Milo Gardner pointed out a crucial problem for the interpretation of
ancient Greek Mathematics: the status of Zero and One, the former almost
completely ignored, the latter substantially excluded from the numbers.
To understand 0 and 1 in ancient Greek Mathematics we would rather
remember that our idea of real number is the result of a long evolution
just began in Greece.
So that for us integer numbers are special real numbers and real numbers
can be written (and approximated) by integers or rational numbers: this
way we melt "integer" and "parts", which were completely distinct in
ancient mathematics. Because the little point between integer and
decimal part splits two radically different universes: "counting" and
"mesuring".
"Counting" was centred on fingers, and then in base 10, measuring was
based on the division of the 'whole': its base was less well defined,
ultimately prevailing 12, both for astronomical reasons (months in a
year) and for its including simpler cases (2, 3, 4, 6).
(Heath in his 'Greek Mathematics' describes a Greek abacus with an
'integer' part suitable for computing in base 10, and a 'fractional'
part suitable for computing in base 12).
To melt these two functions (and these two different universes) was
one of the most troublesome problems for the ancient numerical systems.
Greek Mathematics had even another problem, due to the theoretical
frame of Greek philosophy: they had to theoretically define and classify,
to say "what is number".
Then 1 had to face the double role of "starting" for counting, and
"whole" for measuring, which placed it at the border between multitudes
(numbers) and parts.
To understand the peculiar role of 0 and 1, we must also remember the
original connection between Greek mathematics and Greek language, which
here can not be detailed, but for underlining Plato's analysis in
Sophista 237 c-e, where he strongly relates arithmetical and grammatical
numbers: 1-->singular, 2-->dual, more than 2-->plural; and obviously he
does not find any grammatical number for 0.
The peculiarity of 1 (and partially even of 2) derives then to its being
alternative to the linguistic plural (multitudes=numbers), and the Greek
idiosyncracy for 0 seems just part of their devastating "negative
judgement paradox": <given that an affirmative statement corresponds to
a fact, something that is, we have that a negative statement corresponds
to something which is not; but a statement about what is not, is about
nothing, and hence is impossible>.
My best regards
Luigi Borzacchini