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

part discusses the main characteristic of técnh – its usefulness (cr2simon,
åfélimon).
The invention of calculation (logismó~) put an end to discord (stási~) and in-
creased concord (ômónoia). With the invention of calculation greed (pleonexía)
disappears and equality (ısóta~) arrives, since it is by means of calculation that
we settle our dealings with others. Owing to this the poor receive from the power-
ful and the rich give to those in need, since both believe that owing to calculation
they will have what is fair (tò £son). A standard and a barrier to the unjust, it
averts those who can calculate (ëpistaménou~ logízesqai) from injustice, per-
suading them that they would not be able to stay unexposed when they resort to
calculation, and prevents those who cannot calculate from doing injustice by
showing through calculation their deceit.
As follows from this solemn praise of arithmetic (which is understood here as
the art of calculation), Archytas endeavored to show that
mathe¯mata are at least
no less useful than other técnai. No wonder he relates to the discovery of cal-
culation such important social changes as an increase of concord and an ad-
vance toward greater equality. The progress of knowledge leads to social pro-
gress, whose main criteria – the absence of inner discords, ômónoia and
ısóth~ – are very close to Isocrates’ ideals.
111
Moreover, calculation proves ca-
109
See below, 93 n. 58.
110
See
On Archytas’ Philosophy in three books (D. L. V, 25 No. 92 = fr. 207 Rose) and
Excerpts from Timaeus and the Works of Archytas (No. 94 = fr. 206 Rose). Archytas
is also mentioned in the Aristotelian corpus (
Pol. 1340b 25; Rhet. 1412a 12; Met.
1043a 21;
Probl. 915a 25).
111
The conditions of attaining eÿdaimonía are peace and ômónoia, the latter, in its

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Chapter 2: Science as técnh: theory and history
72
pable of improving people’s moral qualities, keeping them from greed and in-
justice or, at any rate, exposing these vices. Naive as this view of mathematics
may seem, we should not forget that it conforms perfectly with the claims of the
Sophists, who asserted that their lessons made young people not only wiser but
also better, and with the intellectualism of the ethical doctrines of the time in
general. Socrates and Isocrates, Plato and Aristotle shared the conviction that
knowledge makes a man and, accordingly, the society in which he lives, better.
It was about the kind of knowledge that they were at variance. Closer to Archy-
tas was the position of Plato, who believed, to all appearances, that long and
sustained study of mathematics not only strengthens and sharpens a man’s in-
tellect (on which Isocrates and Aristotle also agreed), but also leads him to the
understanding of what is good and accordingly improves his moral qualities.
112
Otherwise the ten years of studying the
mathe¯mata Plato imposed on the future
guardians of the ideal polis would have been spent in vain.
It has long been noted that the first and second parts of Archytas’ fragment,
connected as they are both stylistically and thematically, could hardly have fol-
lowed each other immediately.
113
What, then, could have filled the lacuna be-
tween them? The invention of the
mathe¯mata seems to be the most natural
theme to bridge the gap between the two parts. It is revealing that the first part
deals with the ‘methodology’ of scientific discovery, while the second begins
with the invention of one of the mathematical sciences, the art of calculation
(logismò~ eûreqeí~). In other words, the situation described in the second part
could only have developed
after calculation had been discovered and was the
result of that discovery. Since the circumstances of the discovery are not men-
tioned, one can surmise that the passage left out by the excerpter was related to
this topic, which seems to be perfectly relevant for a work
On Mathematical
Sciences. From the second part of the fragment, extolling the benefits brought
about by the discovery of calculation, it follows that,
before the discovery, so-
cial harmony was not possible. We do not know whether Archytas described
the life preceding this discovery as governed by greed and discord, i.e., whether
he was developing a theory of the origin of culture. We cannot rule out the
possibility that Archytas, like other authors of ‘introductions’ to various téc-
nai, limited himself to a brief digression on the inventors of mathematics.
114
I have already touched here upon the question whether there is any differ-
ence between logismó~ (B 3) and logistik2 (B 4) and whether they referred,
respectively, to practical and theoretical arithmetic. The fact that the second
turn, being the result of ısonomía (Panath. 178, Areop. 21, 69, Nic. 41, 67). See Car-
piglione, J. C. Isocrate, sull’ idea di progresso,
AAN 96 (1985) 247–267, esp. 263f.
112
Burnyeat, M. F. Plato on why mathematics is good for the soul,
Mathematics and
necessity. Proc. of British Academy 103 (2000) 1–81. The same view was held by Ni-
comachus (
Intr. arith., 65.13–16), Ptolemy (Alm., 7.17f.), and Proclus (In Eucl.,
24.4).
113
Blass. De Archytae, 581f.;
DK I, 437n.
114
See above, 51.

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4. Why is mathematics useful?
73