Polytropy

Index to this series

• In the fair city of Callipolis (527c), students who pass their exams will become philosophers at the age of fifty (540a).

• Callipolis itself will come to be, once philosophers seize power in an existing city and throw out everybody over the age of ten (540e).

• This is all said in play (536c), and play is what children must be allowed to do, since (536e)

  the free man ought not to learn any study slavishly. Forced labors performed by the body don’t make the body any worse, but no forced study abides in a soul.

We are reading the latter part of Book VII of Plato’s Republic (Stephanus 521c–41b). We are finishing up the longer account of the education of the aristocrats. We started this in Book VI, after completing the discussion of women and children provoked by Polemarchus at the beginning of Book V.

• The Gods
• Discipline
• To What End
• Play
• Talk
• Hypotheses
• Postulates

The Gods

The discussion came in three waves,

• the first two in the former part of Book V,
• the last one in the latter part of Book V and the former part of Book VI.

At the beginning of the latter part of Book VI then, Socrates announced (502e–3a),

what particularly concerns women and children has been completed, but what concerns the rulers must be pursued as it were from the beginning. We were saying, if you remember, that they must show themselves to be lovers of the city, tested in pleasures and pains, and that they must show that they don’t cast out this conviction in labors or fears or any other reverse. The man who’s unable to be so must be rejected, while the one who emerges altogether pure, like gold tested in fire, must be set up as ruler and be given gifts and prizes both when he is alive and after he has died.

Socrates now elaborates on that last remark with some wordplay (540b):

they go off to the Isles of the Blessed and dwell. The city makes public memorials and sacrifices to them as to demons (ὡς δαίμοσιν), if the Pythia is in accord; if not, as to happy and divine men (ὡς εὐδαίμοσί τε καὶ θείοις).

Socrates invoked Apollo of Delphi in Book IV, near the completion of the shorter account of the city in speech; to the god was left everything else to do with gods, demons, heroes, and the dead (427b). In the Second Wave in Book V, He said the priestess of Delphi might approve the mating of brother and sister (461e).

Through the metaphors of the Sun and Divided Line in the latter part of Book VI, and the Cave in the former part of Book VII, Socrates would seem to have laid out the scheme of all of reality. There are no gods, unless they coalesce in the good that the sun is the image of. Now we are told that the successful among the fifty-year-old students will look on the good itself (τὸ ἀγαθὸν αὐτό 540a), which is “that which provides light for everything.”

What then is the use of the Pythia? She may be a note of reality breaking into the fairy tale told by Socrates. It is a reality that customs don’t just disappear, once you think you have shown them to be irrational.

Discipline

The successful students will be compelled to look on the good. Thus we continue with the theme of necessity, taken up in the first part of Book VII.

The successful students may be women as well as men, as Socrates recalls (540c). Now that they are philosophers, they allowed to spend the greater part of their time in contemplation, while periodically doing their duty, back down in the Cave, as queens and kings, or rather technocrats.

This is all agreed to by Glaucon, who would seem to be much younger than fifty, though he may be about fifty-five when Socrates is finally put to death by the people of Athens.

We are at the second referent in Shorey’s index for the phrase “Philosophers must be kings.” The first was the Third Wave in the latter part of Book V. I looked at both referents when taking up the latter part of Book VI. I then called “chilling” what the philosophers would do when they came to power.

According to Socrates, “All inhabitants above the age of ten they will send out into the fields” (540e). The philosophers will have been selected and trained for this. In the first part of Book V, in the Second Wave, which covered the breeding of children like animals, Socrates said to Glaucon of their parents (466e),

they’ll carry out their campaigns in common, and, besides, they’ll lead all the hardy children to the war, so that, like the children of the other craftsmen, they can see what they’ll have to do in their craft when they are grown up.

Socrates now recalls this, asking Glaucon (537a),

“Don’t you remember,” I said, “that we also said that the children must be led to war on horseback as spectators; and, if it’s safe anywhere, they must be led up near and taste blood, like the puppies?”

“I do remember,” he said.

“Then in all these labors, studies, and fears,” I said, “the boy who shows himself always readiest must be chosen to join a select number.”

Bloom refers to a boy, but the Greek παῖς παιδός can also have feminine gender, so it is perhaps not unsuitable as a source for such words as pediatrics and pedagogy. In any case, παῖς is not used here, but the phrase “the boy who” is Bloom’s interpretation of the masculine relative pronoun ὅς.

To What End

Where is Socrates leading Glaucon, and where is Plato leading us? In the ease with which he accepts Socrates’s horrible plans, Glaucon reminds me of what I must have read in the Next Whole Earth Catalog (1980).

I am not sure where I sat to do the reading. I did not own a copy of the book, and unfortunately the Internet Archive has not got a scan of it (though it has the Last Whole Earth Catalog of 1971). What I read in the book must have been the “short story” of Ron Jones called, remarkably enough for us, “The Third Wave.”

As I read the piece now, I think the story must have been fictionalized, while being based on something: the website attributed to Jones’s students testifies to that. Teaching those students history, asked by one of them how Germans could have claimed ignorance of the Holocaust as it happened, Jones showed them. He started simply enough:

I lectured about the beauty of discipline. How an athlete feels having worked hard and regularly to be successful at a sport. How a ballet dancer or painter works hard to perfect a movement. The dedicated patience of a scientist in pursuit of an Idea … To experience the power of discipline, I invited, no I commanded the class to exercise and use a new seating posture; I described how proper sitting posture assists mandatory concentration and strengthens the will.

With discipline came – improvement, at least by Jones’s account:

Soon everyone in the class began popping up with answers and questions. The involvement level in the class moved from the few who always dominated discussions to the entire class. Even stranger was the gradual improvement in the quality of answers. Everyone seemed to be listening more intently. New people were speaking. Answers started to stretch out as students usually hesitant to speak found support for their effort.

All of that is supposed to have happened in a Monday class. At the end of the next day’s class, Jones gave the students a special greeting, naming it for the same legend that Socrates alludes to in Book V:

I called it the Third Wave salute because the hand resembled a wave about to top over. The idea for the three came from beach lore that waves travel in chains, the third wave being the last and largest of each series.

The high school was in Palo Alto. On Thursday Jones told his students how they were part of something big.

The Third Wave isn’t just an experiment or classroom activity … Across the country teachers like myself have been recruiting and training a youth brigade capable of showing the nation a better society through discipline, community, pride, and action. If we can change the way that school is run, we can change the way that factories, stores, universities and all the other institutions are run. You are a selected group of young people chosen to help in this cause.

On Friday the game was over.

There is no such thing as a national youth movement called the Third Wave. You have been used. Manipulated. Shoved by your own desires into the place you now find yourself. You are no better or worse than the German Nazis we have been studying.

You thought that you were the elect. That you were better than those outside this room. You bargained your freedom for the comfort of discipline and superiority. You chose to accept that group’s will and the big lie over your own conviction.

Socrates could say that sort of thing to Glaucon.

Play

The fifty-year education of the true guardians will culminate in dialectic. It starts with what we call mathematics, though for Socrates this is three:

1. Number theory: “logistic and arithmetic” (λογιστική τε καὶ ἀριθμητική).
2. Plane geometry.
3. Solid geometry.

After these come two more pursuits:

• Astronomy, namely solids or number seen in motion.

• What Shorey calls a counterpart; Bloom, an antistrophe, though Socrates uses not the noun ἡ αντίστροφη, but the adjective ἀντίστροφος -ον: the counterpart of astronomy is the “harmonic” (ἐναρμόνιος -ον), number heard in motion – what we call music, though Socrates gave this term a broader meaning Book II (376e), where ἡ μουσική included λόγοι – tales for Shorey, speeches for Bloom.

The question arises of who should be taught all of this (535a). Socrates says what one would like to believe, unless one is an authoritarian (536d–e):

the study of calculation and geometry and all the preparatory education required for dialectic must be put before them as children, and the instruction must not be given the aspect of a compulsion to learn.

There is however a danger in introducing the young to dialectic: they can treat it as a game, engaging in contradiction for its own sake (539c).

Talk

Regarding the mathematics (531d),

πάντα ταῦτα προοίμια ἐστιν αὐτοῦ τοῦ νόμου ὃν δεῖ μαθεῖν;

all of this is a prelude to the song itself which must be learned (Bloom);

all this is but the preamble of the law itself,
the prelude of the strain that we have to apprehend (Shorey).

Shorey thus gives us two possible interpretations of nomos, as law or song; Bloom picks the latter. The word is traced to the hypothetical Indo-European root *nem-, with meanings

• “dispense, distribute; take” (Beekes);
• “To assign, allot; also, to take” (American Heritage Dictionary).

It is very suggestive that English derivatives include not only numb and nimble, but also number, via the Latin numerus. However, the connection of this last word to the Greek νόμος seems to be only speculative. The Greek noun is derived from the verb νέμω, with meanings as are assigned to the Indo-European root, along with specializations such as “inhabit, pasture, graze.” Thus

• a νομή is a pasture,
• a person who is νομάς, -άδος is pastoral, nomadic.

A standard translation of νόμος is “custom.” I suppose the occupation of certain lands by certain peoples becomes an important custom to respect, even a reason for the establishment of justice, as in Glaucon’s account in Book II. A song or “strain” is a customary sequence of notes, customarily played on certain occasions.

The song whose prelude has been heard is dialectic, but perhaps that term in English is needlessly imposing. We could call it conversation instead. Glaucon exalts it, wanting to be told (532d)

τίς ὁ τρόπος τῆς τοῦ διαλέγεσθαι δυνάμεως,

what is the nature of this faculty of dialectic (Shorey),

what the character of the power of dialectic is (Bloom).

Socrates warns him (533a), in Bloom’s translation, numbering by me:

1. You will no longer be able to follow, my dear Glaucon, although there wouldn’t be any lack of eagerness on my part.
2. But you would no longer be seeing an image of what we are saying, but rather the truth itself, at least as it looks to me.
3. Whether it is really so or not can no longer be properly insisted on. But that there is some such thing to see must be insisted on.

I propose the following interpretation.

1. We speak of following a conversation, but strictly speaking it can only be participated in.

2. If all you do is listen – and this is almost all Glaucon is doing – then indeed all you get is an image, so to speak. The image is words that you can write down in a notebook, regardless of whether you understand them, as in a key passage of Collingwood’s first book, Religion and Philosophy (1916):

   When a man makes a statement about the nature of God (or anything else) he is interested, not in the fact that he is making that statement, but in the belief, or hope, or fancy that it is true. If then the psychologist merely makes a note of the statement and declines to join in the question whether it is true, he is cutting himself off from any kind of real sympathy or participation in the very thing he is studying – this man’s mental life and experiences.

   If you do join in the question, this involves more than quoting the other person, making an image of their words. You have to learn how things really look to the person.

3. There’s no absolute guarantee of truth in what that person sees, be he (or she) Socrates, a mathematician proving a theorem, or anybody else. We do the best we can, out of the conviction that there is a truth to be discovered. This conviction is the most general instance of what Collingwood calls an absolute presupposition.

During this ongoing reading of and blogging about the Republic, I discussed absolute presuppositions most extensively in “Nature”; such posts also constitute a so-called category of this blog – as do posts about what Collingwood says of doing psychology: here the relevant category is named for Collingwood’s phrase, “ceases to be a mind” (which is what happens to the subject of one’s investigations if one does not engage in dialectic with that person).

Hypotheses

Presently Socrates says what provokes Collingwood to write the chapter of An Essay on Metaphysics (1940) called “A Positivistic Misinterpretation of Plato.” For Socrates, there is a hierarchy:

1. The “practical” arts, let us say.
2. Mathematics.
3. Dialectics.

I add corresponding numbers to Socrates’s own words (533b-c):

1. … all the other arts are directed to human opinions and desires, or to generation and composition, or to the care of what is grown or put together.
2. And as for the rest, those that we said do lay hold of something of what is – geometry and the arts following on it – we observe that they do dream about what is; but they haven’t the capacity to see it in full awakeness so long as they use hypotheses and, leaving them untouched, are unable to give an account of them.
3. … only the dialectical way of inquiry (ἡ διαλεκτικὴ μέθοδος) proceeds in this direction, destroying the hypotheses (τὰς ὑποθέσεις ἀναιροῦσα), to the beginning itself in order to make it secure …

The bolded verb here is ἀναιρέω, compounded of ἀνά up and αἱρέω take. Euclid uses the compound ἀφαιρέω, as for the cutting off from a line segment a part equal to a shorter segment. Collingwood himself makes a hypothetical statement about what Socrates says here:

If ‘hypotheses’ are presuppositions (and there is no doubt, I think, that they are), ‘removing hypotheses’ can only mean cancelling presuppositions, that is, ceasing to presuppose them.

Thus (by Collingwood’s plausible account), in Book I, Socrates effects the cancellation of

• Polemarchus’s hypothesis that justice is a skill,
• Thrasymachus’s hypothesis that injustice is a skill.

In mathematics, on the hypothesis that a triangle is right, one can show that the square on the hypotenuse is equal to the squares on the legs. That again is Collingwood’s example. One should learn to do this kind of thing, lest one lack the maturity of the person who (539c)

will imitate the man who’s willing to discuss and consider the truth rather than the one who plays and contradicts for the sake of the game.

Apparently this person should be at least thirty years old. Otherwise there is a danger (539b):

Isn’t it one great precaution not to let them taste of arguments while they are young? I suppose you aren’t unaware that when lads (μειρακίσκοι) get their first taste of them, they misuse them as though it were play, always using them to contradict; and imitating those men by whom they are refuted, they themselves refute others, like puppies enjoying pulling and tearing with argument at those who happen to be near.

If what you are arguing about is only mathematics, presumably reality will calm you down. At least you learn, as Wilfrid Hodges says (and I have already quoted), “one has to accept some facts as given and not up for argument.”

It is not clear that this is the benefit of mathematics that Socrates has in mind. As he works it out with Glaucon, what we think of as one thing, such as a finger, can be also two, such as soft and hard. The example of a finger is Socrates’s (523c), but not the rock with respect to which the finger is soft; nor the clay, hard. Ultimately Glaucon affirms (525a),

For we see the same thing at the same time as both one and as an unlimited multitude.

ἅμα γὰρ ταὐτὸν ὡς ἕν τε ὁρῶμεν καὶ ὡς ἄπειρα τὸ πλῆθος.

This is why the guardians ought to study mathematics – this and that it is useful for war (525b)!

Postulates

It is also not clear that Plato is ready to recognize such hypotheses as Euclid’s five postulates, not to mention the postulates of Archimedes in On the Sphere and the Cylinder (translated by Reviel Netz, 2004):

/1/ That among lines which have the same limits, the straight <line> is the smallest. /2/ And, among the other lines (if, being in a plane, they have the same limits): that such are unequal, when they are both concave in the same direction and either one of them is wholly contained by the other and by the straight having the same limits as itself, or some is contained, and some it has common; and the contained is smaller …

/5/ Further, that among unequal lines … the greater exceeds the smaller by such <a difference> that is capable, added itself to itself, of exceeding everything set forth (of those which are in a ratio to one another).

Archimedes immediately draws a conclusion:

Assuming these it is manifest that if a polygon is inscribed inside a circle, the perimeter of the inscribed polygon is smaller than the circumference of the circle; for each of the sides of the polygon is smaller than the circumference of the circle which is cut by it.

This assertion involves two hypotheses: the small one that a polygon has indeed been inscribed in a circle, and the grand one that Netz numbers 1 among Archimedes’s postulates. Archimedes proceeds to assert,

If a polygon is circumscribed around a circle, the perimeter of the circumscribed polygon is greater than the perimeter of the circle.

In the ensuing demonstration, Archimedes does not appeal explicitly to Postulate 2; presumably you have committed it to memory, not as an image (a sequence of words), but as a conviction.

Would Plato want this postulate to be somehow “removed,” whether by being somehow confirmed or refuted? Gödel has shown us that our mathematics cannot establish its own consistency: can we credit Plato for predicting this?