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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?

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    Some tidbits of information. The greek mathematicians were first and foremost geometers, and rather poor at calculating. There is no such thing as a negative distance in geometry, and there was otherwise no need for negative numbers (it was already hard enough to calculate without them). Negative numbers and the zero came much later, with algebra and better methods of calculation, from India and the middle-east. – Olivier Mar 21 '17 at 14:09
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    They did not "reject" this idea, it never occurred to them (nor to others for centuries after), the types of (geometric) mathematics they were doing did not lead to them. Nor did they reject "irrational numbers", the loose talk of incommensurable ratios as "irrational numbers" in popular books is a modernization that reverses the historical order of causation, see How were irrational numbers that are not constructible accepted by mathematicians? – Conifold Mar 21 '17 at 17:52
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    @Conifold, thanks. When did the distinction between number and magnitude occur relative to the discovery of incommensurable ratios? Am confused by contradictory historical accounts. Some say that Eudoxus through his theory of proportions which came after the discovery of incommensurables drew the distinction between magnitude and number but this is very strange. – Jordan S Mar 29 '17 at 19:18
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    It did not happen like that. Number and magnitude were not considered one and then distinguished. They were originally perceived as two different things, one coming from counting, another from measurement. Pythagoreans originally assumed that all ratios of magnitudes corresponded to ratios of numbers (i.e. of positive integers), before they discovered otherwise with the diagonal and the side of a square. Eudoxus then developed a theory of ratios of magnitudes that did not rely on such a correspondence. It took over a millenium to merge ratios, numbers and magnitudes into a single concept. – Conifold Mar 29 '17 at 19:28
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    I suppose it depends on what "finalized" means. But for all practical purposes it was mostly completed by mid 17th century, the spread of decimals promoted by Stevin in particular played a big part, see the link in my earlier comment. – Conifold Mar 29 '17 at 19:49
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See Euclid's Elements, Book VII, Defs.1&2 :

  1. A unit is that by virtue of which each of the things that exist is called one.

  2. A number is a multitude composed of units.

See also:

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.

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    How does the theory of ratios, e.g., in Eudoxos, relate to "there are no...rational numbers"? For instance, this article by Howard Stein suggests that ratios were seen as relations between quantities (p. 168). Fractional representation of rational numbers could be seen as expressing this view, with the relevant relation being that of division. Is the point that "rational numbers" are merely relational and not "objects" in some more specific sense (e.g., by not possessing some requisite "unity" or something of the sort)? – Dennis Mar 21 '17 at 18:25
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    @Dennis - a ratio between magnitudes was not "perceived" as a number i.e. a "counting-number" that counts a multitude of units. – Mauro ALLEGRANZA Mar 21 '17 at 20:32
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    @Dennis Asaf describes the formal interpretation of everything as sets in modern set theory. Under it "relations" are sets of pairs, and functions are "relations". It has little to do with how the word was understood before 1900-s, or even how it is understood today by non-set-theorists. But even on this interpretation (binary) operations are relations on triples, not pairs. It is not that they used the word "only" for positive integers, but that they were not aware of other systems sufficiently similar to extending the name to them. There was no meaning to (general) "number" as such. – Conifold Mar 21 '17 at 20:57
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    respectfully, i think this is historically innaccurate. the greeks did not have abstract numbers; a number was always a number of something, and numbers of different kinds of things were incommemsurate. so e.g. 2 of apples is not the same as 2 of oramges. a line of length 2 was incommensurate with a region of area 2. they solved this by means of ratios. in both cases the ratio to the (different) unit is 2, and that is what makes it possible to compare them. so ratio is absolutely fundamental. – user20153 Mar 21 '17 at 21:23
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    @Dennis ratio != division. you cannot divide the unit. this is a critical conceptual distinction between greek and arabic mathematics, which does accomodate the fracturing of the unit. – user20153 Mar 21 '17 at 21:30
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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.

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    I'd be interested in what you'd consider a good survey-ish type article or good article/articles on this topic since I'm not too familiar with this early history. I think it would also improve your answer greatly to include some references. Also, I found your comment above about the distinction with respect to "fracturing" interesting. It's something I know a little about on the Greek side, but I wouldn't know where to begin to look for the Arabic perspective. – Dennis Mar 21 '17 at 23:54
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    @Dennis - see link in edited answer. – user20153 Mar 22 '17 at 6:39

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