I recognize myself in these notes by Chris Tiee on tensor analysis—the fever of acquisition, the uses of comparison, and the stray insight that makes the whole book worth it (as the one track does for the whole album):
“One way to learn a lot of mathematics is by reading the first chapters of many books.”—Paul R. Halmos
Ah, the dreaded discussion of texts for tensor analysis. I am addicted to collecting math books (I also often have 10 books checked out from the library simultaneously) and reading the first 20 pages of them. Very occasionally I make it through farther than that. Despite the proliferation of bad tensor analysis texts (some would say all of them are bad), I have to admit I have gleaned everything I have learned about Tensor Analysis from reading these books, collecting the knowledge into a gigantic patchwork. The fact is, each text actually has a gem of insight or two that is not presented in any others. There is much overlap in the bad parts, and some in the good parts too, but of course, it’s always hard to consult so many references, since I often forget the transformation laws on those overlaps. . . not to mention also the transformation laws that tell how the notation changes—regarding this, we have the following
A.1. Joke. Differential geometry is the study of those things invariant under change of notation.
Another problem is also that it’s very hard to strike a balance between being intuitive in the explanations of what these things are—and hence being vague—and also developing a precise, formal theory that is the real deal—hence being obfuscatory.
The quote from Halmos isn’t quite right. It’s actually, “I wish I had read the first ten pages of many more books—a splendid mathematical education can be acquired that way.” As for tensors, a newer book rich in intuition is Dwight Neuenschwander’s Tensor Calculus for Physics.
From Swift’s “apology” to A Tale of a Tub, in its fifth edition (1710):
The greatest part of that book was finished above thirteen years since, 1696, which is eight years before it was published. The author was then young, his invention at the height, and his reading fresh in his head. By the assistance of some thinking, and much conversation, he had endeavoured to strip himself of as many real prejudices as he could; I say real ones, because, under the notion of prejudices, he knew to what dangerous heights some men have proceeded. Thus prepared, he thought the numerous and gross corruptions in Religion and Learning might furnish matter for a satire, that would be useful and diverting. He resolved to proceed in a manner that should be altogether new, the world having been already too long nauseated with endless repetitions upon every subject. The abuses in Religion, he proposed to set forth in the Allegory of the Coats and the three Brothers, which was to make up the body of the discourse. Those in learning he chose to introduce by way of digressions. He was then a young gentleman much in the world, and wrote to the taste of those who were like himself; therefore, in order to allure them, he gave a liberty to his pen, which might not suit with maturer years, or graver characters, and which he could have easily corrected with a very few blots, had he been master of his papers, for a year or two before their publication.
Not that he would have governed his judgment by the ill-placed cavils of the sour, the envious, the stupid, and the tasteless, which he mentions with disdain. He acknowledges there are several youthful sallies, which, from the grave and the wise, may deserve a rebuke. But he desires to be answerable no farther than he is guilty, and that his faults may not be multiplied by the ignorant, the unnatural, and uncharitable applications of those who have neither candour to suppose good meanings, nor palate to distinguish true ones. After which, he will forfeit his life, if any one opinion can be fairly deduced from that book, which is contrary to Religion or Morality.
Preceding this is a satirical list of forthcoming works:
Treatises wrote by the same Author, most of them mentioned in the following Discourses; which will be speedily published.
A Character of the Present Set of Wits in the Island. A panegyrical Essay upon the Number THREE.
A Dissertation upon the principal Productions of Grub-Street.
Lectures upon a Dissection of Human Nature.
A Panegyric upon the World.
An analytical discourse upon Zeal, histori-theophysi-logically considered.
A general History of Ears.
A modest Defense of the Proceedings of the Rabble in all ages.
A Description of the Kingdom of Absurdities.
A Voyage into England, by a Person of Quality in Terra Australis incognita, translated from the Original.
A Critical Essay upon the Art of Canting, philosophically, physically, and musically considered.
And from the long-last opening, after pages of other prefatory material:
WHOEVER hath an Ambition to be heard in a Crowd, must press, and squeeze, and thrust, and climb with indefatigable Pains, till he has exalted himself to a certain Degree of Altitude above them. Now, in all Assemblies, though you wedge them ever so close, we may observe this peculiar Property, that over their Heads there is Room enough, but how to reach it is the difficult Point; it being as hard to get quit of Number, as of Hell.
— Evadere ad auras, Hoc opus, hic labor est.
To this End, the Philosopher’s way in all Ages has been by erecting certain Edifices in the Air; But, whatever Practice and Reputation these kind of Structures have formerly possessed, or may still continue in, not excepting even that of Socrates, when he was suspended in a Basket to help Contemplation, I think, with due Submission, they seem to labour under two Inconveniences. First, That the Foundations being laid too high, they have been often out of Sight, and ever out of Hearing. Secondly, That the Materials, being very transitory, have suffered much from Inclemencies of Air, especially in these North-West regions.
Teacher of Philosophy: […] What do you wish to learn?
Monsieur Jourdain: Everything I can, for I am intensely anxious to be learned; it troubles me that my father and my mother did not see that I was thoroughly grounded in all knowledge when I was young.
Teacher of Philosophy: An admirable sentiment: Nam sine doctrina vita est quasi mortis imago. Doubtless you know Latin and understanding that.
Monsieur Jourdain: Yes, but proceed as thought I did not know it: explain to me what it means.
Teacher of Philosophy: It means, Without knowledge, life is little more than the reflection of death.
Monsieur Jourdain: That Latin is right.
Teacher of Philosophy: Do you not know some of the principles, some of the rudiments of knowledge?
Monsieur Jourdain: Oh! Yes, I know how to read and write.
Teacher of Philosophy: Where would you like us to begin? Would you like me to teach you logic?
Monsieur Jourdain: What is logic?
Teacher of Philosophy: It teaches that which educates the three operations of the mind.
Monsieur Jourdain: What are the three operations of the mind?
Teacher of Philosophy: The first, the second and the third. The first is to have a proper conception of things, by means of universals; the second is to judge accurately, by means of categories; and the third is to draw a conclusion accurately by means of the figures Barbara, Celarent, Durii, Ferio, Baralipton, etc.
Monsieur Jourdain: Those words are regular jawbreakers. That logic does not appeal to me. Let us learn something nicer.
“Nothing. I thought I had part of the Book of Ecclesiastes and maybe a little of Revelation, but I haven’t even that now.”
“The Book of Ecclesiastes would be fine. Where was it?” “Here,” Montag touched his head.
“Ah,” Granger smiled and nodded.
“What’s wrong? Isn’t that all right?” said Montag.
“Better than all right; perfect!” Granger turned to the Reverend.
“Do we have a Book of Ecclesiastes?”
“One. A man named Harris of Youngstown.”
“Montag.” Granger took Montag’s shoulder firmly. “Walk carefully. Guard your health. If anything should happen to Harris, you are the Book of Ecclesiastes. See how important you’ve become in the last minute!”
“But I’ve forgotten!”
“No, nothing’s ever lost. We have ways to shake down your clinkers for you.”
“But I’ve tried to remember!”
“Don’t try. It’ll come when we need it. All of us have photographic memories, but spend a lifetime learning how to block off the things that are really in there. Simmons here has worked on it for twenty years and now we’ve got the method down to where we can recall anything that’s been read once. Would you like, some day, Montag, to read Plato’s Republic?”
“Of course!”
“I am Plato’s Republic. Like to read Marcus Aurelius? Mr. Simmons is Marcus.”
“How do you do?” said Mr. Simmons.
“Hello,” said Montag.
“I want you to meet Jonathan Swift, the author of that evil political book, Gulliver’s Travels! And this other fellow is Charles Darwin, and this one is Schopenhauer, and this one is Einstein, and this one here at my elbow is Mr. Albert Schweitzer, a very kind philosopher indeed. Here we all are, Montag. Aristophanes and Mahatma Gandhi and Gautama Buddha and Confucius and Thomas Love Peacock and Thomas Jefferson and Mr. Lincoln, if you please. We are also Matthew, Mark, Luke, and John.”
Everyone laughed quietly.
“It can’t be,” said Montag.
“It is,” replied Granger, smiling. “We’re book-burners, too. We read the books and burnt them, afraid they’d be found. Micro-filming didn’t pay off; we were always traveling, we didn’t want to bury the film and come back later. Always the chance of discovery. Better to keep it in the old heads, where no one can see it or suspect it. We are all bits and pieces of history and literature and international law, Byron, Tom Paine, Machiavelli, or Christ, it’s here. And the hour is late. And the war’s begun. And we are out here, and the city is there, all wrapped up in its own coat of a thousand colors. What do you think, Montag?”
“I think I was blind trying to do things my way, planting books in firemen’s houses and sending in alarms.”
“You did what you had to do. Carried out on a national scale, it might have worked beautifully. But our way is simpler and, we think, better. All we want to do is keep the knowledge we think we will need, intact and safe. We’re not out to incite or anger anyone yet. For if we are destroyed, the knowledge is dead, perhaps for good. We are model citizens, in our own special way; we walk the old tracks, we lie in the hills at night, and the city people let us be. We’re stopped and searched occasionally, but there’s nothing on our persons to incriminate us. The organization is flexible, very loose, and fragmentary. Some of us have had plastic surgery on our faces and fingerprints. Right now we have a horrible job; we’re waiting for the war to begin and, as quickly, end. It’s not pleasant, but then we’re not in control, we’re the odd minority crying in the wilderness. When the war’s over, perhaps we can be of some use in the world.”
“Do you really think they’ll listen then?”
“If not, we’ll just have to wait. We’ll pass the books on to our children, by word of mouth, and let our children wait, in turn, on the other people. A lot will be lost that way, of course. But you can’t make people listen. They have to come round in their own time, wondering what happened and why the world blew up under them. It can’t last.”
“How many of you are there?”
“Thousands on the roads, the abandoned railtracks, tonight, bums on the outside, libraries inside. It wasn’t planned, at first. Each man had a book he wanted to remember, and did. Then, over a period of twenty years or so, we met each other, traveling, and got the loose network together and set out a plan. The most important single thing we had to pound into ourselves was that we were not important, we mustn’t be pedants; we were not to feel superior to anyone else in the world. We’re nothing more than dust-jackets for books, of no significance otherwise. Some of us live in small towns. Chapter One of Thoreau’s Walden in Green River, Chapter Two in Willow Farm, Maine. Why, there’s one town in Maryland, only twenty-seven people, no bomb’ll ever touch that town, is the complete essays of a man named Bertrand Russell. Pick up that town, almost, and flip the pages, so many pages to a person. And when the war’s over, some day, some year, the books can be written again, the people will be called in, one by one, to recite what they know and we’ll set it up in type until another Dark Age, when we might have to do the whole damn thing over again. But that’s the wonderful thing about man; he never gets so discouraged or disgusted that he gives up doing it all over again, because he knows very well it is important and worth the doing.”
[…]
But now there was a long morning’s walk until noon, and if the men were silent it was because there was everything to think about and much to remember. Perhaps later in the morning, when the sun was up and had warmed them, they would begin to talk, or just say the things they remembered, to be sure they were there, to be absolutely certain that things were safe in them. Montag felt the slow stir of words, the slow simmer.
Paul Samuelson, “Heads I Win, and Tales, You Lose,” in the sixtieth anniversary edition of von Neumann and Morgenstern’s Theory of Games and Economic Behavior:
Contrary to folklore, mathematical ability is not a rare gift uncorrelated with other intellectual abilities: testing demonstrates that the child good with words and logic is most likely to have native potentiality for mathematics also. That schools […] should turn us out ignorant of and resentful of mathematics, is a crime. And not because, in the age of Sputnik and automation, mathematical proficiency is a prerequisite of national prosperity and survival, but rather because of the sheer fun that people miss […]
Bacon’s 55th aphorism in the Novum Organum, translated by Joseph Devey, some two centuries before Darwin on lumpers and splitters:
The greatest and, perhaps, radical distinction between different men’s dispositions for philosophy and the sciences is this, that some are more vigorous and active in observing the differences of things, others in observing their resemblances; for a steady and acute disposition can fix its thoughts, and dwell upon and adhere to a point, through all the refinements of differences, but those that are sublime and discursive recognize and compare even the most delicate and general resemblances; each of them readily falls into excess, by catching either at nice distinctions or shadows of resemblance.
When he was six years old, he could divide two eight-digit numbers in his head and could converse in Ancient Greek. When the six-year-old von Neumann caught his mother staring aimlessly, he asked her, “What are you calculating?”
A mid-15th century manuscript of Martianus Capella’s De nuptiis, Digital Vatican Library, MS Urb. lat. 329, with illumination by the painter Gherardo di Giovanni del Fora
It is a commonplace that medieval education amounted to the teaching of the seven liberal arts, what we call the trivium and quadrivium—grammar, logic (or as it was then called, dialectic), and rhetoric, on the one hand; arithmetic, geometry, astronomy, and music, on the other. But where, exactly, did this classification get codified, and how was it transmitted down through the centuries?
I did not know the answer until I read Charles Homer Haskins’s The Rise of Universitiesa few weeks ago. Cicero speaks of liberal arts, and Quintilian follows him, but the seven disciplines as we know them did not gain currency until Martianus Capella’s early fifth-century allegory De nuptiisPhilologiae et Mercurii (The Marriage of Philology and Mercury), sometimes also called De septem disciplinis and the Satyricon (not to be confused with Petronius’s). In Capella’s rendering, the seven arts, personified as women, are offered as wedding gifts by the gods at the marriage of Mercury and Philology; each offers a long speech describing her domain of study. (The term quadrivium itself appears to have been coined by Boethius, not Capella, and I’m still not certain about trivium.) The manuscript became a standard medieval textbook; it was copied and commented on straight up until the renaissance of the twelfth century, when the Latin-speaking medieval curriculum at last outgrew these rudiments thanks to an influx of translations of Greek texts—especially Aristotle’s—from Arab scholars in Spain. The images in this post are of a mid-15th century manuscript at the Vatican, with illumination and drawings by the painter Gherardo di Giovanni del Fora. (Some other representations of the seven liberal arts are here.)
And yet, in Capella’s hands, these rudiments are not so rudimentary nor as stale as we may imagine—at least, as I imagined. The portraits of the arts are vivid, modern, and cheeky. There is a strong undercurrent of irony in the understated manner of the best classical dialogues. The gods interrupt; the goddesses fire back. Dialectic snarks that she should be forgiven her neologisms since she is asked to speak in Latin rather than Greek. Bacchus, it is said, is “completely unacquainted” with her. The speeches are longer and richer—both stylistically and substantively—than the spare, plodding, sophistical medieval treatise I expected. The language, as far from the sermo humilis of the Christian church fathers as from the later casuistry of the high scholastics, relishes in ulteriority, taking obvious pleasure in finding the most resonant formulation and in saying many things at once. (My title is a case in point: without Dialectic, we read, “nothing follows”—a tiny phrase bursting its semantic seams, so dense is it with significance. I detect at least three different registers of allegorical reference: the concept of logical consequence, according to which one proposition follows from another; the epistemic primacy of logic as method, the tool of all other forms of inquiry; the historical or developmental primacy of logic, in the life of the student and the school—that which must be learned before the higher arts, and a rite of passage one must clear to prove one’s bona fides.) It is a token of our proximity to this past, rather than our distance from it, that I read and delight in it so easily, and find it more familiar than foreign.
Some of my favorite moments so far, from the speeches by Grammar and Dialectic, in the translation by William Harris Stahl and E. L. Burge (1977):
Grammar
Fora’s 15th-century illustration of Grammar
Once again in this little book the Muse prepares her ornaments and wants to tell fabricated stories at first, remember that utility cannot clothe the naked truth; she regards it as a weakness of the poet to make straightforward and undisguised statements, and she brings a light touch to literary style and adds beauty to a page that is already heavily colored. (p. 64)
[…] an old woman indeed of great charm, who said that she had been born in Memphis when Osiris was still king; when she had been a long time in hiding, she was found and brought up by the Cyllenian [Mercury] himself. This woman claimed that in Attica, where she had lived and prospered for the greater part of her life, she moved about in Greek dress; but because of the Latin gods and the Capitol and the race of Mars and descendants of Venus, according to the custom of Romulus she entered the senate of the gods dressed in a Roman cloak. She carried in her hands a polished box, a fine piece of cabinetmaking, which shone on the outside with light from ivory, from which like a skilled physician the woman took our the emblems of wounds that need to be healed. Out of this book she took first a pruning knife with a shining point, with which she said she could prune the fault of pronunciation in children; then they could be restored to health with a certain black powder carried through reeds, a powder which was thought to be made of ash or the ink of cuttlefish. Then she took out a very sharp medicine which she had made of fennelflower and the clippings from a goat’s back, a medicine of purest red color, which she said should be applied to the throat when it was suffering from a bucolic ignorance and was blowing out the vile breaths of a corrupt pronunciation. She showed too a delicious savory, the work of many late nights and vigils, with which she said the harshness of the most unpleasant voice could be made melodious. She also cleaned the windpipes and the lungs by the application of a medicine in which were observed wax smeared on beechwood and a mixture of gallnuts and gum and rolls of the Nilotic plant [papyrus]. Although this poultice was effective in assisting memory and attention, yet by its nature it kept people awake. She also brought out a file fashioned with great skill, which was divided into eight golden parts joined in different ways, and which darted back and forth—with which by gentle rubbing she gradually cleaned dirty teeth and ailments of the tongue and the filth which had been picked up in the town of Soloe [i.e., solecisms]. (p. 64-66)
This phonetics primer, a sort of proto-IPA, comes to a surprising climax:
We utter A with the mouth open, with a single suitable breath.
We make B by the outburst of breath from closed lips.
C is made by the back teeth brought forward over the back of the tongue.
D is made by bringing the tongue against the top teeth.
E is made by a breath with the tongue a little depressed.
F is made by the teeth pressing on the lower lip.
G, by a breath against the palate.
H is made by an exhalation with the throat a little closed.
I is made by a breath with the teeth kept close together.
K is made with the palate against the top of the throat.
L is a soft sound make with the tongue and the palate.
M is a pressing together of the lips.
N is formed by the contact of the tongue on the teeth.
O is made by a breath with the mouth rounded.
P is a forceful exhalation from the lips.
Q is a contraction of the palate with the mouth half-closed.
R is a rough exhalation with the tongue curled against the roof of the mouth.
S is a hissing sound with the teeth in contact.
T is a blow of the tongue against the teeth.
U is made with the mouth almost closed and the lips forward a little.
X is the sibilant combination of C and S.
Y is a breath with the lips close together.
Z was abhorrent to Appius Claudius, because it resembles in its expression the teeth of a corpse. (p. 75)
While Grammar was saying this, and Jupiter and the Delian were urging her forward, Pallas spoke up: “While Literature here is hurrying on to discuss the connection of syllables, she has passed over the historical aspect.” At this objection by the maiden goddess, Grammar in great agitation answered: “I know I must pass over a great deal, so as not to incur the distaste of the blessed by getting entangled in details. So I shall perform my purpose, hastening along the shortest ways, to avoid getting lost, hidden in thick undergrowth or a dense mass of briars.” (p. 76)
When Grammar had said this as if she were merely introducing her subject, Minerva intervened, because of the boredom that had come upon Jove and the celestial senate, and said: “Unless I am mistaken, you are getting ready to go back to the elements and begin telling us about the eight fundamental parts of speech, adding also the causes of solecisms, the barbarisms, and other faults of speech which celebrated poets have discussed at length; you will also discuss tropes, metaplasms, schemata, figures, and all the faults which flow, as it were, from the fountain of embellishment, illustrating either the misconception of the writer who does not understand them or the labored ornamentation of the pedant. If you bring such matters from the elementary school before the celestial senate, you will nip in the bud the goodwill you have won by this display of knowledge. If you were to take up a discussion of rhythm and meter, as you would venture to do with young pupils, Music would surely tear you apart for usurping her office. The teaching you have given us will be well-proportioned and complete if you keep to your own particular subjects and do not cheapen them by commonplace and elementary instruction.” (p. 105)
Dialectic
Fora’s 15th-century illustration of Dialectic, with her hooks and serpent
Into the assembly of the gods came Dialectic, a woman whose weapons are complex and knotty utterances. Without her, nothing follows, and likewise, nothing stands in opposition. She brought with her the elements of speech; and she had ready the school maxim which reminds us that speech consists in words which are ambiguous, and judges nothing as having a standard meaning unless it be combined with other words. Yet, though Aristotle himself pronounce his twice-five categories, and grow pale as he tortures himself in thought; though the sophisms of the Stoics beset and tease the senses, as they wear on their foreheads the horns they never lost; though Chrysippus heap up and consume his own pile, and Carneades match his mental power through the use of hellebore, no honor so great as this has ever befallen any of these sons of men, nor is it chance that so great an honor has fallen to your lot: it is your right, Dialectic, to speak in the realms of the gods, and to act as teacher in the presence of Jove.
So at the Delian’s summons this woman entered, rather pale but very keen-sighted. Her eyes constantly darted about; her intricate coiffure seemed beautifully curled and bound together, and descending by successive stages [editor: “The Latin here, deducti per quosdam consequentes gradus, applies equally well to a logical argument “deduced through certain successive steps” as to Dialectic’s symbolic hairstyle], it so encompassed the shape of her whole head that you could not have detected anything lacking, nor grasped anything excessive [editor: Remigius remarks that this may refer to the requirements of a good definition […] More probably it simply refers to the rigor and completeness of logical argument]. She was wearing the dress and cloak of Athens, it is true, but what she carried in her hands was unexpected, and had been unknown in all the Greek schools. In her left hand she held a snake twined in immense coils; in her right hand a set of patterns [editor: formulae] carefully inscribed on wax tablets, which were adorned with the beauty of contrasting color, was held on the inside by a hidden hook; but since her left hand kept the crafty device of the snake hidden under her cloak, her right hand was offered to one and all. Then if anyone took one of those patterns, he was soon caught on the hook and dragged toward the poisonous coils of the hidden snake, which presently emerged and after first biting the man relentlessly with the venomous points of its sharp teeth then gripped him in its many coils and compelled him to the intended position. If no one wanted to take any of the patterns, Dialectic confronted them with some questions; or secretly stirred the snake to creep up on them until its tight embrace strangled those who were caught and compelled them to accept the will of their interrogator.
Dialectic herself was compact in body, dark in appearance […] and she kept saying things that the majority could not understand. For she claimed that the universal affirmative was diametrically opposed to the particular negative, but that it was possible for them both to be reversed by connecting ambiguous terms to univocal terms [editor: This sentence remains opaque. […]]; she claimed also that she alone discerned what was true from what was false, as if she spoke with assurance of divine inspiration. She said she had been brought up on an Egyptian crag [editor: The original text may, however, have had urbe [city] instead of rupe [crag].] and then had migrated to Attica to the school of Parmenides, and there, while the slanderous report was spread abroad that she was devoted to deceitful trickery, she had taken to herself the greatness of Socrates and Plato.
This was the woman, well-versed in every deceptive argument and glorifying in her many victories, whom the Cyllenian’s two-fold serpent, rising on his staff, tried to lick at, constantly darting its tongues, while the Tritonian’s [Athena’s] Gorgon hissed the the joy of recognition. Meanwhile Bromius [Bacchus], the wittiest of the gods, who was completely unacquainted with her, said […] (pp. 106-108)
[On the darting eyes, I think immediately of Ayn Rand in this interview with Mike Wallace.]
Pallas ordered Dialectic to hand over those items which she had brought to illustrate her sharpness and her deadly sure assertions, and told her to put on an appearance suitable for imparting her skill. Grammar was standing close by when the introduction was completed; but she was afraid to accept the coils and gaping mouth of the slippery serpent. Together with the enticing patterns and the rules fitted with the hook, they were entrusted to the great goddess who had tamed the locks of Medusa. (p. 109)
For assessing virtue as well as practicing it, Jupiter considered the levity of the Greek inferior to the vigor of Romulus, so he ordered her to unfold her field of knowledge in Latin eloquence. Dialectic did not think she could express herself adequately in Latin; but presently her confidence increased, the movements of her eyes were confined to a slight quivering, and, formidable as she had been even before she uttered a word, she began to speak as follows:
“Unless amid the glories of the Latin tongue the learning and labor of my beloved and famous Varro had come to my aid, I could have been found to be a Greek by the test of Latin speech, or else completely uncultivated or even quite barbarous. Indeed, after the golden flow of Plato and the brilliance of Aristotle it was Marcus Terentius’ labors which first enticed me into Latin speech and made it possible for me to express myself throughout the schools of Ausonia. I shall therefore strive to obey my instructions and, without abandoning the Greek order of discussion, I shall not hesitate to express my propositions in the tongue of Laurentum. First, I want you to realize that the toga-clad Romans have not been able to coin a name for me, and that I am called Dialectic just as in Athens: and whatever the other Arts propound is entirely under my authority. Not even Grammar herself, whom you have just heard and approved, nor the lady renowned for the richness of her eloquence [Rhetoric], nor the one who draws various diagrams on the ground with her rod [Geometry], can unfold her subject without using my reasoning. (p. 110)
You should put up with the strangeness of my language, since you have compelled a Greek to treat the subject in Latin. (p. 111)
While Dialectic was holding forth in this way and getting on to matters as complicated as they were obscure, Maia’s son [Mercury] grew impatient and nodded to Pallas, who cut in: “Madam, you speak with great skill; but now stop your exposition before you get entangled in the complexities of your subkect, and its knotty problems exhaust the goodwill of Hymen. You have said in summary all that is fitting from that which learned discussion ahs contributed for the development of the subject in a large volume. A modest spring from deep learning is sufficient; it brings to light things hidden from sight, and avoids tedious discussion, without passing over anything and leaving it unrecognized. The matters that remain are founded on great deceit, and false deception encompasses those who are caught by them, while you prepare sophisms fraught with guile, or seductively make sport with trickeries from which one cannot get free. And when you gradually build up a sorites, or fashion errors which truth condemns, then your sin, your wicked deed, resounds in the ears of the Thunderer, since the lofty denizens of heaven hate everything false in a woman of shame. If you ponder it, what is more cruel than making sport of people? You have had your say, and you will surely become a disreputable and itinerant charlatan if you go on to build up your snares. Away then with shifty profundity, and leave what time remains to your sisters.” (p. 153)
Two weeks ago I read Charles Homer Haskins’s slim volume The Rise of Universities (1923), a charming collection of three lectures—”The Earliest Universities,” “The Mediaeval Professor,” “The Mediaeval Student”—on the birth of universities, especially at Bologna and Paris.
I came to Haskins to get my bearings after the disorientation of discovering, while skimming David Bressoud’s new book Calculus Reordered, that the history of science took an important step forward as early as the early 1300s—centuries before Galileo, et al.—when William Heytesbury and colleagues at Merton College in Oxford clarified the relationship between kinematics and dynamics, giving the first purely mathematical treatment of motion. (Heytesbury’s most important work, the Regulae solvendi sophismata—Rules for Solving Sophisms—seems not to have been translated in full into English.) The dark ages were not quite so dark, after all. Clifford Truesdell sums up the contributions of these so-called Oxford Calculators in his Essays in the History of Mechanics:
The now published sources prove to us, beyond contention, that the main kinematical properties of uniformly accelerated motions, still attributed to Galileo by the physics texts, were discovered and proved by scholars of Merton college. […] In principle, the qualities of Greek physics were replaced, at least for motions, by the numerical quantities that have ruled Western science ever since. The work was quickly diffused into France, Italy, and other parts of Europe. Almost immediately, Giovanni di Casale and Nicole Oresme found how to represent the results by geometrical graphs, introducing the connection between geometry and the physical world that became a second characteristic habit of Western thought.
Contrary to the received image of abortive medieval scholasticism, Haskins paints a portrait of rich intellectual ferment, drawing a great deal more continuity with the present than we usually assume [cf. the dispute over the so-called “continuity thesis” in the history of science]:
The occasion for the rise of universities was a great revival of learning, not that revival of the fourteenth and fifteenth centuries to which the term is usually applied, but an earlier revival, less known though in its way quite as significant, which historians now call the renaissance of the twelfth century. So long as knowledge was limited to the seven liberal arts of the early Middle Ages, there could be no universities, for there was nothing to teach beyond the bare elements of grammar, rhetoric, logic, and the still barer notions of arithmetic, astronomy, geometry, and music, which did duty for an academic curriculum. Between 1100 and 1200, however, there came a great influx of new knowledge into western Europe, partly through Italy and Sicily, but chiefly through the Arab scholars of Spain—the works of Aristotle, Euclid, Ptolemy, and the Greek physicians, the new arithmetic, and those texts of the Roman law which had lain hidden through the Dark Ages. In addition to the elementary propositions of triangle and circle, Europe now had those books of plane and solid geometry which have done duty in schools and colleges ever since; instead of the painful operations with Roman numerals—how painful one can readily see by trying a simple problem of multiplication or division with these characters—it was now possible to work readily with Arabic figures; in the place of Boethius, the “Master of them that know” became the teacher of Europe in logic, metaphysics, and ethics. In law and medicine men now possessed the fullness of ancient learning. This new knowledge burst the bonds of the cathedral and monastery schools and created the learned professions; it drew over mountains and across the narrow seas eager youths who, like Chaucer’s Oxford clerk of a later day, “would gladly learn and gladly teach,” to form in Paris and Bologna those academic gilds which have given us our first and our best definition of a university, a society of masters and scholars.
Later in the book, Haskins notes that this renaissance
added to the store of western knowledge the astronomy of Ptolemy, the complete works of Euclid, and the Aristotelian logic, while at the same time under the head of grammar great stimulus was given to the study and reading of the Latin classics. This classical revival, which is noteworthy and comparatively little known, centered in such cathedral schools as Chartres and Orleans, where the spirit of a real humanism showed itself in an enthusiastic study of ancient authors and in the production of Latin verse of a really remarkable quality. Certain writings of one of these poets, Bishop Hildebert of Le Mans, were even mistaken for “real antiques” by later humanists. Nevertheless, though brilliant, this classical movement was short-lived, crushed in its early youth by the triumph of logic and the more practical studies of law and rhetoric. In the later twelfth century John of Salisbury inveighs against the logicians of his day, with their superficial knowledge of literature; in the university curriculum of the thirteenth century, literary studies have quite disappeared. Toward 1250, when a French poet, Henri d’Andeli, wrote his Battle of the Seven Arts, the classics are already the ancients, fighting a losing battle against the moderns:
Logic has the students,
Whereas Grammar is reduced in numbers.
[…]
Civil Law rode gorgeously
And Canon Law rode haughtily
Ahead of all the other arts.
If the absence of the ancient classics and of vernacular literature is a striking feature of the university curriculum in arts, an equally striking fact is the amount of emphasis placed on logic or dialectic. The earliest university statutes, those of Paris in 1215, require the whole of Aristotle’s logical works, and throughout the Middle Ages these remain the backbone of the arts course, so that Chaucer can speak of the study of logic as synonymous with attendance at a university—
That un-to logik hadde longe y-go.
In a sense this is perfectly just, for logic was not only a major subject of study itself, it pervaded every other subject as a method and gave tone and character to the mediaeval mind. Syllogism, disputation, the orderly marshalling of arguments for and against specific theses, these became the intellectual habit of the age in law and medicine as well as in philosophy and theology. The logic, of course, was Aristotle’s, and the other works of the philosopher soon followed, so that in the Paris course of 1254 we find also the Ethics, the Metaphysics, and the various treatises on natural science which had at first been forbidden to students. To Dante Aristotle had become “the Master of them that know,” by virtue of the universality of his method no less than of his all-embracing learning. “The father of book knowledge and the grandfather of the commentator,” no other writer appealed so strongly as Aristotle to the mediaeval reverence for the text-book and the mediaeval habit of formal thought. Doctrines like the eternity of matter which seemed dangerous to faith were explained away, and great and authoritative systems of theology were built up by the methods of the pagan philosopher. And all idea of literary form disappeared when everything depended on argument alone.
Some choice passages from Renée Neu Watkins’s translation (Waveland Press, 1999) of Leon Battista Alberti’s magnificently shrewd De commodis litterarum atque incommodis (The advantages and disadvantages of books), written in Latin around 1430:
[…] I have meditated and thought long and hard, searching with all my ingenuity for a subject I could treat adequately and which would prove the quality of my intellect, so as to satisfy them if it lay within me. But nothing came to mind that had not been beautifully dealt with by the divinely inspired classical authors, so that no one of our time, however learned, could deal with it better than they, nor did there seem to be some topic left of the kind they had treated, that I could handle well and with grace. The ancients had encompassed all serious and comic material, leaving to us only the opportunity to read them and the obligation to admire. Older contemporaries of ours have seized on a few subjects that lay hidden, perhaps overlooked by the ancients, and have thus gained honor and fame. If one wants glory, however, one must be willing to write something that is not perfect and ideal rather than allow oneself to grow old in erudite silence.
I earnestly beg you, my brother (if I may borrow your own phrasing in Ephebis) to read this little book of ours, correct it according to your own unswerving judgment, and by your emendations kindly make it better and more beautiful.
The life men of learning live is necessarily hard and harsh; by this I mean the ones who, as they should, abandon all other things for the sake of intellectual work. No art, however minor, demands less that total dedication if you want to excel in it. What we know to be true of all arts is most especially true of reading and writing; there is no freedom from striving at any age.
In my experience, however, you won’t find many rich men who think books, let alone the delights of study, are worth the effort.
Who, with a mind occupied by love, will be able to focus whole and steadfast attention on texts? Who can then be fully absorbed in his work, intent on the teachings, ready and able to store up and retain them? Who, when captive to the madness of love, will have the will power and intellectual vigor and enthusiasm to perfect himself in any noble art? Don’t we know how love usually affects people? Sapping energy, corrupting conduct, perverting the intelligence, loading the mind with obsessions, filling the intellect with errors, driving a man to madness: these are its well-known services, the gifts that it bestows.
A brief period away from study has the power to disperse more material than many long hours of application can restore; things placed in memory slip away faster than they can be rememorized or recaptured.
[Cf. the pianist’s quip, variously attributed to Liszt, Rubinstein, and Paderewski: “If I miss one day of practice, I notice; two days, and my friends notice; three days, and everybody notices.”]
When you wish to buy clothes, isn’t it true that your library will say to you: “You owe me that money, I forbid…” If you wish to pursue the hunt, or music, or the martial arts or sports, won’t the books says: “You are stealing this energy from us, we will not bring you fame and reputation!” If you inquire into technical knowledge or painting or three dimensional design, the philosophical disciplines will react strongly: “This is the way you defraud us of your energies. From you we will withhold knowledge of the highest things…!”
If you want to refresh your spirit by a country excursion […] the vocation you have taken up will pull you back from there to books and writing, and if you do not with much labor and long hours devote yourself totally to these, the books themselves will threaten you with shattering disgrace.
But I would not want to obscure the true nature of scholars by concluding that they devote themselves to books with no idea of pleasure. They could not perform such great labors without some notion of pleasure in their minds. There are those who willingly go into mourning, because they take pleasure from being considered very faithful and true to the memory of friendship. Many actions by which we satisfy convention and public opinion seem less painful to us than they really are. The pleasure of study, however, is such that it might better be called pain: sedentary all the time, reading all the time, thinking hard, always alone, renouncing festivities and play. I am not so bitter and hard a man that I would dare call this a pleasant way to live. […] To satisfy the desire to learn is indeed a pleasure, but the very hard work of study and the accompanying anxiety that oppresses the spirit always bring more mental torment than joy. So if we indeed take a certain pleasure in learning, huge cares and labors undermine it. There is a big difference, moreover, between the burden of fighting the intrigues and assault of enemies, which is experienced relatively briefly, and the scholar’s daily anxiety, which is perpetual and immense. For there are innumerable things in books that are supremely worth knowing, nor is it easy to describe how the desire to learn presses upon a scholar. He may participate in difficulty scholarly debates, or explore some elegant, worthy, and learned subject; while he does so, he does not sleep, does not eat, does not rest, and feels almost no satisfaction. The desire to know and to remember it all is constantly gnawing at him. He takes on immense projects, is entangled in an array of possible rhetorical devices, is constantly tense. On top of this, he is always coming across things previously unknown to him: he encounters in his reading adroit, subtle, and clever ideas, finds some unusual illustrative anecdote, or learns new refinements of the power of persuasion; these things provoke in him the desire to learn more, and he is unable to set limits or stop, nor is he granted any peace of mind as long as he has not cleared up every obscurity. Thus, as you see, the scholar is a very complex puzzle himself, and neither physically nor mentally ever, or hardly ever, gets any rest. Bleak solitude, hard labor, endless hours, great anxiety, difficulty questions, total absorption, intense anxiety—as there is no pleasure to be found in this man, so in his whole life there is almost no break in the onslaught of work and worry.
To Alberti’s unremittingly bleak, if otherwise unsurpassed, characterization of the obsessively acquisitive and self-reinforcing anxiety at the heart of bibliophilia (epistemophilia?), one might counterpose a finer appreciation of the compensating blisses and ecstasies: Barthes’s “pleasure of the text,” Feynman’s “pleasure of finding things out,” Morris’s account of “pleasure in the work itself” (which he writes about here and here), something—appropriately, I cannot remember what—in Emerson. See also: Aristotle’s “desire to know,” Wordsworth’s “bliss of solitude,” Joyce’s “luminous silent stasis of aesthetic pleasure.”
