Ed Wall's discussion of number brings in the history of
science as well as the history of math and its intersection
with the philosophy of mathematics. As one of several
references on the history of science aspect, Tobias
Dantzig, wrote:
Number, the language of science, a critical survye written
for the cultured non-mathematician, The MacMilliam Co, 1930.
Logic's intersection with mathematics is also noted by
Gottlob Frege's:
The Foundations of Arithmetic, a logici-mathematical
inguiry into the concept of number, translated by
J.L. Austin, Northwestern U. Press, 1968
-From my point of view, the history of mathematics is
probably the best way to introduce the concept of
number, to the educated mathematician and to the novice
non-mathematician. Available historical threads that can
be considered at consist of:
1. The Western Tradition, and its pre-600 BC foundations
of Egyptian and Babylonian number.
a. Egyptian and Babylonian use of infinite series statements
of rational numbers.
b. Egyptian sources of several finite series statements of
rational numbers (as used for over 3,000 years as
a numeration system).
2. The Classical Greek period, where the arrow of time is
introduced as a linear statement, including the infinite
series of irrational numbers. Note that the abacus view
of number during this period clearly showed a business
aspect that was very different from the science definition.
3. The non-Western Tradition views of cyclical number (and time)
seen from Chinese, Mesoamerican and other science perspectivies.
Modular arithmetic threads are very old, as our base 60 use
of Babylonian numbers provides interesting clues to the past.
4. The algorithmetic view of number, and our modern base 10
decimal system. With the full use of algorithms, introduced
from Persian and Arabic soruces, breaks with the past seem
to have began, as the following have also tended to fog
numbers rich science and mathematical roots.
a. Modern psychologists, like Piaget, have introduced practical
views of number for children 'finger and toes', long applied
only to the world of the abacus, and counting boards (as another
history of computing subject).
b. Modern computers, that first used fixed and floating point,
within a binary number system that was very near the ancient
Egyptian Horus-Eye notation (where infinite series, binary
fractions were rounded off), that have only recently been
resolved by supercomputers' use of polar coordinates, modular
numbers, and little Fermat numbers (speeding up parallel computing
1,000 times).
In summary, Julio's suggestion that the history of mathematics
has little to do with (the modern view of) number, seems out of
place, at least here on HM, right?
Regards to all,
Milo Gardner
Sacramento, Calif.