Re: Explanations

[Timothy Porter <t.porter@bangor.ac.uk>]

Steve says: `Since mathematics is a formal system ...' is it?  I gave a
talk some years ago at the national college of the UK women's institute
and the title I used was:  Mathematics, a human activity.  My point was
that mathematics is done by mathematicians (amongst others).  Until we
find to the contrary, mathematicians are more often than not human (in
the widest sense of the word!!!!!). The form and direction of
mathematical investigation is determined by curiosity, and similar human
emotions., (sometimes also by rivalry, hatred, envy ,  and other ones of
less beauty).

   A (subjective view) good proof convinces the `reader' that the
statement is true. The 'explanation' behind a proof by contradiction
explains  somewhere along the lines: the result is trapped, it cannot
get away, therefore we have it. That is a human judgement and is
sometimes accompanied by the sentiment of `but that argument leaves me
dissatisfied as I do not see why'.  (The level of belief in the use of
contradiction is sometimes an issue but not always.) `Explanation' can
be modelled by a worldview approach, but then you have the problem of
the  teaching situation where the teacher gives an explanation of some
mathematical result, but has to say that the proof has to take a
different route.

In category theory, many proofs are transparent and of the form: what do
we know about the situation, just one fact, so we have to use that....
it works. (I am thinking of classical Yoneda lemma type situations,
since the only elements in hom-sets that we can be sure exist are the
identities.)  A thorough understanding of the proof does give an
explanation of why the result holds. (The problem I have with the
original request for examples is that explanation requires understanding
of the situation so is dependent on the knowledge of the `codomain'/
reader!)


Tim


On 25/04/2011 14:17, ClemsonSteve wrote:
> Quoted from Jean-Pierre Marquis email: "yes, of course, Salmon is
> certainly one of the important contributors to the field [scientific
> explanation]. In mathematics, Paolo Mancosu has been pushing the issue
> for the past 10 years or so, following the paths of Steiner, Resnik and
> a few others."
>
> In science, the issue of explanation has been discussed for at least a
> century. Since mathematics is a formal system and not a physical system,
> we have to be more careful about what *explanation* means. This makes it
> a "worldview" problems? As a constructionist/computationalist I would
> say no constructive, computational proof then there is no explanation.
> Platonist have their own. Is their explanation useful to me? Don't know
> because if I can figure a constructive technique out from the plationic
> technique, I'm good.
>
> steve stevenson
> clemson
>
>

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