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

op. cit., 168f., refers in this connection to Seneca (NQ VI,5.31, VII,25.
4–7, VII,30.5–6) and Pliny (
HN II,15.62); the latter seems to be influenced by
Seneca. “Men who have made their discoveries before us are not our masters, but our
guides. Truth lies open for all; it has not yet been monopolized. And there is plenty of
it left even for posterity to discover.” (Sen.
Ep. 33, 10, cf. 64, 7). It is debatable
whether Seneca’s ideas were born out the experience of the previous progress in
science and technology or derived from Posidonius. For an interesting discussion on
this point and an ample bibliography, see Gauly, B. M.
Senecas Naturales Quaes-
tiones
. Naturphilosophie für die römische Kaiserzeit, Munich 2004, 159ff.
63
Isocrates was born in 436; Archytas, probably about 435/430.
64
On similarities in the understanding of técnh by Isocrates and the author of VM, see
Wilms, H.
Techne und Paideia bei Xenophon und Isokrates, Stuttgart 1995.

---

3. Archytas and Isocrates
61
which was initially oriented toward practical knowledge, in his descriptions of
mathematical disciplines as well:
It seems that arithmetic (logistiká) far excels the other arts (tõn mèn @llãn
tecnõn) in regard to wisdom (sofía), and in particular in treating what it wishes
more clearly than geometry (gewmetriká). And where geometry (gewmetría)
fails, arithmetic accomplishes proofs … (47 B 4).
The terminology of this fragment deserves elucidation. Since it deals with
demonstrations in which arithmetic surpasses geometry itself, what Archytas
means by logistik2 is not practical computation but theoretical arithmetic,
i.e., the theory of number based on deduction. Elsewhere he calls arithmetic
simply @riqmoí, and the four mathematical sciences together maq2mata (B 1).
In the fragment of the treatise Perì maqhmátwn, where the social role of arith-
metic is discussed, it is called logismó~ (B 3). Is there any difference between
logismó~ and logistik2, and does logismó~ refer, accordingly, to practical or
theoretical arithmetic? Leaving this question open for a moment, let us note
that Plato often uses logistik2 and @riqmhtik2 indifferently in this respect;
65
Aristotle also applies the new term, @riqmhtik2, and the old one, logismoí, to
one and the same science.
66
The distinction between
practical logistic and the-
oretical arithmetic is first found in Geminus (Procl. In Eucl., 38.10–12); later it
was taken up by the Neoplatonists, who attributed it, naturally, to Plato.
67
None
of the passages of Plato usually cited in this context, however, suggests this
meaning. This fact was pointed out and explained long ago by J. Klein, who
showed that, in Plato, the difference between logistic and arithmetic comes
down to the former referring mainly to counting and the latter to computation;
both disciplines can be theoretical, as well as practical.
68
As the material of the fifth and fourth centuries shows, we can hardly expect
the names of sciences and their classification to be rigorous and unambiguous.
Neither is there any contradiction in the fact that Archytas treats arithmetic and
geometry as
mathe¯mata (B 1, 3), yet places them elsewhere among técnai (B
4). In his time, the
mathe¯mata, though constituting, among other técnai, a
special group, had not yet become model of ëpist2mh. Under the influence of
the word-formative model of técnh, Archytas in one place even changes the
traditional term gewmetría
69
into gewmetrik3 (técnh).
70
By “(all) other téc-
65
Res. 525a 9, Gorg. 451c 2–5, Tht. 198a 5, Prot. 357a 3, Charm. 165e 6, 166a 5–10.
66
Cf.
Met. 982a 26f., APo 88b 12. logismó~ with reference to theoretical arithmetic,
see also Isoc.
Bus. 23; Xen. Mem. IV,7.8; Pl. Res. 510c 3, 522c 7, 525d 1; Pol. 257a
7. Arithmetic was often referred to as @riqmò~ kaì logismó~: Ps.-Epich. (23 B 56);
Pl.
Res. 522c, Phdr. 274c, Leg. 817e.
67
Olymp.
In Gorg., 31.4f.; Schol. Gorg. 450d–451a–c; Schol. Charm. 165e.
68
Klein, J.
Greek mathematical thought and the origin of algebra, Cambridge 1968,
10ff. (German original:
Q & St 3.1 [1934] 18–105). See also Burkert. L & S, 447
n. 19; Mueller, I. Mathematics and education: Some notes on the Platonic program,
Apeiron 24 (1991) 88ff.
69
Hdt. II, 109; Philol. (44 A 7a); Ar.
Nub. 202, Av. 995.

---

Chapter 2: Science as técnh: theory and history
62
nai” he must have meant not only mathematical sciences (he would have used
the term maq2mata in this case), but also other occupations traditionally re-
lated to this field. Owing to its sofía, arithmetic surpassed all these técnai,
which Archytas considered from the cognitive point of view. In the context of
técnai (crafts, poetry, music, medicine), sofía is usually understood as ‘skill,
craftsmanship, artfulness’ and is often associated with ‘precision’ (@krí-
beia).
71
Archytas transfers this quality from the master to técnh itself, thus
making arithmetic appear more ‘artful’ and, hence, more ‘precise’ than all the
other técnai, including geometry. Arithmetic surpasses the latter in ënárgeia,
i.e., clearness, evidence, and obviousness, which makes it, in comparison, more
demonstrative.
72
This quality of arithmetic is close to what the author of
VM
most of all required of medicine as a técnh: clearness and precision in knowl-
edge (eıdénai tò safé~, katamaqe$n @kribéw~).
73
Isocrates, in his own field,
held similar ideals: písti~ ënarg2~ and @pódeixi~ saf2~ are, for him, the key
notions that characterize the conclusiveness of a statement.
74