p l at o ’ s u n d i v i d a b l e l i n e
Plato’s Undividable Line:
Contradiction and Method in
Republic VI
R ic h a r d F o l e y *
Objects:
A
Forms
B
Lower
Noetics
Mental states:
novhsi~
diavnoia
Intelligible World
C
Physical
Objects
D
Images
(shadows,
reflections)
pivsti~
eijkasiva
Visible World1
1. the overdetermination problem
plato’s divided line presents an immediate challenge to anyone who attempts
to follow the instructions offered in Book VI of the Republic. Although Plato never
explicitly mentions this fact, his instructions entail that the line be divided so that
the middle two subsegments are of equal length. But I will show that, according
to the best interpretation, the two middle subsegments are unequal because they
represent mental states of unequal clarity, and possibly also objects with unequal
degrees of reality. The divided line therefore cannot be partitioned without contradiction. I refer to this challenge as “the overdetermination problem”: the initial
mathematical instructions and the subsequent exposition of the significance of the
1
My argument will not involve any commitments regarding whether the line is drawn vertically
or horizontally, whether with the shadows at the left or right, or whether the longest subsegment corresponds to the Forms or the shadows. Nicholas D. Smith (“Plato’s Divided Line” [“Divided Line”],
Ancient Philosophy 16 [1996]: 25–46) gives an extremely thorough assessment of the alternatives
here. I chose to follow one general tendency in the literature of labeling the section representing the
Forms with the letter ‘A’ and treating it as the longest subsegment because Forms are first in order
of importance. When quoting from other writers, I will convert their nomenclature to mine so that
their meaning will be immediately clear. Regarding the nature of the controversial “lower noetics,” I
do not need to make any ontological commitments, as I will later explain, since the problem I will be
discussing arises regardless of one’s views of the “lower noetic” objects, whether they are Forms, lesser
intelligible objects, or physical objects used as images of Forms.
* Richard Foley is Assistant Professor of Classical Studies at the University of Missouri.
Journal of the History of Philosophy, vol. 46, no. 1 (2008) 1–24
[1]
journal of the history of philosophy 46:1 january 2008
line are individually sufficient to determine how the line is to be divided, yet they
divide the line in contradictory ways.2 The overdetermination problem has been
a perennial concern, and a substantial amount of work has been produced which
attempts to deal with the problem. I will offer a classification of approaches to
the overdetermination problem and show why these approaches are inadequate,
before ultimately arguing for my solution to the problem. Should my solution be
correct, a major point of confusion and contention in the Republic will be resolved.
But first, I will document the conflict between Plato’s instructions for the division
and his subsequent explanation of the analogy’s significance, since only a clear
statement of this conflict will reveal the magnitude of the overdetermination
problem. In showing that there is a problem, I must prove two propositions: first,
that the middle subsegments are equal, and second, that they are unequal. I will
refer to these as “the two halves” of the overdetermination problem, and obviously it is only if both halves are proven that there is a contradiction and hence
a problem at all. There are also two versions of the overdetermination problem.
Since Plato relates the divided line both to this ontology and to his epistemology,
there will be an ontological and an epistemic version of the problem. I will show
that the best reconstruction of Plato’s ontology can avoid the ontological overdetermination problem, but that no reconstruction of his epistemology can avoid
the epistemic overdetermination problem. My ultimate resolution of the problem
involves showing that the overdetermination problem is an intentional feature of
the divided line analogy.
Beginning with the first half of the ontological overdetermination problem,
the procedure Plato describes at 509d6–8—the very first mention of the divided
line—leads to the equality of the lengths of the second and third subsegments.
As an analogy for Plato’s ontology, we are to divide a line unequally, then use the
same ratio (to;n aujto;n lovgon) to divide both of the segments once more, so:
A/B = C/D = (A+B)/(C+D) [509d]
A = BC/D
So substituting for A, we have:
C/D = ((BC/D)+B)/(C+D)
C(C+D)/D = BC/D+B
C(C+D)/D = (BC+BD)/D
C(C+D) = B(C+D)
C=B
The ease with which this equality can be discovered might seem to be an anachronistic product of using an arithmetical method. But contemporary readers were really
at no disadvantage here. For one trained in geometrical methods, the equality is
roughly equally apparent, so a preference between an arithmetical or geometrical
method on this point is largely a matter of convenience.3 I will also later show that
Although I will argue that he fails to provide a solution to the problem, the strongest and clearest
statement of the overdetermination problem is in Robert Brumbaugh, “Plato’s Divided Line,” Review
of Metaphysics 5 (1952): 529–34.
3
I have arithmeticized this proof so it will be more easily grasped by the contemporary reader.
A geometric version of the proof is found in Jacob Klein, A Commentary on Plato’s Meno [Commentary]
(Chapel Hill: University of North Carolina Press, 1965), 119.
2
p l at o ’ s u n d i v i d a b l e l i n e
Plato himself was aware of this equality, and that he expected a significant number
of his readers to become aware of it (though without any explicit statement from
him that the middle subsegments are equal). Of course, readers of any era might
not trouble themselves to discover this equality, but in Plato’s day or ours, only
curiosity and rudimentary mathematical skills are required to discover it.
Moving to the second half of the ontological overdetermination problem,
Plato’s analysis of the analogy suggests that these subsegments are not of equal
length. This half of the overdetermination problem needs to be addressed cautiously because the extent of the problem turns in part upon one’s commitments
regarding the ontology of the lower noetic objects.4 One might claim that the
equality of B and C reflects the fact that the lower noetics are nothing more than
physical objects viewed by mathematicians as images of Forms. A rejection of mathematical “intermediates” would thus explain the equality of the middle segments.
This battle about the existence of mathematical intermediates rages through the
literature of the previous century, and I have no intention of resolving it here.
Instead, I will ultimately show how an epistemic version of the overdetermination problem emerges regardless of one’s views on the intermediates; so denying
a Platonic doctrine of intermediates will at best remove only the ontological
overdetermination problem.
This controversy about intermediates centers on an interpretation of one passage in particular, and interpreting this passage has important repercussions for
how to assess the ontological overdetermination problem.
Then you also know that, although [mathematicians] use visible figures and talk
about them, their thought isn’t directed to these but to those other things that they
are like. The claims they make are about the square itself and the diagonal itself, not
about the diagonal they draw, and similarly with the others. These figures that they
make and draw, of which shadows and reflections in water are images, they now in
turn use as images, in seeking to see those others themselves that one cannot see
except by means of thought.5
Aristotle (Metaphysics 987b14–17) first attributes to Plato the view that there is a
mathematical type of object ontologically intermediate between physical objects
and Forms.6 From this perspective, this passage of the Republic provides an elegant
symmetry: the shadows are copies of physical objects which are copies of the
mathematical intermediates which are themselves copies of Forms. According to
this view, Plato gives a neat, fourfold ontological hierarchy that maps perfectly
onto an epistemic hierarchy. For each type of object, there is a mental state used
to apprehend such objects, eijkasiva, pivsti~, diavnoia, and novhsi~. Sidgwick, Adam,
Grube, Notopoulos, Ross, Raven, Goldschmidt, Robin, Crombie, Davies, Boyle,
and Dreher all endorse a Platonic theory of intermediates, and Ross and Crombie do so explicitly because of the symmetry between epistemic and ontological
4
Smith (“Divided Line,” 32) gives a comprehensive list of the range of possibilities for the ontology of the lower noetics. I find his contention that they are physical objects used as images of Forms
convincing, but my solution to the overdetermination problem will not rest on this view.
5
510d5–11a1. Except where otherwise noted, all quotations from the Republic are from Republic,
trans. G. M. A. Grube and C. D. C. Reeve (Indianapolis: Hackett Publishing Company, 1993).
6
The Complete Works of Aristotle, ed. Jonathan Barnes (Princeton: Princeton University Press,
1995).
journal of the history of philosophy 46:1 january 2008
categories provided by this interpretation.7 On this view, if x is an image of y, then
y is more real and more knowable than x to the same degree. It is plausible to
assume that the unequal division of the subsegments represents these epistemic
and ontological disparities between the objects,8 so the middle two subsegments
are to be divided by the same ratio that holds between the two subsegments of
the visible and between the two subsegments of the intelligible world. Anyone
who attributes a doctrine of mathematical intermediates to Plato, therefore, has
a very serious version of the overdetermination problem: the middle subsegments
must be of unequal length because mathematical intermediates are more real
than physical objects.
It has become increasingly popular to reject the theory of intermediates. However, this rejection has mistakenly been claimed to dissolve the overdetermination problem. On this view, the controversial passage at 510d5–11a1 asserts that
physical objects are indeed used as images, but as images of Forms, not as images
of mathematical intermediates. This interpretation would mean that Plato has
a four-fold hierarchy of mental states, but only a tripartite ontology comprising
images, physical objects, and Forms. The overlap falls precisely in the middle two
sections, where there are two ways of looking at the same type of object. Either
one can have beliefs (pivsti~) about physical objects, or one can use these same
physical objects as images of Forms to think mathematically (diavnoia).
Rejecting the intermediates, however, does not eliminate the overdetermination
problem. This problem reemerges because Plato asserts that the line represents
properties other than just the ontological ones. The inequality of the second
and third subsegments is placed on an epistemic foundation at the conclusion
of Book VI:
Understand that for these four segments, there are these four states which come to
be in the soul: understanding for the highest, thought for the second, belief for the
third, and imaging for the last. Arrange them according to the ratio (tavxon aujta; ajna;
lovgon), considering these mental states to share in clarity to the same extent that it is
the case for those segments to share in truth. (511d6–e4; translation mine)
The most important issue is the correct translation of ‘ajna; lovgon’. What is the
relation between this ratio of mental states and the earlier ontological ratio at
509d6–11d5? One might argue that linguistic considerations should lead us to
7
Henry Sidgwick, “On a Passage in Plato, Republic, B. VI,” Journal of Philology 2 (1869): 96–103
at 103; James Adam, The Republic of Plato [Republic] (Cambridge: Cambridge University Press, [1902]
1929), 159; G. M. A. Grube, Plato’s Thought (Boston: Beacon Press, [1935] 1958), 25; James Anastasios
Notopoulos, “Movement in the Divided Line of Plato’s Republic,” Harvard Studies in Classical Philology
47 (1943): 57–83, at 73; David Ross, Plato’s Theory of Ideas [Plato’s Theory] (Oxford: Oxford University
Press, [1951] 1966), 47–8; J. E. Raven, “Sun, Divided Line, and Cave” [“Sun”], Classical Quarterly NS
3 (1953): 22–32, at 25–26; Victor Goldschmidt, “La Ligne de la République et la Classification des Sciences,” Revue Internationale de Philosophie 32 (1955): 237–55, at 239; Léon Robin, Les Raports de l’Être et
de la Connaissance d’Apres Platon [Les Raports] (Paris: Presses Universitaires de France, 1957), 15; I. M.
Crombie, An Examination of Plato’s Doctrines [Examination] (New York: The Humanities Press, [1962]
1969), 112–13; J. C. Davies, “Plato’s Dialectic: Some Thoughts on the Line,” Orpheus 14 (1967): 3–11,
at 4; A. J. Boyle, “Plato’s Divided Line,” Apeiron 7 (1973): 1–11, at 1–2; John Paul Dreher, “The Driving
Ratio in Plato’s Divided Line” [“Driving Ratio”], Ancient Philosophy 10 (1990): 159–72, at 161–62.
8
There are several ways that this assumption might be rejected, and I will consider them when
discussing the various responses to the overdetermination problem.
p l at o ’ s u n d i v i d a b l e l i n e
believe that they are not the same, and that Socrates is introducing a new ratio
which holds between the mental states. The phrase ‘ajna; lovgon’ is so common that
it need not refer back to any specific antecedent ratio. On this reading, ‘ajna; lovgon’
means that Glaucon is to sort the mental states by a proportion—one equal to the
relative clarity of the mental states. In fact, there is no need to assume that there
is even a single ratio here, since ‘ajna; lovgon’could just mean to arrange the mental
states proportionally, with there being no implication that any two segments are
even related by the same ratio. Such a reading would avoid an epistemic overdetermination problem, because the division of the line would be governed entirely
by the relative clarity of the mental states. With no independent and inconsistent
mathematical statements about the division procedure (as we had at 509d6–11d5),
the division of the mental states line will not be overdetermined.
However, the best reading of 511d6–e4 does identify the mental states ratio
with the initial ontological ratio, for two reasons. First, Socrates explains that the
ratio in question simultaneously represents the truth (ajlhqeiva~) of the objects and
the clarity (safhneiva~) of the mental states. But Socrates had earlier identified the
truth of the objects with the ontological ratio: “Would you be willing to say this, I
said, namely [the line] is divided, as regards truth and untruth [dih/rh'sqai ajlhqeiva/
te kai; mhv], such that just as the opinable is to the knowable, so too the likeness is
to the thing it is like?” (510a8–10, my translation). Since the line was originally
divided by a ratio that expressed the relative truth of the objects subsumed under
each segment, the mental states ratio must be the same, since the clarity of the
mental states ratio is equated with the truth of the objects at 511e2–4. Second,
the position of this passage, located between Socrates’ concluding remarks on
dialectic and his introduction of the allegory of the cave, suggests that it is a summary of the divided line. Serving as a summary, this passage should therefore be
read to refer back to the earlier ratio. It would be odd for Socrates to present a
detailed account of his ontology, follow it with a short, isolated paragraph about
relations between various mental states, only to resume a discussion of his ontology in the allegory of the cave. Admittedly, it is strange that Socrates, thinking
of the ratio first given at 509d would fail to refer to this ratio specifically, instead
using the indefinite ‘ajna; lovgon’. Yet the proximity of the mental states passage
to the lengthy account of the divided line, which runs continuously from 509d6
to 511d5, means that the reader has been holding the ontological ratio in her
mind for some time, and ‘ajna; lovgon’ could easily refer back to this ratio. Now if we
read 511d6–e4 as referring to the earlier ratio, we have one half of the epistemic
version of the overdetermination problem: (A+B)/(C+D) = A/B = C/D, which
entails that B = C. However, it would be possible to identify the ratio of 511e2–4
with that of 509d6–11d5 without producing this set of equalities by asserting that
although the ratio is the same, the terms related to get this ratio are different.
Indeed, nothing at 511d6–e4 shows which mental states are being compared as
regards clarity so as to produce the ratio in question. One could even argue that
the epistemic overdetermination problem can be avoided merely by limiting which
segments are related by the ratio. Yet I think that this way of avoiding the problem
is forced. If the ratio at 509d6–11d5 is the same as that at 511d6–e4, the better
reading is that the ratio is the same because the corresponding mental states are
journal of the history of philosophy 46:1 january 2008
being compared. So for example, the ratio of pivsti~ to eijkasiva is R as regards clarity because pivsti~ stands over physical objects, eijkasiva stands over shadows, and
the ratio of physical objects to shadows is R as regards truth. So the mental states
being compared are the same as the objects that were compared at 509d6–11d5,
(A+B)/(C+D) = A/B = C/D, yielding one half of the epistemic overdetermination
problem (B = C).
The second half of the epistemic overdetermination problem is readily derived
from passages that show that, as regards clarity, Plato believes that diavnoia is superior
to pivsti~. This very passage at 511d6–e4 could itself be taken as evidence for this
point. Plato’s listing of the mental states is placed in a hierarchical order: highest
(ajnwtavtw), second, third, last (teleutaivw)/ . Some writers, including Zeller, Natorp,
and Robin, downplay the importance of the initial division into the intelligible
and visible realms, preferring instead to unify the divided line into a hierarchy
of mental states.9 For such writers, 511d6–e4 plays a crucial role, and it is read
as implying that each adjacent mental faculty is superior to the one beneath it.
If, in addition to emphasizing an epistemic hierarchy where each mental state is
superior to the one beneath it, one takes the use of the singular (ajna; lovgon) literally, we will be left with the line divided into four sections, each longer than the
adjacent one by the same ratio, so that, where R is the arbitrary ratio, R = A/B =
B/C = C/D.10 Assuming, of course, that R ≠ 1, it immediately follows that each
subsegment is a different length.11 The brief passage at 511d6–e4 would include
both parts of the overdetermination problem, by simultaneously connecting the
dividing ratio back to the earlier mathematical stipulations at 509d6–8 (so that
(A+B)/(C+D) = A/B = C/D) and implying that A/B = B/C = C/D. I do think that
this reading is unlikely, given that the ratio in question at 511d6–e4 is the ratio
already specified at 509d6–11d5. Yet even if this passage does not produce two
numerically precise ratios which conflict, it still does imply a conflict because, to
some degree, diavnoia is superior to pivsti~ as regards clarity. Plato arranges the four
mental faculties from the highest to the last, which I read as having an evaluative
9
Eduard Zeller, Philosophie der Griechen, vol. 2-1, (Leipzig: Fues’s Verlag, [1844–52] 1889), 637–38
n. 3; Paul Natorp, Platos Ideenlehre: Eine Einfürung in den Idealismus (Leipzig: Verlag der Dürr’schen
Buchhandlung, 1903), 188–89; Robin, Les Raports, 14–15.
10
There is some debate as to whether this ratio should be viewed as “arbitrary.” Gregory Des Jardins (“How to Divide the Divided Line” [“How to Divide”], Review of Metaphysics 115 [1976]: 483–96),
Dreher (“Driving Ratio”), and Robert Brumbaugh (Plato’s Mathematical Imagination [Imagination]
[Bloomington: Indiana University Press, 1954]) argue that we should divide the line by the golden
ratio, which is when the shorter segment stands to the longer as the longer stands to the whole line. It
will be clear shortly that I have no objection to attributing meanings to the Republic that do not lie on
the surface, yet in the absence of a compelling explanation for why this meaning would have been left
implicit or hidden, I tend to think that we should view the ratio as arbitrary. Balashov offers a detailed
and convincing case for rejecting the golden ratio thesis, Yuri Balashov, “Should Plato’s Divided Line
Be Divided in the Mean and Extreme Ratio?” [“Mean and Extreme”], Ancient Philosophy 14 (1994):
283–95. Nothing in my argument turns on this decision, however.
11
If we preserve a constant ratio as we move from one subsegment to the next, the intelligible
as a whole stands to the visible as a whole as the square of this ratio. Since R > 1, this means that the
intelligible is, comparatively speaking, even more knowable and real than the visible! But, of course,
this conclusion violates Plato’s stipulation that the ratio used to divide the intelligible from the visible
is the same ratio used to divide these two segments into their components. This problem is no surprise
though, since there simply is no mathematically consistent way to unify all that Plato says about the
division procedures. My ultimate claim is that this problem is an intentional feature of the analogy.
p l at o ’ s u n d i v i d a b l e l i n e
connotation. Placing diavnoia second and pivsti~ third connotes that diavnoia is to
some extent superior to pivsti~ as regards clarity.12 So we are left with the problem
that the ratio at 511e2–4 makes the middle segments equal in length, and yet
these two mental faculties are not equal in clarity.
There is another passage which expresses a similar relation between the mental
states: “And you seem to me to call the state of geometers thought but not understanding, thought being intermediate between opinion and understanding”
(511d2–5). The property used to accord thought this intermediate position is
superior clarity (safevsteron). More generally, Plato is quite explicit, particularly
in Book VII, that thought, due to its clarity, is intellectually superior to opinion.
The relative brightness of the sunlight outside the cave to the light of the fire in
the cave is so strong that the transition to the outside world is done only under
compulsion and at the cost of great pain (515e1–4). Since this relative brightness
symbolizes the clarity of the respective mental states, again Plato asserts that diavnoia
is superior to pivsti~ as regards clarity. This assertion cannot be taken to be the final
word, however, since it contradicts Plato’s claim that R = (A+B)/(C+D) = A/B =
C/D. That is, we are forced to confront the overdetermination problem, even if
we reject the doctrine of intermediates, simply because the same problem arises
for the relative clarity of the mental states: the length of the segments represents
the epistemic properties of the mental states, yet although the middle segments
are equal, Plato asserts that diavnoia is clearer than pivsti~.13
This overdetermination problem leads to a series of worries: does Plato himself
know that the procedure he specifies leads to the equality of B and C? I will argue
that Plato’s subsequent restatement of the ratios at 534a offers strong evidence
that he does, but this answer simply generates further questions. How are we to
understand the ontological and epistemological relations between the kinds of
items represented by the middle two subsegments? What is the significance of
Plato’s repeated claim that mathematics serves as an indispensable bridge to the
intelligible world? Plato says that the science of calculation is one of the subjects
that “naturally lead to understanding”; it is “fitted in every way to draw one towards
being” (523a1–3). There are numerous other passages from 522b2–31d4 in which
Plato extols mathematics as uniquely able to direct the mind to the intelligible
world. He even disparages empirical astronomy by contrasting it unfavorably with
an a priori version of the science (529a9–d5). But if B = C, then should not pivsti~
be just as likely as diavnoia to direct one to contemplation of the Forms? These problems are magnified for anyone who endorses a Platonic doctrine of intermediates,
It is possible to claim that there are two different properties at issue. Regarding one property
(presumably clarity), these two faculties are identical, hence merit subsegments of equal length.
Regarding some other property (perhaps appropriate pedagogical stage), the two faculties are asymmetric, hence diavnoia is placed before pivsti~ on a list. I will devote more attention to this suggestion
when considering prominent solutions to the problem, specifically under what I call the “dissolution
interpretation.”
13
Because the overdetermination problem arises regardless of one’s view on the intermediates,
in what follows I will say that dealing with the problem is important in order to unify Plato’s epistemic
views and possibly also ontological views. My point is that an epistemic version of the problem results
merely from the recognition that diavnoia is clearer than pivsti~, but that an ontological version arises
only if one endorses a theory of intermediates where these mathematical objects are the originals of
which physical objects are copies.
12
journal of the history of philosophy 46:1 january 2008
since then there will be an additional ontological variant of the overdetermination
problem. A solution to the overdetermination problem is therefore pressing if we
are to reconcile the divided line analogy with the educational program of Book
VII, or if we are to obtain a consistent view of Plato’s deeper epistemic and possibly
of his ontological commitments.
2. imperfect solutions
The divided line analogy, situated between the sun analogy and the allegory of
the cave, is a passage of central importance for Plato’s metaphysics and epistemology. There are many other issues raised by this crucial passage, but the overdetermination problem has exerted a powerful influence on the development of
interpretations of the divided line. Throughout the literature, there are generally
four ways of interpreting the divided line so as to deal with the overdetermination
problem. My division of the criticism into four groups is based exclusively on this
one issue. I do not mean to suggest that writers grouped together share anything
more than a common stand on this specific issue, and indeed, often they do not.
The extent of this scholarly work reflects the importance of the topic, but if my
analysis of this corpus is correct, we currently lack a solution to a core problem
in Plato’s metaphysics and epistemology. I will first document the magnitude of
the problem by explaining why all of these interpretations are inadequate, before
offering my own solution to the problem.
2.1 The Revisionist Interpretation
One efficient way of dealing with the overdetermination problem is to eliminate
it by revising the text of the Republic. Ast, Stallbaum, and Murphy all seek to revise
the text in a way that will make the initial partition of the line equal, thereby making all four segments equal.14 Nettleship explicitly endorses agnosticism about
whether such a revisionist reading is correct (although his editor, Benson, asserts
that Nettleship should have embraced an unequal ratio because he endorses a
symbolism of progressive mental development as one moves along the line).15
There are several suggestions for how the text could be modified. For example,
Stallbaum reads ‘ajn` i[sa’ for ‘a[nisa’, yielding, ‘Taking a line divided up into two
equal sections’. Ast reads ‘i[sa’ and the F manuscript gives ‘ajn` i[sa’. On the revisionist interpretation, the equality of the middle subsegments presents no special
difficulty because the line is partitioned into four equal quarters, thereby eliminating the overdetermination problem from Plato’s text. Alternately, Murphy claims
that the “words a[nisa tmhvmata only introduce confusion and should probably be
deleted as a gloss that has crept into the text.”16 The “confusion” that Murphy alludes to is presumably the overdetermination problem. One of the fundamental
14
Friedrich Ast, Platonis Politia (Leipzig: B. Schwickerti, 1814); Gottfried Stallbaum, Platonis Dialogos Selectos (Gotha et Erfurt: Guil. Hennings, 1827); N. R. Murphy, “The ‘Simile of Light’ in Plato’s
Republic” [“Simile”], Classical Quarterly (1932): 93–102, at 99 n. 1; and with more conviction in N. R.
Murphy, The Interpretation of Plato’s Republic [Interpretation] (Oxford: Clarendon Press, 1951), 158.
15
Richard Lewis Nettleship, Lectures on the Republic of Plato (New York: St Martin’s Press, [1897]
1961), 236–39 n. 1.
16
Murphy, Interpretation, 158.
p l at o ’ s u n d i v i d a b l e l i n e
problems with the revisionist interpretation is that it now becomes impossible
to attribute any significance to the divided line. Dividing a line into four equal
sections has no analogical significance, and only by preserving a[nisa tmhvmata is it
possible to understand what the line is even doing here. On this interpretation,
the significance of the line is merely to assert that there are four different types
of mental state, sequentially ordered, the relative importance of which is assigned
only in the allegory of the cave.17
The revisionist interpretation is also inconsistent with several other passages
in Plato’s exposition of the divided line. “Would you be willing to say this, I said,
namely it is divided, as regards truth and untruth [dih/rh'sqai ajlhqeiva/ te kai; mhv],
such that just as the opinable stands to the knowable, so too the likeness stands to
the thing it is like?”18 The dividing is to be done in accordance with the truth and
falsity of the objects being divided, and we know that Plato does not take these
to be equal, therefore the line cannot be divided into four equal subsegments.
Additionally, the passage already considered at 511d6–e4 not only generates an
epistemic version of the overdetermination problem, it also shows that the four
subsegments are unequal in length. Plato clearly states that we have four epistemic
capacities, four corresponding types of objects, and the ratio between these capacities and objects correspond to clarity and truth, respectively.
Even confining ourselves to linguistic considerations, the revisionist interpretation has not been in favor recently, with Lafrance offering a powerful refutation.19 Since only one significant manuscript diverges, and ‘a[nisa’ was accepted
in antiquity by Proclus and Plutarch, it is very unlikely that an alternate reading is
correct. One alternative, ‘ajn` i[sa’, even makes for poor Greek.20 Additionally, all
variants of the revisionist interpretation make nonsense of the subsequent claim
that the segments are to be divided using “the same ratio.” If the initial division
were equal, we would expect that Plato would say that the segments are also to
be divided into equal subsections, not that they are to be divided into the same
unspecified ratio. The revisionist interpretation is therefore untenable, and the
overdetermination problem must be confronted.
2.2 The Demarcation Interpretation
Those in our second group—proponents of the demarcation interpretation,
including Jackson, Stocks, A.S. Ferguson, Hardie, Crombie, J. Ferguson, Rose,
Reeve, and possibly Robin and Annas—avoid the overdetermination problem by
claiming that the middle two subsegments were not meant to be compared.21 The
Murphy, “Simile,” 99 n. 1.
510a8–10, translation slightly emended. Grube refers to the “ratio” here, but since the word
‘lovgo~’ does not appear in the Greek, it seems an unfair bias against the revisionist interpretation to
include it.
19
Yvon Lafrance, “Platon et la Géométrie: la Construction de la Ligne en Républic, 509d–11e”
[“Platon”] Dialogue 16 (1977): 425–50, at 435.
20
Benjamin Jowett, The Dialogues of Plato (Oxford: Clarendon Press, [1871] 1953), 307.
21
Henry Jackson, “On Plato’s Republic VI 509d sqq.,” Journal of Philology 10 (1882): 132–50; J. L.
Stocks, “The Divided Line of Plato Rep. VI,” Classical Quarterly 5 (1911): 73–88, at 76–77; A. S. Ferguson, “Plato’s Simile of Light,” Classical Quarterly 15 (1921): 131–52, at 138 n. 3; W. F. R. Hardie, A
Study in Plato (Bristol: Thoemmes Press, [1936] 1993), 56; Crombie, Examination, 112; John Ferguson,
17
18
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general idea of this strategy is to deny that Plato ever compares the middle two
subsegments, and so although the implication of the division procedure is that
these two subsegments are of equal length, and although Plato does usually intend
the relative lengths of the subsegments to correspond to the relative knowablity
and perhaps the relative reality of the corresponding objects, the lack of any
reference to a comparison indicates that, in the specific case of the middle two
subsegments, Plato intends no comparison to be drawn at all. It is always possible
to extend an analogy further than an author intends, and doing so potentially
leads to contradictions, but as long as one is careful to confine the analogy to
the specific comparisons that Plato mentions, there is no danger of generating
the overdetermination problem. This interpretation therefore requires a clear
demarcation between the intended and unintended points of comparison, and
such a demarcation will show that the equality of the middle subsegments can be
dismissed because it falls into the latter category.
Reeve offers the most strident defense of this position:
The propositions of the Line, therefore, symbolize only the things explicitly mentioned,
and none of those characterized as requiring further argument. Consequently, we
cannot infer anything from these proportions about the relative clarity of scientificthought [B] and folk-wisdom [C], or about the relative closeness to the truth of their
accounts. More especially, we cannot infer that they are equally clear, or that their
accounts are equally close to the truth, from the fact that the sections of the Line to
which they are assigned, C and B, must be of equal length. This equality is simply not a
functioning part of the simile, as the green of a shamrock was not a functioning part of Saint
Patrick`s use of it as a simile for the Trinity.22
So although the equality of the middle two subsegments is an implication of the
procedure for dividing the line, the overdetermination problem does not arise.
We are not supposed to compare the lengths of these two subsegments because
Plato himself does not compare the two kinds of mental states represented by
these two subsegments.23 The demarcation interpretation makes the divided line
“Sun, Line, and Cave Again,” Classical Quarterly NS 13 (1963): 188–93, at 188; Lynn Rose, “Plato’s
Divided Line,” Review of Metaphysics 67 (1964): 425–35, at 430; C. D. C. Reeve, Philosopher-Kings
(Princeton: Princeton University Press, 1988), 81; Léon Robin, Platon (Paris: Librairie Félix Alcan,
1938), 110; Julia Annas, An Introduction to Plato’s Republic [Introduction] (New York: Oxford University
Press, 1981), 247.
22
Reeve, Philosopher-Kings, 81; emphasis mine.
23
Robin and Annas also seem to endorse the demarcation interpretation, though the issue is difficult. Robin lists three points of comparison that the line is supposed to represent, and a comparison
between the middle segments is absent from this list, implying that Robin advocates the demarcation
interpretation. The issue is unclear because Robin also frequently speaks about the importance of
hierarchy, and seems to imply that each segment is superior to the one adjacent to it. He is also one of
the most important writers to emphasize the image/object relation that holds between A and B, and
C and D. However, it is unclear if he wants this relation to hold between B and C. Annas is difficult to
classify because she confines her statement that the middle subsegments are equal to a parenthetical
comment, “(Plato does not himself remark that this results in C and B being equal)” (Annas, Introduction, 247). I think it is a mistake to confine this important observation to a mere parenthetical remark.
Another problem is that Annas’s claim strongly suggests that she believes that Plato was aware of the
equality, but simply failed to “remark” on it. This point is not so clear, and requires careful documentation. A proponent of the demarcation interpretation need not say that Plato knew about the equality
of the middle subsegments, only that he intended no comparison between these subsegments. Finally,
Annas manifests no appreciation for the conflict that this equality creates in Plato’s discussion of the
p l at o ’ s u n d i v i d a b l e l i n e
11
an odd analogy by asserting that it has no tacit implications. The divisions of the
line analogically represent only what Plato specifically says that they represent, so
the analogy turns out to be otiose.24 One can dispense with the analogy entirely
merely by attending to the specific relations that Plato explicitly mentions.
The plausibility of the demarcation interpretation hinges on whether Plato
compares the two kinds of objects or mental states represented by the middle
subsegments. If one accepts a Platonic doctrine of intermediates, then we have
already seen that Plato makes just this comparison: “These figures that [students
of mathematics] make and draw, of which shadows and reflections in water are
images, they now in turn use as images, in seeking to see those others themselves
that one cannot see except by means of thought” (510e1–11a1). If this passage is
read as saying that the lower noetics are mathematical objects of which physical
objects are copies, then Plato has rejected the demarcation interpretation simply
by comparing the reality of the middle types of objects. If the demarcation interpretation is to be defended, one must reject the doctrine of intermediates, and
instead read 510e1–11a1 as asserting that mathematicians use physical objects
as images of Forms. On this reading, C and B are not compared, so the success
of this variant of the demarcation interpretation rests on whether there are any
other places where Plato compares B and C.
Again, the conclusion of Book VI offers just such a comparison, and refutes the
demarcation interpretation as well as the revisionist interpretation:
Understand that for these four segments, there are these four states which come to
be in the soul: understanding for the highest, thought for the second, belief for the
third, and imaging for the last. Arrange them according to the ratio, considering
these mental states to share in clarity to the same extent that it is the case for those
segments to share in truth. (511d6–e4; translation mine)
It is difficult to avoid the implication that the ordering of these segments from
highest to last is also supposed to reflect the relative clarity of the mental states.
Understanding is highest (ajnwtavtw) because it is clearest. Placing thought second
and belief third would then imply that thought is superior to belief, and that Plato
does indeed intend a comparison of B and C.
The demarcation interpretation also suffers if one accepts that the divided line
and the allegory of the cave are intended to be close correlates, as Plato seems to
claim at 517a8–b1. The cave explicitly describes the transition between the middle
two stages, from studying the statues in the cave, to seeing the reflections and
shadows outside the cave. Moreover, this transition is described in terms nearly
identical to the other transitions, involving compulsion, blinding light, and pain.
So just as the transition from inside to outside the cave marks an improvement, at
epistemic and possibly ontological relations between the middle two subsegments, so it is somewhat
difficult to classify her stand on the overdetermination problem. I think that her view is that since
Plato did not remark on the equality of the middle subsegments, it was not an relevant feature of the
analogy, which situates Annas in the demarcation camp.
24
There are perhaps some residual functions to the analogy, e.g., one might be able to remember
the various epistemic and ontological relations better by remembering the line analogy. My point here
is simply that according to the demarcation interpretation, Plato has himself already specified all of
the legitimate points of comparison to be drawn between the line analogy and that of which it is an
analogy. The analogy therefore contains no symbolic content that is not explicitly stated by Plato.
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least in the type of knowledge acquired, if not in the reality of the objects studied,
Plato intends a comparison between the middle two subsegments of the line, and
more significantly, he intends B to be longer than C to the extent that liberation
from the cave is motion in a positive epistemic direction. A proponent of the
demarcation interpretation would therefore need either to deny that the cave
should be correlated with the divided line—a claim which is implausible25—or
at least to deny the specific point of comparison I have mentioned. But the latter also seems implausible, since Plato explicitly compares the degrees of clarity
between the middle two stages of the cave, and he intends the line to represent
relative degrees of clarity. Since Plato correlates the line and cave, and provides
a detailed description of the greater clarity achieved when one first leaves the
firelight for the world outside the cave, the demarcation interpretation could
support the contention that Plato intends no comparison between B and C only
by an ad hoc pronouncement that, though parallel, line and cave are not parallel
in this one way.
The final consideration against the demarcation interpretation is the tremendous importance that mathematics has in Plato’s account of philosophical development. The study of mathematics serves as a bridge between physical objects and the
Forms.26 Learning to think mathematically is presented as a necessary condition
for thinking philosophically because mathematics is what leads us from concern
for physical objects to understanding of eternal objects. Once this transition to
eternal objects has been made, it is easier to study the Forms. Plato has indeed
devoted considerable thought to the relation between physical and mathematical
objects, and the transition from thinking about the one to the other constitutes
(according to Plato himself) one of the key moments in the intellectual development of any aspiring philosopher. It follows that the relation of the two middle
sections should receive some attention in any complete account of the division
process. Even if this relation were less important than any of the other relations,
it surely cannot be ignored. In fact, I think that this relation is substantially more
important to Plato than A/B or C/D, considering just how indispensable mathematics is in liberating us from our obsession with the physical world. But even if
there is the slightest comparison of B and C, as indeed there is, the demarcation
interpretation must be rejected.
2.3 The Gaffe Interpretation
The third group of interpreters simply concedes the overdetermination problem, but downplays its importance by claiming that the equality of the middle
subsegments is a minor unintended implication of the division procedure. This
interpretation suffers from a disadvantage from the outset, since it is preferable,
wherever possible, to prefer a consistent, successful interpretation. However, if
the overdetermination problem is conceded, and no solution is found, one might
be forced to accept the gaffe interpretation as the interpretation which is best
Raven, “Sun,” 27–32, offers a defense of the “traditional view” that the line and cave are strongly
parallel.
26
Some relevant passages which offer particularly strong support of this contention include
523a1–3, 525b3–e4, and 526a8–c6.
25
p l at o ’ s u n d i v i d a b l e l i n e
13
able to rescue Plato’s epistemology and perhaps his ontology, while conceding a
minimum of inconsistency. On the gaffe interpretation, then, Plato intended the
subsegment lengths to reflect the epistemological and possibly the ontological relations between the types of objects represented, and he intended the objects of the
middle subsegments to have asymmetric relations, but Plato was simply unaware
that he had given contradictory instructions for the division. This contradiction is
not terribly harmful, since Plato’s meaning is entirely clear if one simply attends
to what he says about the relations between the types of objects.
Adam, Ross, Wedberg, Gaiser, Raven, Cross and Woozley, Findlay, Sayre, and Balashov all fall into this group.27 Ross claims: “The equality of the middle subsections
is an unintended, and perhaps by Plato unnoticed, consequence of what he does
wish to emphasize, that the subsections of each section, and the sections themselves,
stand for objects unequal in reality.”28 Proponents of this interpretation must be
less agnostic than Ross about Plato’s ignorance of the issue. It makes little sense to
say that Plato might have been unaware of the equality of the middle subsegments.
Since the equality of the subsegments produces the overdetermination problem, it
is only by explicitly endorsing Plato’s ignorance of the equality that this interpretation can dismiss the problem as relatively insignificant. Wedberg correctly sees this
point, calling the equality “obviously an unintended feature of the mathematical
symbolism to which no particular significance should be attached.”29
One should be reluctant to attribute ignorance to a writer of Plato’s stature,
though great philosophers can neglect important implications of their claims.
But even this answer does not immediately exonerate Plato; it only prompts the
objection that he should have been aware of the equality. It might not be too serious a flaw in a philosophical work to manifest a lack of concern for mathematical
niceties, but since the divided line is situated as the introduction to the exposition
of an educational program which touts mathematics as the key to gaining philosophical understanding, it would be disastrous to claim that Plato was not aware
of a mathematical property of his line which he should have been able to see. One
is apt to respond that, if a philosopher can attain Plato’s eminence despite a glaring lack of (or at least lack of concern for) basic mathematical skills, then there
can be no serious philosophical need for preparatory training in mathematics.30
By attributing such mathematical lassitude to Plato, this interpretation undercuts
one of the fundamental claims that Plato is trying to make with the divided line.
The overdetermination problem cannot therefore be treated as a minor mistake.
As a mistake it would undercut the core argument of Book VII.
Adam, Republic, 64; Ross, Plato’s Theory, 45–46; Anders Wedberg, Plato’s Philosophy of Mathematics
[Mathematics] (Stockholm: Almqvist & Wiksell, 1955), 102–03; Konrad Gaiser, Platons Ungeschriebene
Lehre (Stuttgart: Ernst Klett Verlag, 1963), 92; J. E. Raven, Plato’s Thought in the Making (Cambridge:
Cambridge University Press, 1965), 145; R. C. Cross and A. D. Woozley, Plato’s Republic (New York:
St Martin’s Press [1964] 1966), 204; J. N. Findlay, Plato: The Written and Unwritten Doctrines (New York:
Humanities Press, 1974), 186; Kenneth Sayre, Plato’s Late Ontology [Late Ontology] (Princeton: Princeton
University Press, 1983), 303 n. 13; Balashov, “Mean and Extreme,” 287 n. 17.
28
Ross, Plato’s Theory, 45.
29
Wedberg, Mathematics, 102–03.
30
The elementary Euclidean proof of the equality is given in Klein, Commentary, 119. This proof
is so simple that it would be difficult to explain Plato’s lack of awareness of the equality, other than by
admitting that he just did not care about mathematics.
27
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The decisive objection to the gaffe interpretation is that there is considerable
evidence that Plato was himself aware of the equality of the middle subsegments.
In Book VII, Plato recapitulates the division of the line, but with a stunning alteration. “And as being is to becoming, so intellect is to opinion, and as intellect
is to opinion, so knowledge is to belief and thought to imaging” (534a3–5). Expressing this in our earlier terminology, (A+B)/(C+D) = A/C = B/D. The crucial
difference, of course, is that Plato has transposed C and B from the original set
of equalities at 509d6–8, viz. (A+B)/(C+D) = A/B = C/D.31 If we combine 534a
with 509d—helping ourselves to the occurrence of (A+B)/(C+D) in both—we
get A/B = C/D = A/C = B/D, which obviously entails that B = C. 32 The equality
now requires absolutely no mathematical manipulation: since Plato first says that
the ratio of intelligible to visible is the same as knowledge to thought, then later
says that the ratio of intelligible to visible is the same as knowledge to belief, it
is obvious that the linear representations of thought and belief must be equal
in length. The simplicity of the inference shows that Plato was almost certainly
aware of the equality. A further consideration leaves no doubt whatsoever. Why
did Plato think that it was permissible to vary the formulation from 509d to 534a
in the way that he did? Plato’s willingness to interchange the position of B and
C in the proportion proves that he recognizes that his original stipulations at
509d entail the equality of these two subsegments. Since the novel formulation
at 534a is consistent with the original formulation at 509d only because B = C, this
is strong evidence that it was the knowledge of the equality which enabled Plato
to adopt this particular reformulation of the ratios.33 Had Plato not known about
the equality, he would not have been able to say of the ratio that it is such that R
= A/B at 509d, and that R = A/C at 534a. Since Plato knew that B = C, the gaffe
interpretation must be rejected. It is not accidental that the middle subsegments
are equal. Any successful resolution of the overdetermination problem will need
to explain why the middle segments are equal, and also need to be compatible
with Plato’s knowledge of this equality. It is this second point that causes the gaffe
31
There is another anomaly, viz. Plato substitutes the term ‘ejpisthvmh’ in place of ‘novhsi~’, and
uses ‘novhsi~’ to refer to the entire intelligible section. These substitutions are consistent with Plato’s
claim that he does not care what words he uses: “But I presume we won’t dispute about a name when
we have so many more important matters to investigate” (533d7–e2). See also Socrates’s reluctance
to argue by pun, 509d2–4. However, it does seem odd that Plato would not only fail to say that this is
why he changed the terminology, but not even mention that he has changed the terminology. Another
similar novelty of this passage is the switch from initially dividing between the intelligible and visible
to dividing between being and becoming, and intellect and opinion.
32
Murphy (Interpretation, 159), Des Jardins (“How to Divide,” 176), Sayre (Late Ontology, 303 n.
13), and Balashov (“Mean and Extreme,” 287) all see that this passage at least suggests that Plato was
aware of the equality. However, none of them is able to explain why Plato would take the trouble to
offer such indirect support without bothering to make his awareness of the equality explicit. Why does
Plato offer such an elliptical hint that he knows B = C? Why not simply state that B = C? My interpretation will explain why Plato limited himself to presenting only indirect evidence for his awareness of
the inequality.
33
One possible misunderstanding should be avoided here. It is, of course, true that if (1) A/B
= C/D, then (2) A/C = B/D for any values of B and C whatsoever; this is just “cross multiplying.”
However, cross multiplication works by multiplying both sides of (1) by the fraction B/C to generate
(2). So if B is not equal to C, then the equalities of (1) will not themselves be equal to the equalities
of (2). Now, since Plato claims that (3) A/B = C/D = R = A/C = B/D, he simply must know that B =
C, otherwise this long string of equalities would not in fact be equal.
p l at o ’ s u n d i v i d a b l e l i n e
15
interpretation to founder. There is now an additional concern as well, viz. why
would Plato vary the formulation at 534a in the way that he does? The variation
is permissible given the equality of B and C, but just to say that it is permissible
still leaves the question of Plato’s motive undecided.
2.4 The Dissolution Interpretation
The fourth group of interpreters argues that the equality of the middle two subsegments is an intended consequence of the divided line, because Plato himself
meant to convey an important similarity between these two types of objects. This
interpretation needs to give a clear account of how to reconcile this equality with
the epistemic and possibly ontological differences between the middle two types
of objects, but if this can be done, then the overdetermination problem simply
dissolves because the equality of the middle subsegments is intended. This interpretation has an attractive elegance: there will be no need for ad hoc revisions to
Plato’s text, no implausible demarcation to the limits of the analogy, and no attribution of error to Plato. Because of this elegance, the dissolution interpretation
has become extremely prominent recently, and it merits careful examination.
Brumbaugh, Fogelin, Des Jardins, Lafrance, and Desjardins endorse this
approach.34 Desjardins, for example, argues that the equality of the middle subsegments is no problem because the two types of objects “are similar: both are
expressible in statements (in contrast to the mere naming of perception), and
both can be justified or unjustified (that is to say, right or wrong, true or false,
in contrast to the infallibility of perceivings).”35 Des Jardins also opts for the dissolution interpretation: “Again, all visible bodies and the maqhvmata . . . are both
countable and measurable, or quantifiable. . . . [V]isible bodies and maqhvmata are
equally of the quantifiable kind.”36 But if the dissolution interpretation is to succeed, at a minimum we need the respect of similarity to hold between B and C,
but not with A or D, otherwise there would be no reason for Plato to reflect this
similarity by making B and C equal, but longer than D and shorter than A. Neither
Desjardins nor Des Jardins offer such a property. The claim that the objects of B
and C “are expressible in statements” might serve to distinguish them from D if
we grant Des Jardins’s claim that D represents non-propositional perceivings, but
surely the objects of A can be expressed in statements. The same problem arises
for Desjardins’s claim that the objects in B and C are true and false, right and
wrong, a property which is shared by the objects of A. Similarly for Des Jardins,
although it is debatable that the objects of the lowest subsegment are not countable, measurable, or quantifiable, it is surely false that the objects of the highest
subsegment fail to meet these conditions. There are also important similarities
between A and B, and between C and D. The former are intelligible objects, the
latter are visible objects, and this similarity would appear to be far more important
than any other similarity between B and C, yet these similarities do not compel
34
Brumbaugh, Imagination, 98; Robert Fogelin, “Three Platonic Analogies,” Philosophical Review 80
(1971): 371–82, at 381–82; Des Jardins, “How to Divide,” 491; Lafrance, “Platon,” 442; Rosemary Desjardins, The Rational Enterprise [Enterprise] (Albany: State University of New York Press, 1990), 176.
35
Desjardins, Enterprise, 176.
36
Des Jardins, “How to Divide,” 491.
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Plato to make their subsegments of equal length. Proponents of this interpretation need to be more careful in selecting a property to explain the equality of the
middle subsegments.
One increasingly popular version of the dissolution interpretation involves
denying the existence of mathematical intermediates, as in Klein, Cooper, Morrison, Bedu-Addo, and Brann.37 The equality of the middle subsegments represents
the ontological fact that both subsegments refer to physical objects, either as objects for pivsti~, or used as images of Forms for diavnoia. On this interpretation,
the ontological issues between the middle two subsegments become sufficiently
confused to stave off the ontological overdetermination problem. In the cave, it
is true that the statues used in the puppet show are copies of “physical objects.”
Then the prisoner upon leaving the cave sees reflections and shadows that are
copies of these same “physical objects.” So both of the middle subsegments deal
with copies of Forms, and hence might be taken to be ontologically on the same
footing. Bedu-Addo argues in just this way.38
Many of the proponents of the dissolution interpretation take the fact that
they can eliminate the ontological version of the overdetermination problem as
evidence for the correctness of their view. I think, however, that by claiming this
as an advantage, they undermine the viability of their interpretation. Since these
interpreters grant that the overdetermination problem is a legitimate worry that
must be resolved, they must admit that their interpretation is undermined by the
fact that the epistemic variant of this very problem has not been eliminated. In
first discussing the problem, I argued that 511d6–e4 correlates the subsegment
lengths not only with the reality of the corresponding objects, but also the truth
and clarity of the corresponding mental faculties. Proponents of the dissolution
interpretation correctly see that the overdetermination problem must be solved,
but they fail to solve the mental states version of the problem. Why are the middle
subsegments equal when diavnoia is superior to pivsti~? If one believes that the
superiority in question is in terms of clarity, then an epistemic version of the
overdetermination problem arises. Smith is the only proponent of the dissolution
interpretation to recognize this point, but in recognizing it, he grants that the
dissolution interpretation does not, in the end, dissolve the overdetermination
problem at all.39 To avoid Smith’s conclusion, one would also need to dissolve the
epistemic version of the problem by rejecting the claim that diavnoia and pivsti~ are
not equally clear. A reconstruction of Plato’s epistemology might be able to accomplish this task. For example, one might say that the physical objects studied by
37
Klein, Commentary, Neil Cooper, “The Importance of diavnoia in Plato’s Theory of Forms,”
Classical Quarterly NS 16 (1966): 65–69, at 67; J. S. Morrison, “Two Unresolved Difficulties in the
Line and Cave,” Phronesis 22 (1977): 212–31, at 223–24; J. T. Bedu-Addo, “Mathematics, Dialectic
and the Good in Republic VI–VII,” Platon 30 (1978): 111–27, at 114; J. T. Bedu-Addo, “diavnoia and
the Images of Forms in Plato’s Republic VI–VII” [“Images”], Platon 31 (1979): 89–109, at 94; Eva T.
H. Brann, “Introduction,” in The Republic, ed. and trans. Raymond Larson (Arlington Heights: AHM
Publishing Corporation, 1979), xxxix.
38
Bedu-Addo, “Images,” 103–04.
39
Smith, “Divided Line,” 40 n. 34. Dissatisfaction with the dissolution interpretation in the epistemic domain ultimately forces Smith to opt for the demarcation interpretation (41–42 n. 35), though
he does not argue for the correctness of this interpretation, and we have already seen its flaws.
p l at o ’ s u n d i v i d a b l e l i n e
17
each are equal representations of their corresponding ideal: the geometer’s drawn
circle represents Circularity itself to the same degree that a just act in the physical world represents Justice itself.40 But as we have seen before, such an epistemic
dissolution will need to explain why this property makes the middle two segments
equal, but longer than D and shorter than A. On this specific suggestion, then,
reflections in water would be a poor copy of physical objects (hence a short subsegment), physical objects and mathematical drawings would be equally good copies
of their corresponding Forms (hence equal segments), and philosophical cognition would produce superior copies of Forms (hence the longest subsegment).
This suggestion is inconsistent with the cave, however, since the analogy of being
reflected in water is used for the world outside the cave. If the mental faculties
are represented by segment lengths that correspond to the degree of clarity to
which they are images of an original, then the lengths of B and D would need to
be equally short, since in the allegory of the cave B is said to be like reflections in
water of objects outside the cave, and D in the line analogy is just said to include
reflections of objects in water. I therefore find no epistemic property that can offer
a consistent explanation of the relative lengths of the divided line.
Even if one could find both ontological and epistemic properties that would
explain the relative lengths of the subsegments, the dissolution interpretation
still fails because it cannot explain why Plato never explicitly discusses the equality of the middle subsegments. If the dissolution interpretation were correct, we
would expect that Plato would make the similarity and dissimilarity explicit, and
refer to the equality of the middle two subsegments as he explains the similarity
of the middle objects, which of course Plato does not do. Why would Plato state
that R = (A+B)/(C+D) = A/B = C/D, but never explicitly state that B = C? It is very
implausible to claim that the equality of the middle subsegments is an important,
non-problematic feature of the analogy, when Plato himself not only never uses
the equality, he never explicitly states his awareness of the equality. Rather oddly,
he only hints elliptically that he knows B = C at 534a. The dissolution interpretation has no explanation for why Plato would be so cryptic about his awareness of
this equality or why he would fail to explain that the analogical significance of
the equality is that the middle sections both contain physical objects. No matter
how one tries to dissolve the overdetermination problem, even if one were to dissolve both ontological and epistemic versions, this interpretation would ascribe
a glaring omission to Plato’s explanation of the significance of the analogy. The
omission is even more serious, given Plato’s thorough description of the transition
to the world outside of the cave. If there is some way in which the light of the fire
is to be identical to the reflections in water outside the cave—some way in which
B and C should be equal in length—why would Plato fail to make this significant
point explicit? Any successful treatment of the overdetermination problem should
explain why Plato intentionally makes B and C equal, yet conceals this intention
from anyone who does not take the trouble to examine 534a carefully.
I am grateful to an anonymous Journal reviewer for this suggestion.
40
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3. intentional contradiction
The proper question is not how to eliminate the overdetermination problem,
but rather, why did Plato include this problem? The key to this question is found
in the passage containing the curious reversal at 534a. Why would Plato change
the formulation at all at 534a? The sudden contrast between subsegments of the
intelligible realm and those of the visible appears anomalous and unmotivated.
The only explanation for this novel characterization would seem to be to reveal
to the careful reader that Plato was aware that his stipulations lead to the equality
of B and C. Plato takes pains to let us know that he knows that his partition procedure entails the equality of the two middle subsegments, which is tremendously
odd. One must begin to suspect that Plato not only intentionally divides the line
in a contradictory way, he also wants his readers to know that he has done so, but
without stating this point explicitly.
The continuation of the passage at 534a is crucial. Because Plato undertakes
a simultaneous discussion of ontology and epistemology at this point, I will need
to introduce subscripts in order to distinguish the two types of divisions.
It will therefore be enough to call the first section knowledge [AE], the second thought
[BE], the third belief [CE], the fourth imaging [DE], just as we did before. The last
two together we call opinion [CE+DE], the other two intellect [AE+BE]. Opinion is
concerned with becoming [CO+DO], intellect with being [AO+BO]. And as being is
to becoming [(AO+BO)/(CO+DO)], so intellect is to opinion [(AE+BE)/(CE+DE)], so
knowledge is to belief [AE/CE] and thought to imaging [BE/DE]. But as for the ratio
between the things these are set over [AO, BO, CO, and DO] and the division of either
the opinable [CE+DE] or the intelligible [AE+BE] section into two, let’s pass them by,
Glaucon, lest they involve us in arguments many times longer than the ones we’ve
already gone through. (533e7–34a8, translation slightly modified)
The last sentence is quite strange, as Robinson has noticed.41 The claim that there
are arguments “many times longer” than the ones already gone through shows
that Plato has a clear idea that something mathematically and philosophically
complicated is lurking behind this recapitulation of the divided line, complicated
enough that the addled Glaucon can only respond that he agrees with Socrates,
“insofar as I’m able to follow.” Plato is surreptitiously hinting that the serious
reader should analyze what these further difficulties might be. Plato states that he
is reluctant to do two things. First, he is unwilling to state the ratio for the things
that the mental states are set over, i.e., he will not state the ratio for AO, BO, CO, and
DO. This demurral is odd, given that Plato states that the ratios for the two major
divisions are equal—(AO+BO)/(CO+DO) = (AE+BE)/(CE+DE)—and that Plato readily
offers the ratios for AE, BE, CE, and DE. It is, of course, possible that the epistemic
subsegments offer no difficulty for division, whereas the ontological subsegments
do offer difficulty. However, it would have been better to have some reason for why
the major divisions are identical, yet the ontological subdivisions are much harder
to draw than the epistemic subdivisions. But Plato’s second cause for reluctance
is even more odd. He is not willing to divide the main two epistemic sections in
two, i.e., he will not break intellect [AE+BE] and opinion [CE+DE] into each of
Richard Robinson, Plato’s Earlier Dialectic (London: Oxford University Press, 1953), 193.
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their two subsegments—a statement that is flatly inconsistent with the equalities
that have just been presented. Plato has just asserted that (AO+BO)/(CO+DO) =
(AE+BE)/(CE+DE) = AE/CE = BE/DE. Though the passage is long and the terminology excessive, it is a straightforward contradiction to state that (AE+BE)/(CE+DE)
= AE/CE = BE/DE, and that the division of intellect and opinion must be passed by.
The former set of equalities just is a statement of how intellect and opinion are
divided into their subsegments. Plato is signaling that the core issue surrounding
the divided line is the issue of contradiction. There is a further issue raised by this
passage. I read Plato’s comment as a goad to pursue an argument that the far too
complacent Glaucon lets slip by. Plato wants the reader to be challenged by this
comment, challenged enough to discover that contradiction lies at the heart of
procedure which specifies how to divide the line. Unlike Glaucon, I do not plan
to ignore these complications.
4. method
Any solution to the overdetermination problem is bound to encounter the difficulty
that Plato makes a conscious decision not to pursue these matters any further, as
Socrates says to Glaucon. So although my solution involves claims that are never
explicitly endorsed by Plato, it has the virtue of filling in all of the blanks in a
way that will be readily recognizable as being in agreement with core elements
of Platonic philosophy—in particular, Plato’s commitment to the importance of
mathematics as propaedeutic to philosophical education and his preference for
discussion over reading as the best philosophical method. Moreover, my interpretation will give a very natural answer to the decisive question of why Plato did not
explicitly mention the equality of the middle two subsegments, or that this feature
generates a contradiction. As a general interpretive principle, it is unacceptable to
impose a hidden meaning on a text without some explanation for why the author
would have hidden it. The dissolution interpretation had particular difficulties
with this principle, since it takes the middle subsegments to be unproblematically
equal, yet provides no explanation for why Plato so carefully hides his awareness
of this equality. However, my interpretation will satisfy this interpretive principle
by offering an explanation of these contradictions along with an explanation of
what Plato would have gained by refraining from offering an explicit statement
of the contradictions or how they are to be resolved.
The principal explanandum is the fact that there is a puzzle here at all. Surely if
Plato so desired, he could have given us a partition of the line that would reflect
his philosophical commitments. If he had wanted to emphasize or minimize the
difference between B and C, he could have done either of these things, as long
as he had been careful to keep the stipulations about the divisions of the line in
agreement with his later claims about the mental states represented by the second
and third subsegments. Since it would be easy to set up a line to reflect the systematic interrelations of all four kinds of knowledge, whatever Plato might have
taken those relations to be, it is of the first importance to determine why Plato
did not avail himself of such an easy and obvious strategy. Why does he allow the
overdetermination problem to emerge at all?
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To resolve this difficulty we need to notice that there are distinct stages that need
to be followed when confronting the overdetermination problem, and quite strikingly, these stages reflect the four divisions of the line itself.42 The overdetermination problem therefore models the progression of types of knowledge represented
by the line. My claim is that as one first reads the divided line analogy, discovers
the overdetermination problem, and then attempts to resolve it, one is likely to
progress sequentially through the mental states and the types of objects that are
presented in the divided line analogy. This claim does not involve any assertion
of necessity, but is a more modest claim about what is psychologically most likely.
People can gain insights in rare or unique ways, and some insights will elude the
grasp of many people, but for anyone who does gain a fairly thorough understanding of the divided line analogy, it is highly likely, given human mental capacities,
that their grasp of the divided line analogy will be governed by the four mental
states of the divided line in the same sequence as they occur in the line.
First, the divided line is grasped as an image. Although images comprise a type
of object that Plato claims is least true and a corresponding mental state that is
least clear, it is important to attain philosophical precision about a type of object
and mental state that is inherently imprecise. Plato asserts that images (eijkone~)
consist of shadows, reflections, and “everything of that sort,” (509e1–10a3). There
are two criteria that place an object into this category: (1) it is an image, (2) the
object of which it is an image is physical. With this precise definition of an image,
it is possible to show that the divided line analogy is initially grasped as an image in
this technical sense. As an analogy for Plato’s ontology, the line does not satisfy this
definition, since although all analogies will by definition meet the first criterion, in
this case, the divided line analogy fails to meet the second. The divided line is an
ontological taxonomy. Although physical objects comprise one of the categories,
it would be wrong to claim that the line is itself a copy of a physical object merely
because the ontology which it represents includes physical objects. But there is
still an important way in which the divided line does meet the second criterion. As
a linguistic description of a divided line, the passage is a poor copy of a physical
line. Draw the divided line: follow Plato’s instructions to produce a diagram of
the line in the physical world. The linguistic description of the divided line (i.e.,
the very words of 509d6–11e5) is a copy of this physical object, which satisfies
both of the criteria for placing this initial grasp of the divided line in the lowest
of the four levels: it is an image of a physical object. Indeed, it is even a relatively
poor copy, which emphasizes Plato’s main contention that the transition from D
and C generates important ontological and epistemic progress. But this claim is
relational in nature: I am claiming a relative superiority of the physical line to the
line as a mere linguistic description of the physical line. To see this relative claim
Robert Hahn (“A Note on Plato’s Divided Line” [“Note”], Journal of the History of Philosophy 21
[1983]: 235–37, at 235) asks to which of the four levels of the divided line the divided line analogy
itself belongs. Hahn places the line among the objects of subsegment B, but I see no need to limit
the analogy in this way. I will show how the line actually takes the reader sequentially through all four
levels. Robert Brumbaugh (Platonic Studies of Greek Philosophy [Albany: State University of New York
Press, 1989], 40) correctly sees that the line can be cognized in a way corresponding to each of the
four sections of the line, though his attempt to correlate these sections with stages in the opening two
books of the Republic (44–49) strikes me as speculative and implausible.
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clearly, one needs to see, first, the way in which the divided line is grasped as a
physical object, and then, why this grasp is superior to the intitial grasp of it as a
mere linguistic description.
My suspicion is that most readers would never have noticed the overdetermination problem at all had they not actually attempted to draw a physical representation of the line. If a reader remains at the lowest level—at the level of mere
written description—her views, in all likelihood, will be contradictory, although
she will not recognize that the divisions of the line and Plato’s comments about
their referents are inconsistent. The result would be to have beliefs which jointly
entail the contradiction B = C and B > C. For most people, it is only when drawing
the line—when we move from the divided line as a linguistic description to the
line as a physical object—that the overdetermination problem becomes apparent.
So, just as moving along the divided line from shadows to images presents us with
objects that are more real and mental states that are more clear, moving from the
divided line as a linguistic description to a physical depiction of the line gives us a
superior grasp of the divided line; it is now possible to see that there is something
potentially contradictory lurking in Plato’s exposition. In fact, the repeated use of
commands in this section—tevmne, tivqei, tavxon—indicates that Plato is encouraging
the reader to draw the line for herself. When drawing freehand, using the same
ratio to divide the line into subsegments makes the middle subsegments very close
to equal, and this fact will be surprising to anyone who believes that pivsti~ is less
clear than diavnoia. It is possible for people already interested in mathematics to
derive the equality of the middle segments directly from geometrical or arithmetical concerns. Once told to divide a line, then re-divide it using the same ratio,
such people would move directly to a mathematical proof of the relations between
the segments, and quickly discover the equality without the intervening aid of a
physical representation of the line. This emphasis on drawing the line, the use of
the imperatives, is not meant for these readers who already embrace a Platonic
commitment to mathematics. But for those readers who do not immediately leap
for their copy of Euclid, a physical depiction of the line is an important step in
identifying the overdetermination problem, which moves us up to the second
subsegment of the divided line, viz. physical objects.43 Seeing the line as a physical
object now makes it easier to see how the initial grasp of the analogy was at the
lowest level of the line. Initially reading Plato’s description of the line, there is
apt to be no recognition of the mathematical and interpretive complexities that
these instructions produce. At this initial stage, this description is grasped as a
mere shadowy copy of the physical line. By calling this stage “initial,” I do not assert that all or even most readers will ever make the transition to a more precise
43
I confess to being a member of this second class of philosophers. In my own case, I discovered
the overdetermination problem drawing the line in front of a class. I simply could not get the middle
two subsegments to look right, since I believed that one should be longer than the other, given what
Plato says about the corresponding types of mental states. My suspicion is that most people would
never notice the overdetermination problem if they were not at this second stage of actually drawing
the line. Of course, a mathematical proof of the equality is common in the literature on the topic,
but a physical drawing of the line is even more common. Drawings of the line feature prominently in
most scholarly works, so it is often difficult to know if these drawings serve merely as a visual aid, or if
it was the act of physically drawing the line that led to a discovery of the equality.
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understanding of Plato’s line. I mean only to claim that this is where most people
begin with their understanding of the line. Of course, this linguistic shadow provides instructions that are sufficient eventually to produce a superior grasp of the
divided line, but my claim is that this superior grasp will usually be gained only by
taking the trouble to draw a physical line. Until this drawing is made, one’s grasp
of the line will be a mere shadow: an imperfect, inconsistent understanding of
the properties of the physical line.
The act of physical depiction itself pushes us beyond the physical world. A
freehand drawing gives a preliminary indication that there is something irregular
about the middle subsegments, but mathematical precision is required to prove
the equality of these subsegments. In proving the overdetermination problem,
the reader must have sufficient mathematical curiosity to examine the stipulations
at 509d closely enough to see that they entail that B = C. Then one looks to see
if Plato was aware of this problem, confirmation of which we find at 534a, again
a recognition of which demands attention to mathematical issues. Proving the
equality of the middle subsegments, and then seeing that Plato himself was aware
of this equality, requires concern for mathematics. Obviously, if one merely drew
the line, one would not know if the middle segments were exactly equal, or even
if the apparent equality was merely an artifact of the specific ratio chosen, or the
length of the line drawn. To know that the middle segments of Plato’s line must
be equal requires a mathematical proof of the equality, since no physical drawing
will ever support the general claim that all lines drawn in this way have the middle
segments equal. Even the most precise physical depiction of the line cannot prove
the equality of the middle segments because any equality might be the product of
the drawing process, not an implication of the division instructions. We therefore
need to move from a physical drawing of the line to a grasp of the mathematical
issues that surround the division procedure, just as the third subsegment of the line
representing mathematics follows the subsegment representing mere belief.44
Lastly, we are led to the thorny question of what the true relation between the
various kinds of knowledge is, given that Plato himself makes contradictory claims.
Does each mental capacity have a domain of objects proper to it alone? Does the
degree of epistemic reliability of the mental faculty reflect the relative degree of
reality of the object? Should we attribute a theory of intermediates to Plato? Of
course, to become entangled in the overdetermination problem, one must first
be committed to considering the mathematical diversions and examples found
in a work of philosophy. Plato frequently points out that the study of mathematics develops the capacity for a priori reflection—the capacity sine qua non of the
philosopher. By including a mathematical puzzle, Plato exhibits his commitment
to the importance of thinking mathematically. But the point is even stronger:
true, mathematics is mental calisthenics for the budding philosopher, but it also a
Both D. Gallop (“Image and Reality in Plato’s Republic,” Archiv für Geschichte der Philosophie 47
[1965]: 113–31, at 119) and Hahn (“Note,” 236) emphasize the connection between the line as a
mathematical analogy and the objects of diavnoia. However, they mistakenly believe that this connection means that the line cannot be related to any of the other three types of object, that it cannot be
a linguistic shadow, a physical object, or a Form. My claim is that most readers will treat the line is all
of these things in sequence as they attain a better understanding of the analogy.
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stepping stone in that it inspires the student to turn toward philosophy. With the
divided line, we find that mathematical curiosity regarding the relation between
B and C ends in philosophical reflection about the true relations between the
four kinds of mental states and their relations to ontological classifications. Plato
presents the divided line in a contradictory fashion because in so doing he forces
the reader to follow the paradigmatic course in the procurement of true wisdom:
from mathematics to philosophy. The inconsistency caused by the overdetermination problem is not an embarrassment to Plato’s claim about the importance of
mathematics, as the gaffe interpretation would imply; instead it is a testament to
Plato’s respect for the discipline.
There is an additional implication of these stages of interpreting the divided
line that bears mentioning. By knowingly giving a contradictory analogy for his
epistemology, and perhaps his ontology as well, Plato reveals that the reader must
ultimately be willing to devote herself to the philosophical enterprise. She cannot
simply read a work like the Republic and hope that enlightenment will be imparted
to her. At best, such knowledge would be true belief based on unreflective acceptance. One of the cornerstones of Platonic epistemology is that knowledge requires
something more than mere acceptance; it requires an active understanding on the
part of the knower. It requires seeing the connections and reasons that make the
belief true. Anything else is, as Plato claims in the Meno, at best like the statues of
Daedalus: prone to flight at the earliest opportunity. What we find in the divided
line is a brilliant attempt to demand the kind of activity from his reader that usually only emerges in hotly contested conversations. Plato constructs the line in
such a way that the reader is forced to interact dialogically with Plato’s writings,
then leave them behind and think for herself. Once it is clear that Plato knows
that he has presented no stable picture of the relations between the four subsegments, so that no mere interpretive strategy can possibly reconcile all parts of the
text, there is nothing left but to rely on one’s own devices. Socrates’s demurral
at 534a—precisely the point that he should be wrapping up his comments about
the line—takes on added meaning. The passage shows that Plato is not willing
to set forth his views on the further complexities that have emerged. It is a task
that he intentionally leaves for his readers, revealing that his final assessment of
the role of the divided line is to force a thoughtful reader to transcend the text.
One significant aspect of the divided line is exactly that Plato refuses to explain
its point. Rather than memorizing the opinions of Plato, Plato demands that we
strike out on our own to identify the interrelations of our various ways of thinking
and the ontological relations between different types of objects. The equality of
the middle two subsegments is a mathematical puzzle, a genuinely Socratic goad
impelling one to independent philosophical thought from humbler mathematical origins.45
I would like to thank Rob Colter, Richard Kraut, Henry Mendell, David Sedley, and two anonymous Journal reviewers for their comments on earlier versions of this paper.
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