Plato’s doctrine of Ideas can be taken in either of two directions. A first direction follows the allegory of the Cave in Republic VII. It suggests that Ideas exist as pure things in a place apart from a world of impure things.
Socrates asks his companion Glaucon to imagine a cave that imprisons a people from birth. Their bodies are fettered so that they must view a screen their entire lives. Another people walks back and forth beside a short wall, just above and behind the prisoners, holding all sorts of puppets and shapes. A fire, a little further back and towards the cave’s exit, projects these objects’ shadows onto the prisoners’ screen. The utterances, and other noises, of the walkers echo from the screen.
The prisoners’ world is one of shadows and reflected sounds. They give names to these shadows and reflections, and praise those among their number that master the names and the recurrence of referents. Freed prisoners must climb, or be dragged, up into the world above. A blinding sun drenches this world, but it slowly illuminates what is: the Ideas, whose shadows and reflections the prisoner’s had merely opined on. They may then come to understand the deceptions, and abbreviation, of life in the cave.
The allegory of the Cave lets us think Ideas as a sort of thing that exists in one place, while other sorts of things exist elsewhere. We can gather the shadows and reflections at one extreme (in the cave) and the Ideas at another (the world above), like a baker separating dry and wet mixtures.
This is a sort of physical way of thinking Ideas. They are singular things that are pure instances of themselves, existing separately to the many impure things. But saying that Ideas are things appears to contradict Plato’s insistence that only a Heraclitean dreamer would, “think that a likeness is not a likeness but rather the thing that it is like”: that a thing is not a thing but rather the Idea that it is an instance of.
A second direction follows the metaphor of the Divided Line in Republic VI. It suggests that Ideas are distinct from, but included in, the world of things.
Socrates asks faithful Glaucon to imagine a line split into unequal lengths to clarify his division of reality into two worlds: the intelligible world of Ideas and the visible world of things and their shadows and reflections. The greater line represents the intelligible world; the lesser line the visible. These two lines are then each split in the same ratio. The four different lines represent different degrees of opacity and clarity within reality as a whole. In the shortest line reality is more opaque; in the longest it is moving toward complete clarity. In the shortest line there are shadows and reflections; in the longest axioms.
This move from two to four is important. Once Socrates makes this move he departs from a metaphysical description towards something epistemic. The four sections no longer only represent the clarity or opacity of reality, but are also four “conditions in the soul”. He names these understanding, thought, belief and the imagination. Understanding and thought work with the Ideas; belief and the imagination opine on things and their reflections.
This way of thinking Ideas is unlike the allegory of the Cave. We have a geometric metaphor, suggesting that Ideas exist in a continuum with the crude elements of reality. They are not in a different place. Instead, the points that break the line belong to a single place. This place represents both reality and the production of correct thought.
We begin with opaque opinions. We then apprehend these in thought, making hypotheses and drawing conclusions from their confused existence. If we can discern the Idea that they participate in, we may then understand what we initially found opaque. The opinions will remain opaque—it is their nature—but we may no longer be disoriented by their chaotic appearance. We may gain mastery over it through the “power of dialectic”.
The existence of these two directions is shown in conflicting interpretations of Plato’s distinction between what is and what is not. Luke Purshouse notes that one line of interpretation [A] says that Plato means what is property x and what is not property x while another line [B] says that Plato means what exists and what does not exist.
If Ideas are what is, then for A Ideas are things that have particular properties completely. They are pure instances of a property. A thing has the property beauty separately to its Idea. A thing is the Idea beauty when it is purified of every other property. But this seems simply to displace the word Idea for Property, adding a third complication to Plato’s theory. We can now ask: what is a property? Saying that Plato is not implying existence and inexistence also contradicts Plato’s use of an alternate phrasing for what is and what is not: “what comes to be and passes away”.
For B, Ideas are what is, while things exist between what is and what is not and thus exist less than Ideas do. Things that we have opinions about do not exist to the same degree as the completely existing Ideas we have knowledge of. B also has the benefit of including the problematic that Plato inherited from the Pre-Socratics; Parmenides, particularly, emphasized the importance of distinguishing between what exists and what does not exist in our investigation of reality.
A can be linked to the Cave and B can be linked to the Divided Line. Ideas are sorts of things for A and they are continuous with, but not, things for B.
The contemporary relevance of Plato’s doctrine of Ideas hangs on the second direction; both on the metaphor of the Divided Line and on insisting on the distinction between what exists and what does not exist.
Take the example of politics.
The political field is opaque. Followers of politicians, parties and parliaments—particularly those invested in psephology—are the picture of Plato’s “lovers of sights and sounds”. Their best representatives are paid to show-off in the opinion pages of daily newspapers. The problem with lovers of sights and sounds is that they deny the existence Ideas. They opt for a certain relativism. There is no Idea that can render politics for intelligible thought. It is meaningless. A mess. A pain. But a pleasure!
Politics is an instance of Plato’s problematic. It exists somewhere between what is and what is not. It is part of the visible world, so it is necessarily more opaque than clear. But, I think, this is Plato’s lesson: surrender to opinion is not our only option.
We can understand “what comes to be and passes away”, if we know the Idea that this unrest participates in. The Divided Line’s double-movement produces this knowledge. If there remain people who do not want to surrender to opinion, then I think it is plausible that Plato’s doctrine of Ideas is worth investigating further.
 Plato, ‘Republic’ in Complete Works (Hackett, 1997), pp 971-1223, see pp 514a-521c. All references to the Republic refer to the Stephanus pagination.
 Plato, ‘Republic’, p 476c.
 Plato, ‘Republic’, pp 509d-511e.
 See: Nicholas P. White, ‘Plato’s Metaphysical Epistemology’ in Richard Kraut (ed), Cambridge Companion to Plato (Cambridge University Press, 1992), pp 277-310, see p 277.
 Plato, ‘Republic’, p 511e.
 Plato, ‘Republic’, p 476c.
 Plato, ‘Republic’, p 511b.
 Luke Purhouse, Plato’s Republic: A Reader’s Guide (Continuum, 2006), pp 79-80. Purshouse attributes the different lines of interpretation to [A] J. Annas, An Introduction to Plato’s Republic (Oxford: Oxford University Press, 1981) and [B] R. Cross and D. Woozley, Plato’s Republic: A Philosophical Commentary (London: MacMillan, 1964).
 Plato, ‘Republic’, p 508d, emphasis added.
 Reginald E. Allen (ed). Greek philosophy: Thales to Aristotle (New York: The Free Press, 1966), pp 11-13.
 Plato, ‘Republic’, p 476b-d.
 Plato, ‘Republic’, p 476c.