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.

Right for the wrong reasons

from George Eliot, Middlemarchwith 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.

No lover of disorder and doubt

William James, The Varieties of Religious Experience (1902):

The wisest of critics is an altering being, subject to the better insight of the morrow, and right at any moment, only “up to date” and “on the whole.” When larger ranges of truth open, it is surely best to be able to open ourselves to their reception, unfettered by our previous pretensions. “Heartily know, when half-gods go, the gods arrive.”

The fact of diverse judgments about religious phenomena is therefore entirely unescapable, whatever may be one’s own desire to attain the irreversible. But apart from that fact, a more fundamental question awaits us, the question whether men’s opinions ought to be expected to be absolutely uniform in this field. Ought all men to have the same religion? Ought they to approve the same fruits and follow the same leadings? Are they so like in their inner needs that, for hard and soft, for proud and humble, for strenuous and lazy, for healthy-minded and despairing, exactly the same religious incentives are required? Or are different functions in the organism of humanity allotted to different types of man, so that some may really be the better for a religion of consolation and reassurance, whilst others are better for one of terror and reproof? It might conceivably be so; and we shall, I think, more and more suspect it to be so as we go on. And if it be so, how can any possible judge or critic help being biased in favor of the religion by which his own needs are best met? He aspires to impartiality; but he is too close to the struggle not to be to some degree a participant, and he is sure to approve most warmly those fruits of piety in others which taste most good and prove most nourishing to him.

I am well aware of how anarchic much of what I say may sound. Expressing myself thus abstractly and briefly, I may seem to despair of the very notion of truth. But I beseech you to reserve your judgment until we see it applied to the details which lie before us. I do indeed disbelieve that we or any other mortal men can attain on a given day to absolutely incorrigible and unimprovable truth about such matters of fact as those with which religions deal. But I reject this dogmatic ideal not out of a perverse delight in intellectual instability. I am no lover of disorder and doubt as such. Rather do I fear to lose truth by this pretension to possess it already wholly. That we can gain more and more of it by moving always in the right direction, I believe as much as any one, and I hope to bring you all to my way of thinking before the termination of these lectures. Till then, do not, I pray you, harden your minds irrevocably against the empiricism which I profess.

A willingness to change his own mind

Gary Marcus, “Happy Birthday, Noam Chomsky,” The New Yorker (2012):

One of Chomsky’s most remarkable traits is his willingness to change his own mind, like Bob Dylan suddenly going electric to the consternation of his early fans. Take for example the distinction that he once made between “deep structure” and “surface structure.” In its crudest form, the notion is that an active sentence (“John loved Mary”) and a passive sentence (“Mary was loved by John”) might seem superficially different; yet they have some important underlying commonality, both in meaning and in their representation in the brain. It’s a neat idea that makes certain very specific claims about how language is represented in our mind, and how sound relates to meaning; decades of linguistic work have been based on it. It also (somewhat uncharacteristically for Chomsky) makes for a perfect sound bite; there are plenty of people who know nothing about linguistics, but have the sense that what he was talking about was “deep versus shallow”; there was an even a Nobel Prize winner, Niels Jerne, who used the metaphor in his Nobel address about language and the immune system. Most people would have lived off a metaphor that good for the rest of their careers; Chomsky has spent the past twenty-five years arguing that he made a mistake. Although the basic metaphor is simple, the distinction between deep structure and surface structure required a great deal of behind-the-scenes technical examination in order to make it work with the complexities of different languages. In its place, Chomsky has recently been trying to develop a simpler, more elegant theory (known as the Minimalist Program) that encompasses the spirit of the original. (Not all of us are convinced about the success of that approach; my own view is that language is irreducibly messy, and that the elegance that Chomsky seeks will not be forthcoming.)

More recently, in a co-written 2002 paper, Chomsky seemed to open a door to a view that he’d long criticized: the idea that the “faculty of language,” as he called it, might draw on parts of the brain that weren’t specialized for language. Up to then, Chomsky had been known in part for idea called “the autonomy of syntax,” which, in crude terms, suggested that grammar was cognitively separate from other aspects of the mind (like our understanding of the world and our desire to eat pizza for dinner). I was so surprised by the dramatic shift that I wrote to him to ask. “A lot of people take [your new] paper to be a renouncing of your earlier arguments.” Was that really the case? His response, as immediate as ever, “As for my own views, they’ve of course evolved over the years. This conception of ‘renouncing beliefs’ is very odd, as if we’re in some kind of religious cult. I ‘renounce beliefs’ practically every time I think about the topics or find out what someone else is thinking.”

Nine academics out of ten never change their mind about anything; most (though there are salient exceptions, like Wittgenstein) lock into a position earlier in their careers and then defend it to the hilt. Chomsky, in contrast, has never stopped critiquing his own theories with the same vigor with which he has criticized others. For fifty years, his search for linguistic truth has been relentless.

The rewards of creative activity

the second paragraph of Allan Forte’s Tonal Harmony in Concept and Practice (1962):

We hold the conviction that the primary aim of serious music study is to illuminate the subject, not to surround it with trivia and bury it beneath detail. At the same time one must realize that a technical approach to music, like a technical approach to any subject, involves specific tasks which are often detailed. These include the learning of a new terminology, the memorizing of certain facts, and the intelligent working out of exercises in order to achieve basic skills. Without these one cannot hope to approach the general concepts essential to the art, nor can one a ain that level of minimal ability which will enable him to enjoy the rewards of creative activity.

Neat, plausible, and wrong

Two versions of the opening of Mencken’s essay “The Divine Afflatus”:

I. As it appears in A Mencken Chrestomathy

Every man who writes, or paints, or composes knows by hard experience that there are days when his ideas flow freely and clearly and days when they are damned up damnably. On his good days, for some reason quite incomprehensible to him, all the processes and operations of his mind take on an amazing ease and slickness. Almost without conscious effort he solves technical problems that have badgered him for weeks. He is full of novel expedients, extraordinary efficiencies, strange cunnings. He has a feeling that he has suddenly and unaccountably broken through a wall, dispersed a fog, got himself out of the dark. So he does a double or triple stint of the best work that he is capable of—maybe of far better work than he has ever been capable of before—and goes to bed impatient for the morrow. And on the morrow he discovers to his consternation that he has become almost idiotic, and quite incapable of any work at all.

This unpleasant experience overtakes poets and contrapuntists, critics and dramatists, painters and sculptors, and also, no doubt, philosophers and journalists; it may even be shared, so far as I know, by advertisement writers and the rev. clergy. The characters that all anatomists of melancholy mark in it are the irregular ebb and flow of the tides, and the impossibility of getting them under any sort of rational control. The brain, as it were, stands to one side and watches itself pitching and tossing, full of agony but essentially helpless. Here the man of creative imagination pays a ghastly price for all his superiorities and immunities; nature takes revenge upon him for dreaming of improvements in the scheme of things. Sitting there in his lonely room, gnawing the handle of his pen, racked by his infernal quest, horribly bedeviled by incessant flashes of itching, toothache, eye-strain and festering conscience—thus tortured, he makes atonement for his crime of having ideas. The normal man, the healthy and honest man, the good citizen and householder—this man, I daresay, knows nothing of all that travail. It is the particular penalty of those who pursue strange butterflies into dark forests, and go fishing in enchanted and for bidden streams.

How are we to account for it? My question, of course, is purely rhetorical. Explanations exist; they have existed for all times, for there is always an easy solution to every human problem—neat, plausible, and wrong. The ancients laid the blame upon the gods: sometimes they were remote and surly, and sometimes they were kind. In the Middle Ages lesser powers took a hand, and one reads of works of art inspired by Our Lady, by the Blessed Saints, by the souls of the departed, and even by the devil. In our own day there are explanations less supernatural but no less fanciful—to wit, the explanation that the whole thing is a matter of pure chance, and not to be resolved into any orderly process—to wit, the explanation that the controlling factor is external circumstance, that the artist happily married to a dutiful wife is thereby inspired—finally, to make an end, the explanation that it is all a question of Freudian complexes, themselves lurking in impenetrable shadows. But all these explanations fail to satisfy the mind that is not to be put off with mere words. Some of them are palpably absurd; others beg the question. The problem of the how remains, even when the problem of the why is disposed of. What is the precise machinery whereby the cerebrum is bestirred to such abnormal activity on one day that it sparkles and splutters like an arc-light, and reduced to such feebleness on another day that it smokes and gutters like a tallow dip?

II. As it appears in Prejudices: Second Series

The suave and oedematous Chesterton, in a late effort to earn
the honorarium of a Chicago newspaper, composed a thousand words
of labored counterblast to what is called inspiration in the arts.
The thing itself, he argued, has little if any actual existence; we
hear so much about it because its alleged coyness and fortuitousness
offer a convenient apology for third-rate work. The man taken in such
third-rate work excuses himself on the ground that he is a helpless
slave of some power that stands outside him, and is quite beyond his
control. On days when it favors him he teems with ideas and creates
masterpieces, but on days when it neglects him he is crippled and
impotent–a fiddle without a bow, an engine without steam, a tire
without air. All this, according to Chesterton, is nonsense. A man who
can really write at all, or paint at all, or compose at all should be
able to do it at almost any time, provided only “he is not drunk or
asleep.”

So far Chesterton. The formula of the argument is simple and familiars
to dispose of a problem all that is necessary is to deny that it
exists. But there are plenty of men, I believe, who find themselves
unable to resolve the difficulty in any such cavalier manner–men whose chief burden and distinction, in fact, is that they do not employ
formulae in their thinking, but are thrown constantly upon industry,
ingenuity and the favor of God. Among such men there remains a good
deal more belief in what is vaguely called inspiration. They know
by hard experience that there are days when their ideas flow freely
and clearly, and days when they are dammed up damnably. Say a man of that sort has a good day. For some reason quite incomprehensible to
him all his mental processes take on an amazing ease and slickness.
Almost without conscious effort he solves technical problems that have
badgered him for weeks. He is full of novel expedients, extraordinary
efficiencies, strange cunnings. He has a feeling that he has suddenly
and unaccountably broken through a wall, dispersed a fog, got himself
out of the dark. So he does a double or triple stint of the best work
that he is capable of—maybe of far better work than he has ever been
capable of before—and goes to bed impatient for the morrow. And on
the morrow he discovers to his consternation that he has become almost idiotic, and quite incapable of any work at all.

I challenge any man who trades in ideas to deny that he has this
experience. The truth is that he has it constantly. It overtakes
poets and contrapuntists, critics and dramatists, philosophers and
journalists; it may even be shared, so far as I know, by advertisement
writers, chautauqua orators and the rev. clergy. The characters that
all anatomists of melancholy mark in it are the irregular ebb and flow
of the tides, and the impossibility of getting them under any sort of
rational control. The brain, as it were, stands to one side and watches
itself pitching and tossing, full of agony but essentially helpless.
Here the man of creative imagination pays a ghastly price for all
his superiorities and immunities; nature takes revenge upon him for
dreaming of improvements in the scheme of things. Sitting there in his
lonely room, gnawing the handle of his pen, racked by his infernal
quest, horribly bedevilled by incessant flashes of itching, toothache,
eye-strain and evil conscience—thus tortured, he makes atonement for
his crime of being intelligent. The normal man, the healthy and honest
man, the good citizen and householder—this man, I daresay, knows
nothing of all that travail. It is reserved especially for artists
and metaphysicians. It is the particular penalty of those who pursue
strange butterflies into dark forests, and go fishing in enchanted and
forbidden streams.

Let us, then, assume that the fact is proved: the nearest poet
is a witness to it. But what of the underlying mystery? How are
we to account for that puckish and inexplicable rise and fall of
inspiration? My questions, of course, are purely rhetorical.
Explanations exist; they have existed for all time; there is always
a well-known solution to every human problem—neat, plausible, and
wrong. The ancients, in the case at bar, laid the blame upon the gods:
sometimes they were remote and surly, and sometimes they were kind. In the Middle Ages lesser powers took a hand in the matter, and so one
reads of works of art inspired by Our Lady, by the Blessed Saints, by
the souls of the departed, and even by the devil. In our own day there
are explanations less super-natural but no less fanciful—to wit,
the explanation that the whole thing is a matter of pure chance, and
not to be resolved into any orderly process—-to wit, the explanation
that the controlling factor is external circumstance, that the artist
happily married to a dutiful wife is thereby inspired–finally, to
make an end, the explanation that it is all a question of Freudian
complexes, themselves lurking in impenetrable shadows. But all of these explanations fail to satisfy the mind that is not to be put off with
mere words. Some of them are palpably absurd; others beg the question. The problem of the how remains, even when the problem of the why is disposed of. What is the precise machinery whereby the cerebrum is bestirred to such abnormal activity on one day that it sparkles and splutters like an arclight, and reduced to such feebleness on another day that it smokes and gutters like a tallow dip?

Supplies are limited

the introductory note to “Further Reading” in Nigel Warburton, Thinking from A to Z:

There are a number of books which purport to give a thorough grounding in critical thinking. Unfortunately many of them demonstrate their authors’ limited abilities to think critically.