are, as
we see, dreamingThe interpreters of Plato must
allow for his Emersonian habit of hitting each nail in turn as hard as he
can. There is no real contradiction between praising mathematics in
comparison with mere loose popular thinking, and disparaging it in
comparison with dialectics. There is no evidence and no probability that
Plato is here proposing a reform of mathematics in the direction of modern
mathematical logic, as has been suggested. Cf. on 527 A. It is the nature of
mathematics to fall short of dialectics. about being, but the clear
waking visionCf. Phileb. 20 B
and on 520 C, p. 143, note g. of it is impossible for them as long as
they leave the assumptions which they employ undisturbed and cannot give any
accountCf. on 531 E. of them. For
where the starting-point is something that the reasoner does not know, and the
conclusion and all that intervenes is a tissue of things not really known,The touch of humor is the expression may be
illustrated by Lucian, Hermotimus 74, where it is used to
justify Lucian's skepticism even of mathematics, and by Hazlitt's remark on
Coleridge, “Excellent talker if you allow him to start from no
premises and come to no conclusion.” what possibility is
there that assentOr
“admission.” Plato thinks of even geometrical reasoning
as a Socratic dialogue. Cf. the exaggeration of this idea by the Epicureans
in Cic.De fin. i. 21 “quae et a falsis initiis
profecta, vera esse non possunt: et si essent vera nihil afferunt quo
iucundius, id est, quo melius viveremus.” Dialectic proceeds
διὰ συγχωρήσεων, the admission of the
interlocutor. Cf. Laws 957 D, Phaedr. 237 C-D,
Gorg. 487 E, Lysis 219 C,
Prot. 350 E, Phileb. 12 A,
Theaet. 162 A, 169 D-E, I 64 C, Rep. 340 B.
But such admissions are not valid unless when challenged they are carried
back to something satisfactory—ἱκανόν—(not necessarily in any given case to the idea
of good). But the mathematician as such peremptorily demands the admission
of his postulates and definitions. Cf. 510 B-D, 511 B. in such cases
can ever be converted into true knowledge or science?”
“None,” said he.“Then,” said I, “is not dialectics the only
process of inquiry that advances in this manner, doing away with hypotheses, up
to the first principle itself in order to find confirmation there? And it is
literally true that when the eye of the soulCf.
on 519 B, p. 138, note a. is sunk