Polytropy

When is a help a hindrance? The Muses have provoked this question. They did this through their agents, the cicadas, who sang around the European Cultural Center of Delphi, during the 11th Panhellenic Logic Symposium, July 12–5, 2017.

My question has two particular instances.

1. At a mathematical conference, can theorems “speak for themselves,” or should their presenters be at pains to help the listener appreciate the results?

2. When the conference is in Greece, even at one of the country’s greatest archeological sites, does this enhance the reading of ancient Greek texts, or is it only a distraction?

Such questions occupy me here. I say the Muses have inspired them. According to the Theogony of Hesiod, Mnemosyne—Memory—bore the Muses to Zeus: he mingled with her for nine nights, and a year later, she produced the nine maidens, called Clio, Euterpe, Thalia, Melpomene, Terpsichore, Erato, Polymnia, Urania, and Calliope. As Socrates continues the story in the Phaedrus (258e–9d), cicadas are those former human beings who so delighted in music, when the Muses were born, that they forget to eat and drink. I recalled the story in the article, “On Chapman’s Homer’s Iliad, Book III.” The bemused humans died, but were resurrected as the insects that sing through hot summer days. The cicadas not only sing: they gossip about us, as they prepare a report on us for the Muses. If the report is favorable, the Muses may inspire us.

Also in the Phaedrus (274c–5b), Socrates tells another story that I brought up in the aforementioned article on Homer:

At Naucratis, in Egypt, was one of the ancient gods of that country, the one whose sacred bird is called the ibis, and the name of the god himself was Theuth (Θεύθ). He it was who invented numbers and arithmetic and geometry and astronomy, also draughts and dice, and, most important of all, letters (γράμματα).

Theuth came to show off his inventions to the god Thamus, who was called Ammon by the Greeks and was king of all Egypt. Concerning letters, said Thamus,

You have invented an elixir (φάρμακον) not of memory (μνήμη), but of reminding (ὑπόμνησις); and you offer your pupils the appearance of wisdom, not true wisdom, for they will read many things without instruction and will therefore seem to know many things, when they are for the most part ignorant and hard to get along with, since they are not wise, but only appear wise.

The written word does not improve our memory, though it may help us to forget. It provides, at best, hypomnesis: a reminder of what we already know. Such would seem to be the doctrine of Theuth, adopted by Socrates as his own.

Whether the doctrine of Theuth is true, or how it is true, is part of the question of the Muses. Is the written word a help or a hindrance? It helps us go about our lives. On the simplest level, the note that we have written in an agenda helps us do what we wanted at the time of writing. And yet if we really wanted to do it, should we not simply remember? Perhaps so: but only because we have become aware of our desire, through the act of expressing it in written form. After the act, we may not need to read what we have written, in order to remember it. Speaking our desire may have been enough: we could have spoken it out loud, with our lungs and vocal cords, or even silently, with our inner voice. On the other hand, using the muscles to write letters on paper, or perhaps even to press letters on a keyboard: this can be understood as speaking in the broadest sense. It is a part of the “original language of total bodily gesture” that Collingwood describes in The Principles of Art (Oxford, 1938, p. 246):

This would have to be a language in which every movement and every stationary poise of every part of the body had the same kind of significance which movements of the vocal organs possess in a spoken language. A person using it would be speaking with every part of himself. Now, in calling this an ‘original’ language, I am not indulging (God forbid) in that kind of a priori archaeology which attempts to reconstruct man’s distant past without any archaeological data. I do not place it in the remote past. I place it in the present. I mean that each one of us, whenever he expresses himself, is doing so with his whole body, and is thus actually talking in this ‘original’ language of total bodily gesture. This may seem absurd. Some peoples, we know, cannot talk without waving their hands and shrugging their shoulders and waving their bodies about, but others can and do. That is no objection to what I am saying. Rigidity is a gesture, no less than movement.

Rigidity is sometimes a gesture of speakers at an academic conference. So is pacing, or simply turning one’s back to the audience so as to read to them from the screen what they can read for themselves. It is unfortunate that projecting words and symbols onto a screen has become the default mode of giving a talk at conferences like the logic symposium in Delphi. In actually writing things out with chalk on a blackboard, one expresses better how one feels about what one is saying. This can help one’s audience.

Evidently Plato did write things down. But he did not write down such lectures as might have been given at the original Academy. When I visited St John’s College as a prospective student, and I sat in on a freshman seminar on the Gorgias, I was frankly dismayed that the students treated Plato’s dialogue, not as a source of doctrine, but as a play. I now think that the students’ approach was correct. Was I in turn indoctrinated by four years at the College? In a footnote in the Introduction of An Essay on Philosophical Method (Oxford, 1933, p. 14, n. 2), Collingwood describes as his own doctrine what would seem to be that of St John’s, namely that the dialogues are an example of how to become liberally educated:

Recent work on Plato, notably that of Professor A. E. Taylor and the late Professor Burnet, has made it impossible any longer to regard the dialogues as essentially statements of a philosophical position or series of positions; but the same authors’ attempt to explain them as essentially studies in the history of thought does not to me at least carry conviction. Whatever else they may be, it seems clear that they were intended as models for the conduct of philosophical discussion, or essays in method. I am not sure whether or no I had the first suggestion of this idea, many years ago, from the conversation of Professor J. A. Smith.

In “NL III: ‘Body As Mind’,” I suggested that Scott Buchanan might have met Collingwood at Oxford. In “The Tradition of Western Philosophy,” I had likened the St John’s College program, founded by Buchanan and Stringfellow Barr, to Collingwood’s understanding of philosophy in An Essay on Philosophical Method.

Returning to the Phaedrus, I note that Plato’s word for elixir is the source for our word pharmacy, as well as for the name of the girl Pharmacea, mentioned earlier in the dialogue (229b–d) as has having been playing with Orithyia when the latter was carried off by Boreas. The rational explanation, belittled by Socrates, is that a blast of wind blew the girl to her death from a rock: after all, Boreas was supposed to be the god of the north wind (called poyraz in Turkish). According to the extant manuscripts of the Phaedrus, the rock from which Orithyia was carried off could have been the Areopagus, rather than somewhere outside of Athens; but scholars seem to think this a gloss. Orithyia was apparently the daughter of Erechtheus, legendary founder of Athens and namesake of the Erechtheum.

A letter is made up of strokes. In Greek, this means a γράμμα (neuter) is made up of γραμμαῖ (plural feminine). The latter can also be the lines in the figures of Euclid. Though essential to modern mathematics, the mathematical symbol should be subject to the same criticism as the letter. It may help to remind us of what we have understood independently; it may also create only the illusion of understanding.

I am dismayed when mathematicians speak over the heads of their audience, or assume that their interests are shared by the audience. Such speakers may have treated an invitation to speak as a prize to be enjoyed, rather than a request for the edification of an entire audience. Speakers may address only their own tribe. At a general mathematics conference, this might mean addressing only the geometers; at an algebra conference, the ring theorists; at a logic conference, the set theorists. Must I conclude that such speakers are interested not in mathematics as such, but only in what they themselves are able to do with it? As G. H. Hardy confesses in A Mathematician’s Apology (§29) of 1940, and as I quoted in “Confessions,”

I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.

Some of us may not outgrow such thoughts.

An incomprehensible speaker may not be showing off his or her erudition to the cognoscenti. The speaker may also just be distracting the audience, so that they will not see that what is being presented is not that big a deal—in the speaker’s own view, which is surely wrong. In one of the first talks that I ever gave, away from my home institution of the time, I was sure that what I had to say was not a big deal. I had not learned that, if you have found a way to make something simple, after studying it for months, this does not mean that others are ready to see the simplicity. My talk was incomprehensible to most of my audience, even though they were supposed to be of my tribe.

Fortunately two tribesmen sat down to tell me so. “Decide whether you want to give good talks,” said one of them. “Evidently many mathematicians have decided that they don’t care; do you want to be one of them?” I suffered black depression for a day or two. I have since then caused this depression in one or two other young mathematicians, as they were trying to start out their own careers. I wanted to think I was only passing along the favor that had been done to me. However, I could have been more discreet about it.

Perhaps not everybody can receive the favor of good advice. Not everybody has—or wants to have—the kind of self-consciousness that Socrates describes, however playfully, as the reason why he is not interested in pursuing a materialistic explanation for the legend of Orithyia and Boreas (Phaedrus 229e–30a):

I have no leisure (σχολή) for [such explanations] at all; and the reason, my friend, is this: I am not yet able, as the Delphic inscription has it, to know myself; so it seems to me ridiculous, when I do not yet know that, to investigate irrelevant things. And so I dismiss these matters and accepting the customary belief about them, as I was saying just now, I investigate not these things, but myself, to know whether I am a monster more complicated and more furious than Typhon or a gentler and simpler creature, to whom a divine and quiet lot is given by nature.

When participants of the Panhellenic symposium had a tour of the ancient site of Delphi, the guide reminded us of the Delphic inscription to which Socrates refers. I assume the inscription cannot be seen today; it did not occur to me to ask.

Knowing oneself is not something that can be commanded by another, any more than accepting the truth of a mathematical theorem can be commanded.

It is grand now to have been to Delphi, home of the famous oracle. And yet melancholy attaches to archeological sites. The romance of life in an imagined past is gone now; can some old stones really revive it? I appreciate mathematics for its independence. The truth of a theorem comes not from an experiment, a vote, or a contest, much less an archeological site, but from one’s own mind. This essential feature of mathematics was observed by the Greeks and passed on to us by Euclid, Archimedes, and Apollonius. The greatest legacy of the Greeks is found in their writing. It is thrilling to visit the Parthenon in Athens, as we did on the way to Delphi; and yet all of the preservation, the restoration, and the repairs of this much-abused structure are for naught, if nobody remembers the dramas enacted in the Theater of Dionysus, below the Acropolis, or the conversations of Socrates in the Athenian Agora, or down in the Piraeus, or beneath a plane tree and the singing cicadas, outside the city.

In Greek literature, the oracle of Delphi appears in Herodotus, first (I.13) as confirming the rule of Gyges over the Lydians of Asia Minor, after he has seized the throne from Candaules with the help of the slain king’s widow. Said to be called Myrsilus by the Greeks, Candaules was of the dynasty of the Heraclidae, that is, the descendents of Heracles or, strictly speaking, “the Asiatic sungod identified with Heracles by the Greeks” (according to a note in the Loeb edition at I.7). Herodotus actually gives the line of descent: Heracles begat Alcaeus, who begat Belus, who begat Ninus, who begat Agron, the first Heraclid king of the Lydians. Concerning the rise to power of the Heraclids, Herodotus says only that this too was confirmed by an oracle, whom he does not name; before the Heraclidae, the Lydians were ruled (as one might expect) by descendents of Lydus, son of Atys, son of Manes. In the time of Atys, during a famine, the Lydians invented games like dice (κύβοι “cubes”), knucklebones (ἀστραγάλοι “astragali”), and ball (σφαῖρα “sphere”): all known games but draughts (πεσσοί), so that they could keep their minds off their hunger (I.94); but this story goes into the distant past, since the Heraclidae themselves “ruled for two and twenty generations, or 505 years, son succeeding father, down to Candaules, son of Myrsus” (I.7).

Such is the account by Herodotus of the Heraclid dynasty of Lydia, up to its last representive, Candaules. His alternative name of Myrsilus has led me to discover a paper whose title is the Halicarnassian historian’s very phrase, “Candaules, whom the Greeks name Myrsilus …” (Greek, Roman, and Byzantine Studies, Vol. 26, No 3 [1985], pp. 229–33). Here J. A. S. Evans reviews various possible sources or parallels for the story of Candaules, his unnamed wife, and Gyges. Evans mentions one scholar who thinks the story cannot be from east of the Greeks. For Evans himself, the best parallel is an eastern one, found in the Biblical Book of Esther. In the first chapter, while “merry with wine,” King Ahasuerus wants Queen Vashti to make an appearance,

to shew the people and the princes her beauty: for she was fair to look on.

An important difference from the story in Herodotus is that Vashti would have been clothed. Candaules wanted his wife seen naked. As I recalled in “Pyrgos Island,” Candaules induced Gyges, his favorite bodyguard, to watch his (Candaules’s) wife as she disrobed before bed. This woman saw Gyges in turn, and thus she saw what disgrace her husband had brought her; for

among the Lydians and most of the foreign peoples it is held great shame that even a man should be seen naked (I.10).

This apparently leads Evans to conclude that Herodotus himself must have invented the story of Candaules.

Does the scholarship like that of Evans help us read Herodotus? If we want to read Herodotus as a novel—and this may be the best way to read—then there is no point in asking where Herodotus gets his stories. We have the stories, and that is enough. On the other hand, one novel may illuminate another, and the Book of Esther can be counted as a novel. The Ahasuerus of Esther may be Xerxes the Great, whose invasion of Greece, Herodotus recounts. More important for present considerations is the plot of the Book of Esther. Vashti refuses to display herself. This is a violation of law. Vashti’s act may give other women the idea of disobeying their husbands. A new queen is sought among the virgins of the empire. One of the options is a Jewish orphan called Esther, who is under the guardianship of her cousin Mordecai. After a twelve-month waiting period (presumably to make sure nobody has got her with child), each virgin is tested. She goes to the king in the evening and leaves in the morning. The king chooses Esther for his new queen. She has not told him she is Jewish. Mordecai overhears a plot by two chamberlains against the king and informs him, through Esther. Mordecai also refuses to bow down to Haman when this man is promoted to high office. In retaliation, Haman arranges for all Jews to be killed. Mordecai asks Esther to intervene with the king. Esther first says it is illegal to see the king unless called, and she has not been called for a month. When Mordecai insists, Esther agrees to break the law, come what may. Her petition to the king is successful. The Jews get to kill all of Haman’s people (who are apparently Amalekites), some 75 thousand of them (9:16). They make the next say “a day of feasting and gladness” (9:17), and they create the feast of Purim to commemorate this. The editors of the Oxford World’s Classics edition of the Bible assert of the Book of Esther,

Whether such novellas were propaganda to encourage Jewish communities to be good civil subjects of the empire or to persuade the imperial authorities of the loyalty of Jewish communities cannot be determined.

Perhaps not. It would seem however that the Book could be used to propagate the doctrine whereby women should obey men, and Jewish women should obey Jewish men, but Jewish men should take shit from nobody.

The story in Herodotus is rather different. Candaules’s wife seizes the initiative and profits from this. She tells Gyges he must die, or else kill Candaules, marry her, and take the throne of Lydia. Gyges chooses the latter. Some Lydians object, but agree to let Delphi settle the question of succession. The Pythia declares that Gyges may rule, though the Heraclidae will have vengeance in the fifth generation. Naturally the proviso is overlooked. Herodotus catalogs what Gyges gives to Delphi, as a thank-offering for the supposedly favorable ruling; and what Midas gave earlier (I.14):

there are very many silver offerings of his there: and besides the silver, he dedicated great store of gold: among which six golden bowls are the offerings chiefly worthy of record. These weigh 30 talents and stand in the treasury of the Corinthians: though in very truth it is the treasury not of the Corinthian people but of Cypselus son of Eetion. This Gyges then was the first foreigner (of our knowledge) who placed offerings at Delphi after the king of Phrygia, Midas son of Gordias. For Midas too made an offering, to wit, the royal seat whereon he sat to give judgment, and a marvelous seat it is; it is set in the same place as the bowls of Gyges. This gold and silver offered by Gyges is called by the Delphians “Gygian” after its dedicator.

Gyges begat Ardys, who begat Sadyattes, who begat Alyattes, who went to war against Miletus, home town of Thales (see “Thales of Miletus”). When the Lydian army burned the Milesian crops, a temple of Athena was also burnt. Alyattes got sick, and he stayed sick, and he asked the Oracle of Delphi why. She would not answer until he repaired the temple (I.19). He did this, and built a second temple to boot, and recovered (I.22). Then he became (I.25)

the second of his family to make an offering to Delphi—and this was a thank-offering for his recovery—of a great silver bowl on a stand of welded iron. This is the most notable among all the offerings at Delphi, and is the work of Glaucus the Chian, the only man of that age who discovered how to weld iron.

Alyattes was succeeded by his son Croesus, who elicited from Delphi the most famous of the oracles, namely that he would destroy a great empire if he attacked the Persians (I.53). The empire he destroyed was his own (I.91). The word of Herodotus, translated as empire, is ἀρχή. This means more fundamentally a beginning or principle, even a mathematical principle; and in my own talk at the conference in Delphi, I suggested a broader interpretation of the oracle: fighting defeats the purpose of mathematics, which is that theorems should be accepted not out of resignation or desperation (or for that matter faith), but out of freely entered conviction.

The oracle given to Croesus may be as useful to us as any maxim. Croesus himself paid dearly for it. Before even receiving it, he paid for it to be favorable to him (I.50–1):

He offered up three thousand beasts from all the kinds fit for sacrifice, and on a great pyre burnt couches covered with gold and silver, golden goblets, and purple cloaks and tunics; by these means he hoped the better to win the aid of the god, to whom he also commanded that every Lydian sacrifice what he could. When the sacrifice was over, he melted down a vast store of gold and made ingots of it, the longer sides of which were of six and the shorter of three palms’ length, and the height was one palm. There were a hundred and seventeen of these. Four of them were of refined gold, each weighing two talents and a half; the rest were of gold with silver alloy, each of two talents’ weight. He also had a figure of a lion made of refined gold, weighing ten talents. When the temple of Delphi was burnt, this lion fell from the ingots which were the base on which it stood; and now it is in the treasury of the Corinthians, but weighs only six talents and a half, for the fire melted away three and a half talents.

When these offerings were ready, Croesus sent them to Delphi, with other gifts besides: namely, two very large bowls, one of gold and one of silver. The golden bowl stood to the right, the silver to the left of the temple entrance. These too were removed about the time of the temple’s burning, and now the golden bowl, which weighs eight and a half talents and twelve minae, is in the treasury of the Clazomenians, and the silver bowl at the corner of the forecourt of the temple. This bowl holds six hundred nine-gallon measures: for the Delphians use it for a mixing-bowl at the feast of the Divine Appearance. It is said by the Delphians to be the work of Theodorus of Samos, and I agree with them, for it seems to me to be of no common workmanship. Moreover, Croesus sent four silver casks, which stand in the treasury of the Corinthians, and dedicated two sprinkling-vessels, one of gold, one of silver. The golden vessel bears the inscription “Given by the Lacedaemonians,” who claim it as their offering. But they are wrong, for this, too, is Croesus’ gift. The inscription was made by a certain Delphian, whose name I know but do not mention, out of his desire to please the Lacedaemonians. The figure of a boy, through whose hand the water runs, is indeed a Lacedaemonian gift; but they did not give either of the sprinkling-vessels. Along with these Croesus sent, besides many other offerings of no great distinction, certain round basins of silver, and a female figure five feet high, which the Delphians assert to be the statue of the woman who was Croesus’ baker. Moreover, he dedicated his own wife’s necklaces and girdles.

All of this could and perhaps did serve as a tour guide for the ancient visitor to Delphi. It may guide also the modern archeologist. The fourth of the 14 rooms of the Delphi Archeological Museum holds findings that may come from kings in Anatolia, as Herodotus describes. I transcribe the text of the sign in the room:

In 1939, many years after the “great excavation” of the sanctuary, an unexpected find supplemented the testimony of ancient writers—particularly that of Herodotus—concerning the fabulous gifts dedicated to Apollo by wealthy rulers of Asia Minor, such as Gyges and Croesus of Lydia and Midas of Phrygia. Beneath the paving stones of the Sacred Way, in front of the Stoa of the Athenians, French archeologists discovered two pits full of objects made of precious material (gold, ivory, silver, copper) dating from the 8th to the 5th century BC. These included fragments of at least three chryselephantine statues, numerous pieces of forged silver sheet from a life-size bull, a great many relief ivory plaques, three exquisite masterpieces of 5th century BC miniature bronzework (exhibited in another hall), as well as humble votive offerings such as weapons and vases, which were found intermingled with earth, charcoal and ashes. The excavation indicated that all these finds had been offerings that were buried after suffering serious damage when the building in which they had been kept up to the mid 5th century BC was destroyed.

Thanks to a years-long restoration project, thousands of fragments from the two pits (known as “repositories”) were reassembled to take the form of the exhibits we see today, which give us a vivid picture of the opulence of the dedications to the sanctuary in the Archaic and early Classical periods. Most of the offerings were produced by Ionian workshops and appear to originate from the wealthy cities of Ionia (Miletus, Ephesus, Samos). Although their original appearance cannot be fully restored, the remains of these chryselephantine statues constitute unique examples of a rare sculptural technique involving a combination of carved ivory and wrought gold, both riveted to a wooden core. According to ancient literary sources, this technique was used in the 6th and 5th centuries BC for cult statues, including those of Athena in the Parthenon and Zeus at Olympia, both of which were created by Phidias.

We may speculate whether some of the objects buried in the pits had been burnt in the very fire that Herodotus mentions. It is a mystery to me why gold would have been buried along with charred ivory, since the gold remained uncorrupted by the fire. I can only suppose that Delphi had so much new gold coming in that previously worked and donated gold need not be preserved. In any case, Herodotus does not mention ivory among the gifts of Midas, Gyges, and Croesus.

Does the discovery of the votive pits at Delphi then tend to confirm the account of Herodotus, or not? The question is of little importance, next to the question of whether, like Croesus on receiving the oracle, we hear only what we want to hear. We ought to listen to what is really being said.

What is really being said in a talk at a conference? What help can, or should, the speaker give to the audience? The speaker may make no effort to show why her or his results should be of interest to anybody else. The interest is taken for granted. Getting results in the first place is hard enough, without the added burden of connecting them to what happens to be of particular interest to some other mathematician. In Delphi, during two hours of lecturing, at least one invited speaker did position his topic within the rest of mathematics. I tried to do something like this in my own contributed talk, within my restriction to twenty minutes. And yet if you don’t automatically see the interest of what is being presented, there may be no ultimate benefit in being coaxed into seeing the interest. I don’t think this is generally so, but it may be so in some cases. In Mathematics: A Very Short Introduction (Oxford, 2002), Timothy Gowers is quite right to adopt the policy that he describes in his Preface:

I do presuppose some interest on the part of the reader rather than trying to drum it up myself. For this reason I have done without anecdotes, cartoons, exclamation marks, jokey chapter titles, or pictures of the Mandelbrot set. I have also avoided topics such as chaos theory and Gödel’s theorem, which have a hold on the public imagination out of proportion to their impact on current mathematical research, and which are in any case well treated in many other books. Instead, I have taken more mundane topics and discussed them in detail in order to show how they can be understood in a more sophisticated way. In other words, I have aimed for depth rather than breadth, and have tried to convey the appeal of mainstream mathematics by letting it speak for itself.

I do question the notion that mathematics can speak for itself. In the book in question, it is Gowers who is speaking; he is speaking for mathematics; he does this without cartoons and “jokey chapter titles”; he is right to do so. In the earlier Logic: A Very Short Introduction (Oxford, 2000), Graham Priest uses cartoons, along with chapter titles like “Names and Quantifiers: Is Nothing Something?” and “Descriptions and Existence: Did the Greeks Worship Zeus?” The book suffers for this, as it suffers for the disclaimer in its final paragraph:

We have been skating over the surface of logic. It has great depths and beauty that one could not even begin to convey in a book of this kind.

This is making excuses. I would be suspicious of a book that tried to promote mathematics this way. Likewise, I would not coax students into pursuing mathematics professionally. The work is difficult and discouraging, and the motivation to do it should come from within oneself.

The written word serves at best as a reminder of what is already within us: this was the argument of Theuth. The written word creates the appearance of knowledge; and the appearance may hinder the creation of the reality. One might make the same objection, directly about images. Certain visualizations should come from within oneself. These may include the visualizations of Delphi made possible by the words of Herodotus. They probably include the visualizations that constitute pornography, at least when the viewer is a child. In “The Terrible Cost of Porn” (The American Conservative, July 12, 2017), Rod Dreher is quite exercised by the availability of hardcore pornography to children whose parents give them smartphones. I mentioned Dreher’s concern in “Community”; I note now that Dreher is also worried that Japanese adults are not having sex. He doesn’t seem happy, though, that Teen Vogue has published “Anal Sex: What You Need to Know,” subtitled “How to do it the RIGHT way.” Myself, I would only question the assertion of writer Gigi Engel, “There is no wrong way to experience sexuality, and no way is better than any other.” Experiencing sexuality could be having somebody else’s sexuality forced upon oneself.

Fairy tales are violent, and it is plausible to me that the imaginary violence of stories is good for children, at least when the stories are consumed in spoken form. Children can then create for themselves the mental images that suit their needs. Graphic depictions supplied from outside are another matter. In what must have been the early 1980s, a young visitor to my aunt and uncle’s property in West Virginia asked whether there were sharks in the North River there. He drew a picture of a shark with sharp bloody teeth, and he scrawled at the bottom, “Mark made Jaws.” His letter J was backwards. My aunt suggested that the boy had been a bit young to see Jaws the movie (which perhaps he had done on VHS).

There are sometimes ways to visualize mathematics. From indeterminate decades ago, I remember reading of a high-school algebra teacher who would explain the identity

by baking a cubical cake for students and slicing it, in three planes, into the eight pieces whose respective dimensions are seen in the right member of the identity. This might be a good visualization, but is of little help for the identity

not to mention the general identity for (a + b)ⁿ (the Binomial Theorem).

In linear algebra, we can define the dimension of a space as the size of a basis, after we have defined bases and proved that all bases of the same space have the same size. The definition of dimension then agrees with the common notion of what an n-dimensional space should be, when n is 1, 2, or 3; but students will just have to learn to accept that higher-dimensional spaces are meaningful and useful, even if the students do not know how to visualize such spaces. Success in mathematics comes from learning what not to try to visualize.

In recent years I have made three-dimensional models of the parabola and hyperbola of Apollonius and the Hexagon Theorem of Pappus. When I read Apollonius at St John’s College, no such models were available, and I did not miss them. I recall a rumor that the College did not want students to see models: it was thought that we should develop our three-dimensional intuitions independently. This may be a fine doctrine; but if students in late adolescence are able to develop such intuitions, it may only be because they have already had experience with sufficiently many three-dimensional objects. When I was young, my interest in mathematics was encouraged by gifts of mathematical puzzles, such as the Soma cube, which (if memory serves) I was given by a friend of my grandparents, when I was five or six years old. Visual aids for older students might help mitigate differences of nurture, if not nature.

Reading War and Peace now (see “War and Talk”), I am impressed by how much I do not remember from reading the novel as a student. I trace this in part to the emotional immaturity of those earlier days. What I had not already felt for myself, was not likely to strike me in the book. In the Agamemnon of Aeschylus, when the Trojan War is over, and the title character comes home, his wife Clytaemnestra kills him for having sacrificed their own daughter Iphigenia to make the winds blow the Greek fleet to Troy. I read and discussed the play in the freshman seminar at St John’s College; but the horror of the sacrifice did not impress me so much as when I saw it depicted on screen, a couple of years later, in the 1977 film of Michael Cacoyannis, Iphigenia. Not that the actual killing of the girl happened on screen; but the emotions involved could be heard in the voices, and read in the faces, of Agamemnon, Clytaemnestra, and their daughter. (I reviewed a curious retelling, called Interruption, of the story of Agamemnon in “35th Istanbul Film Festival, 2016, part 3.”)

The question of helping versus hindering arises finally about mathematical terminology. I have written a paper called “Abscissas and Ordinates” (Journal of Humanistic Mathematics 5, 2015), touching on the use of ordinary words in Greek mathematics. When we use those words in English, they are no longer ordinary words, but have become technical terms. This may help us avoid misleading visualizations; it may also hinder us from forming helpful visualizations for ourselves. In any case, it suggests that we think of mathematics differently than the Greeks did.

In the part of mathematical logic that I learned as recursion theory, but that now seems to be called computability theory, the term oracle is used for something like the Pythian priestess of Delphi: a source of information that we cannot understand. A recursive set of natural numbers is one that we can understand, in the sense of having an algorithm to determine whether (or not) a given number belongs to the set. For example, the set of square numbers is recursive. Not every set of natural numbers can be recursive: one proof is that there are uncountably many sets of natural numbers, but only countably many possible algorithms. There may be an algorithm for settling the question of membership (or not) in a set A, provided membership in a set B can somehow be settled. This would be the case, for example, if the elements of A were the doubles of the elements of B. It is said then that A is Turing-reducible to B. In particular, membership in A can be found by means of an oracle for B.

In my example, B is also Turing-reducible to A. In particular, the two sets are alike recursive or not. To give an example of a non-recursive set, we have consider the notion of an algorithm more carefully. An algorithm for a recursive set is a procedure that, when applied to a particular number, results in an answer to the question of whether the number belongs to the set. It may be that an algorithm gives the answer yes, when this is correct, but in the other case runs forever, without giving the answer no. In the former case, the algorithm is said to halt. The set of numbers where a given algorithm halts is called recursively enumerable. Thus all recursive sets are recursively enumerable. The converse fails, because we can make a list of all algorithms, then form a set C consisting of all numbers n such that the nth algorithm halts at n. This set is recursively enumerable; but its complement D is not. Indeed, suppose the nth algorithm on our list could be used to enumerate the elements of D. The algorithm would halt at n if and only if n were in D, which would mean n was not in C, which would mean the algorithm could not halt at n. This would be absurd. Therefore D cannot be recursively enumerable. Consequently C cannot be recursive. However, C is Turing-reducible to D. If an oracle can tell us whether n belongs to D, then we can infer whether n belongs to C: it belongs, just in case it does not belong to D.

Why should we believe the oracle though? Croesus believed in his interpretation of an oracle, and this led to his downfall. In recursion theory then, is the term oracle well chosen? Does it help one do recursion theory? This would seem to be a good question to ask in Delphi; but I am not aware that any of the recursion-theorists at our meeting raised the question seriously (and I did ask one of them about this). At the ancient site of Delphi, a cult was founded, based on the ultimately irrational notion that direct communion with supernatural knowledge was possible at select geographical locations. Recursion theory is mainly about sets that are not even recursively enumerable. Thus recursion theory is about sets that we cannot understand, even in principle. We can still prove theorems about such sets. This is a paradox.

It is a paradox that the Greeks are revered for rationality, when their religion is founded on the possibility of divine revelations, received at certain geographical locations like Delphi. In the Phaedrus, Socrates himself recognizes that the oracle of Delphi suffers a kind of madness or mania (μανία 244a), and this (according to him) is why prophecy is called “mantic” (μαντικός)—it should really be “manic” (μανικός). In the same spirit, we might say that the former, standard, word is appropriate, since logic in Turkish (which takes the term from Arabic) is mantık. In any case, there are two kinds of madness (265a), one a human disease, the other a “divine release from customary habits,” be it brought by Apollo as at Delphi, by Dionysus, by the Muses, or by Aphrodite and Eros (265a–c). Progress requires such madness.