theorems I know that might bear on the problem, and then reach for my pencil. Generally, you know when the answer is right - it just feels right.

    Sometimes, if I'm not completely sure, I try try to find a counter-example, and if I can't, I get me a beer and relax.

    But... prize-winning mathematicians seem to have the ability to also use theorems that nobody else thought were relevant. I'm thinking here of Andrew Wiles (FLT) and the Russian guy (what's his name?) who solved the Poincare conjucture, for example

    If, however, you want to know how to derive the central axioms of mathematics, don't bother. Some of them are derivable, but most are just axiomatic
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  5. #4 Re: Proofs and Proving 
    Forum Professor river_rat's Avatar
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    Quote Originally Posted by Wilhelm
    How does one go about to prove something in mathematics?
    Well usually a lot of luck mixed with experience in the field, and following your nose - at least for me. Sometimes something just makes sense, the proof kind of falls into your head and sometimes it takes 3 months to solve the problem that when you look back at it you cant believe you could not see it!
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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