Plato’s Divided Line and Cave Allegory

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.

Geordie, my sunshine.

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

https://knowyourmeme.com/memes/platos-allegory-of-the-cave

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.

29 thoughts on “Plato’s Divided Line and Cave Allegory

  1. There’s a lot here, and I’m probably missing a lot of important points. But what sticks out to me, is whether we can know where we really are on the line, or in the cave or outside.

    I’m reminded of a dream I had once, where I realized in the dream that I was dreaming, and then made an effort to wake up. I woke up, and thought I’d succeeded, but a bit later realized things were still wrong. Was I still dreaming? I made a stronger effort, a much more intensive one, and finally succeeded in waking up. At least I think I did. Maybe I’m still in the dream, just at another level.

    Along the same lines, can we ever know whether we’ve really escaped from the cave to true reality? Maybe we’ve escaped from an inner cave, but only to an intermediate one. The outside world itself might just be another cave. And can anyone have justifiable confidence about where they are on the line? We might think we’re at D, but really still be at B with the impression of Dness just a form of that naive realism.

    No idea on the mathematical footnote.

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    • That’s so funny about the dream. I’ve had both kinds—the dream in which I know I’m dreaming and the dream in which I think I’m awake but not entirely sure, then wake up. One time I ‘woke up” four times, and each time felt just as real as the last. Made the rest of the day feel unreal.

      Anyway, I get what you mean about not knowing where we are on the line. I suppose once you intuit the Good, you’ll know it.

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      • Wow, four times. That almost sounds like Inception territory.

        The question is, how do we know we’re intuiting the Good, or just feeling our bias? It seems like we can never be sure, and always have to be open to the possibility that we haven’t arrived yet.

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        • Well, Plato doesn’t say much about the Good itself—he’s very hush hush about it—but once you have ‘seen’ it, it’s supposed to ‘illuminate’ everything else. In other words, it should make clear what all the other forms ‘are good for’, what their purposes (telos) are. I’m guessing by that he means we arrive at something like a definition. If I’m right about that, then the forms will be interconnected in the same way definitions of words refer to a system of other words. If that makes sense. But this is just me speculating and I can’t even remember what exactly I read that gave me this impression.

          Also, I probably should’ve said that by “intuiting” I don’t mean having a gut feeling or suspicion. It’s very far from “I have a feeling he’s lying to me.” It’s more like the sort of intuition that let you know A=A. It’s too fundamental to make an argument for, if that makes sense.

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  2. An excellent post! I’ll have to read it when I’m more awake and functional to have anything useful to say. I’m suffering from a coding binge hangover right now. Brain very fried.

    Do you sing the Sunshine song to Geordie? “You are my sunshine. 🎶🎵🎶 My only sunshine…” 😉

    Liked by 1 person

    • Thanks! I know the post is a bit packed, but as I was writing it, I was thinking of all the times I’ve Googled “plato divided line” and up comes either the same simplistic Cliff’s notes rundown or scholarly articles that take forever to get to the point because every single assertion has to be backed up, every reference cited. I rarely come across really interesting, original interpretations, except in some of articles dealing with the math of the equality of the middle segments. But then, as you know, I can’t really follow the math, so I just assume it all pans out and see what the author has to say.

      And yes, in fact, I do sing that song to Geordie! I also sing him part of “La vie en rose” every night as a lullaby. And many others besides, some of which I make up on the spot. Then there’s this one which I remember from my childhood:

      Liked by 1 person

  3. Good timing! I’m working on, among other things, a renewed engagement with Bradbury’s “Fahrenheit 451” along the lines of Plato’s “Republic,” so this was very helpful, particularly since I’m more familiar with the allegory than the analogy. Good summaries!

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    • Yes! Thanks for pointing this out. The line in the Republic and the ladder in the Symposium are such similar paths that I’ve come to see them as different aspects of the same trajectory towards the Good and the Beautiful. In the divided line you have a focus on the intellect and its objects, whereas in the ladder you learn that desire for immortality is the motivation to acquire higher levels of knowledge. Without desire, there’s be no movement up the line or out of the cave.

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  4. Okay, all very armchair amateur and based almost entirely on what you’ve said and bits I’ve heard along the way. (So, probably febrile nonsense, but when has that ever stopped a comment?)

    I’ll start by saying that one thing I’d never gotten a sense of is “The Good.” What you said about ‘what things are good for’ makes sense. I’d gotten a “good vs evil” sense and couldn’t make head nor tail of it.

    Abstractions in the C slot feels like a sticky point. Both C and D deal with abstractions, and I’m further puzzled by the, on the one hand, consigning mathematics to the C slot but saying they “take their assumptions for granted as self-evident, seeing them as foundations when they’re not.” But proofs are a cornerstone, and math is one of the few human endeavors in which absolute and unquestionable (abstract) truth is possible through mathematical proof.

    I’m further puzzled by Plato being famous for his Platonic solids, which are about as abstract as anything, and I’d always thought belonged in his top category, forms. Yet it seems geometry is in the C slot? That seems wrong. I can see abstractions of the physical entities in B as fitting. Abstracting a circle from real circles, but not seeking the mathematics that underly the notion (the fundamentals of spheres in all dimensions). So I’d agree with you that maybe math straddles C and D.

    For B, how about “justified true belief”? The worldly things we know to be true just because they’ve always been true. (The sun shall rise tomorrow.)

    For A, I thought, “Fashion!” 🙂 As you said, we can’t escape culture, which makes it somewhat real, but fashion definitely is an illusion. (As demonstrated by how rapidly it changes.)

    Does the bonfire in the cave represent greater control by the manipulators? They would have the power of night and day that way. And the source of the light, in some sense, would be false.

    As I’ve mentioned before, Plato’s description of the Divided Line, if taken at face value, requires the B and C segments have equal lengths. The algebra involved is extremely simple, as that footnote shows.

    As I think you know, I wonder if the B and C slots are meant to be equal in size, albeit not in import. I wonder if Plato intended some kind of reset as we cross the major line between opinion and knowledge. We’re at a higher level, which makes C more important — harder to achieve — but in starting over we’re not larger, so to speak.

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  5. Welcome fellow armchair amateur! You’ve come to the right place. 🙂

    —”I’ll start by saying that one thing I’d never gotten a sense of is “The Good.””—Ha! Join the club!

    No, really, Plato was purposefully elusive about what the Good is. He was said to have given a public lecture on the idea of the Good (now lost, of course) for which he was mocked. Apparently laymen saw him as pretty esoteric, even in his own time. From my reading, I gather “The Good” is God. Having a ‘vision’ of the Good is like, or the same as, having a religious experience. That said, The Good is not the same as the Christian conception of God, though there are clearly similarities. Still, getting to that ‘vision’ is not a matter of having blind faith; quite the opposite. You’ve heard the phrase, “The whole is greater than the sum of its parts.” I think the whole is the Good, the parts are the universe and everything in it—including ideas. It’s the first principle or the supreme rational order behind everything. It’s the reason we look.for reasons. 🙂

    —As for Platonic solids, I think essentially you’re right, but it’s complicated. Plato talks about the solids in the Timaeus, and that’s the dialogue in which he gives us some sense of what he means by deducing everything from the Good. So it seems there must be mathematical forms in (C) “dianoia”, and also some manner of doing/thinking about them can put you in (D) “noesis”. Caution to anyone following this thread: This is pretty contentious stuff, the kind scholars will disagree about.

    I’ll quote the relevant section from an article by Debra Nails because I think she answers your question better than I could:

    “The most important reason not to take sensibles as absolutely fundamental, is that they do not cause themselves. They do not have independent existence. It is puzzling, perhaps, that the next higher segment of the line is mathematical – puzzling because it is not immediately obvious how mathematics (invisible, intelligible, eternal, immutable) causes physical objects. In Plato’s telling, the second and third segments of the divided line are equal in length. The physical objects on the second level can be exhaustively described by measurements expressed in numbers. My teacup is not a mere heap of matter but a structured sensible entity about which statements roughly describe its composition, its dimensions, its density, its weight at g, and the wavelengths of light its surface reflects. Were there no such perceptible object, those statements would not be accurate.

    Yet the contents of the third level are represented as more really real than those of the second. The mathematicals (arithmetic and geometry), logic, and the laws of physics are said to be more real, and statements about them more true, than the sensible particulars of the second level and statements about them. Kurt Gödel insisted on it, “To me a Platonism of this kind, also with respect to mathematical concepts, seems to be obvious and its rejection to border on feeble-mindedness.” Werner Heisenberg concurred, “The elementary particles in Plato’s Timaeus are finally not substance but mathematical forms. … In modern quantum theory there can be no doubt that the elementary particles will finally also be mathematical forms but of a much more complicated nature.” Mathematicians of Plato’s time worked with odd, even, and prime numbers, figures, angles, square roots, addition, subtraction, and so on. These are intelligible, eternal, unchanging, and invisible – even if we would now qualify them by conceding that they are limited to base-ten arithmetic and Euclidean geometry. But we’re not done yet: our mathematical operations depend on definitions and axioms that are not themselves proven, so mathematics is not independent. What then accounts for the truths of mathematics?

    Plato’s Socrates in the Republic reasons that a further account is required, that there must be something more fundamental than the formulae manipulated by mathematicians who treat their hypotheses as first principles; that is, there must be an unhypothetical first principle of all that the intellect grasps dialectically, that is causally responsible for everything, forms, numbers, and the sensible objects that mathematicians measure. Once that pinnacle is reached, the fourth and highest segment of the line, the intellect can move back down the divided line and make sound deductions about what was previously understood imperfectly. Not only is my teacup measurable; such forms as being, motion, rest, sameness, and difference characterise it, perhaps the form of the good characterises its function.”

    Above quote from this article: https://www.philosophersmag.com/essays/232-plato

    BTW, earlier in the article, Debra Nails points out that puppies and kittens can tell the difference between shadows and real objects. I found it amusing that we made the same observation about dogs—though, I’m not sure about kittens! 🙂

    —”For B, how about “justified true belief”? The worldly things we know to be true just because they’ve always been true. (The sun shall rise tomorrow.)”

    I don’t think Plato would call the level of knowledge in (B) pistis justified, and the kind of thinking that goes on here can’t be true because it ties down to sense perception. Whereas in (A) eikasia, it’s possible to stumble on the truth through a “good story” or “get it right” by accident without really having knowledge—there’s no insistence that knowledge only comes from the senses here.

    —”Does the bonfire in the cave represent greater control by the manipulators? They would have the power of night and day that way. And the source of the light, in some sense, would be false.”

    Yes! Exactly! Good observation about the source of light from the bonfire being false. What you point out ties into Plato’s theme of things being shadows or copies of something higher.

    —”I wonder if the B and C slots are meant to be equal in size, albeit not in import.”
    Yes! I think you’ve solved the paradox of the equality of the middle lines. Mathematically they’re equal, metaphorically they aren’t equal.

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    • Ha, indeed! Well, if serious philosophers who’ve studied Plato can’t quite parse “the good” I’m not even gonna try. Above my pay grade and all that. I still like what you said about what things are “good for” — I’m just gonna go with that. 😀

      As for math in both the C and D slots, I have in mind an example comparison between an accountant (or other practitioner) [C] and a theoretical mathematician [D]. The former would be expert on the use of math but without understanding its foundations or inner truth. That — the theory and philosophy of math — would be the purview of the latter. Theoretical mathematicians seek the “why” of math.

      The Debra Nails quote raises a question: Is it the idea that a higher level causes a lower level? She writes that sensible objects (B?) “do not cause themselves.” That confused me. She goes on to say C objects are more real than B objects, and that I can understand, although I think the term “real” needs some unpacking there. But in the sense that an abstract circle is more “perfect” than any sensible circle, it is, indeed, more “real.”

      (As an aside, she’s wrong about many of the things she mentioned being confined to base ten arithmetic. The notions of prime numbers, angles, squares, math operations, all transcend mere number bases. Nearly all math does. I think, too, she is wrong to say, “our mathematical operations depend on definitions and axioms that are not themselves proven, so mathematics is not independent.” Theoretical math is all about proving those axioms. Mathematicians don’t like axioms and try very hard to keep them at a minimum. The natural numbers supervene on only two: the idea of zero; the idea of adding one to it.)

      I’ve never been around enough kittens, but I have watched several puppies discover mirrors. Usually a lot of initial commotion until they realize: no smell, no touch,… no dog! I’ve known some dogs (Bentley is one) who understand they can watch you in a mirror — they seem to get the basic principle of reflection. Yet she doesn’t show any interest in her own reflection, and I’m not sure if that’s because she doesn’t recognize herself or just doesn’t care.

      “I don’t think Plato would call the level of knowledge in (B) pistis justified, and the kind of thinking that goes on here can’t be true because it ties down to sense perception.”

      That doesn’t necessarily conflict with the idea of “justified true belief” which I understand to be a formal epistemic concept based on our sensible apprehension and not actually really meaning either “justified” or “true” — the “belief” part is more the emphasis. We consider ourselves “justified” in the “true” belief about (e.g.) the Sun rising because it always has. But it’s not an actual physical truth because the Sun conceivably might not rise.

      “Yes! I think you’ve solved the paradox of the equality of the middle lines.”

      Well, that was easy. What was all the fuss about? 😀 😀 😀

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      • I think your division of practical and theoretical math makes sense. I’d just add I’m not sure “theoretical math” or “philosophy of math” as it exists today (which I don’t know about) is the same as what Plato was talking about, but other than that, I’d have no problem with calling the kind of math that happens in (D) “philosophy of math”.

        —Is it the idea that a higher level causes a lower level? Yeeeeeeeah. I had to think about this one. I probably would’ve gone around the word “cause”—causality is complicated!—in favor of phrases like “makes possible” or “depends on” and the like. To say math causes physical objects, that sounds pretty strange, but I think Debra’s basically right. Maybe it helps to think of Aristotle, who described different kinds of causes (material, formal, efficient, final)…but I didn’t want to get into analyzing causes because Plato doesn’t get into it here. He emphasizes that ultimately, the Good is the cause of all things:

        “The sun provides not only the power of being seen for things seen, but, as I think you will agree, also their generation and growth and nurture, although it is not itself generation…Similarly with things known, you will agree that the good is not only the cause of their becoming known, but the cause that they are, the cause of their state of being, although the good is not itself a state of being but something transcending far beyond it in dignity and power. — The Republic VI (509b)

        I didn’t talk about the analogy of the sun in my post, but think about the way a physical object might cause a shadow. Something is needed for that to happen—light. Light is the emphasis.

        —”But in the sense that an abstract circle is more “perfect” than any sensible circle, it is, indeed, more “real.”

        Exactly. We tend to be what I’m calling “naive realists” who think tangible things are real, ideas are not. Well, Plato’s measuring reality with a very different stick!

        As for what Debra said about math, as you know, I wouldn’t know. But what you said about “proving the axioms” makes me wonder if our theoretical math is what Plato meant for D. You’ll have to tell me:

        “And when I speak of the other division of the intelligible, you will understand me to speak of that other sort of knowledge which reason herself attains by the power of dialectic, using the hypotheses not as first principles, but only as hypotheses – that is to say, as steps and points of departure into a world which is above hypotheses, in order that she may soar beyond them to the first principle of the whole.” (511b)

        —”I’ve known some dogs (Bentley is one) who understand they can watch you in a mirror — they seem to get the basic principle of reflection. Yet she doesn’t show any interest in her own reflection, and I’m not sure if that’s because she doesn’t recognize herself or just doesn’t care.”

        She recognizes herself, but she doesn’t care. How do I know? Well, maybe I don’t KNOW know, but I’m pretty sure that’s just the way dogs are. That said, bet you can spark curiosity in her. I’ll explain…

        Geordie definitely understands how mirrors work too, and he often watches me in the mirror if it’s more convenient than turning his head. It’s not even remotely confusing for him. One time, however, he was playing (killing a stuffed duck) in a new environment and he caught a glimpse of himself in the mirror and ran over to it and barked once, then realized his error. It was a split second error. Hell, I’ve done the same. I’ve seen my own reflection—say, in a shiny glass building—and for a split second thought it was someone else. So if he didn’t recognize himself in the mirror, he’d bark at it. But he doesn’t.

        In fact, a long time ago I started doing something with him called “mirror time”. We sit a few inches away from the mirror and I tell him how pretty he is. “Look at your beautiful big ears!” I touch his ears. “What a pretty nose!” I point to his nose. There’s no real method to this; I just say nice things and whatever comes to mind. He loves it! At first he refused to really look at himself, and he even looked a bit disgusted (but not confused). Nowadays I sometimes catch him checking himself out when he thinks I’m doing something else or not looking. And I don’t mean just a glance. He’ll go right up to the mirror—his face an inch away—and sit down, leisurely taking in his own image. He looks rather ponderous. If I ever get a chance to catch it on video, I will, but fat chance.

        Anyway, if you have a big mirror in the house, you could give it a try with Bentley. Could be a fun activity. And what dog doesn’t love being praised?

        —Justified true belief…I think I know what you mean now. 🙂

        —Yes! I think you’ve solved the paradox of the equality of the middle lines….Well, that was easy. What was all the fuss about? 😀 😀 😀

        The fuss will have to come from someone more invested in their particular way of solving the problem. As for me, I’m open to all!

        Liked by 1 person

        • “He emphasizes that ultimately, the Good is the cause of all things: ‘The sun provides not only the power of being seen for things seen,…'”

          He’s certainly right that the Sun is the source of everything for us. I’m just not sure I’m down with the ontological musings from an era before science was invented. Our understanding of what is is so much more sophisticated now. That may account for why I struggle trying to wrap my head around The Good. It seems a little archaic, maybe?

          “We tend to be what I’m calling “naive realists” who think tangible things are real, ideas are not.”

          My canonical example for a long time has been to ask if unicorns are real? If they’re not, why is the question intelligible? In everyone knows exactly what I mean by “unicorn”, how can they not be real?

          I do think theoretical math fits in D (whereas practical math in C), but I take the point that “theoretical math” in Plato’s time wasn’t anything like it is in our time. But I think the reasoning behind the Platonic solids — 3D, convex, congruent faces, edges, vertices, and angles — qualifies. So, I think, does Pythagoras’s famous geometric proof that the sum of the squares of the two short sides of a right-triangle equal the square of the length of the long side.

          “She recognizes herself, but she doesn’t care. How do I know? Well, maybe I don’t KNOW know, but I’m pretty sure that’s just the way dogs are.”

          Maybe. She knows what I look like, so my reflection in a mirror, that I would expect her to recognize, especially since she can match my movements there and for real. But she would have little reason to know what she looks like, so I’m not sure.

          I love the idea of trying to train her! I’m going to go buy a mirror we can use tomorrow!

          Geordie, from your many descriptions and videos, sounds like a dog on the high end of the intelligence scale. Some of that, I believe, comes from your constant interaction and conversation with him. Dogs that are a deep and daily part of their human’s life seem to achieve an extra level of understanding and behavior.

          I’m going to try using your “mirror time” technique with B’Dog! She’s easy and loveable, but she doesn’t know any games. Fetch bores her after a couple times (if she’ll even deign to fetch once), so I’m always looking for ways to entertain her. About the only thing she loves is walks, but we can only do so much walking. Thanks for the idea!

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          • —”Our understanding of what is is so much more sophisticated now.”
            Ah, but is it right? 🙂

            No, I get what you mean about not being able to wrap your mind around the Good; the idea of a teleologically ordered universe seems archaic to us. On the other hand, science isn’t operating in the same realm.

            —”My canonical example for a long time has been to ask if unicorns are real? If they’re not, why is the question intelligible? In everyone knows exactly what I mean by “unicorn”, how can they not be real?”

            Good questions, a good way to.get people to see ‘real’ in a different light!

            —As for Bentley, “But she would have little reason to know what she looks like, so I’m not sure.”

            Geordie barks and growls at his own image when I open up PhotoBooth on my Mac and get him to look at himself in the screen. He clearly doesn’t recognize himself, at least not at first, although if I wave my arms behind him, he’ll usually get the idea. But there’s no confusion with the mirror. I think it might have something to do with scale; he’s much smaller in the computer screen! So you could try comparing her reaction to her own image in the mirror with some other type of reflection of her own image.

            —Geordie, from your many descriptions and videos, sounds like a dog on the high end of the intelligence scale. Some of that, I believe, comes from your constant interaction and conversation with him.

            I think Geordie’s probably about average on conventional intelligence, probably above average on understanding emotion. And I think you’re right about interaction. Experience counts for a lot. In many ways he’s wiser than when I first got him (though, all bets are off when he gets excited.)

            Have fun with Bentley! It must be nice to have all the fun and none of the responsibility. BTW, if you’re looking for dog treats, Full Moon is the best. Haven’t met a dog who doesn’t love them. I give Geordie the ‘hip and joint’ ones, and he has no idea he’s getting a dose of glucosamine.

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            • I bought a mirror and have been trying to get her interested in her image, but so far she’s supremely disinterested. At first she did look behind it but it’s almost like she does know but isn’t interested. I’ve got eight more days with her, so we’ll see if I can generate more interest or less.

              Full Moon,… I think I’ve seen that in the store. I’ll keep it in mind. It certainly wouldn’t hurt for her to get some vitamins or glucosamine.

              Liked by 1 person

              • It took Geordie a long time before he began looking at himself in the mirror on his own. Also, the mirror I’m using is very big—it’s two mirrored sliding closet doors—and takes up nearly an entire wall. That might make some difference, but who knows. You might try leaving the mirror in a space where you and she hang out a lot and let her get used to it.

                The glucosamine Full Moon treats are a bit difficult to find in the stores, at least around here, but I often see the regular chicken strips in Walgreens, of all places. Of course, you can find anything on Amazon. Anyway, they must put doggie crack in them; Geordie would probably gobble down a whole bag in one sitting if I let him. To slow him down a little, I break them up into smaller pieces and hide the bits in his snuffle mat. You could use a blanket or towel if you think Bentley might like digging around. Tastes better when it’s earned!

                Geordie isn’t terribly interested in conventional dog toys. He doesn’t do fetch. He used to do keep away (I throw the ball, he chases it and runs off to another room with it so that I have to chase him.) He also used to play with squeaker toys, but he’s gone high tech now. I could tell you about different robot toys that have worked out for Geordie, but I don’t know how safe they’d be for her. One thing you could do is take a squeaker toy and cut a slit in it (or a few slits) and put a treat inside. If she gives up on trying to get it out, make the slit bigger. This is the kind of thing that can keep them pretty busy for a while if they’re treat motivated. You could also hide the treats around the house and let her find them. That would require she know ‘sit’ and ‘stay’—kind of tricky to pull off if she doesn’t. Really, it’s tricky even if she does…Geordie often sneaks a peek so he can at least see which rooms I’m hiding the treats in. 🙂

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                • The mirror I bought is big, but not huge. I’ve got it in the living room where we spend most of our time. I’ve seen her glance at herself in it, but that’s about all.

                  I’ve got a Kong toy and a bone with a hole through it. I cram treats in those, and she’s entertained for 20-30 minutes. I’ve done the hiding treats thing since Sam, who was very good about “go hide” (go to the bedroom closet and wait). Bentley isn’t disciplined (her first owner neglected her; her second owner, my friend, gives her all the loves but doesn’t do much training). I’ve had to shut her in the closet, which she hates and, realizing what I’m doing, resists. I tried having her just sit on the bed and it worked! She waited until I called her.

                  Poor thing is in for a few days of cabin fever. The weather turned nasty on us and won’t get nice until Monday. 😦

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                  • I don’t do much training with Geordie either, mainly because he’s clearly already been trained (before I got him), and also because he doesn’t seem to need it. It’s amazing what dogs pick up on all on their own, particularly when they know a game with treats is involved. 🙂

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                    • It is amazing what they pick up on their own. I only see Bentley when I visit BentleyMom or dog-sit, but every time we go for a walk, I try to teach her to heel when we cross the street. (She has one of those nine-meter retractable leashes, so she’s usually out in front catching fresh smells.) I’ve been trying to get her to heel for years now. She always pulls at the now very short lead. But it’s occurred to me recently that maybe she just wants to get across the street which she views as dangerous. (And I just now thought of a way I could test that. See if she’ll heel when we’re on the sidewalk.) It’s also possible all the filth in the street (and salt in the winter) makes her want to cross as fast as possible.

                      She’s a total treat hound, but she gets so impatient when she knows treats are involved that I haven’t had much luck. I was amazed she stayed on the bed long enough for me to hide treats in the living room. I’ll try that again today probably. (Weather turned really nasty on us. Warm enough for wet snow and rain followed by an ugly cold snap (-10 to no higher than +10) that turned everything to ice. Yesterday when we went out, I just stood still and slid 10 feet down my driveway. My aging bones were quivering in fear!

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                    • Not a bad idea to see if she’ll heel on the sidewalk. If the point is to stop her from running out into the street, maybe the thing to do would be to treat-train her to stop and wait for you at the end of the sidewalk before crossing. (Maybe make her sit, then treat, then cross…I dunno. I’m sure there’s a ‘proper’ way to do this, and I bet it’s on the internet). With treats involved, it probably won’t take long for her to get the idea. I knew a dog who was so good at this he could be walked off leash. He’d also wait outside a store for you. I’d never try that with Geordie. Terriers have a mind of their own.

                      That’s too cold to imagine. It reached eighty three here. 🙂

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                    • Terriers do, indeed! Bentley is an American Bully. In fact, her mom did the DNA thing, and she’s purebred! (Given her solid shape, I call her “pure bread!”) So, she is a breed of terrier, and that opinionated mind is a big part of why I love her. (I used to be a lab guy. My dog was a black lab. They are such easy-going dogs. But now that I’ve met not just a terrier, but one technically in the “pit bull” category, I just love that stubborn little “won’t back down” mind!)

                      Thing is, if she knew I had treats with me,… I think she’d be hugely distracted. She’d be totally focused on them. Weather has finally warmed up enough that I can try getting her to heel on the sidewalk.

                      I know what you mean about well-trained dogs, I’ve seen them patiently waiting outside some little store. Sam (my dog) was close to that, and we walked off-lead a lot (when no one was around), but never near a busy street. They don’t really understand cars.

                      Since I’m caring for someone else’s precious dog, I take no risks! I even avoid letting her interact with other dogs because she can be something of a ruffian and a bit unpredictable. Meeting another dog while on leash changes the dynamic. When she has playdates with her dog friends, they run around a fenced in yard and have a great time.

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                    • We won’t see 83 for months! A friend of mine who worked in a call center tried to help a caller from California imagine it. He said she should fill her bathtub with ice, pour in cold water, and stick your arm in that for a while. Or just imagine hanging out in a walk-in freezer for an hour. 😀

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  6. Pingback: Plato’s Divided Line | Logos con carne

  7. This was great. I don’t know much about the Greek Philosophers, or the rest of them for that matter, so this promises to be an enriching experience.
    I’m a believer in the idea that, if you are the smartest guy in the room then you are in the wrong room. I’m glad to be in the right one. I’m hitting the follow button now, and your empire grows apace.

    Liked by 1 person

  8. Pingback: Back to Plato’s Line | Logos con carne

  9. Pingback: Flowers for Algernon: Why the Plato quote? | Diotima's Ladder

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