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

[Michael Barany]

Monroe,

Apologies for suggesting Hoyrup's is a defense of the entire position
I advanced earlier.  His is an excellent study of the appropriation of
the Elements in Early Modern Europe (one of the things for which you
asked for an example) but the chapter does not, as you note, account
for Euclid's later receptions and transformations.

I would suggest that many of the other remarks about Euclid on this
thread do a good job setting up such a consideration.  The difference
would be that instead of faulting Euclid for not having a completely
deductive system one might instead observe that Euclid's system
offered a completely valid form of proof (indeed, the gold standard)
well into the nineteenth century.  My claim that Euclid's interpreters
made him more rigorous comes from the range of translations and
commentaries provided over the last five or so centuries.  One finds
in these an increasing set of explanations, additional illustrations,
and other argumentative devices which were not present in Euclid's
Greece (indeed, Hoyrup does consider earlier non-European
contributions of this sort) but which certainly lend credence and
argumentative force to the demonstrations, and in many cases account
for shortcomings in the deduction.  One need only look at Henry
Billingsley's 1570 treatment of Euclid I.2 (discussed elsewhere on
this thread), where he breaks the problem down into four cases and
re-argues them separately after giving the standard Euclidean proof,
all while claiming that it is the Euclidean proof itself that provides
the ultimate justification.

Now, for the sticky matter of the lawyers.  While my initial statement
was meant to be provocative (and of course without a full argument it
must certainly come off as under-supported), there is an important
observation at the core which is clearer to grasp.  Mathematics and
mathematicians have long maintained their epistemological autonomy
from other forms of knowledge, a split I argue took its present form
in the early nineteenth century, although it had many prior
iterations.  However mathematics does not float independently of its
environs.  The valorization of Euclid in Early Modern Europe had two
effects: it made the Elements a model for truth in other disciplines
(see Hobbes) and it led rhetoricians and others to focus on competing
forms of certainty for those  other disciplines.  As mathematics
developed, so too did these other models of how to know the right and
true (as the parent post for this thread encountered on wikipedia).
As understandings of mathematics and Euclid changed, mathematicians
had to justify and reinforce their art with reference to the standards
around them, including juridical standards and the scientific
standards largely derived from them (see Shapin and Schaffer's famous
Leviathan and the Air-Pump).  Surely it would be a mistake to argue
that the laws of deduction formalized around a century ago apply
purely and retroactively to all of mathematical history---that is the
main point of my comments.

Enjoying this discussion enormously,...

Michael

On Thu, Jul 15, 2010 at 5:29 PM, Monroe Eskew <meskew at math.uci.edu> wrote:
> I have just read the paper by Jens Hoyrup, and I can report that it
> does not advance the same thesis that Michael is advancing.  The
> purpose of the paper is to argue that many medieval and modern-era
> European writers severely downplayed the contributions of Arabic
> civilization to mathematics, while simultaneously stealing many of the
> Arab scholars' ideas with minimal or no attribution.  In the process,
> a myth of purely Greco-European mathematics was constructed, probably
> on the basis of religious, cultural, and racial biases.
>