Euclides and his friend Terspion are mentioned as among those who kept company with Socrates on the day he drank hemlock. After Socrates’ death, Plato stayed with Euclides in Megara, which is a day’s walk from Athens.
Euclides had learned the art of disputation from the writings of Parmenides. After hearing about Socrates, Euclides moved from Megara to Athens and became one of his most zealous disciples. When Athens and Megara fell into one of their periodic conflicts, the Athenians passed a decree forbidding any Megarian from entering Athens on pain of death. Euclides prudently returned to Megara. However, he still came frequently to Athens to visit Socrates. Euclides traveled at night concealed in a long female cloak and veil.
This subterfuge may have led Euclides to formulate the paradox of the veiled figure—also known as the unnoticed man, the hooded man, and the Electra: Socrates knew Euclides but did not know Euclides when disguised. How is this possible? If the veiled figure is identical to Euclides, then the veiled figure has every property that Euclides has: the same eye color, the same number of hairs, the same friends. Since Euclides has the property of being known to Socrates, the veiled figure must also have the property of being known to Socrates.
Paradoxes of knowledge and identity are intensively discussed in Plato’s Theaetetus. Plato depicts Euclides as the chronicler of the philosophical exchange. Socrates’ partners in dialogue are Theodorus, an old eminent mathematician and his gifted sixteen-year-old student Theaetetus. Terspion has been searching for Euclides at the Agora. Terspion finally finds his friend in the street near Euclides’ house in Megara. A somber Euclides explains that earlier in the day he was going down to the harbor and saw Theaetetus. He was badly wounded and was being carried by the army from Corinth to Athens. Euclides advised Theaetetus to convalesce in Megara but Theaetetus was intent on returning home.
Euclides reminds Terspion of Socrates’ prophecy concerning Theaetetus: he would be a great man if he lived. This prediction brings to mind a remarkable conversation between Socrates, Theaetetus, and his mathematics teacher, Theodorus. Euclides witnessed the dialogue and took notes. Subsequently he reconstructed the dialogue with the help of Socrates. Since Terspion is eager to hear the dialogue, Euclides invites him to his home so that they can rest while having the dialogue read to them by one of Euclides’ servants.
Plato casts Euclides as the reconstructor of the dialogue because of Euclides’ interests. Euclides was a strong believer in Socrates’ thesis that all virtues are one thing: knowledge. This thesis stems from a principle of continence: people never wittingly pick an inferior alternative. If you are offered a choice between two figs and one fig, you choose two figs. Since we always aim for the best, people choose evil only when it is in the guise of the good.
Socrates acknowledges that the principle of continence precludes weakness of will. Those who drink wine to excess will sometimes sheepishly concede (between sips) that they know they would be better off abstaining. However, they continue drinking. Socrates takes the same attitude toward overimbibers as most present-day economists: Actions speak louder than words! We should not be misled by the drinker’s lip service to the precepts of others. The drinker’s real preference is revealed by his behavior. People imbibe because that is what they most want to do.
Socrates concedes that people sometimes choose a smaller good that can be immediately obtained over a larger good that would require a wait. He thinks this is due to illusions of perspective. In the late afternoon, your giant shadow appears to have a tiny head. But the head only looks ill proportioned because you are looking at it from the giant’s feet.
Socrates suggests that there are also foreshortening illusions with respect to time. A child might prefer one fig today over two figs tomorrow because one fig now seems like the greater good. As people mature, their knowledge of this illusion weakens its effect. They thereby acquire the virtue of patience. Education reduces other vices. We become less cowardly with respect to snakes after learning that most of them are harmless. As we become more knowledgeable, our steady preference for what is best leads to objectively right choices. All vice is based on ignorance. All virtue is based on knowledge.
In The Republic, Socrates draws administrative corollaries of “virtue is knowledge.” The best choice of a ruler is someone who is most virtuous. Philosophers are the most knowledgeable, therefore philosophers should be kings.
For Socrates, epistemology (the study of knowledge) interlocks with ethics and politics. Epistemology is also related to aesthetics (the study of beauty). Things are beautiful to the extent that they fit their form. A mutilated horse is ugly because it poorly matches the form for horses. A show horse is beautiful because of its fidelity to horsehood. By serving as exemplars, forms are ideals of beauty. Aesthetic appreciation is knowledge of how an object lives up to its form.
Much of Theaetetus is concerned with puzzles about the nature of knowledge in light of certain puzzles about identity. I follow Samuel Wheeler in conjecturing that these are variations of Euclides’ veiled figure paradox. Just as Plato pays tribute to Theaetetus by having him review some of his important mathematical results in the dialogue, Plato pays tribute to Euclides by integrating his paradoxes into the analysis of knowledge.
There are also methodological themes that would have made for bittersweet reading by Euclides. Part of the dialogue summarizes Socrates’ objection to Euclides’ voracious appetite for controversy. Euclides was a contentious man who frequently litigated in civil courts. Socrates disapproved.
Socrates prefers dialectic debate in which both parties cooperate and follow the argument wherever it leads. There is no pressure to obtain a practical result. One has the leisure to linger on an interesting issue. If one side errs, the other side good-naturedly corrects the mistake in a constructive manner. The aim of both sides is a collaborative, sincere pursuit of truth.
Lawyers debate for the sake of persuasion. The truth is irrelevant. Each side in a legal contest is allocated a set amount of time to present his case (as measured by a water clock). So they are always in a hurry and are prevented from pursuing interesting digressions. A lawyer has no hope of nurturing a fresh idea because
. . . his adversary is standing over him, enforcing his rights; the indictment, which in their phraseology is termed the affidavit, is recited at the time: and from this he must not deviate. He is a servant, and is continually disputing about a fellow-servant before his master, who is seated, and has the cause in his hands; the trial is never about some indifferent matter, but always concerns himself; and often the race is for his life. The consequence has been, that he has become keen and shrewd; he has learned how to flatter his master in word and indulge him in deed; but his soul is small and unrighteous.
(Theaetetus 173)
To be persuasive, lawyers act as if they believe what they are asserting. Any lawyer who is ready to lie for his client is also prepared to deceptively argue for him. The obvious way that an argument can be deceitful is through the assertion of premises one does not believe. The more subtle way is to “infer” what one does not believe to follow from those premises (in the hope that the jury will join in the fallacy).
Socrates’ harsh assessment of litigation offended Euclides. He founded his own school in Megara. Euclides does not appear to have toned down his wrangling. The intensity of debate led Timon to say that Euclides had carried the madness of contention from Athens to Megara.
Doctrinally, the Megarians were in close agreement with the Eleatics. Diogenes Laertius reports that Euclides studied the writings of Parmenides and “held that the supreme good is really one, though it has many names, wisdom, God, Mind, and so forth. He rejected all that is contradictory of the good, holding it to be nonexistent.” (1925 ii. 120)
Socrates shares Euclides’ awe of Parmenides. At one point in Theaetetus, Socrates refuses to criticize father Parmenides. Socrates listens placidly to Theodorus’s sour assessment of Parmenides’ opposite, Heraclitus and his devotees:
True to their own treatises, they are in perpetual motion. But their ability to keep to an argument or a question, quietly answering and asking in turn, amounts to less than nothing. Indeed, “less than nothing” fails to do justice to the absence of even the smallest particle of repose in these people. If you ask them a question, they pull from their quivers little oracular phrases and let fly at you with them. And if you ask for an explanation, you are transfixed with another garbled metaphor. You never get anywhere with them—nor do they get anywhere with one another, for that matter; for they take very good care to see that nothing gets settled, either in argument or in their own souls—thinking, I suppose, that this would constitute something stationary; and whatever is stationary they wage war on, and so far as they can banish it altogether from the universe!
(Theaetetus 179 E)
Socrates conjectures that the Heracliteans may be in greater agreement when among themselves. Theodorus insists that each one of them is willful and committed to perpetual discord.
The Heracliteans did have a logical argument for the universality of change. If x is identical to y and x has property F, then y has property F. For instance, if the square root of sixteen is identical to four and four is even, then the square root of sixteen is even. Heraclitus’s point is that this law implies that changing things do not endure through the change. If Socrates when ill is identical to the man who recovered, then every property possessed by the ill Socrates is possessed by the recovered Socrates. But then the healthy Socrates would still be ill. What appears to be a single individual, Socrates, enduring through time, is actually a succession of individuals. The Heracliteans conclude that our ordinary use of identical is loose talk founded on mere resemblance. Socrates before and after his illness are only identical in the way that distinct grains of salt are identical.
Philosophers who have been influenced by Einstein’s physics respond to Heraclitus’s paradox of change by portraying Socrates as a space-time worm. He is a sequence of individual stages. They concede to Heraclitus that Socrates does not endure through time. Instead, he perdures through time. Perdurance is a matter of having parts from different times. A momentary object does not perdure because all of its parts are from the same time. A number does not perdure because it has no temporal parts.
Other philosophers say that Socrates genuinely endures through time because the properties in question have a temporal aspect. The Socrates who had the property of being ill in the morning is identical to the Socrates who does not have the property of being ill in the afternoon.
Heraclitus’s paradox of change uses dynamic individuals as counterexamples to the substitutivity of identicals (the principle that if x = y and x has property F, then so does y). Euclides’ riddle of the veiled figure uses static subjective properties as counterexamples to the same principle. At one and the same time, Euclides has the property of being known by Socrates and lacks the property of being known by Socrates. In addition to having objective properties such as being a man, Euclides has properties that at least partly depend on how people think of him. He can be popular only if many people like him. He can be famous only if many people know him.
The challenge posed by the veiled figure is to explain misidentifications. How can people fail to know true identity statements? Socrates knows Euclides. Euclides is the veiled figure. How could Socrates fail to know that Euclides is the veiled figure?
The paradox of the veiled figure involves an error of omission—failing to believe a true identity statement. Misidentifications can also be errors of commission in which one believes that a true identity statement is false. Before Socrates learned of Euclides’ disguise, he believed it false that the veiled figure was Euclides.
Theaetetus begins with the problem of explaining a more specific misidentification. How can people believe that a false identity statement is true? If someone knows both Socrates and Theaetetus, he will know that Socrates is not Theaetetus. If he does not know them both, then the issue of whether Socrates is Theaetetus will not arise. The thinker will not have the resources even to formulate the false statement. He can only refer to what he knows.
There is certainly some truth to this. Socrates could not have believed the false identity statement “Mencius is Mo Tzu.” Although each of these Chinese philosophers overlapped in time with Socrates, they were too far away to be known by him. The word Mencius would have been meaningless in the mouth of Socrates. Therefore, he could not have had a thought that mixed knowns with unknowns such as “Theaetetus is Mencius.”
The paradoxes of knowledge and identity can be used to challenge Socrates’ assumption that definitions are informative. If the definiens (the terms used to do the defining) says something more than the definiendum (the term being defined), the definition is too broad (like the false definition “Human beings are bipeds”). If the definiens says something less than the definiendum, the definition is too narrow (like the false definition “Human beings are men”). If the definiens says neither more nor less than the definiendum, then the equivalence ensures that the definition is redundant (like the true but trivial “Human beings are human beings”). Thus, all definitions are either false or circular.
The twentieth-century formulation of the paradox is due to C. H. Langford. He was raising a problem with G. E. Moore’s principle that philosophy is mainly a matter of analyzing our concepts. An analysis breaks a concept down into components as in “A brother is a male sibling.” Moore thought that knowledge could be broken down into justified true belief and that rightness could be analyzed as that which produces best consequences. Langford poses a dilemma:
Let us call what is to be analyzed as the analysandum, and let us call that which does the analyzing the analysans. The analysis then states an appropriate relation of equivalence between the analysandum and the analysans. And the paradox of analysis is to the effect that, if the verbal expression representing the analysandum has the same meaning as the verbal expression representing the analysans, the analysis states a bare identity and is trivial; but if the two verbal expressions do not have the same meaning, the analysis is incorrect.
(1968, 323).
The paradox of analysis resembles Meno’s paradox of inquiry. Meno contends that if the inquirer knows enough to identify the correct answer to his question, then he already knows the answer. Langford alleges that an analysis that successfully identifies a concept with its meaning cannot give us knowledge because the identification is trivially correct. A definition can be illuminating only if one were earlier ignorant of an identity statement of the form A = B. If one understands A = B, then one must grasp A and grasp B. But then one will know that A and B are one and the same!
Plato never formulates the paradox of analysis in his dialogues. However, there was a commentator on Plato’s Theaetetus between 50 B.C. and 150 B.C. who displays a rudimentary awareness of the paradox. In the course of explaining a mistaken criticism of a definition, the commentator says
This is a misunderstanding, they say: for the object and the definition are convertible, but the definition does not mean exactly the same as the name. For if one person asked “What is a man?” and the other replied “A rational mortal animal,” just because a rational mortal animal is a man we won’t say that when asked “What is a man?” he replied a “A man.”
(Quoted by Sedley 1993, 136)
The commentator is trying to prevent the principle of the substitutivity of identicals from undermining the informativeness of “Man is a rational mortal animal.” The principle poses the same threat to the informativeness of “Euclides is the veiled figure.” The paradox of analysis is the paradox of the veiled figure as applied to definitional identifications. When the definer says female fox is the meaning of vixen, he is identifying figures in a dark conceptual landscape. We know these identifications are helpful, but we face an unexpectedly good argument for the counteranswer that these identity statements are useless.
I have doubts about whether Euclides himself would have been alarmed by the paradox of analysis. As a Parmenidean, Euclides would solve the problem of the veiled figure by restricting knowledge to the One. Everything that exists is identical to one thing, so it is impossible for there to be distinct things that we could misidentify as being identical. The things in question encompass mental things. Ultimately, there are not many concepts and so there is no opportunity for misidentifications. Like Zeno, Euclides brandishes his paradox as a sword in defense of his master.