Re: Explanations

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

As a good example of a proof giving an explanation using categorical
methods  like the proof of the Seifert-van Kampen Theorem for the
fundamental group in
Crowell, R.~H.
\newblock \enquote{On the van {K}ampen theorem}.
\newblock \emph{Pacific J. Math.} \textbf{9} (1959) 43--50.

It does not quite use modern categorical language but in essence it
proves a colimit theorem by verifying the required universal property.
This then leads to specific calculations.  Previous proofs were
difficult to understand (e.g. van Kampen's account) or restricted to the
simplicial case. The value of the proof was also that it could be
generalised to the groupoid (many base point) case, and to higher
dimensions, using higher homotopy groupoids.


Ronnie Brown



On 22/04/2011 14:55, Graham White wrote:
> 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
>

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