Leonid Zhmud The Origin of the History of Science in Classical Antiquity

Elements.
40
Becker himself dated this theory
to the first half of the fifth century, and van der Waerden to around 500.
41
Though neither of them attributes it directly to Pythagoras, Aristoxenus’ frag-
ment, which they did not take into account, seems to be pointing in this direc-
tion.
42
The four definitions given by Aristoxenus in § 3 are likely to have opened a
Pythagorean arithmetical treatise, a sort of introduction to arithmetic setting
forth, in particular, a theory of even and odd numbers.
43
In Aristoxenus, this
theory appears as a specimen, an example of the ‘science of numbers’ (1 perì
toù~ @riqmoù~ pragmateía) practiced by Pythagoras, especially as being di-
rectly related to his notions of the role of numbers in nature. To judge from the
evidence of Epicharmus, Philolaus, and, in particular, Plato (for whom arith-
metic, as we have noted already, was a science of even and odd), this theory re-
mained quite popular through the whole of the fifth century, even outside the
narrow circle of specialists. In the mathematics contemporary with Plato and
Aristotle, it remained in the background, still functioning as an elementary in-
37
Prot. 357a 3; Gorg. 451 b1, 451c 2; Res. 510c 4; Charm. 166a 5–10; Tht. 198a 6.
38
Becker, O. Die Lehre von Geraden und Ungeraden im IX. Buch der Euklidischen
Elemente,
Q & St B 3 (1934) 533–553 (= Zur Geschichte der griechischen Mathema-
tik, 125–145).
39
Definitions 8–11 relate to the so-called even-odd numbers (cf. Philolaus 44 B 5). The
Pythagorean origin of definitions 6–11 is pointed out in the scholia to Euclid (
Elem.
V, 364.6).
40
The only exception is the ancient demonstration of the incommensurability of the
square’s diagonal with its side (cf. Arist.
APr 41a 26, 50a 37), which figures in sev-
eral manuscript copies of Euclid’s book X as appendix 27 (
Elem. III, 408–410). It
also points to the Pythagorean school, namely, to Hippasus’ discovery of irrational-
ity (Becker. Lehre, 544f., 547; cf. Knorr.
Evolution, 22ff.).
41
Its antiquity is confirmed, in particular, by the quotation from Epicharmus’ comedy
(23 B 2), which seems to reflect the Pythagorean studies in even and odd numbers.
42
Becker, O.
Grundlagen der Mathematik in geschichtlicher Entwicklung, Freiburg
1954, 38; van der Waerden.
Pythagoreer, 392. If Hippasus used the theory of odd and
even to demonstrate the incommensurability of the square’s diagonal with its side,
this theory must indeed go back to Pythagoras’ time.
43
It is worth noting that the later introductions to arithmetic by Nicomachus, Theon,
and Iamblichus arrange the material according to the same pattern: having defined a
unit and a number they proceed to even and odd numbers and their derivatives.

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Chapter 6: The history of arithmetic and the origin of number
224
troduction to arithmetic for laymen. Otherwise it is hard to explain why Plato,
44
Speusippus (fr. 28 Tarán), and Aristotle
45
turn so often to odd and even
numbers, as well as to their derivatives.
46
By the end of the fourth century, after
Eudoxus’ studies in particular, this theory had become a sort of mathematical
rarity that lacked any intrinsic connection with the main body of arithmetical
science. No wonder Euclid placed it at the end of the last of his arithmetical
books (IX, 21–34). From Aristoxenus’ point of view, however, the antiquity and
primitive character of this theory could be an argument for associating it with
Pythagoras as the founder of theoretical arithmetic.
3. The origin of number
Let us turn now to the two versions of the origin of number, or the art of calcu-
lation, mentioned by Aristoxenus (§ 2). The reference to the Egyptian god
Thoth points to Plato’s dialogue