Why the Unit One in Ancient Greece was NOT Considered a Part of the Set of Number

We now understand the integers as abstract objects, but the ancient Greeks understood them as counts of units (the unit, one, was not a number, two was their first) and represented them with lengths of line segments (multiples of some unit line segment). Where we talk of divisibility, Euclid wrote of "measuring," seeing one number (length) a as measuring (dividing) another length b if some integer numbers of segments of length a makes a total length equal to b.

The ancient Greeks also did not have our modern notion of infinity. School children now easily understand lines as infinite, but the ancients were again more concrete (in this regard). For example, they viewed lines as segments that could be extended indefinitely (not something infinite that we view just part of). For this reason Euclid could not have written "there are infinitely many primes," rather he wrote "prime numbers are more than any assigned multitude of prime numbers."