Contrary to folklore

Paul Samuelson, “Heads I Win, and Tales, You Lose,” in the sixtieth anniversary edition of von Neumann and Morgenstern’s Theory of Games and Economic Behavior:

Contrary to folklore, mathematical ability is not a rare gift uncorrelated with other intellectual abilities: testing demonstrates that the child good with words and logic is most likely to have native potentiality for mathematics also. That schools […] should turn us out ignorant of and resentful of mathematics, is a crime. And not because, in the age of Sputnik and automation, mathematical proficiency is a prerequisite of national prosperity and survival, but rather because of the sheer fun that people miss […]

I think of this old title on my shelf: The Sheer Joy of Celestial Mechanics.

This notation sucks!

Paul Votja on Serge Lang:

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.

The acid test of concepts

A striking passage from Frege’s unpublished essay “Boole’s logical calculus and the concept-script,” which he submitted in 1881 first to Zeitschrift für Mathematik und Physik, then the Mathematischen Annalen, 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…”

Some helpful discussions appear in:

  • Jamie Tappenden, “Fruitfulness as a Theme in the Philosophy of Mathematics,” The Journal of Philosophy (2012)
  • 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 Leading Philosophers, Vol. III, Frege’s philosophy of mathematics, edited by Michael Beany and Erich Reck

Recondite but fertile analogies

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.

Visual X

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

H. M. Schey, Div, Grad, Curl and All That

Michio Kuga, Galois’ Dream: Group Theory and Differential Equations

Charles Misner, Kip Thorne, and John Archibald Wheeler, Gravitation

Michael Spivak, A Comprehensive Introduction to Differential Geometry

David Hilbert and Stephan Cohn-Vossen, Geometry and the Imagination

Admirers so few and so languid

Samuel Taylor Coleridge, age 18, to his brother George, with a very green poem setting Euclidean reasoning to verse:

Dear Brother,

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—
—lateral Tri—
—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—
—lateral Tri—
—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
Above another,
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.

Stupid for the rest of the day

From the Wikipedia page on Paul Valéry:

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.

The celestial emporium of benevolent knowledge

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.”

Why Johnny can’t X

David Brin, “Why Johnny Can’t Code,” Salon (2006)

Mike Davidow, Why Johnny Can’t Read and Ivan Can (1977)

Tommy Dreyfus, “Why Johnny Can’t Prove,” Educational Studies in Mathematics (1999)

Rudolf Flesch, Why Johnny Can’t Read (1955) and Why Johnny Still Can’t Read (1981)

Thomas Frank, “Dark Age: Why Johnny Can’t Dissent,” The Baffler (1995)

Konstantin Kakaes, “Why Johnny Can’t Add Without a Calculator,” Salon (2012)

Walter Karp, “Why Johnny Can’t Think,” Harper’s (1985)

William Kilpatrick, Why Johnny Can’t Tell Right from Wrong (1992)

Morris Kline, Why Johnny Can’t Add: The Failure of the New Math (1974)

Myra Linden and Arthur Whimbey, Why Johnny Can’t Write (2012)

Opal Moore, Why Johnny Can’t Learn (1975)

Douglas Rushkoff, “Why Johnny Can’t Program,” Huffington Post (2010)

Arthur Trace, What Ivan Knows that Johnny Doesn’t (1961)