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.
from the preface to Bertrand Russell’s Dictionary of Mind, Matter, and Morals (1952):
I feel considerably honoured that my philosophy should have been thought worthy to be alphabetically anatomized in this dictionary. I have been accused of a habit of changing my opinions in philosophy and, in so far as this is true, the dictionary will enable readers to find it out. I am not myself in any degree ashamed of having changed my opinions. What physicist who was already active in 1900 would dream of boasting that his opinions had not changed during the last half century? In science men change their opinions when new knowledge becomes available, but philosophy in the minds of many is assimilated rather to theology than to science. A theological proclaims eternal truths, the creeds remain unchanged since the Council of Nicaea. Where nobody knows anything, there is no point in changing your mind. But the kind of philosophy that I value and have endeavoured to pursue is scientific in the sense that there is some definite knowledge to be obtained and that new discoveries can make the admission of former error inevitable to any candid mind.
This book ﬁrst arose out of a passage in Borges, out of the laughter that shattered, as I read the passage, all the familiar landmarks of my thought — our thought, the thought that bears the stamp of our age and our geography — breaking up all the ordered surfaces and all the planes with which we are accustomed to tame the wild profusion of existing things, and continuing long afterwards to disturb and threaten with collapse our age-old distinction between the Same and the Other. This passage quotes a “certain Chinese encyclopedia” in which it is written that “animals are divided into: (a) belonging to the Emperor, (b) embalmed, (c) tame, (d) sucking pigs, (e) sirens, (f) fabulous, (g) stray dogs, (h) included in the present classiﬁcation, (i) frenzied, (j) innumerable, (k) drawn with a very ﬁne camelhair brush, (l) et cetera, (m) having just broken the water pitcher, (n) that from a long way off look like flies”. In the wonderment of this taxonomy, the thing we apprehend in one great leap, the thing that, by means of the fable, is demonstrated as the exotic charm of another system of thought, is the limitation of our own, the stark impossibility of thinking that.
But what is it impossible to think, and what kind of impossibility are we faced with here? Each of these strange categories can be assigned a precise meaning and a demonstrable content; some of them do certainly involve fantastic entities — fabulous animals or sirens — but, precisely because it puts them into categories of their own, the Chinese encyclopedia localizes their powers of contagion; it distinguishes carefully between the very real animals (those that are frenzied or have just broken the water pitcher) and those that reside solely in the realm of imagination. The possibility of dangerous mixtures has been exorcized, heraldry and fable have been relegated to their own exalted peaks: no inconceivable amphibious maidens, no clawed wings, no disgusting, squamous epidermis, none of those polymorphous and demoniacal faces, no creatures breathing ﬁre. The quality of monstrosity here does not affect any real body, nor does it produce modiﬁcations of any kind in the bestiary of the imagination; it does not lurk in the depths of any strange power. It would not even be present at all in this classiﬁcation had it not insinuated itself into the empty space, the interstitial blanks separating all these entities from one another. It is not the “fabulous” animals that are impossible, since they are designated as such, but the narrowness of the distance separating them from (and juxtaposing them to) the stray dogs, or the animals that from a long way off look like ﬂies. What transgresses the boundaries of all imagination, of all possible thought, is simply that alphabetical series (a, b, c, d) which links each of those categories to all the others.
Bertrand Russell, “The Uses of Language,” in Human Knowledge: Its Scope and Limits (1948):
Language, like other things of mysterious importance, such as breath, blood, sex and lightning, has been viewed superstitiously ever since men were capable of recording their thoughts. Savages fear to disclose their true name to an enemy, lest he should work evil magic by means of it. Origen assures us that pagan sorcerers could achieve more by using the sacred name Jehovah than by means of the names Zeus, Osiris or Brahma. Familiarity makes us blind to the linguistic emphasis in the Commandment: ‘Thou shalt not take the nameof the Lord in vain.’ The habit of viewing language superstitiously is not yet extinct. ‘In the beginning was the Word’, says our version of St John’s Gospel, and in reading some logical positivists I am tempted to think that their view is represented by this mistranslated text.
Philosophers, being bookish and theoretical folk, have been interested in language chiefly as a means of making statements and conveying information, but this is only one of its purposes, and perhaps not the most primitive. What is the purpose of language to a sergeant-major? On the one hand there is the language of words of command, designed to cause identical simultaneous bodily movements in a number of hearers; on the other hand there is bad language, designed to cause humility in those in whom the expected bodily movements have not been caused. In neither case are words used, except incidentally, to state facts or convey information.