[FOM] Are Proofs in mathematics based on sufficient evidence?

[Monroe Eskew]

On Wed, Jul 14, 2010 at 2:26 AM, Irving <ianellis at iupui.edu> wrote:
> With these distinctions in mind, consider next Russell's conception of
> what Euclid was about.
> In the paper "The Teaching of Euclid" (Mathematical Gazette 2 (May,
> 1902), 165¬-167; reprinted, pp. 467-469, Towards the "Principles of
> Mathematics", 1900–02, edited by Gregory H. Moore (London/New York:
> Routledge, 1993), Volume 3 of The Collected Papers of Bertrand Russell)
> Russell took Euclid seriously to task for the lack of "logical
> excellence" which Euclid was reputed to have presented in his book. The
> point also recurs in the Principles of Mathematics (p. 5) where Russell
> points out the need for rules or principles" of deduction and proceeds
> to offer ten such principles (pp. 4-5, 10-16), including in particular
> "formal implication" or the rule of detachment. We may summarize
> Russell's strong criticisms of Euclid by reminding ourselves of the
> difference between an axiomatic system and a formal deductive system
> and reporting that Russell in essence accuses Euclid of not possessing
> a formal deductive system.

Irving,

I admit not being very familiar with cited writings of Russell, but I
find your interpretation doubtful for a certain reason: Russell could
have made the same criticism of many other more recent mathematicians,
including Gauss, Weierstrass, Cayley, Cauchy, etc.  As far as I am
aware, none of these mathematicians used explicit deduction rules in
their work, yet no reasonable person would accuse them of lacking
"logical excellence."  On the other hand, one can point to certain
arguments in Euclid's Elements which are not valid as ordinary
mathematical arguments (i.e. not valid when charitably formalized into
classical first order logic).  Therefore since he singled out Euclid
he probably meant to refer to these defects rather than the lack of
explicit deduction rules.

I'm not sure how harsh Russell is in his criticism, but myself, I
would not to say that these defects make Euclid's Elements "bad", just
that it is not "excellent."

Best,
Monroe