The linguistic debate confirms that Sumerian numeration
was far more developed by 2,700 BC that Denise S-B
indicates for the birth period of her 'fingers and toes'
version of writing's oldest foundations. For example,
READING THE PAST, MATHEMATICS AND MEASUREMENT, O.A.W.
DILKE shows to me that before 3,000 BC base 60 Sumerians
expressed numbers as an infinite series, with symbols
for, 1, 10, 60 x 10, 60 x 60, and 60 x 60 x 10. It
should also be noted that the earliest weight and
measures need to be factored into the deate. In addition
it has been noted by other writers, that I can not recall
at this time, that Sumerian god symbols were used
for 10, 20, 30, 40, 50 and 60, a point that seemed
to hold back the progress of their mathematical
thinking.
By the late Sumerian period 2,500 - 2,000 BC period,
when writing had fully flowered, the older god symbols
used for 10, 20, 30, 40, 50, 60 numerals were replaced
by abstract cuneiform symbols, with a relative explosion
of the mathematical sciences. And by the Babylonian period,
and the 2,000 BC - 75 AD period, a less cumbersome form
of cuneiform numeration system came into use, with its
infinite series numeration system, that was comparable
to the Egyptian Old Kingdom Horus-Eye.
Specifically, Before 2,150 BC in Egyptian scribes wrote
numbers less than one as:
1 = 1/2 + 1/4 + ... + 1/2n + ...
as many have reported as decimal fractions, a form of
base 2, written in base 10 (with its arithmetic following
a doubling methodology or duplation), as Babylonians
were also able to write during the same time period.
What is more interesting to me is that Sumerian-Babylonian
base 60 and its infinite series, is that by 2,000 BC Egyptians
began to write in a hieratic script finite base 10, as the
Moscow Papyrus reports, a significant improvement over the
infinite series. These facts indicate to me that even if
Sumerian and Elamite scribes were first on the scene with
writing, once Egyptians came into a full understanding
and use of it, phonetics and improved forms of words
and numbers 'quickly' followed, a point that might
suggest that the first origins of writing itself were
based on abstract thinking far more advanced than
'fingers and toes'.
Regards,
Milo Gardner
Sacramento, Calif.