Contrary to folklore, mathematical ability is not a rare gift uncorrelated with other intellectual abilities: testing demonstrates that the child good with words and logic is most likely to have native potentiality for mathematics also. That schools […] should turn us out ignorant of and resentful of mathematics, is a crime. And not because, in the age of Sputnik and automation, mathematical proficiency is a prerequisite of national prosperity and survival, but rather because of the sheer fun that people miss […]
During my time at Yale, I gave two or three graduate courses. Serge always sat in the front row, paying close attention to the point of interrupting me midsentence: “The notation should be functorial with respect to the ideas!” or “This notation sucks!” But, after class he complimented me highly on the lecture.
While on sabbatical at Harvard, he sat in on a course Mazur was giving and often criticized the notation. Eventually they decided to give him a T-shirt which said, “This notation sucks” on it. So one day Barry intentionally tried to get him to say it. He introduced a complex variable Ξ, took its complex conjugate, and divided by the original Ξ. This was written as a vertical fraction, so it looked like eight horizontal lines on the blackboard. He then did a few other similar things, but Serge kept quiet—apparently he didn’t criticize notation unless he knew what the underlying mathematics was about. Eventually Barry had to give up and just present him with the T-shirt.
When he was six years old, he could divide two eight-digit numbers in his head and could converse in Ancient Greek. When the six-year-old von Neumann caught his mother staring aimlessly, he asked her, “What are you calculating?”
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.
The opening of Bertrand Russell’s preface to a 1914 translation of Poincaré’s Science and Method:
Henri Poincaré was, by general agreement, the most eminent scientific man of his generation—more eminent, one is tempted to think, than any man of science now living. From the mere variety of subjects which he illuminated, there is certainly no one who can appreciate critically the whole of his work. Some conception of his amazing comprehensiveness may be derived from the obituary number of the Revue de Métaphysique et de Morale (September 1913), where, in the course of 130 pages, four eminent men—a philosopher, a mathematician, an astronomer, and a physicist—tell in outline the contributions which he made to several subjects. In all we find the same characteristics—swiftness, comprehensiveness, unexampled lucidity, and the perception of recondite but fertile analogies.
Needham’s book is the example par excellence of treating modern analytical material in a more classical, geometric vein; some of these books approach it in that regard, while others just include nice pictures or illustrations. I’m sure I’m forgetting about many texts with beautiful figures; I’ll add to the list as I’m reminded of them.
Tristan Needham, Visual Complex Analysis
Nathan Carter, Visual Group Theory
Martin Weissman, An Illustrated Theory of Numbers
Elias Wegert, Visual Complex Functions
Siegmund Brandt and Hans Dieter Dahmen, The Picture Book of Quantum Mechanics
James Callahan, Advanced Calculus: A Geometric View
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.
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).”]
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.