[hist-analytic] Proving

[Roger Bishop Jones]

On Thursday 07 January 2010 19:48:08 jlsperanza at aol.com wrote:
> D. Frederick:
>
> "When people talk of ‘proof’ they normally mean ‘derivation;’ ifthey
> don’t, they are just mistaken."
>
> I think you are very right on a couple of points.

If not on that one!
The usage which Danny is here speaking of is not standard.
His imputation of error to those mathematician who speak of something as 
proven when we have convincing grounds for belief that there exists a proof of 
it in ZFC is yet another attempt by Danny to impose his own standards on 
others.  Yet another attempt to render useless a part of our normal 
vocabulary.

> 3) I suppose the old mathematicians -- but R. B. Jones should know
> better -- were pretty confused about things. And I _include_ Euclid. (I
> do own the two-volume Thomas edited Greek Mathematics in the Loeb
> Series so should be able to check this out). When the scholastics (to
> think that ´schole´ for the Greeks was "otium" is a joke seeing that
> monks were and really _are_ into ´converting´ people) talked of
>
>           Q. E. D.
>
> -- i.e. quod erat demonstrandum, this is possibly empty flatus vocis,
> for what we need is a phrase like ¨quod erat probandum". I should
> revise the Latin for this. And also the Greek for "proving".

You may find that the distinction which Danny is trying to make (putting aside 
his terminological totalitarianism), is not so very far removed from that made 
by Aristotle between demonstrative proof and the other kind.
Not sure now what he called the other kind, was it dialectical?
Anyway, in the former one reasons only from necessary premises, and in the 
latter from premises which might not be necessary, and might not be true.
Admittedly its not the same, for Danny's conception of mere proof requires 
that the premises be "self-evident" which term he interprets so strongly as to 
be a species of "vacuous name".
Whatever its relationship to Danny's position, this (demonstrative) probably 
is a reasonably close ancient predecessor to the usage of proof by modern 
mathematicians (though probably not the usage mathematical logicians, which is 
more definitely divergent from Danny's usage).  A crucial difference here being 
that the requirement that the premises be necessarily true is weaker than the 
requirement that they be self-evident, and one may be satisfied of this is one 
is willing to accept the axioms of ZFC as "implicit definitions" of its domain 
of discourse.

Danny's extreme is of course also discoverable in ancient Greece, if not in 
Euclid or Aristotle.  The classic exponents were the Pyrrhoneans.

Roger Jones