“This at
least,” said I, “will not be disputed by those who have even
a slight acquaintance with geometry, that this science is in direct
contradiction with the language employed in it by its adepts.Geometry (and mathematics) is inevitably less
abstract than dialectics. But the special purpose of the Platonic education
values mathematics chiefly as a discipline in abstraction. Cf. on 523 A, p.
152, note b; and Titchener, A Beginner's Psychology, pp.
265-266: “There are probably a good many of us whose abstract idea
of ‘triangle’ is simply a mental picture of the little
equilateral triangle that stands for the word in text-books of
geometry.” There have been some attempts to prove (that of Mr. F.
M. Cornford in Mind,April 1932,
is the most recent) that Plato, if he could not anticipate in detail the
modern reduction of mathematics to logic, did postulate something like it as
an ideal, the realization of which would abolish his own sharp distinction
between mathematics and dialectic. The argument rests on a remote and
strained interpretation of two or three texts of the
Republic(cf. e.g. 511 and 533 B-D) which, naturally
interpreted, merely affirm the general inferiority of the mathematical
method and the intermediate position for education of mathematics as a
propaedeutic to dialectics. Plato's purpose throughout is not to exhort
mathematicians as such to question their initiatory postulates, but to mark
definitely the boundaries between the mathematical and other sciences and
pure dialectics or philosophy. The distinction is a true and useful one
today. Aristotle often refers to it with no hint that it could not be
abolished by a new and different kind of mathematics. And it is uncritical
to read that intention into Plato's words. He may have contributed, and
doubtless did contribute, in other ways to the improvement and precision of
mathematical logic. But he had no idea of doing away with the fundamental
difference that made dialectics and not mathematics the coping-stone of the
higher education—science as such does not question its first
principles and dialectic does. Cf. 533 B-534 E.”
“How so?” he said. “Their language is most
ludicrous,The very etymology of
“geometry” implies the absurd practical conception of
the science. Cf. Epin. 990 Cγελοῖον
ὄνομα. though they cannot help it,Cf. Polit. 302 E,
Laws 757 E, 818 B, Phileb. 62 B,
Tim. 69 D, and also on 494 A. The word ἀναγκαίως has been variously misunderstood and
mistranslated. It simply means that geometers are compelled to use the
language of sense perception though they are thinking of abstractions
(ideas) of which sense images are only approximations. for they speak
as if they were doing somethingCf.
Aristot.Met.
1051 a 22εὑρίσκεται δὲ
καὶ τὰ διαγράμματα ἐνεργείᾳ: διαιροῦντες γὰρ
εὑρίσκουσιν, “geometrical constructions, too, are
discovered by an actualization, because it is by dividing that we discover
them.” (Loeb tr.) and as if all their words were directed
towards action. For all their talkFor φθεγγόμενοι cf. on 505 C, p. 89, note g.
is of squaring and applyingCf. Thompson on
Meno 87 A. and adding and the like,E. Hoffmann, Der gegenwärtige
Stand der Platonforschung, p. 1091 (Anhang, Zeller,
Plato, 5th ed.), misunderstands the passage when he says:
“Die Abneigung Platons, dem Ideellen irgendwie einen dynamischen
Charakter zuzuschreiben, zeigt sich sogar in terminologischen Andeutungen;
so verbietet er Republ. 527 A für die Mathematik
jede Anwendung dynamischer Termini wie τετραγωνίζειν,
παρατείνειν, προστιθέναι” Plato does not forbid
the use of such terms but merely recognizes their inadequacy to express the
true nature and purpose of geometry. whereas in fact