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

De loc. in hom. 46; De affect. 45.
98
Jouanna.
De l’art, 187. For a similar argument, see Dissoi logoi 6, 11. Aristotle, as
well as Archytas, admitted the possibility of an accidental discovery: in their experi-
ments, it was not wit but chance that made the poets discover how to produce such
effects in their plots (
Poet. 1454a 10f.). See also Protr. fr. 11 Ross: sumbaíh mèn
gàr Àn kaì @pò túch~ ti @gaqón.
99
See above, 58f.
100
VM 2, De arte 9. See also: Isoc. Antid. 83 (the new is not easy to find) and Ad Dem.
18–19 (it is easier to learn what has already been discovered).
101
This point was insisted upon both by Isocrates (
Panath. 208–209) and Plato (Res.

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3. Archytas and Isocrates
69
and wrote about, offered better opportunities for progress, both general and in-
dividual, than medicine, which deals with a multitude of individual cases and
relies on practical experience, not on general rules alone. This is probably what
accounts for the disagreement between the mathematician and the doctor on
how easily new knowledge is discovered and assimilated. As a sharp observa-
tion in one of the Hippocratic treatises shows (
De loc. in hom. 41), doctors were
well aware of the difference between a system of clear and well-defined rules,
on the one hand, and medical knowledge as such, on the other:
Medicine cannot be learned quickly because it is impossible to create any estab-
lished principle in it (kaqesthkó~ ti sófisma), the way that a person who learns
writing according to one system that people teach understands everything; for all
who understand writing in the same way do so because the same symbol does not
sometimes become opposite, but is always steadfastly the same and not subject to
chance. Medicine, on the other hand, does not do the same thing at this moment
and the next, and it does opposite things to the same person, and at that things that
are self-contradictory.
102
Then the author gives a number of examples showing how similar means can
lead to opposite results and different means to similar results (41–44). As an an-
tithesis to medicine, the Hippocratic cites the generally known rules of writing,
but this idea could easily be illustrated by the example of mathematics as well.
In
Nicomachean Ethics, Aristotle, characterizing medicine in practically the
same terms,
103
remarks that the difference between scientific knowledge (ëpi-
st2mh) and practical reason (frónhsi~) is manifest, in particular, from the fol-
lowing fact (1142a 11–20):
While young men become geometricians and mathematicians and wise in matters
like these, it is thought that a young man of practical wisdom cannot be found.
The cause is that such wisdom is concerned not only with universals but with par-
ticulars, which become familiar from experience, for it is length of time that gives
experience; indeed one might ask this question too, why a boy (pa$~) can become
a mathematician, but not a wise man (sofó~) or a natural scientist (fusikó~). Is
it because the objects of mathematics exist by abstraction, while the first prin-
ciples of these other subjects come from experience, and because the young men
have no conviction about the latter but merely use the proper language, while the
essence of mathematical objects is plain enough to them?
104
455b 7f.). On the rapid progress in the acquisition of knowledge by those who con-
versed with Socrates, see Pl.
Tht. 150d–151a (qaumastòn Ôson ëpididónte~ …
pálin ëpididóasi).
102
Transl. by P. Potter. On the importance of kairó~ in medicine, see De loc. in hom.
44.
103
It has no fixed principle (oÿdèn êsthkò~ Écei, 1104a 4f.), the physicians must tà
prò~ tòn kairòn skope$n (ibid.); what a feverish patient generally benefits from
may not prove useful in each particular case (1180b 9).
104
Transl. by J. Barnes. In
EE, Aristotle illustrates this idea with the example of a fa-
mous mathematician: “Hippocrates was a geometer, but in other respects was

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Chapter 2: Science as técnh: theory and history
70
An important parallel to the motifs of scientific progress and the ease of
learning mathematics is found in
Protrepticus, written when Aristotle was at
the Academy. (Let us note that here Aristotle brings philosophy and mathemat-
ics together, rather than opposing them to each other.) Asserting that the ac-
quisition of philosophical knowledge is possible, useful, and (comparatively)
easy, Aristotle supports the ease of learning philosophy by the following argu-
ments: those who pursue it get no reward from men to spur them, yet their prog-
ress in exact knowledge is more rapid compared with their success in other téc-
nai.
105
In the passages parallel to this text, Iamblichus speaks of the rapid prog-
ress in both philosophy and mathematics,
106
while Proclus mentions mathemat-
ics alone.
107
Considering the passage from the
Nicomachean Ethics quoted
above and several passages in the
Metaphysics (981b 13–22, 982b 22f.) close
in meaning to
Protrepticus, we can safely surmise that while discussing the
(relative) ease of acquiring exact knowledge and the resulting rapid progress of
theoretical sciences, Aristotle referred not to philosophy alone, but to mathe-
matics as well.
108
thought silly and foolish, and once on a voyage was robbed of much money by the
custom collectors of Byzantium, owing to his silliness, as we are told.” (1247a
17f. = 42 A 2, transl. by J. Barnes).
105
tò gàr m2te misqoñ parà tõn @nqrøpwn ginoménou to$~ filosofoñsi, di’ Ön
suntónw~ oÛtw~ Àn diapon2seian, polú te proeménou~ eı~ tà~ Álla~ técna~
Ômw~
ëx ölígou crónou qéonta~ parelhluqénai ta$~ @kribeíai~, shme$ón moi
doke$ t4~ perì t3n filosofían e£nai ®+stønh~ (Iambl. Protr., 40.19–20 = Protr.
fr. 5 Ross = B 55 Düring). The rapid progress in mathematics was also mentioned in
an early Academic treatise, see below, 87ff.
106
Iambl.
De comm. math. sc., 83.6–22 = Protr. fr. 8 Ross = C 55:2 Düring: tosoñton
dè nñn proelhlúqasin ëk mikrõn @formõn
ën ëlacístœ crónœ zhtoñnte~ oÎ
te perì t3n gewmetrían kaì toù~ lógou~ kaì tà~ Álla~ paideía~, Ôson oÿdèn
Êteron géno~ ën oÿdemi* tõn tecnõn. Cf. mikrà~ @formá~ in Arist. Cael. 292a
15.
107
Procl.
In Eucl., 28.13–22 = Protr. fr. 5 Ross = C 52:2 Düring: tò mhdenò~ misqoñ
prokeiménou to$~ zhtoñsin Ômw~
ën ölígœ crónœ tosaúthn ëpídosin t3n tõn
maqhmátwn qewrían labe$n.
108
It is revealing that in Isocrates’ response to the
Protrepticus it is mathematics that is
in question (see below, 74f.). Another important parallel is the following passage in
Plato: lightly esteemed as the studies in solid geometry are by the multitude and
hampered by the ignorance of their students as to the true reasons for pursuing them,
they nevertheless in the face of all these obstacles force their way ahead by their in-
herent charm (
Res. 528b 6f. = Protr. C 55:1 Düring). Aristotle shifts the accent a
little: at present mathematics and philosophy make more rapid progress than all the
other técnai, though the studies in the latter are morally and materially stimulated,
while those preoccupied with theoretical knowledge are rather hampered than en-
couraged (see above, n. 105–107).

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4. Why is mathematics useful?
71
4. Why is mathematics useful?
Before going on to the second part of Archytas’ passage, let us note the obvious
similarity of the ideas stated in it, not only to the Sophists’ theory of técnh and
the Hippocratics’ notions of their own science, but also to Plato’s and Aris-
totle’s views on theoretical knowledge. Archytas’ influence on Plato is beyond
all doubt,
109
and so is Aristotle’s familiarity with Archytas’ works.
110
Of course,
the classical theory of técnh was known to Plato and Aristotle independently of
Archytas. His role as intermediary is more likely reflected in the fact that the
passage from science as técnh to science as ëpist2mh took place under the
decisive influence of the
mathe¯mata in which, in his generation, Archytas was
the major expert. Substituting themselves for técnh, the mathe¯mata became the
standard toward which the Academy and the Lyceum were oriented while they
created the new model of science as exact, certain, and irrefutable knowledge,
i.e., ëpist2mh. From this point of view, Archytas as mathematician was far
more important than Archytas as philosopher, the author of
On Mathematical
Sciences. Nevertheless, the influence of this work on the development of the
new model of science cannot be ruled out completely.
Unlike the first, methodological part of Archytas’ fragment B 3, its second