The versions of the Liar paradox which use empirical predicates already point up one major aspect of the problem: many, probably most, of our ordinary assertions about truth and falsity are liable, if the empirical facts are extremely unfavorable, to exhibit paradoxical features. Consider the ordinary statement, made by Jones:
(1) Most (i.e., a majority) of Nixon’s assertions about Watergate are false.
Clearly, nothing is intrinsically wrong with (1), nor is it ill-formed. Ordinarily the truth value of (1) will be ascertainable through an enumeration of Nixon’s Watergate-related assertions, and an assessment of each for truth or falsity. Suppose, however, that Nixon’s assertions about Watergate are evenly balanced between the true and the false, except for one problematic case,
(2) Everything Jones says about Watergate is true.
Suppose, in addition, that (1) is Jones’s sole assertion about Watergate, or alternatively, that all his Watergate-related assertions except perhaps (1) are true. Then it requires little expertise to show that (1) and (2) are both paradoxical: they are true if and only if they are false.
The example of (1) points up an important lesson: it would be fruitless to look for an intrinsic criterion that will enable us to sieve out—as meaningless, or ill-formed—those sentences which lead to paradox. (1) is, indeed, the paradigm of an ordinary assertion involving the notion of falsity; just such assertions were characteristic of our recent political debate. Yet no syntactic or semantic feature of (1) guarantees that it is unparadoxical. Under the assumptions of the previous paragraph, (1) leads to paradox. Whether such assumptions hold depends on the empirical facts about Nixon’s (and other) utterances, not on anything intrinsic to the syntax and semantics of (1). (Even the subtlest experts may not be able to avoid utterances leading to paradox. It is said that Russell once asked Moore whether he always told the truth, and that he regarded Moore’s negative reply as the sole falsehood Moore had ever produced. Surely no one had a keener nose for paradox than Russell. Yet he apparently failed to realize that if, as he thought, all Moore’s other utterances were true, Moore’s negative reply was not simply false but paradoxical.) The moral: an adequate theory must allow our statements involving the notion of truth to be risky: they risk being paradoxical if the empirical facts are extremely (and unexpectedly) unfavorable. There can be no syntactic or semantic “sieve” that will winnow out the “bad” cases while preserving the “good” ones.
In the Second Analogy Kant expresses in a number of ways the thought that the order of perceptions of htose objective states of affairs the succession of one upon the other of which constitutes an objective change is—as, in the sense examined and with the qualifications mentioned, we see it is—a necessary order. The order of perceptions is characterized not only as a necessary, but as a determined order, an order to which our apprehension is bound down, or which we are compelled to observe. These may all perhaps be admitted as legitimate ways of expressing the denial of order-indifference. But from this point the argument proceeds by a non sequitur of numbing grossness.
A striking passage from Frege’s unpublished essay “Boole’s logical calculus and the concept-script,” which he submitted in 1881 first to Zeitschrift für Mathematik und Physik, then the MathematischenAnnalen, and finally the Zeitschrift für Philosophie und philophische Kritik, but was rejected each time:
I now return once more to the examples mentioned earlier, so as to point out the sort of concept formation that is to be seen in those accounts. The fourth example gives us the concept of a multiple of 4 […]. The eighth example gives us the concept of the congruence of two numbers with respect to a modulus, the 13th that of the continuity of a function at a point etc. All these concepts have been developed in science and have proved their fruitfulness. For this reason what we may discover in them has a far higher claim on our attention than anything that our everyday trains of thought might offer. For fruitfulness is the acid test of concepts, and scientific workshops the true field of study for logic.
This is pp. 32-33 in the translation by Peter Long and Roger White, with the assistance of Raymond Hargreaves, in Posthumous Writings, edited by Hans Hermes et al (Blackwell, 1979).
This remains one of the lesser appreciated themes in Frege’s work. The standard fare in a philosophy major mentions sense and reference, concept and anti-psychologism, but I’d bet even very few working philosophers are familiar with these suggestive passages on fruitfulness. Of course, every time I read the word I can only think of Keats, on autumn: “Season of mists and mellow fruitfulness…”
Jamie Tappenden, “The Mathematical and Logical Background to Analytic Philosophy,” The Oxford Handbook of the History of Analytic Philosophy
Jamie Tappenden, “Extending knowledge and ‘fruitful concepts’: Fregean themes in the philosophy of mathematics,” in Gottlob Frege: Critical Assessments of LeadingPhilosophers, Vol. III, Frege’s philosophy of mathematics, edited by Michael Beany and Erich Reck
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):
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)
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.
Samuel Taylor Coleridge, age 18, to his brother George, with a very green poem setting Euclidean reasoning to verse:
I have often been surprising that Mathematics, the quintessence of Truth, should have found admirers so few and so languid. Frequent consideration and minute scrutiny have at length unravelled the case; viz. that though Reason is feasted, Imagination is starved; whilst Reason is luxuriating in its proper Paradise, Imagination is wearily travelling on a dreary desert. To assist Reason by the stimulus of Imagination is the design of the following production. In the execution of it much may be objectionable. The verse (particularly in the introduction of the ode) may be accused of unwarrantable liberties, but they are liberties equally homogeneal with the exactness of Mathematical disquisition, and the boldness of Pindaric daring. I have three strong champions to defend me against the attacks of Criticism; the Novelty, the Difficulty, and the Utility of the work. I may justly plume myself, that I first have drawn the nymph Mathesis from the visionary caves of abstracted Idea, and caused her to unite with Harmony. The first-born of this Union I now present to you; with interested motived indeed—as I expect to receive in return the more valuable offspring of your Muse.
This is now—this was erst,
Proposition the first—and Problem the first.
On a given finite line
which must no way incline;
To describe an equi—
—A, N, G, E, L, E.
Now let A. B.
Be the given line
Which must no way incline;
The great Mathematician
Makes the Requisition,
That we describe an Equi—
—angle on it:
Aid us Reason—aid us Wit!
From the centre A. at the distance A. B.
Describe the circle B. C. D.
At the distance B. A. from B. the centre
The round A. C. E. to describe boldly venture.
(Third postulate see.)
And from the point C.
In which the circles make a pother
Cutting and slashing one another,
Bid the straight lines a journeying go.
C. A. C. B. those lines will show
To the points, which by A. B. are reckon’d,
And postulate the second
For authority ye know.
A. B. C.
Triumphant shall be
An Equilateral Triangle,
Not Peter Pindar carp, nor Zoilus can wrangle.
Because the point A. is the centre
Of the circular B. C. D.
And because the point B. is the centre
Of the circular A. C. E.
A. C. to A. B. and B. C. to B. A.
Harmoniously equal must forever stay;
Then C. A. and B. C.
Both extend the kind hand
To the basis A. B,
Unambitiously join’d in Equality’s Band.
But to the same powers, when two powers are equal
My mind forebodes the sequel;
My mind does some celestial impulse teach,
And equalizes each to each.
Thus C. A. with B. C. strikes the same sure alliance.
That C. A. and B. C. had with A. B. before
And in mutual affiance
None attempting to soar
The unanimous three
C. A. and B. C. and A. B.
All are equal, each to his brother,
Preserving the balance of power so true:
Ah! the like would the proud Autocratix do!
At taxes impending not Britain would tremble,
Nor Prussia struggle her fear to dissemble;
Nor the Mah’met-sprung wight
The great Mussulman
Would stain his Divan
With Urine the soft-flowing daughter of Fright.
But rein your stallion in, too daring Nine!
Should Empires bloat the scientific line?
Or with dishevell’d hair all madly do ye run
For transport that your task is done?
For done it is—the cause is tried!
And Proposition, gentle maid,
Who soothly ask’d stern Demonstration’s aid,
Has prov’d her right, and A. B. C.
Of angles three
Is shown to be of equal side;
And now our weary stead to rest in fine,
‘Tis raised upon A. B. the straight, the given line.
from George Eliot, Middlemarch, with a funny rhythmic echo of the bromide “all good things come to an end, but diamonds are forever” in the second sentence:
Miss Brooke argued from words and dispositions not less unhesitatingly than other young ladies of her age. Signs are small measurable things, but interpretations are illimitable, and in girls of sweet, ardent nature, every sign is apt to conjure up wonder, hope, belief, vast as a sky, and colored by a diffused thimbleful of matter in the shape of knowledge. They are not always too grossly deceived; for Sinbad himself may have fallen by good-luck on a true description, and wrong reasoning sometimes lands poor mortals in right conclusions: starting a long way off the true point, and proceeding by loops and zigzags, we now and then arrive just where we ought to be. Because Miss Brooke was hasty in her trust, it is not therefore clear that Mr. Casaubon was unworthy of it.
We hear a lot about being right for the wrong reasons, but not so much about being wrong for the right reasons—arguably just as common, if not more so, and perhaps less of a sin. As for being wrong for the wrong reasons, that is still not so bad as being “not even wrong.”
If we care to be scholastic, we might map this fourfold way onto the apparatus of informal logic. If we fudge Eliot’s focus on “conclusions” and take rightness instead to be a matter of having given true premises, then to be right for right reasons is to be sound; to be wrong for right reasons is to be valid but unsound; to be right for wrong reasons is to be invalid and epistemically lucky; and to be wrong for wrong reasons is simply to be a user of Twitter.
If we care, instead, to be cancelled, we might look to the work of heterodox philosopher Donald Rumsfeld, who took his cue from analytical chemistry. In this typology there are known knowns, known unknowns, unknown knowns, and unknown unknowns. (Rumsfeld himself is an instance of the third.) He thus extends the great philosophical tradition of drawing squares, from Plato and Aristotle to Levi-Strauss.
The method has become so popular it has since been taken up by statisticians.
The effect of a lecture depends upon the habits of the listener; because we expect the language to which we are accustomed, and anything beyond this seems not to be on the same level, but somewhat strange and unintelligible on account of its unfamiliarity; for it is the familiar that is intelligible. The powerful effect of familiarity is clearly shown by the laws, in which the fanciful and puerile survivals prevail [i.e., “in which fanciful and childish elements prevail”—ML], through force of habit, against our recognition of them. Thus some people will not accept the statements of a speaker unless he gives a mathematical proof; others will not unless he makes use of illustrations; others expect to have a poet adduced as witness. Again, some require exactness in everything, while others are annoyed by it, either because they cannot follow the reasoning or because of its pettiness; for there is something about exactness which seems to some people to be mean, no less in an argument than in a business transaction.
Hence one must have been already trained how to take each kind of argument, because it is absurd to seek simultaneously for knowledge and for the method of obtaining it; and neither is easy to acquire. Mathematical accuracy is not to be demanded in everything, but only in things which do not contain matter. Hence this method is not that of natural science, because presumably all nature is concerned with matter. Hence we should first inquire what nature is; for in this way it will become clear what the objects of natural science are [and whether it belongs to one science or more than one to study the causes and principles of things].
The effect which lectures produce on a hearer depends on his habits; for we demand the language we are accustomed to, and that which is different from this seems not in keeping but somewhat unintelligible and foreign because of its unwontedness. For the customary is more intelligible. The force of habit is shown by the laws, in whose case, with regard to the legendary and childish elements in them, habit has more influence than our knowledge about them. Some people do not listen to a speaker unless he speaks mathematically, others unless he gives instances, while others expect him to cite a poet as witness. And some want to have everything done accurately, while others are annoyed by accuracy, either because they cannot follow the connexion of thought or because they regard it as pettifoggery. For accuracy has something of this character, so that as in trade so in argument some people think it mean. Therefore one must be already trained to know how to take each sort of argument, since it is absurd to seek at the same time knowledge and the way of attaining knowledge; and neither is easy to get.
The minute accuracy of mathematics is not to be demanded in all cases, but only in the case of things which have no matter. Therefore its method is not that of natural science; for presumably all nature has matter. Hence we must inquire first what nature is: for thus we shall also see what natural sciences treats of [and whether it belongs to one science or to more to investigate the causes and the principles of things].
The Greek for “exactness” here is ἀκριβολογία, akribologia, a combination of λόγος and the adjective ἀκριβής (exact, precise, scrupulous, methodical).
In extant ancient Greek literature the word often carries a negative valence: excessive precision, pedantry. It appears perhaps most famously in Book 1 of Plato’s Republic at 340e, where Thrasymachus accuses Socrates of too much akribologia: “ὥστε κατὰ τὸν ἀκριβῆ λόγον, ἐπειδὴ καὶ σὺ ἀκριβολογῇ, …” consequently, according to precise speech, since you too demand precision... (In Paul Shorey’s translation, “so that, speaking precisely, since you are such a stickler for precision.” Shorey’s footnote: “For the invidious associations of ἀκριβολογία (1) in money dealings, (2) in argument, cf. Aristotle Met. 995 a 11, Cratylus 415 A, Lysias vii. 12, Antiphon B 3, Demosthenes. xxiii. 148, Timon in Diogenes Laertius ii. 19.”)
Erasmus makes reference to this line of Thrasymachus in the first, slimmer edition of his adages, Collectanea Adagiorum, published in Paris in 1500. In the translation by John Grant, based on the slightly revised and updated 1506 edition, the discussion appears at 335:
335. Ad vivum. Summo iure / To the quick. By the letter of the law
The meaning is “right to the skin.” We use the expression to refer to actions that are conducted with utmost precision, as when we pursue something with excessive keenness. In Plato Thrasymachus calls Socrates a false accuser, meaning a pettifogger, because he applies a very narrow interpretation to what has been said, distorting the sense of the words whose meaning is clear rather than showing how somewhat carelessly expressed words can be given a better sense. He adds, “Therefore, according to your precise mode of interpretation (since you cut right to the quick), no craftsman can make a mistake.” Similar to this is what Cicero says when defending Caecina: “All the others turn to that way of speaking when they think they have a fair and good defense to make in a case. If, however, there is a wrangling about words and phrases, and, as the saying goes, the letter of the law is applied, they are in the habit of using such fine words as “fair” and “good” to counter such wickedness.” To fight Summo iure “By the letter of the law” means to cut back the laws to the quick and to apply a very narrow interpretation. From this we get “Extreme right is extreme wrong.”
Sachiko Kusukawa traces this lineage in her essay “Ad Vivum Images and Knowledge of Nature in Early Modern Europe.” As she points out, Cicero uses the phrase ad vivum in his dialogue De Amicitia,On Friendship, translated here by W. A. Falconer:
This, however, I do feel first of all—that friendship cannot exist except among good men; nor do I go into that too deeply [neque in ad vivum reseco], as is done by those who, in discussing this point with more than usual accuracy [subtilius], and it may be correctly, but with too little view to practical results, say that no one is good unless he is wise. We may grant that; but they understand wisdom to be a thing such as no mortal man has yet attained. I, however, am bound to look at things as they are in the experience of everyday life and not as they are in fancy or in hope.
Erasmus makes reference to Cicero’s usage of ad vivum resecare in the longer edition of his adages, at II.4.13:
M. Tullius lib. De amicitia Ad vivum resecare dixit pro eo, quod est rem exactius, quam sat est, ac morosius excutere: Sed hoc, inquit, primum sentio, nisi in bonis amicitiam esse non posse. Neque id ad vivum reseco, ut illi, qui haec subtilius disserunt. Mutuo sumpta metaphora a tonsoribus capillos aut ungues resecantibus, nam ii saepenumero molesti sunt, dum nimium diligentes esse student. Idem in libris De finibus dixit pressius agere pro exactius et accuratius. Plautus in Bacchidibus sub persona Chrysali: Tondebo auro usque ad vivam cutem. Et hoc ipsum tondere pro deludere Graecis in proverbio est.
Kusukawa gives the translation of “mutuo sumpta…”: “the image is borrowed from hairdressers as they cut short one’s hair or finger-nails; for they are often tiresome with their efforts to be needlessly precise.” As she notes, Plato’s use of akribologei
had been translated by the Florentine scholar Marisilio Ficino (1433–1499) into “ad vivum resecas (you speak needlessly precisely).” This is probably the reason why Erasmus connected the two classical passages of Cicero and Plato, but ad vivum was by no means a fixed Latin counterpart to akribologia, a word that was also found in Aristotle’s Metaphysics. [She quotes what I quote above.] In the medieval translation by William of Moerbeke (ca. 1220–1235–ca. 1286), the Greek phrase akribologia had been left untranslated, but the Byzantine humanist John Argyropoulos (1415–1487) rendered this mathematical akribologia as “exacta discussio mathematicorum (mathematicians’ exacting examination).” Another point made in the passage above [that is, Aristotle] is that overattention to detail in discussion or transaction was deemed “mean” (illiberales in Argyropoulos’s translation). The mean-spiritedness of excessive precision was carried over to the phrase “exigere ad vivum,” which Erasmus identified as a characteristic harshness (rigor). Ad vivum thus meant something like “to the bare bones” in English, with the negative sense of verbatim, an overattention to the strict sense of a word or to the letter of the law which reflected some meanness in spirit. Erasmus, whose ambition was to educate his audience to speak and write with the rhetorical flair of the classical authors, had little positive to say about this sense of ad vivum.
On the anomalous quality of Book 2 of the Metaphysics, see Werner Jaeger, Aristotle: Fundamentals of the History of His Development, p. 169:
Aristotle’s literary executors were philosophers. They would have given much to be able to construct, out of the precious papers that they found, as true a picture as possible of the whole intellectual system of ‘first philosophy’ as Aristotle had intended it to be; but their desire was thwarted by the incomplete and disparate character of the material. For one thing is certain; the editors themselves did not believe that with the order which they established they were giving posterity the complete course of lectures on metaphysics. They realized that they were offering an unsatisfactory makeshift, which was all that the condition of their materials allowed. The postscript to the introductory book, the so-called little α, comes after big Α simply because they did not know where else to put it. It is a remnant of notes taken at a lecture by Pasicles, a nephew of Aristotle’s disciple Eudemus of Rhodes. [With footnote: “Asclepius, in his commentary on the Metaphysics (p. 4 l. 20, in Hayduck), refers this information, which reached him as a tradition handed down m the Peripatetic school, to Α; but this is a confusion. His account must come from notes taken at a lecture by Ammonius, and obviously he misheard. The true account is given by the scholiast on little α in the codex Parisinus (cf. Ent. Met. Arist., p. 114).”]
From the translators’ preface of the first English edition (1685) of Arnauld’s La Logique ou l’art du penser (1662):
The Common Treatises of Logic are almost without number, and while every Author strives to add something of his own, sometimes little to the purpose, sometimes altogether from the matter, the Art is become, not only Obscure and Tedious, but in a great measure Impertinent and Useless.
Thus the Schoolmen may be said to have clogg’d and fetter’d Reason, which ought to be free as Air, and plain as Demonstration itself, with vain misapplications of this Art to Notion and Nicety, while they make use of it only to maintain litigious Cavils and wrangling Disputes. So that indeed the common Logics are but as so many Counterscarps to shelter the obstinate and vain-glorious, that disdain Submission and Convincement, and therefore retire within their Fortifications of difficult Terms, wrap themselves up in Quirk and Suttlety, and so escape from Reason in the Clouds and Mists of their own Raising.
Barthes, Le Plaisir du Texte, 1973, translated by Richard Miller:
Imagine someone (a kind of Monsieur Teste in reverse) who abolishes within himself all barriers, all classes, all exclusions, not by syncretism but by simple disregard of that old specter: logical contradiction; who mixes every language, even those said to be incompatible; who silently accepts every charge of illogicality, of incongruity; who remains passive in the face of Socratic irony (leading the interlocutor to the supreme disgrace: self-contradiction) and legal terrorism (how much penal evidence is based on a psychology of consistency!). Such a man would be the mockery of our society: court, school, asylum, polite conversation would cast him out: who endures contradiction without shame? Now this anti-hero exists: he is the reader of the text at the moment he takes his pleasure.