Pythagoras and the Pythagoreans
8
Three points
: generator of a triangle of dimension two
Four points
: generator of a tetrahedron, of dimension three.
The sum of these is ten and represents all dimensions. Note the ab-
straction of concept. This is quite an intellectual distance from “fingers
and toes”.
Classification of numbers.
The distinction between even and odd
numbers certainly dates to Pythagoras. From Philolaus, we learn that
“...number is of two special kinds, odd and even, with a
third, even-odd, arising from a mixture of the two; and of
each kind there are many forms.”
And these, even and odd, correspond to the usual definitions, though
expressed in unusual way
9
. But
even-odd
means a product of two and
odd number, though later it is an even time an odd number. Other
subdivisions of even numbers
10
are reported by Nicomachus (a neo-
Pythagorean
∼
100 A.D.).
•
even-even
—
2
n
•
even-odd
—
2(2
m
+ 1)
•
odd-even
—
2
n
+1
(2
m
+ 1)
Originally (our) number 2, the dyad, was not considered even,
though Aristotle refers to it as the only even prime. This particular
direction of mathematics, though it is based upon the earliest ideas
of factoring, was eventually abandoned as not useful, though even and
odd numbers and especially prime numbers play a major role in modern
number theory.
Prime
or
incomposite
numbers and
secondary
or
composite
numbers
are defined in Philolaus:
9
Nicomachus of Gerase (
ß
100 CE) gives as ancient the de
Þ
nition that an
even
number is
that which can be divided in to two equal parts and into two unequal part (except two), but
however divided the parts must be of the same type (i.e. both even or both odd).
10
Bear in mind that there is no zero extant at this time. Note, the “experimentation” with
de
Þ
nition. The same goes on today. De
Þ
nitions and directions of approach are in a continual
ß
ux, then and now.