Valéry’s most striking achievement is perhaps his monumental intellectual diary, called the Cahiers (Notebooks). Early every morning of his adult life, he contributed something to the Cahiers, prompting him to write: “Having dedicated those hours to the life of the mind, I thereby earn the right to be stupid for the rest of the day.”
The subjects of his Cahiers entries often were, surprisingly, reflections on science and mathematics. In fact, arcane topics in these domains appear to have commanded far more of his considered attention than his celebrated poetry. The Cahiers also contain the first drafts of many aphorisms he later included in his books. To date, the Cahiers have been published in their entirety only as photostatic reproductions, and only since 1980 have they begun to receive scholarly scrutiny. The Cahiers have been translated into English in five volumes published by Peter Lang with the title Cahiers/Notebooks.
Last week, for the first time in fourteen and a half years, I visited the used bookstore I used to frequent in high school.
It hardly has changed. The walls are the same hunter green, the shelves still crammed with more books, and more good books, than one expects to find there. I saw and smelled my way back into a life I had forgotten I could remember. There was the section where I first read Prufrock in an original edition, the music of the twenty-two-year-old Eliot writing about a middle-aged man making more an impression of sound than sense to my seventeen-year-old ears, the words issuing, in part, some promise of the learning and the language Cambridge would hold for me; there was the table in the back corner where I first played chess, hardly worse than I do today, but at least then I won.
And though I did not buy my copy of A Portrait of the Artist as a Young Man there—instead I used an ugly, serviceable edition from Barnes & Noble, paired with Dubliners, the pages of which are filled with overwrought teenage marginalia I now wince to read—suddenly I found myself thinking of the last paragraph of this familiar passage as I stood there as I might have when my mother was still alive, a boy wondering what it meant to be both an artist and a young man, seduced by the surprise of the scholastic secret, the knowledge of words, and as I quoted the words to myself, that phrase I would take as a credo, I recalled that moment of first being taken in by the idea of beauty, reading these pages to myself in that solitary salvific silence Stephen describes, paying no mind to Lynch, and knowing that cardiac condition, which I still know, however blushingly:
—To finish what I was saying about beauty, said Stephen, the most satisfying relations of the sensible must therefore correspond to the necessary phases of artistic apprehension. Find these and you find the qualities of universal beauty. Aquinas says: Ad pulcritudinem tria requiruntur integritas, consonantia, claritas. I translate it so: Three things are needed for beauty, wholeness, harmony, and radiance. Do these correspond to the phases of apprehension? Are you following?
—Of course, I am, said Lynch. If you think I have an excrementatious intelligence run after Donovan and ask him to listen to you.
Stephen pointed to a basket which a butcher’s boy had slung inverted on his head.
—Look at that basket, he said.
—I see it, said Lynch.
—In order to see that basket, said Stephen, your mind first of all separates the basket from the rest of the visible universe which is not the basket. The first phase of apprehension is a bounding line drawn about the object to be apprehended. An esthetic image is presented to us either in space or in time. What is audible is presented in time, what is visible is presented in space. But temporal or spatial, the esthetic image is first luminously apprehended as shelfbounded and self-contained upon the immeasurable background of space or time which is not it. You approach it as one thing. You apprehend its wholeness. That is integritas.
—Bull’s eye! said Lynch, laughing. Go on.
—Then, said Stephen, you pass from point to point, led by its formal lines; you apprehend it as balanced part against part within it limits; you feel the rhythm of its structure. In other words, the synthesis of immediate perception is followed by the analysis of apprehension. Having first felt that it is one thing you feel now that it is a thing. You apprehend it as complex, multiple, divisible, separable, made up of its parts, the result of its parts and their sum, harmonious. That is consonantia.
—Bull’s eye again! said Lynch, wittily. Tell me now what is claritas and you win the cigar.
—The connotation of the word, Stephen said, is rather vague. Aquinas uses a term which seems to be inexact. It baffled me for a long time. It would lead you to believe that he had in mind symbolism or idealism, the supreme quality of beauty being a light from some other world, the idea of which the matter was but the shadow, the reality of which it was but the symbol. I thought he might mean that claritas was the artistic discovery and representation of the divine purpose in anything or a force of generalization which would make the esthetic image a universal one, make it outshine proper conditions. But that is literary talk. I understand it so. When you have apprehended that basket as one thing and have then analyzed it according to its form and apprehended it as a thing you make the only synthesis which is logically and esthetically permissible. You see that it is that thing which it is and no other thing. The radiance of which he speaks is the scholastic quidditas, the whatness of a thing. This supreme quality is felt by the artist when the esthetic image is first conceived in his imagination. The mind in that mysterious instant Shelley likened beautifully to a fading coal. The instant wherein that supreme quality of beauty, the clear radiance of the esthetic image, is apprehended luminously by the mind which has been arrested by its wholeness and fascinated by its harmony is the luminous silent stasis of esthetic pleasure, a spiritual state very like to that cardiac condition which the Italian physiologist Luigi Galvani, using a phrase almost as beautiful as Shelley’s, called the enchantment of the heart.
from Borges’s essay “El idioma analítico de John Wilkins,” published in Otras Inquisiciones, translated by Ruth L. C. Simms, later made famous by the opening of Foucault’s Les mots et les choses:
At one time or another, we have all suffered through those unappealable debates in which a lady, with copious interjections and anacolutha, swears that the word luna is more (or less) expressive than the word moon. Apart from the self-evident observation that the monosyllable moon may be more appropriate to represent a very simple object than the disyllabic word luna, nothing can be contributed to such discussions. After the compound words and derivatives have been taken away, all the languages in the world (not excluding Johann Martin Schleyer’s volapük and Peano’s romance-like interlingua) are equally inexpressive. There is no edition of the Royal Spanish Academy Grammar that does not ponder “the envied treasure of picturesque, happy and expressive words in the very rich Spanish language,” but that is merely an uncorroborated boast. Every few years the Royal Academy issues a dictionary to define Spanish expressions. In the universal language conceived by Wilkins around the middle of the seventeenth century each word defines itself. Descartes had already noted in a letter dated November, 1629, that by using the decimal system of numeration we could learn in a single day to name all quantities to infinity, and to write them in a new language, the language of numbers. He also proposed the formation of a similar, general language that would organize and contain all human thought. Around 1664 John Wilkins began to undertake that task.
Wilkins divided the universe into forty categories or classes, which were then subdivisible into differences, subdivisible in turn into species. To each class he assigned a monosyllable of two letters; to each difference, a consonant; to each species, a vowel. For example, de means element; deb, the first of the elements, fire; deba, a portion of the element of fire, a flame. In a similar language invented by Letellier (1850) a means animal; ab, mammalian; abi, herbivorous; abiv, equine; abo, carnivorus; aboj, feline; aboje, cat; etc. In the language of Bonifacio Sotos Ochando ( 1845) imaba means building; imaca, brothel; imafe, hospital; imafo, pesthouse; imari, house; imaru, country estate; imede, pillar; imedo, post; imego, floor; imela, ceiling; imogo, window; bire, bookbinder, birer, to bind books. (I found this in a book published in Buenos Aires in 1886: the Curso de lengua universal by Dr. Pedro Mata.)
The words of John Wilkins’s analytical language are not stupid arbitrary symbols; every letter is meaningful, as the letters of the Holy Scriptures were meaningful for the cabalists. Mauthner observes that children could learn Wilkins’s language without knowing that it was artificial; later, in school, they would discover that it was also a universal key and a secret encyclopedia.
After defining Wilkins’s procedure, one must examine a problem that is impossible or difficult to postpone: the meaning of the fortieth table, on which the language is based. Consider the eighth category, which deals with stones. Wilkins divides them into the following classifications: ordinary (Hint, gravel, slate); intermediate (marble, amber, coral); precious (pearl, opal); transparent (amethyst, sapphire); and insoluble (coal, clay, and arsenic). The ninth category is almost as alarming as the eighth. It reveals that metals can be imperfect (vermilion, quicksilver); artificial (bronze, brass); recremental (filings, rust); and natural (gold, tin, copper). The whale appears in the sixteenth category: it is a viviparous, oblong fish. These ambiguities, redundances, and deficiencies recall those attributed by Dr. Franz Kuhn to a certain Chinese encyclopedia entitled Celestial Emporium of Benevolent Knowledge. On those remote pages it is written that animals are divided into (a) those that belong to the Emperor, (b) embalmed ones, (c) those that are trained, (d) suckling pigs, (e) mermaids, (f) fabulous ones, (g) stray dogs, (h) those that are included in this classification, (i) those that tremble as if they were mad, (j) innumerable ones, (k) those drawn with a very fine camel’s hair brush, (l) others, (m) those that have just broken a flower vase, (n) those that resemble flies from a distance. The Bibliographical Institute of Brussels also resorts to chaos: it has parceled the universe into 1,000 subdivisions: Number 262 corresponds to the Pope; Number 282, to the Roman Catholic Church; Number 263, to the Lord’s Day; Number 268, to Sunday schools; Number 298, to Mormonism; and Number 294, to Brahmanism, Buddhism, Shintoism, and Taoism. It also tolerates heterogeneous subdivisions, for example, Number 179: “Cruelty to animals. Protection of animals. Moral Implications of duelling and suicide. Various vices and defects. Various virtues and qualities.”
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 first book of Quintilian’s Institutio Oratoria, translated by Harold Edgeworth Butler:
For almost all others who have written on the art of oratory have started with the assumption that their readers were perfect in all other branches of education and that their own task was merely to put the finishing touches to their rhetorical training; this is due to the fact that they either despised the preliminary stages of education or thought that they were not their concern, since the duties of the different branches of education are distinct one from another, or else, and this is nearer the truth, because they had no hope of making a remunerative display of their talent in dealing with subjects, which, although necessary, are far from being showy: just as in architecture it is the superstructure and not the foundations which attracts the eye. I on the other hand hold that the art of oratory includes all that is essential for the training of an orator, and that it is impossible to reach the summit in any subject unless we have first passed through all the elementary stages. I shall not therefore refuse to stoop to the consideration of those minor details, neglect of which may result in there being no opportunity for more important things, and propose to mould the studies of my orator from infancy, on the assumption that his whole education has been entrusted to my charge.
These two branches of knowledge were, as Cicero has clearly shown,1 so closely united, not merely in theory but in practice, that the same men were regarded as uniting the qualifications of orator and philosopher. Subsequently this single branch of study split up into its component parts, and thanks to the indolence of its professors was regarded as consisting of several distinct subjects. As soon as speaking became a means of livelihood and the practice of making an evil use of the blessings of eloquence came into vogue, those who had a reputation for eloquence ceased to study moral philosophy, and ethics, thus abandoned by the orators, became the prey of weaker intellects. As a consequence certain persons, disdaining the toil of learning to speak well, returned to the task of forming character and establishing rules of life and kept to themselves what is, if we must make a division, the better part of philosophy, but presumptuously laid claim to the sole possession of the title of philosopher, a distinction which neither the greatest generals nor the most famous statesmen and administrators have ever dared to claim for themselves. For they preferred the performance to the promise of great deeds. I am ready to admit that many of the old philosophers inculcated the most excellent principles and practised what they preached. But in our own day the name of philosopher has too often been the mask for the worst vices. For their attempt has not been to win the name of philosopher by virtue and the earnest search for wisdom; instead they have sought to disguise the depravity of their characters by the assumption of a stern and austere mien accompanied by the wearing of a garb differing from that of their fellow men. Now as a matter of fact we all of us frequently handle those themes which philosophy claims for its own. Who, short of being an utter villain, does not speak of justice, equity and virtue? Who (and even common country-folk are no exception) does not make some inquiry into the causes of natural phenomena? As for the special uses and distinctions of words, they should be a subject of study common to all who give any thought to the meaning of language. Let our ideal orator then be such as to have a genuine title to the name of philosopher: it is not sufficient that he should be blameless in point of character (for I cannot agree with those who hold this opinion): he must also be a thorough master of the science and the art of speaking, to an extent that perhaps no orator has yet attained.
Perfect eloquence is assuredly a reality, which is not beyond the reach of human intellect. Even if we fail to reach it, those whose aspirations are highest, will attain to greater heights than those who abandon themselves to premature despair of ever reaching the goal and halt at the very foot of the ascent.
In the course of these discussions I shall deal in its proper place with the method of teaching by which students will acquire not merely a knowledge of those things to which the name of art is restricted by certain theorists, and will not only come to understand the laws of rhetoric, but will acquire that which will increase their powers of speech and nourish their eloquence. For as a rule the result of the dry textbooks on the art of rhetoric is that by straining after excessive subtlety they impair and cripple all the nobler elements of style, exhaust the lifeblood of the imagination and leave but the bare bones, which, while it is right and necessary that they should exist and be bound each to each by their respective ligaments, require a covering of flesh as well.
from Sextus Empiricus, Outlines of Pyrrhonism, book 1, section 12, translated by Benson Mates:
We always say that as regards belief (i.e., dogma) the Pyrrhonist’s goal is ataraxia, and that as regards things that are unavoidable it is having moderate pathè. For when the Pyrrhonist set out to philosophize with the aim of assessing his phantasiai – that is, of determining which are true and which are false so as to achieve ataraxia – he landed in a controversy between positions of equal strength, and, being unable to resolve it, he suspended judgment. But while he was thus suspending judgment there followed by chance the sought-after ataraxia as regards belief. For the person who believes that something is by nature good or bad is constantly upset; when he does not possess the things that seem to be good, he thinks he is being tormented by things that are by nature bad, and he chases after the things he supposes to be good; then, when he gets these, he fails into still more torments because of irrational and immoderate exultation, and, fearing any change, he does absolutely everything in order not to lose the things that seem to him good. But the person who takes no position as to what is by nature good or bad neither avoids nor pursues intensely. As a result, he achieves ataraxia. Indeed, what happened to the Pyrrhonist is just like what is told of Apelles the painter. For it is said that once upon a time, when he was painting a horse and wished to depict the horse’s froth, he failed so completely that he gave up and threw his sponge at the picture – the sponge on which he used to wipe the paints from his brush – and that in striking the picture the sponge produced the desired effect. So, too, the Pyrrhonists were hoping to achieve ataraxia by resolving the anomaly of phenomena and noumena, and, being unable to do this, they suspended judgment. But then, by chance as it were, when they were suspending judgment the ataraxia followed, as a shadow follows the body. We do not suppose, of course, that the Pyrrhonist is wholly untroubled, but we do say that he is troubled only by things unavoidable. For we agree that sometimes he is cold and thirsty and has various feelings like those. But even in such cases, whereas ordinary people are affected by two circumstances – namely by the pathé themselves and not less by its seeming that these conditions are by nature bad – the Pyrrhonist, by eliminating the additional belief that all these things are naturally bad, gets off more moderately here as well. Because of this we say that as regards belief the Pyrrhonist’s goal is ataraxia, but in regard to things unavoidable it is having moderate pathé.
“[…] Are you suggesting that one cannot sometimes say what is true?” What I am suggesting is that “Because it is true” is not a reason or basis for saying something; and I am suggesting that there must, in grammar, be reasons for what you say, or be a point in your saying of something, if what you say is to be comprehensible. We can understand what the words mean apart from understanding why you say them; but apart from understanding the point of you saying them we cannot understand what you mean.
A charming discussion of what should be called the fundamental theorem of computation theory, in Epstein and Carnielli, Computability: Computable Functions, Logic, and the Foundations of Mathematics (2008):
We have studied one formalization of the notion of computability. In succeeding chapters we will study two more: recursive functions and functions representable in a formal system.
The Most Amazing Fact All the attempts at formalizing the intuitive notion of computable function yield exactly the same class of functions.
So if a function is Turing machine computable, it can also be computed in any of the other systems described in Chapter 8.E. This is a mathematical fact which requires a proof. […] Odifreddi, 1989 establishes all the equivalences. […]
The Most Amazing Fact is stated about an extensional class of functions, but it can be stated constructively: Any computation procedure for any of the attempts at formalizing the intuitive notion of computable function can be translated into any other formalization in such a way that the two formalizations have the same outputs for the same inputs.
In 1936, even before these equivalences were established, Church said,
We now define the notion, already discussed, of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers (or of a lambda-definable function of positive integers). This definition is thought to be justified by the considerations which follow, so far as positive justification can ever be obtained for the selection of a formal definition to correspond to an intuitive notion.
So we have
Church’s Thesis: A function is computable iff it is lambda-definable.
This is a nonmathematical thesis: it equates an intuitive notion (computability) with a precise, formal one (lambda-definability). By our amazing fact this thesis is equivalent to
A function is computable iff it is Turing machine computable.
Turing devised his machines in a conscious attempt to capture in simplest terms what computability is. That his model turned out to give the same class of functions as Church’s (as established by Turing in the paper cited above) was strong evidence that it was the “right” class. Later we will consider some criticisms of Church’s Thesis in that the notion of computability should coincide with either a larger or a small class than the Turing machine computable ones.