What are Philolaos' “even-odd” numbers?

Number, indeed, has two proper kinds (ιδια ειδη), odd and even, and a third mixed together from both, the even-odd(αρτιοπέριττον). Of each of the two kinds there are many shapes, of which each thing itself gives signs.

(Philolaus F5 = Stobaeus, Eclogae 1.21.7c; tr. Huffman 1993)

The quote is from Horky's Plato and Pythagoreanism, 2013, (p.141), it is commented twice further in text. On p185 the author writes " the so-called even-odd (αρτιοπέριττον), which appears to refer to the “one.” Aristotle, too, knew this fragment, and he is explicit in associating the “one” with the “even-odd” class and stating that it is derived from the “even” and the “odd.” On p190, disappointingly, he says "however It is beyond the scope of this study to examine in more detail the significance of the concept of “mixture” of Forms or classes".

The “one” is not the number preceding two which Ancient greeks apparently did not accept as a number. Philolaos quote might suggest that it is set apart as it has no shape but I am tempted to read it as suggesting that the mixed numbers have multiple shapes, while the numbers from the proper kinds have each just one.

So what are the 3 kinds of numbers? The odd numbers are perhaps the ones that we also call odd, (starting with 3), while the even numbers are seen as, so to say, 'strictly even', the powers of 2, that is 2,4,8,16 and the rest, that is 6,10,12 etc. as even-odd. References and comments (or should this be moved to Philosophy SE)?

• According to Liddell & Scott, perseus.tufts.edu/hopper/…, an even-odd is two times an odd number. – Michael E2 Nov 19 '17 at 0:35
• Thanks but LS sends us back to F5 and Aristotle's "one" (Arist.Fr.199); to Plutarch (?= Plu.2.1139f ) and what Ph.1.3?. – sand1 Nov 19 '17 at 9:25
• First, let me say that while I am familiar with a couple of relevant texts, I don't have a comprehensive knowledge of the area, and thus my opinions should not be thoroughly relied upon. Perhaps you're suggesting LS on even-odd has been superseded (plausible). I thought you were asking just about the definitions of three terms. I'm not sure of the context, the Pythagoreans or broadly "Ancient greeks." The Ancient greeks were a heterogeneous bunch, and perhaps the Pythagoreans were, too. For instance, Euclid gives a different classification of numbers (even-times odd etc.)..... – Michael E2 Nov 19 '17 at 15:13
• ...The LS reference to an Aristotle fragment, which can be tracked down, seems at odds with the definition in LS. The fragment raises more questions than it answers, which is the way with fragments, I suppose. It seems to me that Aristotle might be criticizing the Pythagorean concept of even-odd, as opposed to stating or explaining it. – Michael E2 Nov 19 '17 at 15:13
• I'd guess he means the classification scheme given by Euclid, book 7, definitions 8-10. Euclid classifies numbers according to this scheme in book 9, propositions 32-34. – Marius Kempe Nov 19 '17 at 19:00

Aristotle comments on the Pythagorean theory in Met, Book I (A), 986a14-986a22:

these thinkers also consider that number is the principle both as matter for things and as forming their modifications and states, and hold that the elements of number are the even and the odd, and of these the former is unlimited, and the latter limited; and the $$1$$ proceeds from both of these (for it is both even and odd), and number from the $$1$$; and the whole heaven, as has been said, is numbers.

But see also Thomas Heath, A History of Greek Mathematics. Volume I (1921), page 71:

The explanation of this strange view might apparently be that the unit, being the principle of all number, even as well as odd, cannott itself be odd and must therefore be called even-odd.

There is, however, another explanation, attributed by Theon of Smyrne to Aristotle, to the effect that the unit when added to an even number makes an odd number, but when added to an odd number makes an even number: which could not be the case if it did not partake of both species.

Philolaus' Fr.5 is discussed at lenght in: Carl Huffman, Philolaus of Croton: Pythagorean and Presocratic, Cambridge UP (1993), page 178-on. There are various interpretations, including the possibility of an interpolation.

See page 190:

In summary, I believe that the even-odd is a derived class of numbers whose first member is, as the ancient tradition indicates, the one, but which also includes numbers that consist of even and odd numbers combined in ratios (e.g. $$2 : 1 , 4 : 3$$, and $$3 : 2$$ ) . This class of numbers corresponds to the third class of things in Fr.2, which consists of members that are harmonized from both limiting and unlimited constituents. The even-odd numbers are the numbers by which these harmonized things are known. This connection of course remains conjectural, but I believe that it is a plausible way to make sense of both Fr.2 and Fr.5 of Philolaus and Aristotle's testimony that there was a connection between the even - odd dichotomy and the unlimited - limited dichotomy, although my suggestion does not identify the two as Aristotle does.

• Using today terms,one could say that for the ancient Greeks, before the discovery of irrationals, numbers were integers>1, even or odd, and rationals. So, including 1 among the even-odd makes sense but not the explanation given by Theon and others, as adding any odd number changes parity. – sand1 Nov 21 '17 at 16:48
• @sand1 - Strictly speaking, for ancient Greeks numbers where only naturals. Then they have ratios between magnitudes, and magnitudes can be "measured" by numbers. You can see the post: irrationality-of-the-square-root-of-2. – Mauro ALLEGRANZA Nov 21 '17 at 16:50
• Notice also that for Euclid 'irrational' (ἄλογος – a better translation is 'ratioless') means something different from the modern 'irrational' – Euclid's term for his equivalent of the modern concept is 'incommensurable'. (See the definitions of book 10 of the Elements.) – Marius Kempe Nov 21 '17 at 21:15
• Also, in Euclid a number doesn't measure a magnitude – only objects that have a ratio to each other can measure each other (strictly, if the measuring one is equal to or less than the measured one). – Marius Kempe Nov 22 '17 at 14:06