Education and Plato's Parable of the Cave
Everyone knows that Plato is deeply interested in education. In one way or another, nearly all of his dialogues are concerned with it -- what it can accomplish and how it can accomplish it, who is qualified to impart and receive it, why it is valuable, and so forth. My emphasis in this essay is on Plato's views of education as they unfold in one extended passage in the Republic. Here, in a series of images and commentary on these images, Plato develops some of the central points necessary to sustain the outlandish claim that
unless communities have philosophers as kings... or the people who are currently called kings and rulers practice philosophy with enough integrity... there can be no end to political troubles... or even to human troubles in general. (473C11-D5)
The right sort of ruler can only be produced by careful and systematic education, as Plato continually reminds us (see, e.g., 487A7-8, 490C8-493A2, 497A3-D2, 502C9-504A1). Concern with this sort of education gives rise to the famous images of the sun, the divided line, and the cave. Although I am primarily concerned in this essay with the cave image and its educational implications, occasionally this will require a look at the sun and the line and other contiguous passages in the Republic. The analogy of the divided line is brought in to shed light on the sun simile (509D1-5), which is itself introduced as a way of clarifying the role of the good (508B12-C2). Plato himself insists that the cave image must be "applied" to the discussion that leads up to it (517B1-2), and he continues to draw out the implications of the cave image until quite late in Book VII (see 532A1-533D4).
- I -
The cave image is offered as "an analogy for the human condition -- for our education or lack of it" (514A1-2). Imagine prisoners in a cave, chained and unable to turn their heads; as a result they see only what is directly in front of them. What they see are shadows cast by objects behind them which are illuminated by firelight further behind and above them. The objects are carried along and extend above a low wall behind the prisoners. The bearers of the objects are hidden behind the wall and so cast no shadows; but occasionally they speak, and the echoes of these words reach the prisoners and seem to come from the shadows. The prisoners can talk among themselves, and they naturally assume that the names they use apply to what they see and hear -- the shadows passing in front of them. Socrates offers a grim assessment of their plight: "The shadows of artifacts constitute the only reality people in this situation would recognize" (515C1-2).
As Glaucon observes, this is a weird image, and these are weird prisoners. Nevertheless, Socrates says, they are like us (homoious hêmin, 515A5). Of course we do not really spend our time chained and looking helplessly at shadows produced by those intent on deceiving us. Yet for Plato something about our condition makes the cave an apt image.
The prisoners see only shadows, and these shadows are cast by artifacts, likenesses of animals and people (514B8-515C2). So the prisoners are, in Plato's view, at least two removes from truth or reality, although they do not realize this and would object if the suggestion were made to them (515C8-D7). If they were freed and made to turn around towards the firelight, the prisoners would be dazzled and unable to make out the objects that cast the shadows on the wall (515C4-D1). If they were compelled to look directly at the fire, this would hurt their eyes, and they'd probably prefer to go back to the comfortable and familiar darkness of their prison (515E1-5). If they were forced out of the cave entirely, out into the sunlight, this would be even more painful, and objects outside the cave would be even harder for them to make out (515E7-516A3). Gradually, however, their eyes would grow used to the light and they would start to discern shadows, then reflections, then maybe even the objects themselves (516A5-B2).
Plato tells us explicitly how to unpack some of the details of this image. First, the cave is the region accessible to sight or perception (517B1-2). A few pages earlier, in the sun simile, Plato distinguishes between the visible realm and the intelligible realm, between things grasped by perception and things grasped by reasoning or intelligence (507D8-509D4). The visible realm comprises ordinary perceptible things; the intelligible realm comprises what Plato calls the forms or ideas. The bound prisoner -- and by implication the ordinary uneducated person -- has no access to intelligible forms. In fact, he has no idea there are such things. Worse yet, his access is not to perceptible things themselves, but only to shadows of those things. He may be exceedingly good at identifying these shadows, better even than someone who has been freed and has seen the artifacts responsible for casting the shadows and knows how the shadows were cast (516C8-D7). Still, like the sight-lover Plato discusses earlier in Book V (475D1-480A13), his epistemic horizons are limited.
Second, the world outside and above the cave is the intelligible region (517B4), accessible not to perception but to reasoning. The objects here are more real or true than the artifacts in the cave, since they are the originals of which the artifacts are likenesses (515D1-7).
Third, the upward journey out of the cave into daylight is the soul's ascent to the intelligible realm (517B4-5). Having distinguished these realms earlier in the sun simile and said something about their relations in the divided line analogy, Plato now explicitly intimates that one can move from one realm to the other. This is precisely the movement to be effected by Platonic education -- although what is being moved is not the eye but the soul. I shall turn shortly to the nature of this movement and how Plato thinks it is best accomplished.
Before turning to the process, however, recall briefly what Plato sees as the end result of such movement, the epistemic condition of the philosopher-ruler. Such a person Plato is willing to credit with understanding (epistêmê). Such a person has a secure grasp of the forms, not just in the abstract but as they manifest themselves in things around us (402B5-C8, 520C4-5). Such a person's view of things is synoptic: he "sees things whole" or "has a unified view of things." It is Plato's bold claim that only when such people are allowed to rule will a community flourish. The stakes involved are very high, and the value of any process that can reliably produce such people is obviously very great.
- II -
Perhaps the first thing we notice about the prisoners in the cave is that they are looking in the wrong direction. Their bonds prevent them from turning their heads away from the rear wall of the cave, and what they need to see is behind their heads (514B1-2, 515A9-B1). The first step in the journey out of the cave is to stand up and turn around towards the firelight (periagein, 515C7), and the first impulse of the freed prisoners upon being made to look towards the firelight is to turn back towards the familiar shadows (apostrephein, 515E2). This notion of orientation is central to Plato's idea of education: he later describes real education as the art of orientation (technê... tês periagogês, 518D3-4) and the educator's task as that of turning souls around (metastrephein, 518D5).
This is to be contrasted with what Plato presents as a common practice of educators, who "claim to introduce knowledge into a soul which doesn't have it, as if they were introducing sight into eyes which are blind" (518B6-C2). Such a view of education neglects the fact that the power to learn and the organ with which to do so is present in everyone (518C4-6, 519A3-B6, 527D6-E3, 530B6-C1). Education, Plato remarks,
should be... the art of orientation. Educators should devise the simplest and most effective methods of turning souls around. It shouldn't be the art of implanting sight in the organ, but should proceed on the understanding that the organ already has the capacity, but is improperly aligned and isn't facing the right way. (518D3-7)
Plato refers to this power to learn as phronêsis or intelligence at 518E2, where he goes on to say that it is useful and beneficial, or useless and harmful, depending on its orientation (518E4-519A1). Part of Platonic education, then, consists in reorienting this neutral capacity of intelligence, directing it away from one sort of object and towards another.
But this is only part of the story. Plato goes on to remark about the soul of the bad but clever person that
if this aspect of that kind of person is hammered at from an early age, until the inevitable consequences of incarnation have been knocked off -- the leaden weights, so to speak, which are grafted on to it as a result of eating and similar pleasures and indulgences and which turn the sight of the soul downwards -- if it sheds these weights and is reoriented towards the truth, then (and we're talking about the same organ and the same people) it would see the truth just as clearly as it sees the objects it faces at the moment. (519A8-B5)
This person's desires, at least as much as his intelligence, account for his condition. There is an important cognitive dimension to Platonic education, but there is an equally important affective or desiderative dimension. Consider the money-lover Plato describes in Book IX:
His reasoning and spirited parts... are made to sit on the ground on either side of the king's feet [i.e. his appetitive part]. The only calculations and researches he allows his reasoning part to make are concerned with how to start with a little money and increase it, the only admiration and respect he allows his spirited part to feel are for wealth and wealthy people, and he restricts his ambition to the acquisition of money and to any means towards that end. (553D1-7)
Analogous claims are made about the honor-loving person (475A9-B2, 549C2-550B7) and the wisdom-loving person or philosopher (581B5-7, 475B8-C8). So the educator must be concerned not only with the reorienting of intelligence, but with desires as well. The reorientation of desire is accomplished by proper use of traditional mousikê and gymnastikê, discussed at length in Books II-III, and it is clearly presupposed by the educational program outlined in the pages following the cave image. I shall discuss it briefly before turning to the cognitive dimension in the next section of this essay.
Focusing entirely on reorienting the student's intelligence runs the risk of ignoring fully two-thirds of the student's soul, since like everyone he is a compound of three parts. As Plato observes in Book IX,
the correspondence between the three classes into which the community was divided and the threefold division of everyone's soul provides the basis for a further argument .... Each of the three mental parts has its own particular pleasure, so that there are three kinds of pleasure as well. The same would also go for desires and motivations. (580D2-8)
This claim -- that each part of the soul has its own particular pleasure -- is essential to understanding the sort of reorientation Plato wants to effect by means of education. In the Republic, people are sorted into classes (producers, guardians, rulers) according to which part of their soul motivates or rules them. The appetitive part is described as money-loving and gain-loving (philochrêmaton, 580E5; philokerdes, 581C4), and its principal concerns are the pleasures of food, drink, and sex (439D6-7). The spirited part is honor-loving (philonikon, 58162, C4) and focuses on the pleasures of competition, with doing what is noble and avoiding what is base. The reasoning part is wisdom-loving (philosophon, 581B9, C4) and is "entirely directed at every moment towards knowing the truth of things" (581B5). Since the goal of Platonic education is to produce philosophers, we need to know how best to bring people whose primary desires may be for food or drink, or for good reputation, to the state where their primary desires are for wisdom and truth.
Central to Plato's conception of desire and its educability is the idea that
anyone whose predilection tends strongly in a single direction has correspondingly less desire for other things, like a stream whose flow has been diverted into another channel.... So when a person's desires are channeled towards learning and so on, that person is concerned with the pleasure the soul feels of its own accord, and has nothing to do with the pleasures which reach the soul through the agency of the body. (485D6-12)
Like a closed hydraulic system, each person has a definite amount of desiderative energy, and if that energy is expended on one sort of object, it is not available to expend on another sort of object. Reorienting the soul involves rechannelling desires, diverting them away from one sort of object and towards another. How is this done?
The process starts long before the student begins his mathematical studies, much earlier in the Republic (Books II-III, 374E10-412B1). Plato disparages this "primary education" at 522A2-B1 -- "there's nothing in it which can lead a student towards the kind of goal [we're] after at the moment" -- but it is clearly presupposed by the mathematics-intensive curriculum he goes on to outline. In part, this is a matter of exercise for the body (gymnastikê) and cultural studies for the soul (mousikê) (376E3-4), although even gymnastikê is primarily aimed at the soul (410B5-412A2). It may even be the most important stage in the educational process:
The most important stage of any enterprise is the beginning, especially when something young and sensitive is involved.... That's when most of its formation takes place, and it absorbs every impression anyone wants to stamp upon it. (377A12-B3)
The stories we tell, the poetry we read, the music and songs we play and sing, can instill in the young student's soul good rhythm, harmony, grace, a disciplined and good character, and love of beauty (see 376E9-403C8, especially 400D1-403C7). Mousikê, cultural education, is important, Plato maintains, because
rhythm and harmony sink more deeply into the soul than anything else and affect it more powerfully than anything else and bring grace in their train. For someone who is given a correct education, their product is grace; but in the opposite situation it is inelegance. A proper cultural education would enable a person to be very quick at noticing defects and flaws in the construction or nature of things.... He'd find offensive the things he ought to find offensive. Fine things would be appreciated and enjoyed by him, and he'd accept them into his soul as nourishment and would therefore become truly good; even when young, however, and still incapable of rationally understanding why, he would rightly condemn and loathe contemptible things. And then the rational soul would be greeted like an old friend when it did arrive, because anyone with this upbringing would be more closely affiliated with rationality than anyone else. (401D5-402A4)
Stories and songs affect the student's desires, and do so in ways that do not rely on reasoning. Music does so especially directly: "rhythm and harmony sink more deeply into the soul than anything else," and can produce "grace" (euschêmosunê), which clearly involves harmony among the student's desires (401D5-6; cf. 410A7-9, 423E4-5, 424E5-425A6). Stories can contribute to this end as well by providing the student with models to imitate. Plato takes the power of imitation very seriously:
Any imitative roles [children] do take on must... be appropriate ones. They should imitate people who are courageous, self-disciplined, just, and generous and should play only those kinds of parts; but they should neither do nor be good at imitating anything mean-spirited or otherwise contemptible, in case the harvest they reap from imitation is reality.... If repeated imitation continues much past childhood, it becomes habitual and ingrained and has an effect on a person's body, voice, and soul. (395C2-D2)
Songs and stories bring about changes in parts of the soul other than the reasoning part, which is why Plato dismisses them at 522A2-B1 as having nothing to offer the educational program described in Book VII. But it is clear that Plato believes that without the order and grace they provide, this program would have little chance of success.
Another aspect of early upbringing presupposed by the educational program of Book VII is gymnastikê, physical training (403C9-410B3), proper diet and regimen and a variety of competitive games. It is clear from 410B5-412A2 that even gymnastikê is primarily aimed at the soul; the educator's aim here is to work a change analogous to that brought about by mousikê:
The goal he aims for with this physical exercise and effort is the spirited part of [the student's] nature. This is what he wants to wake up. (410B5-6)
The educator "wakes up" the spirited part of the student's soul (egeirein; the same verb is used when Jesus tells his disciples to go out and raise the dead, Matthew 10:8). Plato's other similes for what the educator must do here include stretching and relaxing strings on a lyre (410D8-E2, 412A4-7), forging iron (411A9-B2), and feeding and starving (411C4-D5). The goal is to avoid the excessive docility and softness produced by exclusive attention to mousikê and the intractable brutishness and hardness produced by exclusive attention to gymnastikê. The idea is to enable the reasoning and spirited parts to work together in the management of the appetitive part: together the former can be "put in charge of" or "guard" or "watch over" the appetitive part (prostatein, 442A5; phulattein, 442B6; têrein, 442A7). Ideally, there should be concord and attunement among the parts (442C10-11, 589B3-4). The best blend of mousikê and gymnastikê produces the "harmonious" and "docile and orderly" temperament called wisdom-loving or philosophical (410E1-3, 411C5; see also 441E8-442A2).
By describing this temperament as philosophical, Plato is not claiming that such people have genuine understanding (epistêmê) or that by virtue of their cultural and physical training they are equipped to grasp the forms. Such training can produce habits of inner harmony and grace, not understanding, In the terms of the cave image and the discussion that follows it, mousikê and gymnastikê can help to free a chained prisoner and turn him around towards the firelight to see the objects that cast the shadows. But since neither is concerned with the reasoning part of the soul or with what is intelligible, they cannot help the prisoner out of the cave. They leave the student at the bottom of the rough steep slope, still focused on perceptible things. He has not begun the upward journey out of the cave into daylight which Plato likens to the soul's ascent to the intelligible realm (517B4-5). So, indispensable as they are, they are only the beginning of the student's educational journey.
- III -
The next stage of the upward journey, which helps the prisoner up out of the cave and into the light of day, and which enables him to see real things by the light of the sun, is a more straightforwardly cognitive process. Plato says a great deal about the cognitive dimension of education in the pages that follow the cave image. He outlines a curriculum that progresses through arithmetic, plane and solid geometry, astronomy, and harmonics (522C6-534D1). These mathematical studies occupy prospective guardians for ten years, from age twenty to age thirty (537B8-D2). Although he criticizes the ways these subjects are commonly taught, Plato insists that properly pursued they can lead the people who study them "up to the light" (521C2) and contribute to "the reorientation of a soul from a kind of twilight to true daylight" (521C6-7). It is this program of study that helps the student out of the cave into the outside world, and equips him to see and understand the things outside, the heavenly bodies, even the sun itself.
Why does mathematics play such a large role in Plato's curriculum? Scholars frequently cite Pythagorean influences, and this is no doubt part of the story. Aristotle reports that Plato "follows [the Pythagoreans] in many things" (Metaphysics 1.5, 987a30), and Pythagorean ideas figure largely in several of Plato's dialogues, including the Republic. But relatively little is known about early Pythagorean views, and so this is of limited use in understanding the role of mathematics in Plato's educational program.
The point of the mathematical curriculum is to focus the student's soul on the intelligible realm, to wean him away from reliance on and preoccupation with what is perceptible. This is clear from what leads up to the discussion of the mathematical sciences. The first thing we need in order to jar the student's soul out of its complacency are experiences that call upon the intellect (porakalein, 523A10), that force the student to think in terms of intelligible forms (anogkadzein, 524A6, C7). The clearest examples of such thought-provoking experiences are those involving number (524D7-525A5). For example, in the perceptible realm one (whole) is also many (parts); to avoid confusion, the student is forced to distinguish things as they appear to the eyes from things as they are in themselves. This is beyond the grasp of perception and requires thought and reasoning. Hence the mathematical sciences are essential to the reorientation of the student's soul, and this is why number and counting are said to have a "consummate ability to attract one towards reality" (523A2-3).
This also explains why Plato is unimpressed by the way the mathematical sciences are commonly studied. Students must be delivered from reliance on perception and made to employ purely intellectual processes (526B1-3, 537D5-8, 522C6-525B3). Any approach to the mathematical sciences that doesn't serve this end, Plato summarily rejects (see 526E1-527B11, 527D5-E2, 529A1-C3, 530E5-531C4). But a bit later on, in a somewhat more positive statement, he qualifies that rejection when he says that
engaging in all the subjects we've been discussing has some relevance to our purposes, and all that effort isn't wasted, if the work takes one to the common ground of affinity between the subjects, and enables one to work out how they are all related to one another; otherwise it's a waste of time. (531C9-D4)
The mathematical sciences are all propadeutic to a discipline Plato calls dialectic, and it is this "common ground of affinity between the subjects" that the dialectician is concerned to work out. He has a synoptic or comprehensive view of the relationships the mathematical sciences have to each other and to reality (537C1-2). The task of dialectic is to work out, with regard to all things, what each of them is in itself, to give an account of each thing (533B1-3, 534B3-4). Plato is willing to say that someone who has such a synoptic view has understanding (epistêmê) of things that the rest of us can only grasp by belief or opinion (see 601E7-602A1, 534B8-C5, 520C1-6, 402B5-C8).
How does dialectic accomplish this task? Unfortunately, Plato never answers this question directly. When Glaucon asks Socrates to "tell us the ins and outs of the ability to do dialectic, and how many different types of it there are, and what methods it employs" (532D8-E1), he is unhelpfully told that neither he nor Socrates is up to that task. There are some clues, however. The first comes with Plato's observation that dialectic "uproots" the things it takes for granted (anairein, 533C8, often translated "do away with" or "destroy"):
dialectic is the only field of inquiry... whose quest for certainty causes it to uproot the things it takes for granted in the course of its journey, which takes it towards an actual starting-point. (533C7-D1)
Recall the divided line analogy. There Plato refuses to allow that mathematicians have understanding (nous); he calls their condition "thinking" (dianoia), which is "the intermediate state between believing (doxa) and knowing (nous)" (511D2-5). Why? Because they
take for granted things like numerical oddness and evenness, the geometrical figures, the three kinds of angle, and any other things of that sort which are relevant to a given subject. They act as if they know about these things, treat them as basic, and don't feel any further need to explain them either to themselves or to anyone else, on the grounds that there is nothing unclear about them. They make them the starting-points for their subsequent investigations, which end after a coherent chain of reasoning at the point they'd set out to reach in their research. (510C2-D3)
Mathematicians deal with forms, but their investigations start too far in and take too much for granted. They treat their concepts -- what it is to be a square or diagonal and so on in themselves (510D7-511A1) -- as basic, and don't bother to explain them. As a result, Plato says, they
are evidently dreaming about reality. There's no chance of their having a conscious glimpse of reality as long as they refuse to disturb the things they take for granted and remain incapable of explaining them. For if your starting-point is unknown, and your end-point and intermediate stages are woven together out of unknown material, there may be coherence, but knowledge is completely out of the question. (533B8-C5)
Dialectic, on the other hand, uproots what it takes for granted in the course of its journey. It doesn't leave these things "undisturbed" (akinêtos, 533C2), but tries to explain or give an account of them (533C2-3, 510C6-8). It isn't that the dialectician takes nothing for granted; it is rather that, unlike the mathematicians Plato criticizes, he sees that what he takes for granted in the course of his inquiries itself requires explanation.
This is where the second clue comes in. To explain or give an account of something, whether a complex psychological state like justice or a simple object like a wagon or a shuttle or a lump of clay, is to make clear what it is. One does this by relating it to other things, saying exactly how it is like some things and unlike others. (Think of the way Plato gives an account of justice or morality in the Republic.) This very general idea is used consistently throughout Plato's dialogues, and it is made the centerpiece of discussion in late dialogues such as the Theaetetus, Sophist, and Statesman. It is not enough to give an account, however; one must defend it as well:
If someone is incapable of arguing for the separation and distinction of the form of goodness from everything else, and cannot, so to speak, fight all the objections one by one and refute them (responding to them resolutely by referring to the reality of things, rather than to people's beliefs), and can't see it all through to the end without his position suffering a fall -- if you find someone to be in this state, you'll deny that he has knowledge of goodness itself or, in general, of anything good at all. (534B8-C5)
The point here is perfectly general, and nothing hangs on the choice of goodness as an example. Platonic dialectic is the distillation of a familiar process, conversing (dialegesthai), the give and take of question and answer. Genuine understanding is articulate, or at any rate articulable: Plato is unwilling to credit anyone with understanding who cannot give and defend an account of that which he claims to understand.
This concern for account-giving and explaining what one's inquiries take for granted makes it clear why, despite his sometimes disparaging remarks about current practice, Plato is clearly impressed by the geometry of his day. Tradition has it that he placed over the door to the Academy an inscription saying "Let no one unskilled in geometry enter." What must have impressed Plato about geometry is the way in which particular truths can be related to others to form a system. Someone with genuine understanding of geometry -- someone who has really mastered the material presented in Euclid's Elements, e.g. -- knows much more than a lot of particular truths. He understands a system. Not only does he know that certain things are true and how those things depend for their truth on other things; he sees the ways in which these particular truths fit together to form a rationally ordered whole. His view of the subject matter is synoptic. The dialectician-philosopher has a similarly synoptic view, but one that encompasses not only geometry but all areas of inquiry.
Plato's conception of the position of dialectic vis-à-vis the sciences is similar to one expressed much more recently by Wilfrid Sellars, who writes that
The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term.... What is characteristic of philosophy is not a special subject-matter, but the aim of knowing one's way around with respect to the subject-matters of all the special disciplines.... It is therefore the "eye on the whole" which distinguishes the philosophical enterprise.
This ideal, at least as Plato develops it, where this synoptic understanding is supposed to reside in single individuals, may strike us as naive or unrealistic. Interestingly, in later dialogues such as the Laws, Plato himself seems much less confident about the possibility of identifying this sort of comprehensive understanding and about the advisability of putting people alleged to possess it in positions of power (see 875C6-D5, 945B3-948B2). But the underlying idea that genuine understanding involves what Sellers calls the "eye on the whole" characterizes Plato's latest dialogues as well as the Republic (see Sophist 252E9-254D5, Statesman 280A8-285C2, Philebus 17C11-19A2, Laws 875B1-D5).
- IV -
As we have seen, the purpose of Platonic education is to free the soul of the things that turn its sight downward and to reorient it towards the truth (519A8-B5). Such education is liberating. It is also liberal -- Plato insists that studies in the mathematical sciences not be "compulsory" (epanagkês), on the grounds that "compulsory intellectual work never remains in the soul" (536D5-E4). Aristotle suggests a similar distinction in Politics VIII between the "liberal sciences" (eleutheriai epistêmai) and those he calls vulgar (banausos) (Politics 1337b4-21). The latter are undertaken because they are useful and necessary; the former contribute to one's happiness by making it possible for one to do something worthwhile with one's leisure time. It is doubtful whether Plato's discussion of education in the Republic had any direct influence on Aristotle's discussion in Politics VIII, but together these views exerted a profound influence on subsequent thinking about education.
Yet the outlines of Plato's curriculum are certainly not original. The Pythagorean Archytas of Tarentum, whom Plato befriended on one of his Italian journeys, says that
[The mathematicians] have given us clear knowledge about the speed of the stars and their risings and settings, about geometry, arithmetic, sphaeric [i.e. astronomy], and last and especially music. For these studies seem to be related. (Fragment 1 in Diels-Kranz)
This fourfold division of mathematics into arithmetic, geometry, astronomy, and music theory is a Pythagorean commonplace. Aristotle says (Metaphysics 986a15-18) that the Pythagoreans believed that things are made up of numbers. But if numbers and points and lines are real constituents of the world, then arithmetic and music (concerned with numbers and their relations) and geometry and astronomy (concerned with points, lines, and figures and their relations) are essential for understanding things. Arithmetic provides the basis for music, geometry for astronomy -- and we have a rationale for a Pythagorean mathematical "curriculum." Plato adapts this rationale (and the fourfold division) to suit his own purposes, and in the generations after Plato it was codified in many Pythagorean and Neoplatonist works. Nicomachus of Gerasa, a Pythagorean compiler of the second century, distinguishes the same "four ways" (tessares methodoi) in his Introduction to Arithmetic. Around the same time the Middle Platonist Theon of Smyrna, in his handbook Mathematical Knowledge Useful for the Understanding of Plato, begins by offering the same fourfold division of the mathematical sciences. It is clear from works of Plutarch, Galen, and others that by this time the fourfold division of mathematics was hardening into a curriculum.
Later Greek and (especially) Roman authors, in efforts to produce workable and uniform courses of study for young men, distinguished between the trivium (grammar, rhetoric, and logic) and the quadrivium (arithmetic, geometry, astronomy, and music theory). In the hands of Roman authors such as Varro, Cicero, and Seneca these studies became the artes liberales, subjects suited for the education of a Roman citizen, studied not for straightforwardly practical reasons but for the sake of knowledge itself. Varro's Disciplinae, for example, composed sometime around 40 B.C., is an influential encyclopedia of the (in this case) nine liberal arts -- grammar, dialectic, rhetoric, arithmetic, geometry, astrology, music, medicine, and architecture. Later writers reclassified the last two arts as mechanical or "servile." Two later works are especially important for their influence on educational thinking. Martianus Capella's The Marriage of Philology and Mercury, composed in the first third of the fifth century, is an allegory in which the seven liberal arts (in the person of seven bridesmaids) give accounts of themselves to the guests at a heavenly wedding feast. Capella's work was enormously popular as a school text in the Middle Ages. Another of the most widely read books in the Middle Ages was Isidore of Seville's Etymologies, composed around A.D. 630. Book III contained a wealth of information -- much of it garbled or incomplete -- about the quadrivium. The popularity of Martianus and Isidore's works as school texts attests to the fact that by the seventh century A.D. the seven liberal arts had become a standard educational curriculum. They became the organizing principle even for the education of monks, as Cassiodorus's Institutiones (ca. 550) makes clear.
Most later medieval university education was constructed around these ideas, although the quality of quadrivial education varied enormously. The thirteenth-century universities of Oxford and Paris were divided into four faculties -- arts, medicine, law, and theology -- and the arts faculty were expected to provide the basis and preparation for the three "higher" faculty. At Paris, students of medicine and theology were required to have preliminary training in the arts, which involved mastery of the trivium and quadrivium. This arrangement persisted well into the Renaissance, although in practice the teaching of mathematics and science was often an empty scholastic shell of its original Pythagorean-Platonic self. But the structure of the medieval university, and of its modern and contemporary descendants, attests to the power of this Platonic educational vision.
That it is Plato's vision is acknowledged by many medieval sources. Artistic depictions of the disciplines and their relations are a good source of evidence here. The seven liberal arts are represented by seven bridesmaids in Capella's allegory, and there are any number of early medieval illuminations in which the trivial and quadrivial arts are so depicted (see the frontispiece to this issue). Plato, the architect, and Socrates, his teacher, are central -- as in our frontispiece. It is Plato who most compellingly established the curriculum that incorporates the special "arts" embodied in the figure. It is his educational vision that formed, and arguably still forms, the basis for much liberal arts education.
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