# ancient concepts and modern concepts

Is there an extant published expository account, comprehensible to all mathematicians, of the conceptual differences between ancient Greek mathematical concepts and modern ones?

I have in mind things like this:

• Euclid (I'll need to look between the covers of a book to be sure whether this is right . . . .) didn't know how to multiply more than three numbers because no more than three lines can be mutually orthogonal; but he did know how to find the smallest number measured by more than three numbers (what today we would call the LCM);
• Consequently (?) he didn't know about factoring numbers into primes (for example, $90= 2\cdot3\cdot3\cdot5$ is not the LCM of its prime factors) (and that's why he stopped short of stating, let alone proving, the uniqueness of prime factorizations), but of course they did know that every number is "measured by" at least one prime number;
• (Maybe?) Euclid did not consider $1$ to be a number;
• The ancient Greeks had no concept of real number. They had a concept of congruence of line segments, so that they could say that one line segment goes into another between $6$ and $7$ times, and the remainder goes into the shorter segment between $2$ and $3$ times, etc. etc., so they knew what it meant to say the ratio of the length of segment A to that of segment B is the same as the ratio of the length of segment C to segment D. They even knew what it means to say the ratio of lengths A to B is the same as the ratio of areas E to F, and similarly volumes. But they did not make the mistake of knowing whether a particular area is less than a particular length. Modern mathematicians seem to make that mistake by saying those are real numbers; I think modern physicists may avoid that error.
• They did not have a concept of irrational number (since they didn't have a concept of real number), but they knew what it meant to say that two line segments have no common measure, and how to prove it in some cases (e.g. no segment can be laid end-to-end some number (= cardinality) of times to make the length of the side of a square and some other number of times to make the diagonal).
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When I wrote "comprehensible to all mathematicians", I had in mind that it would not be the sort of thing that is comprehensible to historians of mathematics but would go over the heads of mere mathematicians. That sort of thing exists, just as there are things comprehensible to mathematicians that would go over the heads of mere historians of mathematics. – Michael Hardy May 8 '12 at 18:28
Ptolemy in particular does computations that are very much not in the spirit of Elements, admittedly substantially later. So do Heron, and Diophantus. It is not possible to characterize "ancient Greek mathematics" in any simple way. We are dealing with a time period longer than the span from medieval times to 2012. Almost all of the early work, and much of the later work,has disappeared. – André Nicolas May 8 '12 at 18:51
@Michael The way you have written the above gives the impression that I stated "P because Q". But I did not. Could you please remove my name from that sentence (or link to what I actually wrote). Thanks. – Bill Dubuque May 8 '12 at 18:55
Dear Michael, I have a memory that somewhere in Bourbaki there is a discussion of the Greek point of view on (what we would call) real numbers, and in particular of Eudoxus's theory of proportions. I would guess that this was written by Dieudonne (who I think wrote most of the historical discussions in Bourbaki). Regards, – Matt E May 9 '12 at 0:55
P.S. I hope that someone will see the preceding comment and pin down the precise reference. – Matt E May 9 '12 at 0:55