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.

I.

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!

II.

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.

III.

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.

IV.

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)

Quite an everyday occurrence

from Huygens and Barrow, Newton and Hooke, Vladimir Arnold, translated by Eric J. F. Primrose (1989):

Hooke was a poor man and began work as an assistant to Boyle (who is now well known thanks to the Boyle-Mariotte law discovered by Hooke). Subsequently Hooke began working in the recently established Royal Society (that is, the English Academy of Sciences) as Curator. The duties of the Curator of the Royal Society were very onerous. According to his contract, at every session of the Society (and they occurred every week except for the summer vacation) he had to demonstrate three or four experiments proving the new laws of nature.

Hooke held the post of Curator for forty years, and all that time he carried out his duties thoroughly. Of course, there was no condition in the contract that all the laws to be demonstrated had to be devised by him. He was allowed to read books, correspond with other scientists, and to be interested in their discoveries. He was only required to verify whether their statements were true and to convince the Royal Society that some law was reliably established. For this it was necessary to prove this law experimentally and demonstrate the appropriate experiment. This was Hooke’s official activity.

[…]

At that time it was easy to carry out fundamental discoveries, and large numbers of them were carried out. Huygens, for example, improved the telescope, looked at Saturn and discovered its ring, and Hooke discovered the red spot on Jupiter. At that time discoveries were not unusual events, they were not registered, not patented, as they are now, they were quite an everyday occurrence. (This was the case not only in the natural sciences. Mathematical discoveries at that time also poured forth as if from a horn of plenty.)

But Hooke never had enough time to dwell on any of his discoveries and develop it in detail, since in the following week he needed to demonstrate new laws. So in the whole manifold of Hooke’s achievements his discoveries appeared somewhat incomplete, and sometimes when he was in a hurry he made assertions that he could not justify accurately and with mathematical rigour.

[…]

Holding the chair at Cambridge, Newton earned considerably more (200 pounds a year), and the farm that he had inherited, which he leased out and where the famous apple tree grew, gave him roughly the same income. Despite the fact that Newton was quite well off, he did not want to spend any money on the publication of the book, so he sent the Principia to the Royal Society, which decided to publish the book at its own expense. But the Society had no money, so the manuscript lay there until Halley (who was the son of a rich soap manufacturer) published it on his own account. Halley took on himself all the trouble of publishing the book, and even read the proofs himself. Newton, in correspondence at this time, called it “Your book”…

An engine of discovery

from the preface to Cell Biology by the Numbers, Ron Milo and Rob Phillips:

One of the great traditions in biology’s more quantitative partner sciences such as chemistry and physics is the value placed on centralized, curated quantitative data. Whether thinking about the astronomical data that describes the motions of planets or the thermal and electrical conductivities of materials, the numbers themselves are a central part of the factual and conceptual backdrop for these fields.  Indeed, often the act of trying to explain why numbers have the values they do ends up being an engine of discovery.

A requirement for genuine expertise

David Foster Wallace in conversation with Dave Eggers, The Believer, 2003:

We live today in a world where most of the really important developments in everything from math and physics and astronomy to public policy and psychology and classical music are so extremely abstract and technically complex and context-dependent that it’s next to impossible for the ordinary citizen to feel that they (the developments) have much relevance to her actual life. Where even people in two closely related sub-sub-specialties have a hard time communicating with each other because their respective s-s-s’s require so much special training and knowledge. And so on. Which is one reason why pop-technical writing might have value (beyond just a regular book-market $-value), as part of the larger frontier of clear, lucid, unpatronizing technical communication. It might be that one of the really significant problems of today’s culture involves finding ways for educated people to talk meaningfully with one another across the divides of radical specialization. That sounds a bit gooey, but I think there’s some truth to it. And it’s not just the polymer chemist talking to the semiotician, but people with special expertise acquiring the ability to talk meaningfully to us, meaning ordinary schmoes. Practical examples: Think of the thrill of finding a smart, competent IT technician who can also explain what she’s doing in such a way that you feel like you understand what went wrong with your computer and how you might even fix the problem yourself if it comes up again. Or an oncologist who can communicate clearly and humanly with you and your wife about what the available treatments for her stage-two neoplasm are, and about how the different treatments actually work, and exactly what the plusses and minuses of each one are. If you’re like me, you practically drop and hug the ankles of technical specialists like this, when you find them. As of now, of course, they’re rare. What they have is a particular kind of genius that’s not really part of their specific area of expertise as such areas are usually defined and taught. There’s not really even a good univocal word for this kind of genius—which might be significant. Maybe there should be a word; maybe being able to communicate with people outside one’s area of expertise should be taught, and talked about, and considered as a requirement for genuine expertise.… Anyway, that’s the sort of stuff I think your question is nibbling at the edges of, and it’s interesting as hell.