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

[Monroe Eskew]

On Sat, Jul 17, 2010 at 1:00 AM, Irving <ianellis at iupui.edu> wrote:
> In responding to the issue of whether Russell's criticisms of Euclid,
> to the extent that they it is an axiomatic system rather than a formal
> deductive system, or, as Russell asserted, not rigorously logical,
> might also be applied to the work of other great mathematicians,
> including those listed by Monroe, namely Gauss, Weierstrass, Cayley,
> Cauchy, etc., I would begin by noting that, unlike Peano in the
> Arithmetices principia to be a formal deductive system, or the claims
> which were made on behalf of Euclid's Elements for its being THE
> exemplary model of rigorous logical proof, mathematicians such as
> Gauss, et alia, were not claiming to devise either formal deductive
> systems or even axiomatic systems, but were, in the case for example,
> of an Euler or a Gauss, working on solving specific mathematical
> problems, and it has been recognized that much of their work was
> "computational" (or, in the 18th century, taken as a synonym,
> "algorithmic").

They may have done a large number of computations, but these
mathematicians also thought creatively, had new ideas, etc.  For
example could you call Cayley's introduction of the general concept of
a group and he proof that any finite group is embeddable into a
permutation group, "mere computation"?  Similar things come to mind
regarding the fundamental theorem of algebra, Gaussian distribution,
Gaussian curvature, etc.  There is some great mathematics there, not
merely computation, though it's not of a foundational nature.  And its
all rigorous, but not of it is done in a formal deduction system.

I believe that if Euclid had given correct informal proofs from
Hilbert's axioms, but still failed to specify any formal inference
rules, Russell would not have found any fault with the Elements.  His
criticisms of I.1 and I.4 seem to be mathematical rather than
meta-mathematical points:  Euclid did not have any continuity axioms,
and the superposition technique simply does not follow from anything
prior.