A Short Piece Featured In (Re)constructing “A Is A” by O.G. Rose

Geometry, Astronomy, and Platonic Forms As Ordering Principles

O.G. Rose
13 min readMar 1, 2022

On understanding “forms” like “invisible guardrails” in which things “be-come” themselves.

Photo by Patrick McManaman

The second half of Book VII in The Republic mainly consists of Plato laying out the educational program he believes Philosopher Kings need to undergo in order to take their place as good rulers of the city. Socrates is speaking with Glaucon, who asks about the subject of astronomy and if that should be taught; in that context, Socrates says something of note:

‘I can’t conceive of any subject making the soul look upward except one concerned with that which is, and that which is invisible. If anyone attempts to learn something about sensible things […] he never learns anything […] even if he studies lying on his back on the ground or floating on it in the sea, his soul is looking not up but down.’¹

Plato, through Socrates, suggests that learning about things which change is practically to learn nothing at all, because if I learn that “x is y” but x changes into z a moment later, then learning “x is y” was practically learning nothing. Change and temporality are major challenges to learning, and Plato suggests that we should focus on “forms” to solve and overcome this problem. But what does this mean?

Plato’s project can be viewed as a response to the problem of change, as similarly Aristotle’s Metaphysics is a response to his observations about change throughout the Physics. Both take on the problem in different ways, and admittedly my understanding of Plato here might “bring him more down to earth” than is justified. Still, understanding Plato in the way I’ll now describe helps me get a hold on what he means by “forms,” though I ask for your forgiveness if this interpretation is completely off.


When Plato discusses “sensible things,” he means things which undergo change; again, for Plato, to learn about something which changes is to learn knowledge which quickly ceases to be knowledge at all. For Plato, learning the fact “Sarah is smiling” is hardly learning, because Sarah a moment from now won’t be smiling: we “observed” more than “learned,” per se. Instead, learning how smiling in general is possible, what smiling signifies, what kind of person Sarah is to determine why she smiles — all of this is far more useful in Plato’s mind, for these are “principles” behind smiling and Sarah’s person. No, these “principles” are not necessarily “the best” things to focus on and learn, but I think we’re starting to get closer to what Plato means by “form” in focusing on them.

Audio Summary

Plato claims that truth ‘must be grasped by reason and thought, not by sight,’ which suggests that “if we can see it, it’s not worth learning.”² What does this mean? Well, again, Plato seems to hold a critical distinction between “observation” and “learning,” which is to say that “observing Sarah smiling” is not the same as “learning that Sarah is smiling”; in fact, Plato seems like he would want us to deconstruct all uses of the word “learning” which suggest “learning is merely observation.” Even the phrase, “I learn Sarah is smiling” is questionable, for it sounds like “memorizing the fact that Sarah is smiling,” which can be associated with “the memorization games” down in the Cave which entertain and trap the prisoners. Perhaps we can talk about “learning why” or “learning how” Sarah is smiling, but “learning that” Sarah is smiling is not a good use of the word “learning.” For Plato, we don’t really learn about what’s visible, only observe it.

I don’t think Plato means to suggest that observation is necessarily bad (after all, it’s unavoidable), but what concerns Plato is our tendency to replace “learning about invisible realities” with “observing visible realities.” “Observing that Sarah is smiling” feels like “learning that Sarah is smiling,” and so it’s easy to think of learning as merely observing. But there is a difference, as hopefully “The Critique of Pure Observation” by O.G. Rose makes clear. Thinking they are identical, we can convince ourselves that we are thinking and learning when we are only observing, as we can convince ourselves we are “only observing” when we engage in empirical studies, as if it’s possible to “observe” without doing so within a conceptual framework. Plato’s distinction between “observation” and ‘learning/thinking” can prove useful to correct both of these mistakes, though admittedly Plato may not be a fan of phenomenology, which I think is critical and useful (but that’s another topic for another time).

Learning why Sarah is smiling today will likely prove to be useless knowledge tomorrow, and so Plato would not have us focus on this unless, that is, we are trying to determine a “constant principle” behind why Sarah smiles and when. What seems open to debate is “how much” Plato would have us “observe” in order to help us figure out underlying principles and rules, or if Plato thinks all forms can be determined without any observation at all. Sometimes, it seems like the only truths Plato thinks are worth knowing are the ones we can conclude entirely on our own (without any “data” or “input” from observation), and other times it seems like Plato wants a dialectic between “observation” and “internal consideration” — I have no idea. Favoring phenomenology, I prefer the dialectical understanding of Plato, but I am no expert on his work. Regardless though, I think it’s true that Plato emphasizes “learning,” and “learning” for Plato involves the invisible. To pull a section from “How Does Anyone Leave Plato’s Cave?” by O.G. Rose, Plato tells us:

‘Therefore, calculation, geometry, and all the preliminary education required for dialectic must be offered to the future rulers in childhood, and not in the shape of compulsory learning either […] no free person should learn anything like a slave. Force bodily labor does no harm to the body, but nothing taught by force stays in the soul […] use play instead. That way you’ll also see better what each of them is naturally fitted for.’³

“Calculation” and “geometry” are “studies of the invisible,” per se, for by them Plato seems to mean the internal process by which we arrive at truth (philosophy, dialectics) and understand the rules and principles according to which shapes are themselves. Plato also loves math, for math may describe an “invisible world” according to which the world functions, and even if it doesn’t, mathematical training makes students better at “internal calculation.” For Plato, it’s all about “the invisible” (to put it generally), because the ways and reasons “changing things” develop like they do is relative to things or “paths” which are “invisible” (which would include justice, goodness, truth, and ethics in general).

As I understand them, “Platonic forms” are the invisible rules and principles things follow to “be(come) themselves”: they are the “guardrails,” per se, in which things “become” on their way into “coming into themselves,” their “full being.” Cups “become” within different “guardrails” than say cats, and that’s why cups don’t become cats or cats become cups. Forms set the parameters of development, and they are why everything can always be changing and yet the universe not be chaotic and still intelligible. Personally, to reference a point regarding Aristotle I make in “(Re)construction,” I see no reason why we can’t think of “Platonic forms” in terms of DNA and “natural habits” (versus “natural laws”). Now, I’m not claiming they are DNA and “natural habits,” but I am saying that thinking about “forms” like DNA and “natural habits” can be helpful, even if ultimately erroneous.


The main section I want to focus on to help us understand “forms” in Plato explores the topic of “astronomy,” which Plato seems to like but also dislike. Focusing on it, I think it helps unveil Plato’s view on what constitutes “real learning”:

Glaucon: ‘[W]hat did you mean when you said that astronomy must be learned in a different way from the way in which it is learned at present if it is to be a useful subject for our purposes?’

Socrates: ‘We should consider the decorations in the sky to be the most beautiful and most exact of visible things, seeing that they’re embroidered on a visible surface. But we should consider their motions to fall far short of the true ones […] [W]e should use the embroidery in the sky as a model in the study of other things. If someone experienced in geometry were to come upon plans very carefully drawn and worked out by Daedalus or some other craftsman or artist, he’d consider them to be very finely executed, but he’d think it ridiculous to examine them seriously in order to find the truth in them about the equal, the double, or any other ratio […] [L]et’s study astronomy by means of problems, as we do geometry, and leave the things in the sky alone.’⁴

I find this exchange very strange but also telling. Plato seems to be saying that “the movement of the heavens” is too perfect to warrant serious study. How odd: I thought “Platonic forms” were perfect? If anything is worth studying, wouldn’t this fit the bill? Well, only if we understand “Platonic forms” as “perfections,” but once we correct our understanding of the word “perfection,” we might start to get a better grasp on Plato’s thought.

Plato is fine with studying astronomy to the degree we learn about “invisible geometric movements” like orbits, the study of which would require mathematics, but it also seems that he’s not that interested in pursuing this line of thought, because how will learning “the invisible truths about the heavens” help us run the Republic? The discussion between Socrates and Glaucon is occurring in the context of an inquiry into the nature of education involving Philosopher Kings, and the point of that educational program is to make the most of the Republic. Here, Plato suggests that learning about astronomy without geometry is meaningless, and Plato even suggests that learning geometry in this context isn’t that great either.

For Plato, we don’t really learn from studying the heavens, because the heavens are perfect and taking care of themselves just fine. They move, but they don’t really change, while we below on earth find ourselves amongst movement and change. To study the heavens is to study too easy a problem, per se: the variables are set and frozen in ways different from what we deal with on earth. We need to learn how to see “unchanging principles” in the middle of movement and change, not up above where there is only movement: again, that makes our task too easy. We could almost say that Plato wants us to make scientific discoveries “out in the field,” not just “in the observatory,” though he’s okay with us doing both. However, he might be concerned that the ease of “staying in the observatory” could tempt us to stay there and never risk walking under a sky from which rain can fall, thus his concerns with astronomy.

Astronomy seems too perfect for Plato, which suggests that for Plato learning is not simply about “discovering the perfect,” as perhaps we’ve been led to believe. There is something about the perfect which indeed describes Plato, but the word “perfect” might mean something different here than what we’ve been led to believe. Now, I’m more confident about what I’m about to claim regarding Aristotle versus Plato, but I think it also applies to him. If not, forgive my error, but hopefully the idea itself still proves valuable.


If we accept the premise that things are “toward” the good (based perhaps on the idea that “nothing is orientated and organized relative to what it believes is bad for it,” though this isn’t to say that bad consequences still can’t result), it follows that what ultimately organizes, animates, and shapes entities would be “the most perfect version of that thing.” This would be a state in which a thing is most “fittingly itself”: the word “perfect” here doesn’t necessarily have to be taken as disembodied, heavenly, or divine (though Plato does seem to ascribe to such a realm).

A wrench is “perfect” if it fixes a broken engine that we need running; my thumb is “perfect” if it helps me hold my coffee mug; my mouth is “perfect” if I can eat and talk with it. “Perfect” is relative to task, situation, and/or purpose: it’s meaningless to discuss “perfect” outside this framing. What is a perfect wrench if not the wrench which “gets the job done?” A golden one without any blemishes? Well, if it looks nice and I can’t use it, who cares? Also, the fact the wrench is made of gold might make me hesitant to use it, because what if I damage it? And so on — beyond a particular situation and “purpose,” the word “perfect” holds an unclear meaning. Sure, we can discuss a “perfect realm” like Heaven where every tear is wiped away, but that is a different matter entirely from the question of when and how a wrench is “perfect” in my everyday life. In finitude and experience, “perfect” seems tied to purpose.

Following this line of thought, we can conclude that “a perfect cup” is one that “does what cups are made to do,” and — here’s the main move for Plato — all cups are “toward,” and trying to become, what they were made to do, of which “the perfect cup” does. “Towardness” plus “time” equals “becoming,” so every cup (in being in time) is “becoming” “the perfect cup,” and the “perfect cup” is a being versus a becoming, because it is the end point at which all cups are trying to reach (“end points” don’t need to go anywhere and aren’t “toward” any state beyond them). This doesn’t mean every cup actually becomes “the perfect cup” (or that some “realm of perfect cups” necessarily exists), but only that “the perfect cup” is what they are all “toward” and “emerge” relative to, regardless if “the perfect cup” actually exists somewhere in finitude or not. If this logic of a cup all follows, we can then claim that “lowercase-everything is ‘toward’ and ‘becoming’ uppercase-Everything,” which is to say all “being is becoming Being” (regardless, again, if some “Platonic Realm” ultimately exists or not).

Now, it seems to me that Plato may practically locate all “forms” in the mind, precisely because only humans can contemplate and understand a situation in order to determine what is needed to “perfectly” address that situation (or to determine when a “cat” is fully able to be itself, according to a standard we “recognize”). Like mathematics, a question is left hanging on if Platonic forms are “realized” or “discovered,” but, regardless, understanding only occurs within the mind (it’s also possible that some forms are “realized,” say in a cat, while other forms are “created,” say in a wrench — I’m not sure).

If we think forms are “realized,” then readings favoring a “Platonic Realm” are much stronger; if forms are “created,” then the reverse is the case. Regardless though, “forms” play a role in mental processes either way, as does mathematics, regardless if math is ultimately “constructed” or “self-existing.” On these grounds, I think we can perhaps find use in Plato even if we cannot accept his forms as “discovered.” Practically speaking, forms are things we experience mentally, and they practically wouldn’t exist if we didn’t think. For this reason, I think it is fair to strongly associate “Platonic forms” with mental processes, especially if ultimately it is indeterminable if forms, like math, are “created” or “discovered.”

It takes a mind to determine if a certain object with a certain shape is a “wrench,” as it also takes a mind to determine that this certain object with a certain shape could be used to solve the problem of a broken engine. A mind is needed to connect everything together — nature won’t bring together a “wrench” with an engine, nor understand how the wrench can fix the engine so that the engine can “work” and be itself — and a mind is needed to determine “how” the object with x shape is to be applied to the engine to fix the engine. In this way, “forms” are mental: for us, they practically do not exist in “nature” or on their own without us (unless we believe that “mind” is somehow “part of nature,” say in panpsychism, but that is another topic for another time).


For Plato, “forms” can only be learned, not observed, and they are more like the “orbits” which planets follow than they are a “heavenly realm” which we are all trying to access. Each cup is “becoming” itself like a planet following its orbit: the trajectory and “toward-ness” is simply part of what makes things themselves and not something else.⁵ As balls fall when released, following “the natural habit” of gravity, so things are “toward” becoming their perfect selves. No, there’s no guarantee they get there or “realize” perfection, and without the help of us and our minds, natural causality won’t help things much, but still the “toward-ness” is considerable reality.⁶

We can observe cups, but we can only learn about “toward-ness” and Cups, and so it is regarding forms and “toward-ness” that Plato claims “true education” consists. We don’t need education for “grasping” cups, only our eyes, and an education concerned with “sensory things” won’t offer us much more than what we can provide ourselves. Additionally, what we “observe” about a cup today might not be true tomorrow, so what have we gained? For Plato, nothing: education only occurs for those who “keep their eyes on the unseen.”





¹The Republic. Translated by G.M.A Grube. Indianapolis, Indiana. Hackett Publishing Company, 1992: 201.

²The Republic. Translated by G.M.A Grube. Indianapolis, Indiana. Hackett Publishing Company, 1992: 201.

³The Republic. Translated by G.M.A Grube. Indianapolis, Indiana. Hackett Publishing Company, 1992: 208.

⁴The Republic. Translated by G.M.A Grube. Indianapolis, Indiana. Hackett Publishing Company, 1992: 201–202.

⁵I am tempted to say “forms” and “toward-ness” are identical, but I’m not so sure.

⁶This begs the question though if everything has “form” or only that which humans think about, and I’m not sure if Plato addresses this point. Perhaps he does, but I’m not sure. If we align “forms” with DNA, this problem can be addressed, but there might be a lot more to it which is worth exploring at another time.




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O.G. Rose

Iowa. Broken Pencil. Allegory. Write Launch. Ponder. Pidgeonholes. W&M. Poydras. Toho. ellipsis. O:JA&L. West Trade. UNO. Pushcart. https://linktr.ee/ogrose