[FOM] Are proofs in mathematics based on sufficient evidence?
meskew at math.uci.edu
Thu Jul 15 12:29:00 EDT 2010
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
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
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.
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?
> 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.
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[FOM] Je, uthibitisho katika hisabati unategemea uthibitisho wa kutosha? "Ms. J.M.H.E.R.T. (Mwanzoni mwa mwaka wa 2010) aliandika: ""Nimesoma makala ya J.H.E.R.T. na ninaweza kuripoti kwamba haisemi juu ya nadharia ileile ambayo Michael anaendelea nayo." "Lengo la karatasi hii ni kusema kwamba waandishi wengi wa Ulaya wa enzi za kati na za kisasa walipunguza sana mchango wa ustaarabu wa Kiarabu kwa hisabati, wakati huo huo wakiba mawazo mengi ya wasomi wa Kiarabu na sifa ndogo au hakuna.""" Katika mchakato huo, hadithi ya hisabati ya Kigiriki-Ulaya tu ilijengwa, labda kwa msingi wa upendeleo wa kidini, kitamaduni, na kikabila. Karatasi ya Hoyrup haionyeshi mbinu za kisasa au itikadi za ukali kwa wanasheria wa Ulaya wa enzi ya kisasa. "Kama ushahidi wa kihistoria, yeye anataja tends kusaidia counter-thesis kwamba hisabati ya kisasa inatokana na hisabati ya nyakati zilizopita, na Wamisri, Wahindi, Byzantines, Phoenicians, Wagiriki, Waarabu, na Wayahudi wote kufanya michango muhimu. """ Si wanahisabati Wagiriki tu bali wengine wengi kutoka mataifa mengi walifanyiza mageuzi ya hesabu. "Hayo ni kwa sababu Howe anasema, ""Kama ilivyo katika karne ya 17, hesabu za Ulaya zilikuwa zimepotoka kabisa kutoka kwa kanuni za kale, mtu anahitaji tu kufikiria mwelekeo wa ushindi wa kutokuwa na ukali katika matibabu ya infinitesimals, na juu ya ushirikiano wa hisabati na falsafa ya majaribio.""" Hii ina maana kwamba Hoyrup haamini uthibitisho mkali kuwa uvumbuzi wa kisasa ulioongozwa na wanasheria, bali kitu kilicho katika kitabu cha mwisho cha Euclid, na hatutaka kusema kwamba tunamwona Euclid akifanya uthibitisho mkali kwa sababu ya tafsiri ya kisasa ya Elements. "Hii inahitaji moja ya mambo mawili: 1) Nakala ya ""Elements"" ilibadilishwa na kupotoshwa kwa makusudi ili ifanane na viwango vinavyobadilika vya ukali." Hoyrup hawasemi hivyo. "Hata ingawa ""Elements"" imepangwa katika sentensi zilizohesabiwa, na zile za baadaye zinatokana na hitimisho la zile za awali, na kwamba Euclid hutumia lugha ambayo inaonekana kuwa imepangwa kama taarifa za nadharia, ikiwa-basi-sillogisms, mfano wa kuwepo, reductio ad absurdum, nk, maneno yake na muundo wake zilimaanisha vitu tofauti kabisa kwa wakala wake na kwa kweli hazikuwa hoja za kimantiki au uthibitisho." Kwa njia fulani, tunapoona hoja za akili katika Euclid, ni tu ajali ya tafsiri au kulazimisha maana yetu wenyewe juu ya wingi wa wahusika. Hoyrup haitoi madai kama hayo. "Julayi 14, 2010: Michael Barney, mwandishi wa ""The New York Times"" aliandika: ""Je, unaweza kunionyesha chanzo kizuri ambacho kinatoa hoja kwa ajili ya nadharia hii, ambacho hutoa mifano nzuri ya maandishi ya kile unachokiita ""Wed"" ambayo ni maandishi yanayotokana na maana ya awali?" "Jens Hoerup, mwandishi wa ""The Formation of a Myth: Greek Mathematics - Our Mathematics,"" alitoa maoni yake juu ya suala hili katika kitabu chake cha 1996 ""L'Europe Mathématique: Mathematical Europe.""" """Kutoka kwa Catarina, mapendekezo ya kitabu cha Netz ni mazuri, kwa kweli hayazungumzi jinsi jiometri ya Kigiriki (na kupunguzwa kwa ujumla) imewekwa na Wazungu, lakini inajenga upya toleo la kile ambacho kinaweza kuhesabiwa kama ""maana ya awali"" ambayo inaonekana kuwa ya kigeni kwa mtu yeyote na mtazamo wa kawaida wa leo wa Euclid." Maelezo zaidi kuhusu FOM