NICHOLAS D. SMITHsubsection above that one with the originals of these images. He then
offers his fi rst explanation of the signifi cance of the image/original re-
lation between these two subsections:
Would you, then, be willing to say that in respect of truth and untruth,
the division is in this proportion: As the believable is to the knowable,
so the likeness is to what it is like? (510a8– 10)Now, the relationship between the believables and the knowables Plato
makes in t his pa ssage, which is said to cor respond to t he relat ionship be -
tween the images and originals in the two lower subsections of the line,
can only refer to the contents of the entire lower main segment (both
of the two lowest subsections, that is), and those of the entire upper seg-
ment (both subsections), respectively. “Believables” (identifi ed in book
5 with the sensibles) surely do not belong (under this description, at any
rate) above the main division of the line, initially said to divide the in-
telligible from the sensible domains. Hence, the “believable/knowable”
distinction cannot be a way to characterize the relationship between the
lower and upper subsections of the upper segment of the line. Similarly,
Plato should not here be understood as comparing one of the lower with
one of the upper subsections of the line, for he has yet to explicate any-
thing about either of the upper subsections. The “believable/knowable”
distinction, accordingly, must be understood as a way to characterize
the fi rst proportion of the two main segments of the line, which the
proportion between the two lowest subsections, which Socrates had just
explained, is supposed to replicate.^9
If I am right about what the “believable/knowable” distinction is
supposed to show, however, it follows that even though the mathemati-
cians are contrasted negatively with the dialecticians, they are included
from the very beginning of the line simile in the section associated with
the “knowables.” So on this ground, too, it seems clear enough that
Plato regards the mathematicians as employing the cognitive power of
knowledge (the power naturally related to the “knowables”). There may
be some cognitive defi ciency in such studies, accordingly, but that de-
fi ciency is not to be characterized as a lack of—so much as a defi cient
employment of—knowledge.
What, then, is the defi ciency of mathematical method relative
to the dialectical method? Plato answers this question explicitly, and
the explanation in no way denies the possession or use of the power of
knowledge to the mathematicians. Instead, their defi ciency, relative to
the dialecticians, is just that they continue to rely on the use of images,
of which the dialecticians have no further need, and they also continue