The post below is a way to record a passage in the Euthyphro where Socrates say something true and important about mathematics. The passage is on a list of Platonic passages that I recently found, having written it in a notebook on May 23, 2018. The other passages are in the Republic; here they are, for the record, with some indication of why they are worth noting (translations are Shorey’s, originally from 1930 and 1935 in the old Loeb edition):

  1. “Glaucon … said … ‘do you agree that there is a kind of good which we would choose to possess, not from desire for its after effects, but welcoming it for its own sake? …’ ‘I recognize that kind,’ said I” (357B). I have recognized learning, as at the Nesin Mathematics Village, as being of such a kind.

  2. “ ‘Have you ever noticed this, that natural reckoners (λογιστικοί) are by nature quick in virtually all their studies (μαθήματα)? And the slow, if they are trained and drilled in this … all improve and become quicker than they were?’ ‘It is so,’ he said” (526B). Whether this is true or not, it may be generally believed, and it is propaganda for mathematics.

  3. “ ‘Our present mistake,’ said I, ‘and the disesteem that has in consequence fallen on philosophy are, as I said before [495 C ff.], caused by the unfitness of her associates and wooers …’ ” (535 C). In writing “On Gödel’s Incompleteness Theorem,” I expressed my dismay over the attitude (and attendant misunderstanding) of one contemporary professional philosopher.

  4. “ ‘Now, all this study of reckoning (λογισμοί) and geometry (γεωμετριοί) and all the preliminary studies that are indistinguishable preparation for dialectics must be presented to them while still young, not in the form of compulsory instruction.’ ‘Why so?’ ‘Because,’ said I, ‘a free soul ought not to pursue any study slavishly; for while bodily labours performed under constraint do not harm the body, nothing that is learned under compulsion stays with the mind’ ” (536D–E). Orwell may not necessarily disagree when he writes in “ ‘Such, such were the joys …,’ ” “I doubt whether classical education ever has been or can be successfully carried on without corporal punishment. The boys themselves believed in its efficacy.” I considered this passage in §4 of “Abscissas and Ordinates” in the Journal of Humanistic Mathematics (January, 2015). I have admired Pirsig’s argument against compulsory education, but I wrote about a third way in “All You Need Is Love.”

Now I turn to the Euthyphro, which is assigned the alternative title On Holiness. The dialogue’s Wikipedia article may reflect its reception in contemporary philosophy departments; my concerns may differ.

Politics may be different, but you cannot come to blows over mathematics. Ayşe points this out in her courtroom statement:

… matematikte kavga edemezseniz, varsa hatayı söylersiniz, karşınızdaki düzeltebilirse düzeltir, düzeltemezse iddiasını geri çeker, ya da sizin hata dediğiniz şey hata değildir, bu sefer de siz hatalı olduğunuzu kabul edersiniz.

… you cannot fight in mathematics: if you see an error, you say so; if the other person can correct it, they do so; if they cannot, they withdraw their claim; or perhaps what you called a mistake was not, and then you accept that you were in error.

Socrates makes a similar point, in the dialogue of Plato called the Euthyphro (7B):

ἆρ᾽ ἂν εἰ διαφεροίμεθα ἐγώ τε καὶ σὺ περὶ ἀριθμοῦ ὁπότερα πλείω, ἡ περὶ τούτων διαφορὰ ἐχθροὺς ἂν ἡμᾶς ποιοῖ καὶ ὀργίζεσθαι ἀλλήλοις, ἢ ἐπὶ λογισμὸν ἐλθόντες περί γε τῶν τοιούτων ταχὺ ἂν ἀπαλλαγεῖμεν;

If you and I were to disagree about number, for instance, which of two numbers were the greater, would the disagreement about these matters make us enemies and make us angry with each other, or should we not quickly settle it by resorting to arithmetic?

“Of course we should,” says Euthyphro. Of course.

The translator here is Harold North Fowler, in the Loeb edition of Plato, Volume I, first published in 1914. (My printing is from 1982; a new edition, with new translators, was published in 2017.) Jowett’s earlier translation (reprinted in the Plato volume of the Great Books of the Western World, 1952) is similar:

Suppose for example that you and I, my good friend, differ about a number; do differences of this sort make us enemies and set us at variance with one another? Do we not go at once to arithmetic, and put an end to them by a sum?

The word sum seems to be Jowett’s interpolation. What Jowett and Fowler translate as arithmetic is in Greek λογισμός, for which the first meaning given by Liddell and Scott is counting; their first quotation is from Thucydides (4.122), about counting days to determine whether a city revolted against the Athenians before or after they concluded an armistice with the Spartans.

The word λογισμός is evidently connected to λόγος and thus to our logic as well as words like epistemology. Further back, there is a connection to the Latin origin of the stem of collecting. This last word itself describes the supposed meaning of the Indo-European root *leg-. A meaning that developed from collecting is speaking, which is part of the meaning of recounting and telling. Another meaning of telling is counting. As William Strunk says, in a passage emphasized by E. B. White (and which nicely uses the subjunctive mood):

Vigorous writing is concise … This requires not that the writer make all sentences short or avoid detail and treat subjects only in outline, but that every word tell.

Every word should tell; every word should count.

We never fight over which of two numbers is greater; we just count them up and see. So says Socrates, and Euthyphro agrees. One may however wonder whether Socrates himself, or Plato, has a counterargument in mind.

Some of us do argue over which of two numbers is greater. They could be numbers of persons attending different American Presidential inaugurations. Arguing may be appropriate here. At least, refusal to agree would be appropriate, when a President asserts, contrary to the evidence, that his inauguration was the biggest ever. Unfortunately his people have parroted the lie, as if accepting the dictum that 2 + 2 = 5.

The rhetorical point of Socrates remains sound. Everybody will agree that there are things that cannot be argued about, because there is a clear method for resolving disputes. Socrates gives also the example of comparing objects by weighing them (7C).

There are however matters over which people contend bitterly (7C–D):

περὶ τίνος δὲ δὴ διενεχθέντες καὶ ἐπὶ τίνα κρίσιν οὐ δυνάμενοι ἀφικέσθαι ἐχθροί γε ἂν ἀλλήλοις εἶμεν καὶ ὀργιζοίμεθα; ἴσως οὐ πρόχειρόν σοί ἐστιν, ἀλλ᾽ ἐμοῦ λέγοντος σκόπει εἰ τάδε ἐστὶ τό τε δίκαιον καὶ τὸ ἄδικον καὶ καλὸν καὶ αἰσχρὸν καὶ ἀγαθὸν καὶ κακόν.

But about what would a disagreement be, which we could not settle and which would cause us to be enemies and be angry with one another? Perhaps you cannot give an answer offhand; but let me suggest it. Is it not about right and wrong, and noble and disgraceful, and good and bad?

Euthyphro agrees with Socrates here. Socrates takes him to have agreed with more: that the topics mentioned are the only ones that cause persons to become enemies. Since the gods are sometimes at odds with one another, as we see in Homer and Hesiod, they must therefore disagree about what is good and bad. Since Euthyphro has said that the holy is dear to the gods, and the unholy not, the same thing can be both holy and unholy.

For Socrates, this is a problem, because Euthyphro has given an affirmative answer to the question (5D),

ἢ οὐ ταὐτόν ἐστιν ἐν πάσῃ πράξει τὸ ὅσιον αὐτὸ αὑτῷ, καὶ τὸ ἀνόσιον αὖ τοῦ μὲν ὁσίου παντὸς ἐναντίον, αὐτὸ δὲ αὑτῷ ὅμοιον καὶ ἔχον μίαν τινὰ ἰδέαν κατὰ τὴν ἀνοσιότητα πᾶν ὅτιπερ ἂν μέλλῃ ἀνόσιον εἶναι;

Is not holiness always the same with itself in every action, and, on the other hand, is not unholiness the opposite of all holiness, always the same with itself and whatever is to be unholy possessing some one characteristic quality?

We should not be so ready to agree. According to Fowler’s Introduction,

The purpose of the [Euthyphro] is in part to inculcate correct methods of thinking, more especially the dialectical method …

Instruction in methods of thinking may perhaps seem needless to modern readers; even they, however, may find it interesting, and in Plato’s times it was undoubtedly necessary.

When recognized as having been written on the eve of what would turn out to be the Great War, such progressive words become naïve or foolish, unless intended ironically. Every child born on this planet will need lessons in thinking, though many of the lessons may not be explicitly delivered. When a young couple spend their time staring silently at their mobiles, like the pair next to me in the food court of the local shopping mall the other day, perhaps their baby is not getting a good lesson.

Does Fowler agree with Socrates that holiness must be some one thing, and unholiness some one other thing? If we are Moderns, we do not agree with Socrates, at least by Collingwood’s account in The New Leviathan (1942); for we have adopted the principle of the limited objective.

31. 62. Ancient sciences aimed at an unlimited objective. They defined their aims by asking questions like: ‘What is Nature?’ ‘What is Man?’ ‘What is Justice?’ ‘What is Virtue?’ A question of this sort was to be answered by a definition of the thing. From this definition, which had to state the ‘essence’ of the thing defined, implications could be derived, each implication being the statement of some ‘property’.

To seek an “essence” was to try to answer a question of the form, “What is x?”

31. 67. To a question in this form, for example: ‘What is Nature?’ modern science answers: ‘I do not know. What the essence of nature is nobody knows, and nobody need care. When they asked that question the Greeks were asking a question too vague to be precisely answered.’

31. 68. Limit your objective. Take time seriously. Aim at interpreting not, as the Greeks did, any and every fact in the natural world, but only those which you think need be interpreted, or can be interpreted (the two things are not, after all, so very different); now, choose where to begin your attack. Select the problems that call for immediate attention. Resolve to let the rest wait.

Socrates’s problem is that Meletus has indicted him for impiety (5C). Euthyphro is indicting his own father for murder: the murder of Euthyphro’s hired man, who got drunk and murdered one of the enslaved men of the family. Euthyphro’s father tied up the killer and threw him in a ditch, then sent to Athens for advice on what to do next. Meanwhile the man died of exposure.

Euthyphro is progressive, like Fowler perhaps. Most persons would not charge kin with murder of non-kin. Socrates is like most persons here, but Euthyphro is not (4B).

It is ridiculous, Socrates, that you think it matters whether the man who was killed was a stranger or relative, and do not see that the only thing to consider is whether the action of the slayer was justified or not, and that if it was justified one ought to let him alone, and if not, one ought to proceed against him, even if he share one’s hearth and eat at one’s table.

Euthyphro recognizes the principle of equality before the law, and he is going to stick with it. On September 24, 2016, I was applauded for quoting the Universal Declaration of Human Rights on this principle, during my talk about Thales in the ruined Roman theater of Miletus. Persons or things can be equal without being the same, and this is important for ancient Greek mathematics, as in some theorems that are attributed to Thales:

  1. A diameter divides a circle into two equal halves.
  2. The base angles of an isosceles triangle are equal.
  3. Intersecting straight lines create equal vertical angles.

For Euthyphro, respecting the principle of equality is piety. It does not matter that his father was himself trying to do the pious thing by sending to Athens for an ἐξηγητής, an exegete, somebody whom Liddell and Scott define as an “expounder, interpreter … at Athens, of sacred rites or customs, modes of burial, expiation, etc., spiritual director.” Cited passages illustrating this meaning are from the Euthyphro itself.

In going after his father, Euthyphro is a man of principle. He has also tradition on his side: the mythical tradition whereby Cronus castrated Uranus (thus giving birth to Aphrodite, as recalled incidentally in “Antitheses”), and Zeus imprisoned Cronus.

The tradition would seem to continue, here in the land from which the Greek gods were “stolen”—by the perverse account of a young tour guide in Antalya, whose ideas may have had some tenuous connection to the Blue Anatolia Thesis (Mavi Anadolu Tezi). Not finding a good source for the Thesis in English, I quote and translate words of Murat Belge (Birikim, October 2006):

Önce “tez”den başlayalım: Halikarnas Balıkçısı’nın (Cevat Şakir) işlediği kadarıyla bu tez büyük ölçüde Batı Anadolu’yla, İonia ile sınırlıdır. Cevat Şakir, İonia ile bugünkü Yunanistan arasında büyük kültürel-entellektüel farklar olduğuna inanmıştır. Ona göre, Tales’ten Demokritos’a maddeci Yunan felsefesi İonia’nın ürünüdür, Anadolu’da yaşayan filozoflar tarafından geliştirilmiştir. Ama bu demokratik felsefe, Yunan karasında Sokrates ve Platon’un elinde idealist ve totaliter bir felsefeye dönüştürülmüştür.

First let’s begin with the “thesis”: as far as the Fisherman of Halicarnassus (Cevat Şakir) worked it out, it is mainly limited to Western Anatolia, to Ionia. Cevat Şakir believed there were great cultural-intellectual differences between Ionia and what is today Greece. According to him, materialist Greek philosophy, from Thales to Democritus, is a product of Ionia, developed by philosophers living in Anatolia. But this democratic philosophy was turned into an idealist, totalitarian philosophy on Greek soil, in the hand of Socrates and Plato.

Freya Stark addresses the same geographical distinction in Ionia: A Quest (1954; Tauris Parke, 2010), which I quoted at greater length towards the end of the Thales article:

Curiosity ought to increase as one gets older … Whatever it was, the Ionian curiosity gave a twist for ever to the rudder of time. It was the attribute of happiness and virtue. To look for the causes of it is a hopeless quest in Greece itself; the miracle appears there, perfect, finished and inexplicable. But in Asia Minor there may be a chance, where Thales of Miletus, ‘having learnt geometry from the Egyptians, was the first to inscribe a right-angled triangle in a circle, whereupon he sacrificed an ox.’

Eleven years after founding the Turkish Republic in Anatolia and Thrace, Mustafa Kemal took the name Atatürk, “Father Turk.” The current ruler of the people called Turks would seem to be working out a damnatio memoriae, as by razing the Atatürk Culture Center on Taksim Square, as well as planning the demolition of Atatürk Airport (if the new Istanbul Airport can be finished).

In the Republic, Socrates rejects the tradition that has come to be called Oedipal (377D–8A):

Hesiod and Homer … methinks, composed false stories which they told and still tell to mankind … There is, first of all, the greatest lie about the things of greatest concernment, which was no pretty invention of him who told how Uranus did what Hesiod says he did to Cronos, and how Cronos in turn took his revenge; and then there are the doings and sufferings at the hands of his son. Even if they were true I should not think that they ought to be thus lightly told to thoughtless young persons.

Euthyphro is a thoughtless young person, who thinks he knows everything worth knowing. Socrates brings him to confusion on this point, just as he (Socrates) is getting ready to be tried for doing this kind of thing to others. It may be safer to teach young persons mathematics.

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