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

On Arithmetic does not seem to raise any seri-
ous doubts. The onus, at any rate, lies with those who state the contrary. Rash as
it is to base conjectures on the subject of the book on a single fragment, the
work seems to have been of a popular philosophical, rather than historical char-
acter. It is also clear that its material did not repeat that of Eudemus’
History of
Arithmetic, though certain themes treated in both of them, as well as the posi-
tions of their authors, could well coincide.
In § 1, relying on a certain tradition (doke$), Aristoxenus tells us that Pytha-
goras highly valued the science of numbers (1 perì toù~ @riqmoù~ pragma-
teía)
23
and advanced it; he turned arithmetic into a theoretical science by sep-
arating it from practical need (1 tõn ëmpórwn creía). Obvious similarities
between this passage and Eudemus’
Catalogue seem to reflect some common
notions of the development of exact sciences (5.5). First of all, familiar from
the
Catalogue (In Eucl., 64.17f.) and characteristic of the Peripatetics in gen-
eral is the contrast between creía as the primary impulse towards acquiring
knowledge and pragmateía, the scientific discipline created by Pythagoras
from the study of numbers. Furthermore, the verb proagage$n, which belongs,
as we remember, to the semantic group meaning ‘progress’, occurs twice in
21
On Stobaeus’ working method, see Mansfeld, Runia.
Aëtiana, 196ff.
22
Aristoxenus’ treatise
On Music contained, apart from theoretical material, much in-
formation on various musical discoveries and their authors (fr. 78–81, 83).
23
pragmateía, in the sense of ‘scientific discipline’ or ‘branch of science’, is re-
peatedly found in Aristoxenus (
Elem. harm., 5.6, 6.1, 7.5 etc.).

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Chapter 6: The history of arithmetic and the origin of number
220
Eudemus (ibid., 67.7, 67.22). And, last but not least, Aristoxenus associates
with Pythagoras the same transformation of arithmetic into a theoretical sci-
ence as Eudemus does with geometry (eı~ sc4ma paideía~ ëleuqérou meté-
sthsen). We should recall that Aristotle attributed to Pythagoras the study of
mathematical sciences, in particular numbers; in
Metaphysics, he says that the
Pythagoreans were the first to advance mathematics.
24
While Eudemus limited his
History of Geometry to purely scientific prob-
lems, Aristoxenus, following Aristotle, thought it important to note the connec-
tion between the Pythagorean study of numbers and the philosophical and
scientific ideas of this school: according to him, Pythagoras likened all things to
numbers, “for number contains all else as well, and there is a lógo~ between all
the numbers”.
25
The similarity between Aristoxenus’ words and Aristotle’s
Metaphysics 985b 23f. was noted in particular by Frank and Burkert, who in-
terpreted it to the effect that Aristoxenus, lacking any independent evidence on
Pythagoras as a mathematician, simply quoted
Metaphysics, while substituting
‘Pythagoras’ for ‘the Pythagoreans’.
26
However, the parallels with Eudemus’
Catalogue and Aristotle’s fr. 191, quoted above, give this similarity a different
meaning: all the three testimonies reflect the notion, common to the Lyceum,
that Pythagoras made a decisive contribution to transformation of mathematics
into a theoretical science.
Besides, one cannot fail to note the essential differences between the Pytha-
gorean ‘number philosophy’ in Aristotle’s interpretation, on the one hand, and
Aristoxenus’ understanding of the resemblance between things and numbers,
on the other. What Aristoxenus meant by tá te Álla that number has (Écei) is
not wholly clear, but his reference to lógo~ existing between all numbers
seems to indicate that he understands the Pythagorean tradition of relationship
between things and numbers in the same epistemological sense as Philolaus,
for whom “all the things that are known have number, for without it is imposs-
ible to understand or to know anything” (44 B 4).
27
Aristotle, who indeed men-
tioned certain ômoiømata between numbers and things perceived by the Py-
thagoreans (
Met. 985b 28f.), meanwhile inclined, rather, to an ontological in-
terpretation, according to which mathematical principles are, at the same time,
the principles of all being (985b 25, 986a 16,
etc.). The connection of an odd
number, which has a beginning, a middle, and an end, with medical prognostics
(§ 4), as mentioned by Aristoxenus, differs substantially from the numerical
metaphysics Aristotle imposed on the Pythagoreans.
28
It is revealing that while
24
Puqagóra~ … diepone$to perì tà maq2mata kaì toù~ @riqmoú~ (fr. 191 Rose);
tõn maqhmátwn âyámenoi prõtoi tañtá te pro2gagon (Met. 985b 23f.).
25
Cf. “In numbers they seemed to see many resemblances to the things that exist and
come into being – more than in fire and earth and water” (Arist.