4.0939 History of Mathematics (2/64)

Elaine Brennan & Allen Renear (EDITORS@BROWNVM.BITNET)
Fri, 25 Jan 91 00:04:11 EST

Humanist Discussion Group, Vol. 4, No. 0939. Friday, 25 Jan 1991.

(1) Date: Wed, 23 Jan 91 22:58:33 -0500 (32 lines)
From: David Durand <dgd@cs.bu.edu>
Subject: 4.0932 Multiculturalism: Math

(2) Date: Thu, 24 Jan 91 15:56:41 EST (32 lines)
From: dthel@conncoll.bitnet
Subject: Pythagoras & multiculturalism

(1) --------------------------------------------------------------------
Date: Wed, 23 Jan 91 22:58:33 -0500
From: David Durand <dgd@cs.bu.edu>
Subject: 4.0932 Multiculturalism: Math

In response to the comments on the history of mathematics that have been
contributed to the debate on multiculturalism, I have a few points to
offer. This is based on memories from my 10-years past History of
Mathematics classes at Brown. (A wonderful department, unique in the
country, and alas, dues to be disbanded as the current professors
retire). As I recall, the earliest Pythagorean triangles (3,4,5 and
some others) are attested in Sumerian or Akkadian texts, intended for
the use of surveyors. -- I could maybe dig up citations on these if
they are needed. The Egyptians had similar tables. The Greek
innovation was a proof that the pythagorean relation held for ALL right
triangles. The Chinese did independently (as far as is known) discover
a proof of the Pythagorean theorem, but I am completely unsure as to the

While this account that I am repeating may be biased, the mathematical
texts extant from the middle eastern cultures do not support assertions
of supressed mathematical speculation, since they are generally
extremely concrete (appropriate to clay texts I suppose) in the phrasing
and approach to such problems. At least in the ones we examined in
class, they were always presented as word problems about quantities of
earth, or areas of land. While there may have been a hidden tradition
of the sort required for such historical claims to make sense,
intellectual honesty requires that we find some level of factual support
for our assertions. At least in this area, I think such support is

(2) --------------------------------------------------------------38----
Date: Thu, 24 Jan 91 15:56:41 EST
From: dthel@conncoll.bitnet
Subject: Pythagoras & multiculturalism

I believe I can respond to part of David Kelly's query on the
predecessors of the Pythagorean theorem. Babylonian mathematical
tablets from the Old Babylonian period (1800-1600 BC) demonstrate that
the Babylonians were acquainted with the problem of determining the
length of the diagonal of a square, i.e. the hypotenuse of a triangle.
A tablet in the Yale Babylonian Collection has the figure of a square
with a diagonal drawn through it, and it notes the length of the sides
and of the diagonal in cuneiform characters, using a sexagesimal number
base. The advantage of their numbering system was that place values
could be expressed. Other tablets from the same period list a series of
numbers and their approximate square roots (approximate obviously
because they are the squares of irrational numbers) -- a handbook of
service- able values in short. Both of these tablets are illustrated
(Plates 6 and 7) in Otto Neugebauer's The Exact Sciences in Antiquity,
2nd edit. In general, what seems to distinguish Greek mathematics and
indeed pre-Hellenistic science in general, from Babylonian and Egytpian
science and mathematics is that the latter do not show any interest in a
single (theoretical) expression of a problem. They do however have
close knowledge of the actual numbers needed for the solution of various
problems. Neugebauer claims that the Babylonians knew the Pythagorean
theorem, but to my admittedly limited knowledge, no such theoretical
knowledge exists among the Babylonians. The Babylonians gave
serviceable approximations of irrational numbers, but never a
theoretical expression. The theorem is ascribed to Pythagoras; that of
course is another problem, since the Pythagoreans ascribed all their
statements and discoveries to Pythagoras himself. We therefore do not
know at what time the theorem was actaully put forth. What does seem
clear however, is that knowledge of the irrationality of a square's
diagonal was known at least a thousand years before the Pythagorean
Greeks existed. However it was Babylon's achievem ent, not Egypt's.

Dirk t.D.Held, Classics, Connecticut College.