Yesterday, Luigi submitted a most interesting view of Aristotle,
as Aristotle may have viewed infinity and finite systems (of thought).
During Luigi's discussion Archimedes' method was introduced associated
with Eudoxus, a point that I would like to expand upon before commenting
on Aristotle and a view of Greek infinite and finite number systems (as
one system of thought).
To re-introduce Archimedes' method, it should be recalled that Luigi
classified it as a method of exhaustion, a point that appears to be
only partly true as reported by Dijksterhuis. Using the following 1/4th
geometric series, as a method that Archimedes has been reported to have
exactly summed up, by a finite series, the areas of certain geometric
shapes (if I am reading history correctly), the following are both an
arithmetic and algebraic statements:
4A/3 = A + B + C + D + E + E/4 (equation 1.0),
where A can be any number, even pi, and that B = A/4, C = B/4, D = C/4
and E = D/4, that may best be numbered equation 1.4, based on the
following infinite series discussion.
Note that an infinite series is related to equation 1.4 (as renumbered
from 1.0), by replacing A for all the B, C, D, and E terms/letters, or,
4A/3 = A + A/4 + A/16 + A/64 + A/256 + ... (equation 2.4)
It should also be noted that Archimedes and Eudoxus very well may have
generalized equation 2.4, for any mod m geometric series, as given by
the implied modern mathematical induction facts:
5A/4 = A + A/5 + A/25 + A/125 + ... (equation 2.5)
6A/5 = A + A/6 + A/36 + ... (equation 2.6)
.
.
.
mA A A A
---- = -------- + ------ + ... + ------ + ... (equation 2.m)
(m-1) (m -1)^0 (m -1)^1 (m -1)^n
Even more interesting, from an modern arithmetic point of view, is
that the last term of equation 1.0, E/3, can be substituted for any
of equations 2.4 - 2.m, beginning in the 2-term, creating a finite
series, numbered 1.1 - 1.m. One central historical question that is
stated. Did Archimedes and/or Eudoxus actually think in a manner that
was equivalent equations 1.1- 1.m, based on equations 2.1 - 2.m?
If the pending fresh translation of the recently purchased $2,000,000
Christie's document, on which Dijksterhuis gained his view of equation
1.0 (based on Heilberg's translation), confirms the above suggestion,
then a point concerning Aristotle and his view of infinite and finite
numbers can be put forward.
That is, Aristotle seemed to have used finite objects as well as infinite
objects, as elements of his physical arithmetic. Was Aristotle drawing
a parallel to Plato's Academy and its view of number that tended to
stress the finite side, or was he trying to draw an analogy to Eudoxian
arithmetic?
Again, I thank Luigi very much for his most interesting HM post.
Regards,
Milo Gardner
Sacramento, Calif.