I would like to take this opportunity to congratulate Ronald Calinger
for his new history of mathematics textbook, "Conceptual History of
Mathematics", as outlined here on HM.
I look forward to reading its broad review of ancient Near East,
Chinese, Indian (Vedic), Islamic and Mayan mathematics before 1500 AD.
Ancient mathematics in all its beauty does cover a wider swath than
the two pedagogies that I mentioned a short while ago.
The Dresden Codex and its beautify modular arithmetic, stated in an
astronomical context, predicted the cycles of the heavens much as
Vedic scholars were doing, as Oystein Ore cited Brahmaguta's use of
the Chinese Remainder Theorem (about the same time). Mayan arithmetic
used a wonderful position number system, one based on mod 4, mod 5,
that is commonly summarized as only base 20. George Sanchez' 1961
book "Arithmetic in Maya" covers an abacus register aspect Mayan
writing, much as George G. Joseph ("Crest of the Peacock") cited was
used by the Inca, and their base 10 quipu, in a manner that I highly
recommend.
Indeed, Ronald has taken a wide geographical view of the history of
mathematics, implying other pedagogies, well beyond the additive
arithmetic of Sumer, and the abstract mathematics of pre-Socratic
Greece, that I suggested, a short time ago.
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Yesterday's outline of Calinger's "Conceptual History of Mathematics"
high lights two pedagogical points that may spark a debate. Ronald's
book takes two common historical positions that can be historically
re-evaluated on two levels.
First, is the singular focus on the "... writing and metrology in
ancient Sumer", and "the mathematical achievements of Ancient
Mesopotamia"; and second, the quick transition to "the beginnings
of theoretical mathematics in Pre-Socratic Greece".
Historians have long assumed that writing was first found in Sumer,
when two other nearby regions may have been equal or even previous
'inventors' of writing. Denise Schmandt-Bessert's analysis of Sumer
writing includes the limited use of additive trading 'token'
containers that first reported writing symbols around 3,000 BC.
Yet, in nearby ancient Persia, these same class of token containers
existed, about the same time. Where is an equal analysis of this
culture's use of early writing?
As has been discussed here on HM, several times, writing has been
found in Egypt prior to 3,000 BC, maybe as early as 3,400 BC.
Research along this thread has only begun. When is research going
to be completed?
Several of you may ask, why is it important to challenge the long
held position of Sumer's initial origin of writing? First, the
additive scope of Denise's research (assuming that writing developed
over a long period of time) leaves out the strong possibility that
abstract mathematics developed before writing, and writing developed
over a short period of time (the Gelb thesis).
I find it intriguing that abstract mathematics (plausibly proto-number
theory) may have been the actual birth mother of writing in all three
regions of greater Mesopotamia.
Second, to transition to the Egyptian proto-number theory issue, a
common thread of my HM posts, theoretical forms of rational number
conversion to unit fraction series clearly formed the basis of Greek
numeration. Yet, Egyptian scribal math has since the 1920's has not
been formally considered as using proto-number theory. That is, what
problem is there in considering a culture other than Greece for the
first use of abstract mathematics, when even the Encyclopedia for
the History and Philosophy of Mathematics (1992) states that the RMP
used abstract mathematics?
Finally, as an introduction to the history of mathematics, textbook
authors now have the opportunity to use a wide array of pedagogical
positions, beyond the two standard ones of additive Sumer writing and
theoretical Greek mathematics. Opening up their course outlines to
potential conclusions of on-going research, or research that clearly
should take place is not a radical step, right? That is, does anyone
else wish to consider challenging the above two common pedagogical
threads, with a comment or two? For example, does Ronald Calinger's
book cover the above research points in some manner, and then for
time's sake, or some other set of reasons settle for using the
standard two pedagogies?
Regards to all,
Milo Gardner