Re: Explanations

[Graham White <graham@eecs.qmul.ac.uk>]

And the folklore is (I haven't checked this in a proper history book)
that Gauss proved quadratic reciprocity numerous times because he didn't
consider the proofs sufficiently explanatory. It's certainly true that
modern proofs (i.e. those using the methods of algebraic number theory)
generalise it, and thereby explain, for example, what it is about the
rationals, and the number two, that makes primes in the rationals obey
quadratic reciprocity. I think one conclusion here is that, if you say
"explanatory", I am entitled to answer "so what do you want explained?"

Another point is this: there are lots of combinatorial
identities of the form

big ugly formula_1 = big ugly formula_2

which can be proved directly (for example, by induction
and a lot of algebra), but where the proof is utterly unilluminating.
And in many cases there are more conceptual proofs which people
generally find more illuminating (depending on taste, of course).

Graham

-------- Forwarded Message --------
> From: peasthope@shaw.ca
> Reply-to: peasthope@shaw.ca
> To: categories@mta.ca
> Cc: peasthope@shaw.ca
> Subject: categories: Re: Explanations
> Date: Thu, 21 Apr 2011 11:09:36 -0800
>
> Fred & all,
>
>> My goodness! I'd turn that question around: is there any proof (apart
>> from an "indirect" proof, or "proof by contradiction") that one would
>> *not* "consider as being explanatory in this sense?"
>
> Speaking as a novice: yes, certainly.  Isn't it a question of degree?  Some
> proofs explain beautifully while others are clear as mud; most are
> between.  Ideally a proof shouldn't depend upon natural language but
> most do.  Striking sometimes how changing a few words of a sentence
> can make a concept obvious rather than nebulous.
>
> For example, I've proven some of the power laws for map objects.  There
> should be a way to reduce the definition of a map object and the power
> laws to analogues in arithmetic.  Still eludes me.  My proofs have yet to
> help.  So my understanding is incomplete and my power law proofs are
> poor.
>
> Best regards,                      ... Peter E.
>
> --
> Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
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