Polytropy

Constituting the latter part of Book I, the second of the Republic readings features the only sustained contribution of Thrasymachus, who argues that, if it can be pursued perfectly, injustice is superior to justice.

According to my electronic search of Jowett’s translation, confirmed with a search of Bloom’s, Socrates will mention Thrasymachus and his argument in Books II, VI, VIII, and IX, but the man himself will speak again only near the beginning of Book V, to agree with the demand of Polemarchus, Adeimantus, and Glaucon that Socrates explain the community of women and children in the imaginary city that Socrates will have been describing.

Below I shall review the current reading in the following sections.

• The objection of Thrasymachus (336b–8b)
• How Thrasymachus defines justice (338c–9e)
• Polemarchus and Cleitophon on the dilemma of Thrasymachus (340a–c)
• The possibility of error (340c–2e)
• Sheep and shepherd (343a–4c)
• Rule as chore (344d–7a)
• Debate or dialectic (347a–8b)
• Justice as virtuous and wise (348b–50d)
• Justice as everything else (350d–end)

Socrates shows to my satisfaction that the argument of Thrasymachus is incoherent. Although Thrasymachus agrees verbally with the refutation, Socrates knows he is not really convinced.

Why should he be? How can a verbal argument change an opinion? Mathematics shows that such change is possible. It can still be difficult.

I see the difficulty in memories of my own development. The memories are called up by the words of Wilfrid Hodges in his article “An Editor Recalls Some Hopeless Papers” (The Bulletin of Symbolic Logic, Mar., 1998, Vol. 4, No. 1, pp. 1–16). You can obtain the article from the author’s website, as an appendix to the slides of a talk called “Some things a logician would like to know about human reasoning” (August 2001). Hodges observes of Cantor’s diagonal argument (namely “the one proving that the set of real numbers and the set of natural numbers have different cardinalities”),

This argument is often the first mathematical argument that people meet in which the conclusion bears no relation to anything in their practical experience or their visual imagination. Compare it with two other simple facts of cardinal arithmetic. First, m × n = n × m. We can see what this amounts to by thinking of a rectangle with one side of length m and one side of length n. The picture points to the right formal argument when m and n are finite, and exactly the same argument works when they are infinite. Or second, 1 + ω = ω. We don’t meet ω [namely the set {0, 1, 2, …} of natural numbers] in our everyday life, but we can see how to prove the inequality by moving each number along by one …

But then we come to Cantor’s result, and all intuition fails us …

I resisted Cantor’s argument myself, when I read about it in some popular book in high school. My intuition failed to grant me the possibility that one infinite set could really be larger than another in Cantor’s sense. Also Cantor was denying me the possibility of finding a one-to-one correspondence between any two infinite sets. I did not like being told that I could not do something.

In our Republic reading now, Socrates denies Thrasymachus the possibility of finding happiness through seizing absolute power in a city.

The objection of Thrasymachus (336b–8b)

I said that we would hear from Thrasymachus again only in Book V. He will say there (450b):

χρυσοχοήσοντας οἴει τούσδε νῦν ἐνθάδε ἀφῖχθαι, ἀλλ᾽ οὐ λόγων ἀκουσομένους;

do you suppose this company has come here to prospect for gold and not to listen to discussions? (Shorey)

do you suppose these men have come here to look for fool’s gold and not to listen to arguments? (Bloom)

Both translators explain that the verb χρυσοχοέω means literally to smelt ore to extract gold. Bloom says engaging in this would be a wild goose chase. Shorey says only, “The expression was proverbial and was explained by an obscure anecdote.” He does give references, one being to “commentators on Herodotus iii. 102.” The cited passage is about the harvesting of gold from the desert sand that is dug up by the ants of India, which are “not as big as dogs but bigger than foxes.”

It could be that, in Book V, Thrasymachus will be referring ironically to the current reading. He makes here a bold entrance, complaining about, among other things, how Socrates and Polemarchus have been deferring to one another. Letting A stand for this deferring, I analyze Socrates’s response as a command of the form “Do not suppose that B would imply not-A, but that C and A are true” (336e):

μὴ γὰρ δὴ οἴου,

• εἰ μὲν χρυσίον ἐζητοῦμεν,
  • οὐκ ἄν ποτε ἡμᾶς ἑκόντας εἶναι ὑποκατακλίνεσθαι ἀλλήλοις ἐν τῇ ζητήσει καὶ διαφθείρειν τὴν εὕρεσιν αὐτοῦ,
• δικαιοσύνην δὲ ζητοῦντας, πρᾶγμα πολλῶν χρυσίων τιμιώτερον,
  • ἔπειθ᾽ οὕτως ἀνοήτως ὑπείκειν ἀλλήλοις καὶ οὐ σπουδάζειν ὅτι μάλιστα φανῆναι αὐτό.

Shorey preserves the logic:

For you surely must not suppose

• that while if our quest were for gold
  • we would never willingly truckle to one another and make concessions in the search and so spoil our chances of finding it, yet
• that when we are searching for justice, a thing more precious than much fine gold,
  • we should then be so foolish as to give way to one another and not rather do our serious best to have it discovered.

Bloom “simplifies” the logic, or violates it, for the sake of readability:

• If we were searching for gold
  • we would never willingly make way for one another and ruin our chances of finding it;

so don’t suppose that

• when we are seeking for justice, a thing more precious than a great deal of gold,
  • we would ever give in to one another and not be as serious as we can be about bringing it to light.

The dialogue of Socrates and Polemarchus has been annoying Thrasymachus in two ways:

1. Socrates avoids saying what he thinks justice is.
2. The account that he induces Polemarchus to give is misguided.

Therefore Thrasymachus makes two demands:

1. That Socrates say what justice is.

2. That he say it properly:

   And see to it you don’t tell me that it is the needful, or the helpful, or the profitable, or the gainful, or the advantageous; but tell me clearly and precisely what you mean (Bloom).

   And don’t you be telling me that it is that which ought to be, or the beneficial or the profitable or the gainful or the advantageous, but express clearly and precisely whatever you say (Shorey).

   καὶ ὅπως μοι μὴ ἐρεῖς ὅτι τὸ [336δ] δέον ἐστὶν μηδ᾽ ὅτι τὸ ὠφέλιμον μηδ᾽ ὅτι τὸ λυσιτελοῦν μηδ᾽ ὅτι τὸ κερδαλέον μηδ᾽ ὅτι τὸ συμφέρον, ἀλλὰ σαφῶς μοι καὶ ἀκριβῶς λέγε ὅτι ἂν λέγῃς.

Socrates understands either that Thrasymachus really wants to speak for himself, or that he can be goaded into doing so. When Socrates cannot accept Thrasymachus’s restriction, Thrasymachus boasts that he can.

He also suggests that Socrates will deserve a punishment for not being able to. Socrates agrees: the punishment of learning. Thus Socrates rejects the idea that punishment is something bad.

Thus Socrates disagrees with a common understanding of punishment, by the account of Gregg D. Caruso, in dialogue with Daniel C. Dennett in “On free will: Daniel Dennett and Gregg Caruso go head to head,” in Aeon:

Consider, for example, the various justifications one could give for punishing wrongdoers. One justification, the one that dominates our legal system, is to say that they deserve it. This retributive justification for punishment maintains that punishment of a wrongdoer is justified for the reason that he/she deserves something bad to happen to them just because they have knowingly done wrong. Such a justification is purely backward-looking. For the retributivist, it is the basic desert attached to the criminal’s immoral action alone that provides the justification for punishment … free-will skeptics typically point out that the impositions of sanctions serve purposes other than punishment of the guilty: it can also be justified by its role in incapacitating, rehabilitating and deterring offenders.

The speaker here is a professional philosopher who describes himself as “a free-will skeptic.” I might have expected Caruso to have learned from Plato, if nowhere else, that a just punishment is not something bad, but something good. It is precisely what achieves the “rehabilitating” that Caruso refers to. The process may be painful, just as education can be.

If Thrasymachus can give the better account of justice, he wants Socrates to pay a fine. In an echo or foreshadowing of the Apology, Glaucon offers to pay. Socrates himself can offer only praise.

How Thrasymachus defines justice (338c–9e)

Socrates does not offer praise after Thrasymachus says,

I affirm that the just is nothing else than the advantage of the stronger.

φημὶ γὰρ ἐγὼ εἶναι τὸ δίκαιον οὐκ ἄλλο τι ἢ τὸ τοῦ κρείττονος συμφέρον.

Socrates has a question, an objection, and another question:

1. Polydamas the pancratiast is stronger than we are, and beef is advantageous for him; does that mean beef would be advantageous for us?
2. The term “advantage” is one of those that Thrasymachus forbade Socrates to use.
3. What is advantageous when the stronger make mistakes?

Polydamas is supposedly son of Nicias, whose son Niceratus was in the company with Polymarchus that detained Socrates and Glaucon; but I see no scholarly assertion that the two sons were brothers. Shorey does have a note to the passage about Polydamas, but the note refers only to the beef-eating:

The unwholesomeness of this diet for the ordinary man proves nothing for Plato’s alleged vegetarianism. The Athenians ate but little meat.

Perhaps some people read the Republic the way Osama bin Laden would the Quran, as the crudest kind of guide to life. In an excerpt from her memoir in the Guardian (12 July 2004), the sister-in-law of bin Laden recalls the treatment of his infant son during a visit to Taif:

On one very hot day, Abdallah commenced howling, and kept it up for hours. He was thirsty and Najwah kept trying to feed him water with a teaspoon, but it was obvious he was far too small to manage to drink properly from a spoon. Najia was gulping water from her baby bottle constantly and I offered it to Najwah.

“Take it, he’s thirsty,” I told her, but Najwah wouldn’t take the bottle. She was almost crying herself. “He doesn’t want the water,” she kept saying. “He won’t take the spoon.”

My mother-in-law, known to me always as Om Yeslam, had to explain that Osama didn’t want the baby to use a bottle. (Following the Koran, Islamic scholars lay firm emphasis on the mother’s duty to breastfeed their children; bottle-feeding is regarded by some Muslim people as a decadent western practice.)

Thus Carmen bin Laden. For a Westerner though, not just being vegetarian yourself, but forcing your children to be fruitarian, would make you like Osama. In the former part of Book V, the Republic will impose on children the discipline of not knowing their parents.

Meanwhile, Thrasymachus spells out the meaning of the qualification of “advantage” in his definition of justice. Whether the state be tyranny, democracy, or aristocracy, it has a ruling class, and justice is the prevailing of their word.

Shorey compares this with Pascal’s commodité du souverain. Pascal lists this definition with two others in the ninth fragment of la liasse Misère. People say variously that justice is

• the authority of the legislator,
• the interest of the sovereign,
• present custom.

Pascal either chooses the last definition, prescriptively, or judges it to be the best, descriptively. Here are his own words, in Sellier 94, Lafuma 60, Brunschvicg 294, which is in the first Pascal reading on this blog:

Il y a sans doute des lois naturelles, mais cette belle raison corrompue a tout corrompu …

De cette confusion arrive que

• l’un dit que l’essence de la justice est l’autorité du législateur,
• l’autre la commodité du souverain,
• l’autre la coutume présente.

Et c’est le plus sûr. Rien, suivant la seule raison, n’est juste de soi, tout branle avec le temps. La coutume fait toute l’équité, par cette seule raison qu’elle est reçue. C’est le fondement mystique de son autorité; qui la ramènera à son principe l’anéantit. Rien n’est si fautif que ces lois qui redressent les fautes. Qui leur obéit parce qu’elles sont justes, obéit à la justice qu’il imagine, mais non pas à l’essence de la loi, elle est toute ramassée en soi. Elle est loi et rien davantage.

You should obey the established law, or custom, not because you judge it to be just, but just because it has been established. Perhaps Pascal thinks that goes for the sovereign as well as the subject. Then Thrasymachus disagrees. He does agree that justice is simple, and this is why he makes fun of Socrates for treating it as something deep. Obviously justice means obeying the laws promulgated by the ruling class of one’s city. But this will turn out to be too simple too.

Polemarchus and Cleitophon on the dilemma of Thrasymachus (340a–c)

Polemarchus and Cleitophon debate what Thrasymachus has said. The former thinks Thrasymachus has made two contradictory propositions:

• It is just to obey the stronger, even though they may mistakenly command what is disadvantageous to them.
• The just is the advantage of the stronger.

Cleitophon wants to resolve the contradiction by redefining what is advantageous to the rulers as what seems to them to be so. Polemarchus objects that Thrasymachus did not do this.

It doesn’t matter, says Socrates, quite sensibly; Thrasymachus is allowed to clarify or change his meaning.

The possibility of error (340c–2e)

Having granted the possibility that rulers can go wrong, Thrasymachus is lost. Either the stronger is not always the better or more deserving, or else it is not always clear who is stronger in the first place.

Thrasymachus effectively takes the latter option by bringing in the example of an art, as Socrates is wont to do. The doctor is not truly being a doctor when he makes a mistake.

Thrasymachus must now admit to Socrates that neither is the doctor truly being a doctor when he is trying to make money.

An art as such has no defect. Therefore any advantage it seeks is only for what it governs, as medicine governs the body.

Therefore the true ruler never considers his own advantage in place of that of his subjects.

Sheep and shepherd (343a–4c)

Thrasymachus says that’s right, with an adjustment. The just person serves the advantage of somebody else, namely the person who is stronger and rules, as the shepherd does over the sheep.

Apparently Thrasymachus wants to forestall the observation that being unjust may get you punished. With perfect injustice, you get only the best of everything.

Rule as chore (344d–7a)

Thrasymachus thinks rulers of cities get desirable things, and this means it’s good to be a ruler.

On the contrary, says Socrates, it’s bad, or at least unpleasant. Moreover, the art of ruling is different from the art of money-making. Thus the art of ruling does not provide for its own benefit, and this is why those who are ruled must give their rulers compensation, in the form of money, honor, or forgiveness of the penalty of not ruling.

Debate or dialectic (347a–8b)

Glaucon does not understand this penalty. Socrates explains that it is having to be ruled by worse men than oneself. In a city of good men, the citizens would fight not to rule.

Glaucon is like Thrasymachus in sticking with what he believes, regardless of the argument; for despite all of the goods that come from being perfectly unjust, Glaucon chooses justice.

How is the question to be settled of whether justice or injustice is preferable? There are two options.

1. In a debate between the two sides, a third party would have to judge who has the better argument.
2. Proper dialectic (but this word is not used) leads naturally to agreement.

I note that mathematical questions are not settled by debate; there’s no “going head to head” as Dennett and Caruso supposedly did for the Aeon article that I quoted earlier.

Glaucon chooses dialectic.

Justice as virtuous and wise (348b–50d)

For Thrasymachus, perfect injustice is more profitable than justice, and the perfectly unjust are good and prudent. Perfection here is an important qualification:

You, perhaps, suppose I am talking about cutpurses. Now, such things, too, are profitable, when one gets away with them; but they aren’t worth mentioning compared to those I was just talking about.

Thrasymachus recognizes that he cannot always get what he wants. One reason may be the threat of the institutions of justice, but he imagines that this threat is external to his desires and can in principle be removed. This is why he can accept the proposition that Socrates offers him, that injustice is “in the camp of virtue and wisdom.”

Thrasymachus is thus completely reversing “customary usage,” and according to Socrates,

you seem really not to be joking now, but to be speaking the truth as it seems to you.

Thrasymachus neither confirms nor denies. But now Socrates makes the argument that causes Thrasymachus to blush, even though, according to Shorey at least, the argument is fallacious.

The key verb will be πλέον ἔχω or πλεονεκτέω. Beekes confirms in his etymological dictionary that the latter form comes from the former. The root sense then is of having more. Bloom translates the verb as “to get the better of”; for Shorey, it means “overreaching, getting more than your share” (strictly he’s talking about the noun πλεονεξία). The unjust person wants to overreach with everybody; the just, only with the unjust. In this way, the just person resembles the skilled person. The skilled person is good and wise, and therefore – says Socrates – so is the just person.

The first element of πλέον ἔχω / πλεονεκτέω is from πλείων, which Beekes derives from the Indo-European root *pleh₁- “full,” but which is construed as the comparative of πολύς from *p(e)lh₁-u- “many.” The Greek words give such English words as “pleonastic” and “polygon” respectively. The online American Heritage Dictionary Indo-European Roots Appendix lists these words under the single root *pelə-¹, along with “full” and “fulfil” and, through Latin, “plenum” and “replete.” In the older edition of the appendix in the Grolier International Dictionary (1981), the Indo-European root is given as *pel-⁸.

To see the argument of Socrates in more detail, let us look at what Thrasymachus respectively denies and affirms:

ὁ δίκαιος τοῦ δικαίου δοκεῖ τί σοι ἂν ἐθέλειν πλέον ἔχειν; (349b)

Do you think the just man would want to overreach or exceed another just man? (Shorey)

In your opinion would the just man be willing to get the better of the just man in anything? (Bloom)

οὐκοῦν καὶ ἀδίκου γε ἀνθρώπου τε καὶ πράξεως ὁ ἄδικος πλεονεκτήσει καὶ ἁμιλλήσεται ὡς ἁπάντων πλεῖστον αὐτὸς λάβῃ; (349c)

Then the unjust man will overreach and outdo also both the unjust man and the unjust action, and all his endeavor will be to get the most in everything for himself. (Shorey)

Then will the unjust man also get the better of the unjust human being and action, and will he struggle to take most of all for himself? (Bloom)

Here is the conclusion (350c):

1. ἔοικεν ἄρα, ἦν δ᾽ ἐγώ, ὁ μὲν δίκαιος τῷ σοφῷ καὶ ἀγαθῷ, ὁ δὲ ἄδικος τῷ κακῷ καὶ ἀμαθεῖ. κινδυνεύει.

2. ἀλλὰ μὴν ὡμολογοῦμεν, ᾧ γε ὅμοιος ἑκάτερος εἴη, τοιοῦτον καὶ ἑκάτερον εἶναι. ὡμολογοῦμεν γάρ.

3. ὁ μὲν ἄρα δίκαιος ἡμῖν ἀναπέφανται ὢν ἀγαθός τε καὶ σοφός, ὁ δὲ ἄδικος ἀμαθής τε καὶ κακός.

In Shorey’s version:

“Then the just man is like the wise and good, and the unjust is like the bad and the ignoramus.” “It seems likely.”

“But furthermore we agreed that each is such as that to which he is like.” “Yes, we did.”

“Then the just man has turned out on our hands to be good and wise and the unjust man bad and ignorant.”

Thus there is a syllogism:

1. The just man is like the wise and good; the unjust, the bad and ignorant.

2. That to which each thing is similar, each thing is such as.

3. Therefore the just man is such as the good and wise; the unjust, the ignorant and bad.

Implicitly, to be like something and to be similar to it are the same thing. Also, to be such as the good and wise is to be good and wise. Shorey himself abstracts a general principle, to which he objects. Here is his note on Socrates’s original suggestion (to which Thrasymachus agrees) of the major premiss, τοιοῦτος ἄρα ἐστὶν ἑκάτερος αὐτῶν οἷσπερ ἔοικεν “Then each is such as that to which he is like” (349d):

The assumption that a thing is what it is like is put as an inference from Thrasymachus’s ready admission that the unjust man is wise and good and is like the wise and good. Jevons says in “Substitution of Similars”; “Whatever is true of a thing is true of its like.” But practical logic requires the qualification “in respect of their likeness.” Socrates, however, argues that since the good [just] man is like the good craftsman in not overreaching, and the good craftsman is good, therefore the just man is good. The conclusion is sound, and the analogy may have a basis of psychological truth; but the argument is a verbal fallacy.

I have corrected the Perseus version to agree with the print version. Shorey’s concern seems to be that if we accept Socrates’s argument, then we should accept an argument such as the following, that the bad craftsman is good, since he resembles the good craftsman in being a craftsman. However, we are not doing mathematics. In particular, Thrasymachus has already denied that the bad craftsman is a craftsman at all.

It is true that, in mathematics, by some definitions, a partial ordering may not be an ordering. However, this example does not parallel the assertion that the bad craftsman is not a craftsman. Every ordering is a partial ordering, although presumably not every craftsman is a bad craftsman.

Justice as everything else (350d–end)

Justice is now virtue and wisdom. Questions remain, the first one being whether injustice can still be mighty or strong (ἰσχυρός, ά, όν). Thrasymachus could make a speech, but Socrates would disallow it, and so Thrasymachus will only answer Socrates’s questions.

Strength needs cooperation, even within a single person; and this requires justice.

Everything (such as a horse or an eye) has a proper work and therefore a proper virtue. The work of the soul is living, and doing this work well requires the virtue of justice. Thus the life of the just man will be good, happy, and blessed.

Therefore, finally, injustice is never more profitable than justice.

This section is the longest of those that I have made of this reading; however, the argument seems to be the easiest. I think Collingwood uses the same argument to dispel such a dream as Thrasymachus has, that it would be great to be perfectly unjust. The following is from Collingwood’s first published work, “The Devil,” in a compilation called On Prayer (1916):

The Devil is generally regarded as being not only entirely bad, but the cause of all evil: the absolute evil will, as God is the absolute good will. But a very little reflexion shows that this is impossible. Good cannot contradict good, just as truth cannot contradict truth; but two errors may conflict, and so may two crimes. Two good men can only quarrel in so far as their goodness is fragmentary and incomplete; but there is no security that two absolutely bad men would agree. The reverse is true; they can only agree so far as they set a limit to their badness, and each undertakes not to thwart and cheat the other. Every really good thing in the world harmonises with every other; but [463] evil is at variance not only with good but with other evils. If two thieves quarrel over their plunder, a wrong is done whichever gets it, but no one Devil can will both these wrongs. The idea of a Devil as a person who wills all actual and possible evil, then, contradicts itself, and no amount of psychological evidence or mythological explanation can make it a conceivable idea.

Socrates himself says in the Phaedrus (255b):

οὐ γὰρ δήποτε εἵμαρται κακὸν κακῷ φίλον
οὐδ᾽ ἀγαθὸν μὴ φίλον ἀγαθῷ εἶναι.

For it is allotted that neither bad ever be friend to bad,
nor good not be friend to good.

Harold North Fowler’s translation in the Loeb edition turns the double negative into a positive:

For it is a law of fate that evil can never be a friend to evil
and that good must always be a friend to good.

Collingwood will allude to the same argument, twenty-six years after “The Devil,” in The New Leviathan (1942):

41. 76. For barbarism implies not only a quarrel between any barbarist and any civilized man; it also implies a quarrel between anyone barbarist and any other; and that any state of harmony between them is merely this quarrel suspended.

Thus barbarism fails in the long run. Of course, we all die in the long run too.