# Did ancient Greek mathematicians consider numbers independently of geometry?

I am currently reading Oliver Bryne's edition of Euclid's Elements, and in The Elements many arithmetic propositions are proved geometrically, and it feels to me that numbers are always treated as something that is tied to a geometrical object (be it the length of a segment, the area of a shape, or the angle between two lines) and never as entities independent of a geometrical realisation. For example, Euclid never seems to bring up the product of more than three quantities, which would indicate that his idea of the product was rooted in its geometric interpretation (as well as that he was probably limiting himself to only three dimensions).

Euclid also categorises quantities as being of one of four "kinds" : linear, superficial, solid, and angular, and he never adds two quantities of different kinds together, indicating that he really thought of them as distinct in nature. To me, this is very reminiscent of dimensional analysis, but it leads me to wonder if Euclid thought that quantities consisted of a number and a kind (or a number and a unit, in modern terms), or if the number itself was of a certain kind. In other words, if presented with a segment of length $$2$$ and a square of area $$2$$, would both $$2$$s have been considered the same or different?

Furthermore, I know that Pythagorean scholars represented integers as dots arranged in triangles, rectangles, and other shapes, which allowed to think about them in simple geometrical ways.

This leads me to ask myself if Greek mathematicians of the Antiquity ever considered numbers as independent mathematical objects, which were not necessarily geometrical quantities. The use of dots by Pythagoreans makes me think that it could be the case, but still they did arrange them in geometrical shapes, so I am still unclear.

(I am not just asking if Euclid or Pythagoreans ever did consider numbers as such, but if there was a point in the vast period that is Greek Antiquity where numbers were considered to be separable from geometry.)

• Hi. What word did you mean to put where "conure" is? Also, you need more paragraph breaks. May 30 '21 at 19:11
• Oops, I meant "conjure", but I just replaced it by "bring up" because I thought it would be more clear anyway. May 30 '21 at 21:01
• "and he never adds two quantities of different kinds together, indicating that he really thought of them as distinct in nature." Just like we don't add horses and apples. Besides, "area" and "angle" are different; it's impossible to add them. Jun 1 '21 at 4:30
• Eh, even a product of 3 numbers is really just chained 2-number products. We only write $a \times b \times c$ because $(a \times b) \times c == a \times (b \times c)$, so the parentheses aren't necessary. Jun 1 '21 at 16:40
• @chepner Yes, but when Euclid refers to products, he's not really refering to products of numbers as I understand, rather he's talking about "products of segments" which yield surfaces. It struck me as significant because it suggests that either he does not assign a number to the length of a segment, or that he refuses to do some geometric operation which he thinks has no real analog (so the product of 4 lines would be excluded because it would need 4 dimensions to illustrate), or both. Jun 2 '21 at 8:31

The answer is yes. There was a split. First of all, for the Greek mathematics (and very long after them) "numbers" were integers. "Rational numbers" were called fractions, and no concept of real number existed. Therefore, mathematics was essentially split into two independent areas: arithmetic and geometry. Diophantus wrote on arithmetic, he never mentions geometric interpretation of his problems, and it is not known whether he was aware of any such interpretation. (By modern nomenclature his research subject belongs to algebraic geometry). Apollonius wrote on geometry, and never mentions numbers. (From the modern point of view, he is another founder of algebraic geometry). Euclid wrote on both subjects, but his arithmetic books are separate from his geometric books, and there is little interaction.

People like Euclid and Archimedes had of course a good intuitive grasp of the concept of real number, and they had a theory of proportions when discussing such thing as lengths and areas, and what we call irrational numbers. This theory of proportions was used up to 18th century.

Remark. I recommend to read Euclid with a good modern commentary written by a mathematician, rather than an historian. The best one on my point of view is by Robin Hartshorne.

• I see, so more precisely, numbers (in the sense that mathematicians of this era understood the word "number" as) were not just considered independent from geometry, but separate from it alltogether. They belonged to the realm of arithmetic, which was separate from that of geometry. Also yes, I've been wanting to find a historical commentary of the Elements in order to understant it better as a piece of history. May 31 '21 at 8:02
• How do you qualify the proof of the irrationality of $\sqrt{2}$ by (ancient) Greek mathematicians (Theaetetus) as found in Plato's Republic? May 31 '21 at 12:57
• @Xi'an: for the Greeks, this meant that "there is no number whose square is 2". Concerning the proof, there were at least 2different proofs: one by geometry and one by arithmetic. May 31 '21 at 14:52
• @Thomas.M: Robin Hartshorne, Geometry: Euclid and beyond. Undergraduate Texts in Mathematics. Berlin: Springer. xi, 526 p. (2000) May 31 '21 at 14:55
• @AlexandreEremenko: Is that book a summary presentation or does it go through the actual original text and comments it thoroughly?
– Jim
Jun 2 '21 at 8:12

If you consider Diophantus "ancient" then the answer is "no". In his "Arithmetic" numbers are not necessarily related to geometry or physics. For Pythagoras, indeed, numbers did not exist without geometry or physics interpretation.