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

[Michael Barany]

Monroe,

Your experiences with Euclid represent one form of a remarkably common
theme in the history of European mathematics.  Of course, there are
many interpretations of this history, but one I find compelling holds
that beginning in the late 15th century Euclid's Elements was
effectively detached from most of the meanings it may have had in
earlier eras (particularly with respect to truth and rigor) and
re-appropriated to provide a historical basis for new claims regarding
the right way to pursue knowledge.  The interesting byproduct of this
transformation was that Euclid's Elements became more rigorous through
the cascading attempts to translate and teach the text over the
ensuing centuries.  But "rigorous" is always a debatable term, and
different generations of translators and teachers attached different
meanings to their prized geometric text. Thus, in the seventeenth
century Hobbes used one interpretation of Euclid's Elements to claim
that he had successfully (and rigorously!) doubled the cube while his
opponents used their version of Euclidean rigor to claim he had done
no such thing.  In the eighteenth century (as Joan Richards and
several others argue) Euclidean rigor was often seen to be more stale
and pedantic than a valuable method for mathematics.  The early
nineteenth century saw several waves of reinterpretation of the
Elements which culminated (for many) in Weierstrass's school of
analytic rigor, followed by the versions of abstract algebra and logic
more familiar on this e-list.

It may be interesting to recall that Cauchy himself was a
mathematician in a family of French lawyers,...

Smiles,

Michael

On Sat, Jul 10, 2010 at 12:18 PM, Monroe Eskew <meskew at math.uci.edu> wrote:
> On Fri, Jul 9, 2010 at 2:43 PM, Michael Barany
> <michael.barany at tellurideassociation.org> wrote:
>>
>> If one goes back 300 years (or even fewer) one finds the very active
>> conflation of proofs of different sorts, with juridical proof usually
>> taken to be the standard against which others are compared.  The idea
>> of scientific proof had significant origins in debates about law and
>> rhetoric (see Shapiro, 1986, `To a Moral Certainty': Theories of
>> Knowledge and Anglo-American Juries 1600--1850, Hastings Law Journal
>> 38:153--193), and the mathematical notion of proof we have today
>> certainly has a lot to do with eighteenth and nineteenth century
>> adaptations of these ideas.
>>
>
> I find this surprising, given the relatively high level of rigor found
> in Euclid.  To put it in context, in my high school education, the
> most rigorous arguments I encountered were not in the mathematics
> classroom but rather in the debate team.  However when I read some
> Euclid as a college freshman, I saw a whole new level of care, rigor,
> and reliance on first principles.  Of course modern mathematics is a
> big step above Euclid, but one gets the impression (both from reading
> the math and the history) that its methods are descendants of those of
> the ancient Greek mathematicians, rather than modern-era lawyers.
>
> Monroe
>
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