No knower is an island

Woodblock depicting the island of Bensalem from Bacon’s New Atlantis

Popper on “Crusonian science” in The Open Society and Its Enemies (1945), a particularly vivid and memorable illustration of what has become a central area of research in academic philosophy:

Two aspects of the method of the natural sciences are of importance in this connection. Together they constitute what I may term the ‘public character of scientific method’. First, there is something approaching free criticism. A scientist may offer his theory with the full conviction that it is unassailable. But this will not impress his fellow-scientists and competitors; rather it challenges them: they know that the scientific attitude means criticizing everything, and they are little deterred even by authorities. Secondly, scientists try to avoid talking at cross-purposes. (I may remind the reader that I am speaking of the natural sciences, but a part of modern economics may be included.) They try very seriously to speak one and the same language, even if they use different mother tongues. In the natural sciences this is achieved by recognizing experience as the impartial arbiter of their controversies. When speaking of ‘experience’ I have in mind experience of a ‘public’ character, like observations, and experiments, as opposed to experience in the sense of more ‘private’ aesthetic or religious experience; and an experience is ‘public’ if everybody who takes the trouble can repeat it. In order to avoid speaking at cross-purposes, scientists try to express their theories in such a form that they can be tested, i.e. refuted (or else corroborated) by such experience.

This is what constitutes scientific objectivity. Everyone who has learned the technique of understanding and testing scientific theories can repeat the experiment and judge for himself. In spite of this, there will always be some who come to judgements which are partial, or even cranky. This cannot be helped, and it does not seriously disturb the working of the various social institutions which have been designed to further scientific objectivity and criticism; for instance the laboratories, the scientific periodicals, the congresses. This aspect of scientific method shows what can be achieved by institutions designed to make public control possible, and by the open expression of public opinion, even if this is limited to a circle of specialists. Only political power, when it is used to suppress free criticism, or when it fails to protect it, can impair the functioning of these institutions, on which all progress, scientific, technological, and political, ultimately depends.

In order to elucidate further still this sadly neglected aspect of scientific method, we may consider the idea that it is advisable to characterize science by its methods rather than by its results. Let us first assume that a clairvoyant produces a book by dreaming it, or perhaps by automatic writing. Let us assume, further, that years later as a result of recent and revolutionary scientific discoveries, a great scientist (who has never seen that book) produces one precisely the same. Or to put it differently we assume that the clairvoyant ‘saw’ a scientific book which could not then have been produced by a scientist owing to the fact that many relevant discoveries were still unknown at that date. We now ask : is it advisable to say that the clairvoyant produced a scientific book? We may assume that, if submitted at the time to the judgement of competent scientists, it would have been described as partly ununderstandable, and partly fantastic; thus we shall have to say that the clairvoyant’s book was not when written a scientific work, since it was not the result of scientific method. I shall call such a result, which, though in agreement with some scientific results, is not the product of scientific method, a piece of ‘revealed science’.

In order to apply these considerations to the problem of the publicity of scientific method, let us assume that Robinson Crusoe succeeded in building on his island physical and chemical laboratories, astronomical observatories, etc., and in writing a great number of papers, based throughout on observation and experiment. Let us even assume that he had unlimited time at his disposal, and that he succeeded in constructing and in describing scientific systems which actually coincide with the results accepted at present by our own scientists. Considering the character of this Crusonian science, some people will be inclined, at first sight, to assert that it is real science and not ‘revealed science’. And, no doubt, it is very much more like science than the scientific book which was revealed to the clairvoyant, for Robinson Crusoe applied a good deal of scientific method. And yet, I assert that this Crusonian science is still of the ‘revealed’ kind; that there is an element of scientific method missing, and consequently, that the fact that Crusoe arrived at our results is nearly as accidental and miraculous as it was in the case of the clairvoyant. For there is nobody but himself to check his results; nobody but himself to correct those prejudices which are the unavoidable consequence of his peculiar mental history; nobody to help him to get rid of that strange blindness concerning the inherent possibilities of our own results which is a consequence of the fact that most of them are reached through comparatively irrelevant approaches. And concerning his scientific papers, it is only in attempts to explain his work to somebody who has not done it that he can acquire the discipline of clear and reasoned communication which too is part of scientific method. In one point—a comparatively unimportant one—is the ‘revealed’ character of the Crusonian science particularly obvious; I mean Crusoe’s discovery of his ‘personal equation’ (for we must assume that he made this discovery), of the characteristic personal reaction-time affecting his astronomical observations. Of course it is conceivable that he discovered, say, changes in his reaction-time, and that he was led, in this way, to make allowances for it. But if we compare this way of finding out about reaction-time, with the way in which it was discovered in ‘public’ science—through the contradiction between the results of various observers—then the ‘revealed’ character of Robinson Crusoe’s science becomes manifest.

To sum up these considerations, it may be said that what we call ‘scientific objectivity’ is not a product of the individual scientist’s impartiality, but a product of the social or public character of scientific method; and the individual scientist’s impartiality is, so far as it exists, not the source but rather the result of this socially or institutionally organized objectivity of science.

A historically minded philosophy syllabus on this subject—what has been meant by the “social character” of knowledge—might run through Socratic dialogue; Descartes on self-knowledge; Hegel and the dialectical turn; Freud and the psychoanalytic turn (as rupture of Cartesianism); Marx and the ideological turn (as rupture of autonomous liberal subject); Peirce, Dewey, and other pragmatists on communities of inquiry; Kuhn, Lakatos, and midcentury philosophy of science (normal science, research programs); more contemporary logical puzzles over private language, self-knowledge, and common knowledge; externalism in epistemology, philosophy of language, and philosophy of mind; social construction and the science wars (post-Sokal); Foucault, Ian Hacking, and historical ontology; the careers of “social epistemology” (compare “standpoint epistemology”) as new research programs; the rise of sociology and especially political economy of science; epistemic scrutiny of mathematical practice and new anxieties over mathematical knowledge (post-Four Color Theorem); and contemporary work on democracy and epistemology.

Writ large, this story is often as much about the self as it is about knowledge. On one side there is inwardness, individuality, privacy, personality, property, logic, and pure reason (or at least the romantic artist, the Crusonian pure reasoner); on the other there is outwardness, community, public life, impersonality, the commons, conversation, and the dialogic imagination.

See also

Elizabeth Anderson, “The Epistemology of Democracy”

Michael Brady and Miranda Fricker, The Epistemic Life of Groups

Helen Longino, Science as Social Knowledge

The illusion of having dealt with it adequately

Robert Warshow on the New Yorker in Partisan Review (“Melancholy to the End,” vol. 14, no. 1, 1947):

The New Yorker at its best provides the intelligent and cultured college graduate with the most comfortable and least compromising attitude he can assume toward capitalist society without being forced into actual conflict. It rejects the vulgarity and inhumanity of the public world of politics and business and provincial morality, and it sets up in opposition to this a private and pseudo-aristocratic world of good humor, intelligence, and good taste. Its good taste has always been questionable, to be sure, but the vulgarity of the New Yorker is at least more subdued and less persistent than the ordinary vulgarity of journalism.

The New Yorker has always dealt with experience not by trying to understand it but by prescribing the attitude to be adopted toward it. This makes it possible to feel intelligent without thinking, and it is a way of making everything tolerable, for the assumption of a suitable attitude toward experience can give one the illusion of having dealt with it adequately.

Knowers on a limited scale

From Matthew Price’s review of Alfred Kazin’s diaries in Bookforum:

Cook questions Kazin’s liberal anti-communism—the Soviet Union appalled him—but Kazin would savagely rebuke the neoconservatives. He went to town on the snobbish Mr. Sammler’s Planet in the New York Review of Books, calling it a “normal political novel of our day, didactic to a fault.” (Bellow accused Kazin of slander: “You were saying that its author was a wickedly deluded lunatic.”) After an evening with Irving Kristol and other ex-leftists in 1969, Kazin bitterly ruminated in his journal: “They are all such specialists, such knowers on a limited scale, such professors impaled on their own bitterness. They have to be right . . . the world can go to hell, but they are right.” Kazin despised people who knew that they knew; he often knew that he didn’t know.

Russian poetry and the culture of memory

from Robert Chandler’s introduction to the Penguin Book of Russian Poetry:

A friend—a well-read poet and editor—once told me how astonished he had been to discover, many years after first reading him, the Mayakovsky—the Poet of the Russian Revolution—always wrote in rhyme and metre. My friend does not know Russian and all the translations he had seen were in free verse. And he had taken it for granted that a revolutionary poet would want to be free of traditional form… Russian poetry, however, has developed differently from the poetry of most other European countries.


In most of Europe the invention of print made it seem less important that a work of literature be easy to commit to memory. The decline of a magical or religious worldview also did much to encourage the rise of prose and the decline of poetry. Russia, however, has never seen the full emergence of a rational and secular culture—the official ethos of the Soviet era, though avowedly secular, was supremely irrational—and poetry has, throughout most of the last two hundred years and in most social milieus, retained its importance. Almost all Russian see Pushkin, rather than Tolstoy or Dostoevsky, as their greatest writer.


As for such poets as Osip Mandelstam, Anna Akhmatova and others disaffected with the new reality, they were soon living in what Akhmatova called a “pre-Gutenberg” age. They could no longer publish their own poems and it was dangerous to write them down. Akhmatova’s Lydia Chukovskaya (1907–96), has described how writers would memorize one another’s works. Akhmatova would write out a poem on a scrap of paper, a visitor would read it and Akhmatova would burn the paper. “It was like a ritual,” Chukovskaya says. “Hands, matches, an ashtray. A ritual beautiful and bitter.” Mandelstam died in a prison camp in 1938. Had his handling of rhyme, metre and other formal devices been less perfect, his widow might have been unable to preserve his work in her memory and much might have been lost.

Russian poetry has been forced, again and again, to return to its oral origins. This is especially evident with regard to the Gulag. There are many accounts of how people survived, and helped their fellow-prisoners to survive, through reciting poetry. The poet and ethnographer Nina Gagen-Torn has written how, in 1937, she and a cellmate were between them able to recite most of Nikolay Nekrasov’s Russian Women, a poem of at least two thousand lines about two aristocratic women, who, in 1826, chose to follow their husbands—participants in the failed December Revolt—to exile in Siberia. Ten years later, imprisoned for a second time, Gagen-Torn recited Blok, Pushkin, Nekrasov, Mandelstam, Gumilyov and Tyutchev. Every day her cellmates would ask her to recite more. Afterwards it was (in her words) “as if someone had cleaned the dust from the window with a damp sponge—everybody’s eyes now seemed clearer.” Gagen-Torn goes on to reflect on the role of rhythm: “The shamans knew that rhythm gives one power over spirits. He who had power over rhythm in the magic dance would become a shaman, an intermediary between spirits and people; he who lacked this power would fly head over heels into madness. Poetry, like the shaman’s bells, leads people into the spaces of ‘the seventh sky.'”

Fructified in sand


On the bleakness of my lot
Bloom I strove to raise.
Late, my acre of a rock
Yielded grape and maize.

Soil of flint if steadfast tilled
Will reward the hand;
Seed of palm by Lybian sun
Fructified in sand.

Already too long nauseated

From Swift’s “apology” to A Tale of a Tub, in its fifth edition (1710):

The greatest part of that book was finished above thirteen years since, 1696, which is eight years before it was published. The author was then young, his invention at the height, and his reading fresh in his head. By the assistance of some thinking, and much conversation, he had endeavoured to strip himself of as many real prejudices as he could; I say real ones, because, under the notion of prejudices, he knew to what dangerous heights some men have proceeded. Thus prepared, he thought the numerous and gross corruptions in Religion and Learning might furnish matter for a satire, that would be useful and diverting. He resolved to proceed in a manner that should be altogether new, the world having been already too long nauseated with endless repetitions upon every subject. The abuses in Religion, he proposed to set forth in the Allegory of the Coats and the three Brothers, which was to make up the body of the discourse. Those in learning he chose to introduce by way of digressions. He was then a young gentleman much in the world, and wrote to the taste of those who were like himself; therefore, in order to allure them, he gave a liberty to his pen, which might not suit with maturer years, or graver characters, and which he could have easily corrected with a very few blots, had he been master of his papers, for a year or two before their publication.

Not that he would have governed his judgment by the ill-placed cavils of the sour, the envious, the stupid, and the tasteless, which he mentions with disdain. He acknowledges there are several youthful sallies, which, from the grave and the wise, may deserve a rebuke. But he desires to be answerable no farther than he is guilty, and that his faults may not be multiplied by the ignorant, the unnatural, and uncharitable applications of those who have neither candour to suppose good meanings, nor palate to distinguish true ones. After which, he will forfeit his life, if any one opinion can be fairly deduced from that book, which is contrary to Religion or Morality.

Preceding this is a satirical list of forthcoming works:

Treatises wrote by the same Author, most of them mentioned in the following Discourses; which will be speedily published.

A Character of the Present Set of Wits in the Island.
A panegyrical Essay upon the Number THREE.
A Dissertation upon the principal Productions of
Lectures upon a Dissection of Human Nature.
A Panegyric upon the World.
An analytical discourse upon Zeal,
histori-theophysi-logically considered.
A general History of
A modest Defense of the Proceedings of the
Rabble in all ages.
A Description of the Kingdom of
A Voyage into
England, by a Person of Quality in Terra Australis incognita, translated from the Original.
A Critical Essay upon the Art of
Canting, philosophically, physically, and musically considered.

And from the long-last opening, after pages of other prefatory material:

WHOEVER hath an Ambition to be heard in a Crowd, must press, and squeeze, and thrust, and climb with indefatigable Pains, till he has exalted himself to a certain Degree of Altitude above them. Now, in all Assemblies, though you wedge them ever so close, we may observe this peculiar Property, that over their Heads there is Room enough, but how to reach it is the difficult Point; it being as hard to get quit of Number, as of Hell.

Evadere ad auras,
Hoc opus, hic labor est.

To this End, the Philosopher’s way in all Ages has been by erecting certain Edifices in the Air; But, whatever Practice and Reputation these kind of Structures have formerly possessed, or may still continue in, not excepting even that of Socrates, when he was suspended in a Basket to help Contemplation, I think, with due Submission, they seem to labour under two Inconveniences. First, That the Foundations being laid too high, they have been often out of Sight, and ever out of Hearing. Secondly, That the Materials, being very transitory, have suffered much from Inclemencies of Air, especially in these North-West regions.

The two cultures of integration theory

As a Math Stack Exchange question recently reminded me, every good calculus student eventually asks what the dx in integration really means—and every good calculus teacher has her own answer. But the answers are often confused.

The mess arises in part because the question can be interpreted syntactically or semantically. As a piece of notation, it is essentially a bookkeeping device, indicating the variable with respect to which the integration is carried out—as Leibniz, who gave us the symbol, well understood. In symbolic computation the formal string y dx thus means something quite different than x dx. The semantic question is even more interesting. As a matter of mathematical substance, the dx generalizes in two directions: in one direction it denotes a measure, and in another direction it denotes a differential form. These are the two cultures of integration theory. Experts speak both languages—and can translate fluently between them—but we do a very bad job at putting the two in conversation in the undergraduate curriculum.

The difference shows up not just in the sort of object we integrate (a function in the former case, a form in the latter), but also in the sort of object we integrate over. Measure-theoretic integration works over (certain) bare sets, but form-theoretic integration requires more structure: a set plus an orientation. Integrating a function over the unit disc is measure theory; integrating a form over the unit disc oriented counterclockwise is exterior calculus. The one theory is static: it is a theory of content, how much stuff is in a set. The other theory is dynamic: it is a theory of flux, how stuff is flowing through a set. The difference is also visible in how the two integrals transform. For measures, the change of variables theorem uses the absolute value of the Jacobian. For forms, the theorem drops the absolute value.

This distinction is important both conceptually (it leads to very different theories) and historically (the ideas emerged quite independently). But it tends to be glossed over in the typical undergraduate calculus sequence, where the various types of integrals are run together rather than carefully distinguished. (Witness the ambiguity of the phrase “surface integral.”) And it is all but lost once manifolds become the central object of study. In most cases, given the goal of generalizing vector calculus or developing de Rham cohomology, the tendency is to shortchange the measure-theoretic perspective, reducing the meaning of “calculus on manifolds” simply to the exterior calculus. This trajectory makes it look like Stokes’s theorem is the ultimate culmination of the basic ideas of derivative and integral, the Whiggish consummation the concepts have been building to all along.

Not that the theory of forms isn’t deep and important. But it is easy to leave the typical undergraduate sequence having forgotten entirely about measure, and we may come away thinking the only integration possible on a manifold is integration of forms. Indeed one could be forgiven for asking whether a differential geometer even needs to know the name Lebesgue. The impression is only reinforced when the diligent student of calculus is dropped into a first course in measure. There one encounters a totally new language, beginning with sigma algebras. Eventually connections come into view, but at first blush the subject appears to have little to do with calculus, and nothing to do with manifolds.

This state of affairs is unfortunate, but also easy to remedy. The distinction is already apparent in the two basic types of surface integrals discussed in any multivariable calculus class: the integral of a function with respect to surface area (which does not require an orientation), and the flux integral of a vector field through a surface (which does require an orientation). Even more basically, it’s apparent in the two basic types of line integrals: the integral of a function with respect to arc length (which, again, does not require an orientation), and the work integral of a vector field along a curve (which does require an orientation). Emphasizing the differences between these two types of integrals would not only help resolve conceptual confusion; it would convey a fuller awareness of what “calculus on curved spaces” can mean and anticipate a wider range of mathematical territory.

Inside Euclidean space, the measure-theoretic ideas of arc length and surface area lead to notions such as surface measure (using Jacobians in a chart) and Hausdorff measure (for more complicated sets), generalizing Lebesgue measure to handle sets of lower dimension. This is the beginning of geometric measure theory, motivated historically by problems in the calculus of variations. Outside Euclidean space, where tools can’t be imported from the ambient space, these idea lead to the (intrinsic) notion of a density on a smooth manifold. (A related notion is what Ted Frankel calls a pseudoform in The Geometry of Physics.) Such densities always exist (by a partition of unity argument), but on a Riemannian manifold, a natural density is determined by compatibility with the metric: the Riemannian density, also known as the Riemannian volume element. It thus makes sense to talk about volume—and the integrals of functions with respect to volume—on any Riemannian manifold, orientable or not. (As it should: the Möbius band has an area, of course, even though it is not orientable.) Arc length, area, and volume elements aren’t forms; they aren’t even linear! They’re densities.

This understanding is not impossible to acquire, but it is harder than it needs to be. The few references that discuss these matters clearly and explicitly are all relatively advanced. The easiest fix, I think, is to linger a little longer over the special status of arclength and surface area integrals in the calculus sequence, taking time to preview these two rich cultures of integration.


On the two types of integral

Terry Tao, “Differential Forms and Integration”

“Integration of forms and integration on a measure space”, Math Stack Exchange

On surface measure and its relation to Hausdorff measure

John Benedetto and Wojciech Czaja, Integration and Modern Analysis

Gerald Folland, Real Analysis: Modern Techniques and Their Applications, sections 11.2-11.4 (theorem 11.25 gives the relationship between surface measure and Hausdorff measure)

Francesco Maggi, Sets of Finite Perimeter and Geometric Variational Problems, chapter 11 (theorem 11.3 gives the relationship between surface measure and Hausdorff measure)

Kennan Smith, Primer of Modern Analysis, chapter 15 (section 6 gives the relationship between surface measure and Hausdorff measure)

Daniel Stroock, Essentials of Integration Theory for Analysis (section 5.2 constructs surface measure, and section 8.3.4 shows the relationship with Hausdorff measure)

Michael E. Taylor, Measure Theory and Integration

On densities

John Hubbard and Barbara Hubbard, Vector Calculus, Linear Algebra, and Differential Forms, Chapter 5 (like Munkres, defines Riemannian density and surface measure for submanifolds of Euclidean space, without identifying them as such)

Folland Real Analysis, section 11.4

Serge Lang, Differential and Riemannian Manifolds

John M. Lee, Introduction to Smooth Manifolds, 2nd edition, pp. 427-434 (includes a discussion of the divergence theorem)

Lynn Loomis and Shlomo Sternberg, Advanced Calculus, revised edition, chapter 10 (includes the divergence theorem and an implicit discussion of the surface area problem)

James Munkres, Analysis on Manifolds, chapter 5 (defines Riemannian density and surface measure for submanifolds of Euclidean space, without identifying them as such)

Liviu I. Nicolaescu, Lectures on the Geometry of Manifolds, section 3.4

“The ds which appears in an integral with respect to arclength is not a 1-form. What is it?”MathOverflow

“Why do I need densities to integrate on a non-orientable manifold?”, MathOverflow

“Lebesgue measure theory vs. differential forms?”, Math Stack Exchange

Nothing but a little marrow

From Rabelais’s prologue to the first book of Gargantua and Pantagruel, translated in 1653 by Sir Thomas Urquhart:

Most noble and illustrious drinkers, and you thrice precious pockified blades (for to you, and none else, do I dedicate my writings), Alcibiades, in that dialogue of Plato’s, which is entitled The Banquet, whilst he was setting forth the praises of his schoolmaster Socrates (without all question the prince of philosophers), amongst other discourses to that purpose, said that he resembled the Silenes. Silenes of old were little boxes, like those we now may see in the shops of apothecaries, painted on the outside with wanton toyish figures, as harpies, satyrs, bridled geese, horned hares, saddled ducks, flying goats, thiller harts, and other such-like counterfeited pictures at discretion, to excite people unto laughter, as Silenus himself, who was the foster-father of good Bacchus, was wont to do; but within those capricious caskets were carefully preserved and kept many rich jewels and fine drugs, such as balm, ambergris, amomon, musk, civet, with several kinds of precious stones, and other things of great price. Just such another thing was Socrates. For to have eyed his outside, and esteemed of him by his exterior appearance, you would not have given the peel of an onion for him, so deformed he was in body, and ridiculous in his gesture. He had a sharp pointed nose, with the look of a bull, and countenance of a fool: he was in his carriage simple, boorish in his apparel, in fortune poor, unhappy in his wives, unfit for all offices in the commonwealth, always laughing, tippling, and merrily carousing to everyone, with continual gibes and jeers, the better by those means to conceal his divine knowledge. Now, opening this box you would have found within it a heavenly and inestimable drug, a more than human understanding, an admirable virtue, matchless learning, invincible courage, unimitable sobriety, certain contentment of mind, perfect assurance, and an incredible misregard of all that for which men commonly do so much watch, run, sail, fight, travel, toil and turmoil themselves.

Whereunto (in your opinion) doth this little flourish of a preamble tend? For so much as you, my good disciples, and some other jolly fools of ease and leisure, reading the pleasant titles of some books of our invention, as Gargantua, Pantagruel, Whippot (Fessepinte.), the Dignity of Codpieces, of Pease and Bacon with a Commentary, &c., are too ready to judge that there is nothing in them but jests, mockeries, lascivious discourse, and recreative lies; because the outside (which is the title) is usually, without any farther inquiry, entertained with scoffing and derision. But truly it is very unbeseeming to make so slight account of the works of men, seeing yourselves avouch that it is not the habit makes the monk, many being monasterially accoutred, who inwardly are nothing less than monachal, and that there are of those that wear Spanish capes, who have but little of the valour of Spaniards in them. Therefore is it, that you must open the book, and seriously consider of the matter treated in it. Then shall you find that it containeth things of far higher value than the box did promise; that is to say, that the subject thereof is not so foolish as by the title at the first sight it would appear to be.

And put the case, that in the literal sense you meet with purposes merry and solacious enough, and consequently very correspondent to their inscriptions, yet must not you stop there as at the melody of the charming syrens, but endeavour to interpret that in a sublimer sense which possibly you intended to have spoken in the jollity of your heart. Did you ever pick the lock of a cupboard to steal a bottle of wine out of it? Tell me truly, and, if you did, call to mind the countenance which then you had. Or, did you ever see a dog with a marrowbone in his mouth,—the beast of all other, says Plato, lib. 2, de Republica, the most philosophical? If you have seen him, you might have remarked with what devotion and circumspectness he wards and watcheth it: with what care he keeps it: how fervently he holds it: how prudently he gobbets it: with what affection he breaks it: and with what diligence he sucks it. To what end all this? What moveth him to take all these pains? What are the hopes of his labour? What doth he expect to reap thereby? Nothing but a little marrow. True it is, that this little is more savoury and delicious than the great quantities of other sorts of meat, because the marrow (as Galen testifieth, 5. facult. nat. & 11. de usu partium) is a nourishment most perfectly elaboured by nature.

In imitation of this dog, it becomes you to be wise, to smell, feel and have in estimation these fair goodly books, stuffed with high conceptions, which, though seemingly easy in the pursuit, are in the cope and encounter somewhat difficult. And then, like him, you must, by a sedulous lecture, and frequent meditation, break the bone, and suck out the marrow,—that is, my allegorical sense, or the things I to myself propose to be signified by these Pythagorical symbols, with assured hope, that in so doing you will at last attain to be both well-advised and valiant by the reading of them: for in the perusal of this treatise you shall find another kind of taste, and a doctrine of a more profound and abstruse consideration, which will disclose unto you the most glorious sacraments and dreadful mysteries, as well in what concerneth your religion, as matters of the public state, and life economical.

Yes, but proceed as though I did not know it

From Moliere’s Le Bourgeois gentilhomme, translated by A. R. Waller:

Teacher of Philosophy: […] What do you wish to learn?

Monsieur Jourdain: Everything I can, for I am intensely anxious to be learned; it troubles me that my father and my mother did not see that I was thoroughly grounded in all knowledge when I was young.

Teacher of Philosophy: An admirable sentiment: Nam sine doctrina vita est quasi mortis imago. Doubtless you know Latin and understanding that.

Monsieur Jourdain: Yes, but proceed as thought I did not know it: explain to me what it means.

Teacher of Philosophy: It means, Without knowledge, life is little more than the reflection of death.

Monsieur Jourdain: That Latin is right.

Teacher of Philosophy: Do you not know some of the principles, some of the rudiments of knowledge?

Monsieur Jourdain: Oh! Yes, I know how to read and write.

Teacher of Philosophy: Where would you like us to begin? Would you like me to teach you logic?

Monsieur Jourdain: What is logic?

Teacher of Philosophy: It teaches that which educates the three operations of the mind.

Monsieur Jourdain: What are the three operations of the mind?

Teacher of Philosophy: The first, the second and the third. The first is to have a proper conception of things, by means of universals; the second is to judge accurately, by means of categories; and the third is to draw a conclusion accurately by means of the figures Barbara, Celarent, Durii, Ferio, Baralipton, etc.

Monsieur Jourdain: Those words are regular jawbreakers. That logic does not appeal to me. Let us learn something nicer.

I must set it to music

From Berlioz’s Evenings in the Orchestra (1854), translated by Charles E. Roche, pp. 314-315:

Beethoven, when carried away by the subject of Leonora, or
Conjugal Love, saw in it only the sentiments it gave him the
opportunity of expressing, and never took into account the sombre
monotony of the spectacle It presents. The libretto, of French
origin, was first set to music in Paris by Gavaux. Later an Italian
opera was made out of it for Paer, and it was after having heard
in Vienna the music of the latter’s Leonora that Beethoven had
the naive cruelty to say to him: “The subject of your opera pleases
me; I must set it to music.”

It was not the only quip Ludwig had in him. There’s also this, from Anton Schindler’s biography Beethoven as I Knew Him:

On New Year’s Day, 1823, Beethoven, his nephew, and the author were sitting at their noon meal, when the master was handed a New Year’s card from his brother who lived close by. The card was signed, “Johann van Beethoven, Land-owner.” Immediately the master wrote on the back of the card, “Ludwig van Beethoven, Brain-owner,” and returned it to him.