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

[Monroe Eskew]

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.

Hoyrup's paper makes no attribution of modern techniques or ideals of
rigor to modern-era European jurists.  Instead, the historical
evidence he cites tends to support the counter-thesis that modern
mathematics comes from mathematics of times past, with Egyptians,
Indians, Byzantines, Phoenicians, Greeks, Arabs, and Jews all making
crucial contributions.  Not only the Greek mathematicians but many
others from many nations shaped the evolution of mathematics.

Near the end of the text, Hoyrup says:

"Seventeenth century European mathematics might well have found itself
to deviate radically from the Ancient canons— one need only think of
the victorious non-rigorous trend in the treatment of infinitesimals,
and on the integration of mathematics with experimental philosophy."

This implies that Hoyrup does not believe rigorous proof to be a
modern invention inspired by lawyers but rather something present in
ancient work.

Lastly, he does not make any claim to the effect that we only see
Euclid as doing (somewhat) rigorous proofs because of a modern
reinterpretation of the Elements.  This claim would require one of two
possibilities:

1) That text of the Elements was at some point deliberately changed
and falsified in order to fit in with evolving standards of rigor.
Hoyrup makes no such accusation.

2) That even though the Elements is structured in numbered
propositions, with the later ones drawing upon the conclusions of the
earlier ones, and that Euclid consistently uses language that appears
to be structured as statements of hypotheses, if-then syllogisms,
existential instantiation, reductio ad absurdum, etc., his words and
their structure meant totally different things to his contemporaries
and were not actually logical arguments or proofs.  That somehow when
we see clever reasoning in Euclid, it is merely an accident of
translation or an imposition of our own meaning on a malleable mass of
characters.  Hoyrup does not advance such a claim.

Best,
Monroe



On Wed, Jul 14, 2010 at 3:12 PM, Michael Barany
<michael.barany at tellurideassociation.org> wrote:
>> Can you point me towards a good source that argues for this thesis,
>> which provides good textual examples/evidence of what you refer to as
>> the text becoming detached from the original meaning?
>
> Monroe,
>
> The paper that introduced me to this kind of historiography was a
> somewhat obscure one by Jens Hoyrup called  "The formation of a myth:
> Greek mathematics---our mathematics" from a bilingual volume from 1996
> titled L'Europe mathematique / Mathematical Europe.  Catarina's
> recommendation of Netz's volume is a good one,... it doesn't really
> talk about how Greek geometry (and deduction in general) has been
> rendered by Europeans, but it reconstructs a version of what might
> count as its "original meaning" which appears quite alien to anyone
> with a conventional present-day view of Euclid.