Today I ran across Clawson's latest, 1999, math book, "Mathematical
Sorcery". It takes several fundamental positions that needs to be
challenged.
For example, Clawson takes three, among several other, pedagogical
positions, as I summarize (in my own words).
1. Abstract base 60 mathematics emerged in 8000 BC, well before Sumer
and Babylon existed. The West's first writing system apparently
emerged in Sumer, based on the Denise Schmandt Besserat's token
thesis.
2. Egyptian fraction mathematics, represented by
3/7 = 1/3 + 1/11 + 1/231
was an awkward notation, and therefore Egyptian mathematics could
not have developed beyond the basic (additive) arithmetic level.
Whatever abstract thought that did exist in Egypt probably was
diffused from Sumer/Babylon.
3. Greeks were great mathematicians, developing and/or using the
highest form of mathematics in the Ancient Near East.
There are three major problems with Clawson's pedagogical positions.
1. Writing may not have emerged first in Sumer. This point is in
historical limbo, currently under review in several respects, as
mentioned on this listserver several times. One alternative is
that Egyptian writing emerged 400+ years before Sumer's writing,
a proposal that is very new; but one that has hard archeological
evidence upholding the hypothesis. Another alternative is an
ancient Persian culture, one that also used tokens.
2. Egyptian fractions, as one way to write numbers, was not an
awkward notation, for the ancients (moderns find the system
awkward, primarily because they have not taken the time to study
the ancient documents that record the system). It was used for
generally writing rational numbers in exact and concise ways,
within a finite number system. For example, ancient Egyptians
and Greeks both used the system, while maintaining the ability
to write within infinite series, as needed. It is clear that
Greeks offered only minor improvements to the finite system,
and both cultures would have written
3/7 = 1/4 + 1/7 + 1/28
based on a tabular construction. One table follows that makes
my point.
2/7 = 1/4 + 1/28
3/7 = 1/4 + 1/7 + 1/28
4/7 = 1/2 + 1/14
5/7 = 1/2 + 1/7 + 1/14
6/7 = 1/2 + 1/4 + 1/14 + 1/28
I have only a modern notion where Clawson found the
3/7 = 1/3 + 1/11 + 1/231 series.
This 3/7th unit fraction series looks made up more from a
modern Fibonacci greedy algorithm view of 1200 AD history,
suggested by Sylvester in the 1890's, rather than copied from
a historical 300 BC or 1650 BC text.
3. I agree with Clawson that Greeks were wonderful mathematicians.
However, Greeks used the same Egyptian fraction numeration system
that Clawson suggested limited Egyptian mathematics. How can this
be? Did the development of geometry, or another form of number
theory (arithmoi) occur outside of the Egyptian fraction arithmetic
system?
Could be that base 60 number theory provided the number theory
that Greeks followed, as implied by Clawson? Only late Greek
scholars, like Ptolemy, formally used the base 60, and then only
in astronomical calculations, as best as I have seen.
I would appreciate an answer or two to the above points. For sure
Clawson has taken a very common set of pedagogical positions, based
on over 100 years of History of Science suggestions. However,
Clawson's apparent lack of concern for pending changes in
1. a confirmed date, and culture that first developed writing,
2. a lack of understand of Egyptian fractions as a method to
concisely write any rational number,
and,
3. the confirmed Egyptian mathematics that Greeks based their work
does require a comment, from time to time. Simply assuming that
Greeks first invented all the number theory, out of thin air,
needed to develop their geometry and other mathematics is a
rather old, and worn out pedagogy,
does surprise me.
Regards to all,
Milo Gardner