The Ancient Greeks famously rejected the conception of irrational numbers or rather refused to treat them as numbers - they regarded them as geometrical magnitudes. While I understand why this was the case with the irrationals, I do not understand: why did they not regard negative numbers as numbers? What was their philosophical reason for rejecting the idea of negative numbers?
See Euclid's Elements, Book VII, Defs.1&2 :
A unit is that by virtue of which each of the things that exist is called one.
A number is a multitude composed of units.
- Thomas Heath, A History of Greek Mathematics : Vol.I (1921), page 69 :
Aristotle observes that the One is reasonably regarded as not being itself a number, because a measure is not the thing measured, but the measure or the One is the beginning (or principle) of number. But note that for Greek math the only numbers are the natural ones and they must be distinguished from magnitudes : a segment, a square, ... which are "measured by" numbers.
In ancient Greek mathematics there are two different types of "basic" entities: numbers and magnitudes; there are no negative or rational numbers, but only magnitudes measurable with multiples of a suitable unit one.
Numbers (arithmós) are used for counting some number of things taken as uniform when counted; they are counted as “objects.” That word which is pronounced last in counting off or numbering, gives the “counting-number”, the arithmos [see Plato, Theaetetus, 198c ].
Thus the arithmos indicates a definite number of definite things. It proclaims that there are precisely so and so many of these things.
See also Euclid, Book X: units are counted while magnitudes are measured.
Ans see Euclid, Book VII:
The less of two unequal numbers [...] being continually subtracted from the greater...
Subtraction is used always this way: the greater "minus" the less. In this way, no negative quantities will be produced.
the simple answer is that the greek concept of number was based on multitudes and magnitudes, and a "negative" multitude/magnitude makes no sense. negative numbers arose when the arabs came along thinking about numbers in terms of accounts: positive means credit, negative means debt. zero means balance.
a good analysis of greek thinking about number etc. is at https://doi.org/10.1006/hmat.1996.0038. For arabic thinking, I'm afraid you'll have to learn Arabic, there is no good translation of Al-Khwaramzi's Kitab al-Jabr in English.