After all this time blogging, I can’t believe I haven’t written about Plato’s divided line analogy, the cave allegory‘s sophisticated cousin. It’s deep, it’s mystifying, it’s what makes The Republic The Republic.
I don’t plan to explain the divided line in a scholarly fashion—there’s plenty of that kind of thing around. In other words, I’m not gonna back up every claim I make, because that would take forever and this is a blog. So if you’re reading this after Googling “plato divided line” because you’ve got a paper for a philosophy class due tomorrow, consider this a heads up—if you regurgitate what I say here as fact, you’d better hope your professor is extraordinarily lenient.
Let’s dive in.
In this part of the Republic, just before the famous cave allegory in Book VII, Plato has Socrates discussing with Glaucon the difference between the visible and intelligible realms. (Glaucon, Plato brother, is a Pythagorean. That’s important because the divided line is a mathematical proportion):
“Let us represent them as a divided line, partitioned into two unequal segments, one to denote the visual and the other the intelligible order. Then, using the same ratio as before, subdivide each of the segments. Let the relative length of these subdivisions serve as indicators of the relative clarity of perception all along the line.”—Plato’s Republic Book VI, 509d. Translated by Richard W. Sterling and William C. Scott
Here’s a diagram of the divided line, as I conceive of it:
Scholars have written entire books about this analogy, even just portions of it, so while I’m presenting you with this fairly simple-looking diagram, keep in mind that explaining what each of these ontological and epistemological realms means and how they’re related to each other is far from a simple matter. Also, my terms for the segments are interpretations based on my reading of Plato. Other terms you might come across for these segments…well, take your pick:
Levels of Knowledge
Section D: Noesis: what I call wisdom, others might call reason, authentic knowledge, philosophical reasoning, rational insight, dialectical knowledge. I’d actually like to have two terms here. I’d reserve the term “wisdom” for the philosopher who has intuited the idea of The Good and is going back down through the lower levels to show how all the forms are related to one another, and I’d call the stage prior to insight into The Good “dialectical reasoning”. “Wisdom” is something I’m using in a technical way here. It’s not simply knowledge gained through life experience, though it does have that flavor. It’s about knowing the purpose or function (telos) of all the forms (ideas) and how they are related to one another. I suspect you can’t fully know any single form without complete knowledge of the system of forms as a whole. The Good is that which unifies and makes possible the entire system. Consider it this way: Dialectical reasoning (pre-insight into the The Good) will feel something like trying to put together a puzzle without being able to look at the picture on the box. Wisdom—post-insight into The Good—will be like having the picture on the box right in front of you.
Section C: Dianoia: what I call critical thinking, others might call understanding, geometric analysis, calculation, mathematical reflection, formal reasoning. Critical thinking is not a term I’m super happy with. Plato mostly gives mathematical examples, but leaves the category open to other rigorous disciplines that rely to some degree on tangible objects (like drawings, figures and graphs) and take their assumptions for granted as self-evident, seeing them as foundations when they’re not.
Section B: Pistis: what I call naive realism, others might call belief, faith, trust, common sense, sense-belief, conviction, sensory inspection, sensory or empirical observation. I’m going with “naive realism” because I think it captures the epistemic attitude of having an unwarranted conviction that knowledge can only come through observation of the so-called real world of tangible, concrete objects. This is not empiricism in a scientific sense—physics is definitely far too theoretical and mathematical to belong in this category—and so I don’t like the term ’empirical observation’ here. To me that sounds awfully Humean, and I don’t think Plato intended for this attitude to be quite so rigorous. On the other hand, I wouldn’t call pistis “common sense” either because your idea of common sense might be vastly different from mine. Besides, common sense seems to be a title more fitting of the lower section.
What do I mean by naive realism then? It’s what happens when you see past social conventions, religious and political propaganda and end up with a hard-boiled “what you see is what you get” outlook.
There’s a partial unveiling that happens at this level, but for those undergoing it, it’s revolutionary. When you pull back the curtain and see what the shadows really are, you’ll feel lied to, manipulated by religious and political leaders who’ve used shadows (images, propaganda) to gain power over you. You make no distinction between their myths and the ideas behind their myths; it’s all invented cloud, conjured vapor. “What is real then?” someone asks you. With a gleam in your eye you knock on the table and say, “That’s what’s real. The rest is bullshit.” When someone like Socrates comes around asking for a definition of justice, you think he’s either a complete idiot or he’s trying to become a powerful puppet master himself. The more he talks, the more you think it’s the latter. Finally you can’t take any more of his nonsense. You butt in and shout, “Justice doesn’t really exist, you moron! It’s just an idea the powerful use to keep fools like you in check.” Or you insist that “Justice is the interest of the stronger”—a clever way of saying the same thing. In fact, this is the position Thrasymachus tries to defend in the Republic.
But perhaps by taking all ideas to be fictions, Thrasymachus has thrown the baby out with the bathwater.
Section A: Eikasia: what I call uncritical acceptance others might call picture thinking, imagination, conjecture, superficial inspection. I don’t know about you, but “picture thinking” doesn’t mean much to me. “Imagination” feels too creative for this category, too self-aware. Same goes for “conjecture.” But I’m not gonna dig my heels in over “uncritical acceptance”—”superficial inspection” might be just as good. Whatever you call it, this is the epistemic state where you mistake the image for what the image is an image of.
So…when you watch TV (shadows), you think what’s happening on-screen (the cave wall) is really happening? But who the hell does that? Even Geordie knows the difference between TV and reality (except when a nature show’s on, he does sometimes sniff around the speakers and look behind the TV to make sure there’s no lion’s butt sticking out.)
Maybe we shouldn’t take images in too strict a sense. Yes, images are important, but let’s also think in terms of icons, movies, memes, art, music, politics, trends—culture. Culture is the reflection of reality that we mistake for reality. Culture is what politicians try to tap into to manipulate voters, what movements and organizations try to influence to garner support for their causes. Culture, and those who partake in it, inhabit eikasia. As we know, culture is unavoidable. It’s everywhere, and it’s virtually inescapable. The top three segments on the divided line tend to get a lot of attention, but eikasia describes how the majority of people live their lives.
I’ve made eikasia sound like a place where the masses smoke their opium, but it’s also what you might call a safe space to be in, at least in certain circumstances. If you’re fortunate enough to live in a good society, then uncritical acceptance of the norm is not so bad. It’s possible to accidentally get things right, though getting it right isn’t the same thing as possessing knowledge. For instance, Cephalus, the old man who appears at the beginning of the Republic, engages with Socrates very briefly (and poorly) before excusing himself to attend to religious rites. He’s an example of someone who accidentally gets it right. He’s good-natured and financially moderately well-off (Interesting thing to note: Plato echoes the math behind the divided line in giving us a description of the old guy’s inheritance and how it’s been managed through the generations). Because Cephalus is old, and because he seems to possess a natural inclination to be a good, upstanding fellow, Socrates does something which might seem out-of-character—he lets Cephalus run away from the debate without ‘doing him the service’ of publicly humiliating him. What’s more, when Cephalus’ son, Polymarchus, takes his place, Socrates is fairly gentle with him too. Although it appears both father and son possess a feeble, hand-me-down moral outlook, an uncritical acceptance of prevailing religious and social conventions, it’s because they are basically good people that Socrates seems to think it’s best to leave them—particularly the old guy—in peace. Better to get it right and not know why than to get it wrong and think you’ve got it right. A little bit of knowledge is a dangerous thing.
My thoughts on the line as a whole: Until you reach the idea of the Good (at the highest point on the line), you don’t have complete knowledge. At stages B and C, you’ll think you know what you don’t yet know. (You’ve heard that before, right? Socrates’ famous line: I know that I know nothing. It’s like that, except the opposite: I think I know, but I don’t.) At stage A (eikasia, uncritical acceptance), you aren’t even thinking about what you know or don’t know; you’re letting someone else do the thinking for you. However, in each of those cases, A, B, and C, you’re mistaking something for something else.
In the realm of D (noesis), however, the philosopher’s assumptions are not taken for granted, but are seen as hypotheses and springboards to knowledge of the Good.
Levels of Things Known
D: What I call pure ideas, others might call forms, ideals
C: What I call abstractions, others might call mathematical objects, mathematical idealizations. I like “abstractions” because it gives you the sense of something lifted out of the physical world without really belonging to it.
This ontology of this segment of the line is to me the most baffling. It could be that only certain mathematical objects belong to this realm and other mathematical objects belong to the one above. Some scholars believe “mathematical intermediates” are the objects of dianoia; I have no clear idea of what those are. I haven’t made up my mind on this. I’m inclined to think all mathematical objects are forms/ideas, and dianoia is simply an inferior mode of apprehending them. But if that’s the case, the one-to-one correspondence of knowledge to things known that Plato seems to be implying falls apart. So…who knows.
B: What I call concrete, tangible things, others might call physical objects, objects of vision or perception, sensible objects
A: What I call myths/fictions, others might call images, appearances, shadows, illusions
The Cave Allegory and the Divided Line
Unless you’ve been living in a cave, you’ve heard of the cave allegory. In it, the prisoner who manages to exit the cave and see the light outside then has a moral duty to return to the cave’s darkness to inform the others. To be sure, convincing them to leave the cave will be a damn near impossible feat. The images displayed on the cave wall constitute their reality; they’ve never seen each other or themselves or the bonfire and object-manipulators behind them which makes their shadowy reality possible. They think this philosopher who can’t tell one shadow from another is out of his mind.
But wait a minute. Does seeing the manmade bonfire inside the cave as well as the natural sunlight outside make the freed cave-dweller a philosopher? From what we’ve learned from the divided line, no, it doesn’t. It’s too easy to forget that the one who escapes the cave is, as Plato says, “like” a philosopher, but not really a philosopher. My theory is, Plato wrote the allegory of the cave for a specific audience: naive realists, those who inhabit the realm of pistis.
So let’s look at the cave allegory from the point of view of the naive realist. The naive realist is likely to think of himself as the enlightened philosopher who’s left the cave of ignorance. After all, he’s had that inner revolution I talked about earlier, he knows that painful feeling of ‘seeing the light’ and realizing he’s been duped all his life. He’ll recognize the imprisoned cave dwellers instantly; he used to be one of them. He sees their uncritical acceptance of myths and propaganda, as well as the powerful leaders standing behind them, manufacturing their shadowy reality. He’s bound to think, at least at first, that Plato’s saying the visible world outside the cave is the ultimate reality because that’s what he, the naive realist, believes. The philosopher who has exited the cave has seen the light of truth. Great. Fantastic. Welcome to the real world.
However, our naive realist might reflect on what Plato wrote and wonder: but if the sunny world outside the cave represents reality, what does the bonfire represent? What’s up with the bonfire? Why did Plato bother to describe some very deep subterranean dwelling illuminated by a manmade bonfire when he could have just written about a shallow cave where the puppet masters stand at the entrance making shadows with the sunlight shining directly in? The bonfire seems unnecessary, repetitive. The more thoughtful naive realist (or the one with a term paper due tomorrow) loses sleep over that superfluous bonfire. Maybe, just maybe, the more thoughtful naive realist comes to realize that the light from the bonfire isn’t just some bizarre detail to ignore, but instead is meant to represent the sunlight shining in through the kitchen window right now. The bonfire is sunlight in what Plato calls the visible world. But wait…that would mean…the visible world is not really outside the cave…WTF?
Cue the divided line diagram. The divided line diagram is written for—you guessed it—the mathematician or critical thinker who inhabits the realm of dianoia. And you better believe there’s a WTF? moment in store for the critical thinker too: from Plato’s description of the divided line (the quote above), the lengths of the middle segments must be equal. (For those of you who are mathematically inclined, check out the footnote on page one of this article.) So what does it mean?
You tell me.
No seriously. You tell me. What do you think?
BTW, if you’re wondering, where’s the story for the eikasians? Well, they get one too! It’s at the end of the Republic and it’s called the Myth of Er.