Cannot one say what is true?

Stanley Cavell, The Claim of Reason, pp. 205–206:

“[…] Are you suggesting that one cannot sometimes say what is true?” What I am suggesting is that “Because it is true” is not a reason or basis for saying something; and I am suggesting that there must, in grammar, be reasons for what you say, or be a point in your saying of something, if what you say is to be comprehensible. We can understand what the words mean apart from understanding why you say them; but apart from understanding the point of you saying them we cannot understand what you mean.

The most amazing fact

A charming discussion of what should be called the fundamental theorem of computation theory, in Epstein and Carnielli, Computability: Computable Functions, Logic, and the Foundations of Mathematics (2008):

We have studied one formalization of the notion of computability. In succeeding chapters we will study two more: recursive functions and functions representable in a formal system.

The Most Amazing Fact
All the attempts at formalizing the intuitive notion of computable function yield exactly the same class of functions.

So if a function is Turing machine computable, it can also be computed in any of the other systems described in Chapter 8.E. This is a mathematical fact which requires a proof. […] Odifreddi, 1989 establishes all the equivalences. […]

The Most Amazing Fact is stated about an extensional class of functions, but it can be stated constructively: Any computation procedure for any of the attempts at formalizing the intuitive notion of computable function can be translated into any other formalization in such a way that the two formalizations have the same outputs for the same inputs.

In 1936, even before these equivalences were established, Church said,

We now define the notion, already discussed, of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers (or of a lambda-definable function of positive integers). This definition is thought to be justified by the considerations which follow, so far as positive justification can ever be obtained for the selection of a formal definition to correspond to an intuitive notion.

So we have

Church’s Thesis: A function is computable iff it is lambda-definable.

This is a nonmathematical thesis: it equates an intuitive notion (computability) with a precise, formal one (lambda-definability). By our amazing fact this thesis is equivalent to

A function is computable iff it is Turing machine computable.

Turing devised his machines in a conscious attempt to capture in simplest terms what computability is. That his model turned out to give the same class of functions as Church’s (as established by Turing in the paper cited above) was strong evidence that it was the “right” class. Later we will consider some criticisms of Church’s Thesis in that the notion of computability should coincide with either a larger or a small class than the Turing machine computable ones.

The two cultures of statistical modeling

Peter Norvig, “On Chomsky and the Two Cultures of Statistical Learning” (2011):

At the Brains, Minds, and Machines symposium held during MIT’s 150th birthday party, Technology Review reports that Prof. Noam Chomsky

derided researchers in machine learning who use purely statistical methods to produce behavior that mimics something in the world, but who don’t try to understand the meaning of that behavior.

The transcript is now available, so let’s quote Chomsky himself:

It’s true there’s been a lot of work on trying to apply statistical models to various linguistic problems. I think there have been some successes, but a lot of failures. There is a notion of success … which I think is novel in the history of science. It interprets success as approximating unanalyzed data.

This essay discusses what Chomsky said, speculates on what he might have meant, and tries to determine the truth and importance of his claims. Chomsky’s remarks were in response to Steven Pinker’s question about the success of probabilistic models trained with statistical methods.

  1. What did Chomsky mean, and is he right?
  2. What is a statistical model?
  3. How successful are statistical language models?
  4. Is there anything like their notion of success in the history of science?
  5. What doesn’t Chomsky like about statistical models?

The abstract of Leo Breiman, “Statistical Modeling: The Two Cultures” in Statistical Science (2001):

There are two cultures in the use of statistical modeling to reach conclusions from data. One assumes that the data are generated by a given stochastic data model. The other uses algorithmic models and treats the data mechanism as unknown. The statistical community has been committed to the almost exclusive use of data models. This commitment has led to irrelevant theory, questionable conclusions, and has kept statisticians from working on a large range of interesting current problems. Algorithmic modeling, both in theory and practice, has developed rapidly in fields outside statistics. It can be used both on large complex data sets and as a more accurate and informative alternative to data modeling on smaller data sets. If our goal as a field is to use data to solve problems, then we need to move away from exclusive dependence on data models and adopt a more diverse set of tools.

“White box” machine learning

From “A White-Box Machine Learning Approach for Revealing Antibiotic Mechanisms of Action,” Cell 177, 1–13 (2019):

Data-driven machine learning activities are poised to transform biological discovery and the treatment of human disease (Camacho et al., 2018, Wainberg et al., 2018, Webb, 2018, Yu et al., 2018a); however, existing techniques for extracting biological information from large datasets frequently encode relationships between perturbation and phenotype in opaque “black-boxes” that are mechanistically uninterpretable and, consequently, can only identify correlative as opposed to causal relationships (Ching et al., 2018). In natural systems, biological molecules are biochemically organized in networks of complex interactions underlying observable phenotypes; biological network models may therefore harbor the potential to provide mechanistic structure to machine learning activities, yielding transparent “white-box” causal insights (Camacho et al., 2018, Yu et al., 2018b).

Chemical and genetic screens are workhorses in modern drug discovery but frequently suffer from poor (1%–3%) hit rates (Roses, 2008). Such low hit rates often underpower the bioinformatic analyses used for causal inference because of limitations in biological information content. Experimentally validated network models possess the potential to expand the biological information content of sparse screening data; however, biological screening experiments are typically performed independently from network modeling activities, limiting subsequent analyses to either post hoc bioinformatic enrichment from screening hits or experimental validation of existing models. Therefore, there is a need to develop biological discovery approaches that integrate biochemical screens with network modeling and advanced data analysis techniques to enhance our understanding of complex drug mechanisms (Camacho et al., 2018, Wainberg et al., 2018, Xie et al., 2017). Here we develop one such approach and apply it to understanding antibiotic mechanisms of action.


Machine learning aims to generate predictive models from sets of training data; such activities are typically comprised of three parts: input data, output data, and the predictive model trained to compute output data from input data (Figure 1A; Camacho et al., 2018). Although modern machine learning methods can assemble high-fidelity input-output associations from training data, the functions comprising the resulting trained models often do not possess tangible biochemical analogs, rendering them mechanistically uninterpretable. Consequently, predictive models generated by such (black-box) machine learning activities are unable to provide direct mechanistic insights into how biological molecules are interacting to give rise to observed phenomena. To address this limitation, we developed a white-box machine learning approach, leveraging carefully curated biological network models to mechanistically link input and output data (Yu et al., 2018b).

h/t Anne Trafton of MIT News, “Painting a Fuller Picture of How Antibiotics Act”:

Markus Covert, an associate professor of bioengineering at Stanford University, says the study is an important step toward showing that machine learning can be used to uncover the biological mechanisms that link inputs and outputs.

“Biology, especially for medical applications, is all about mechanism,” says Covert, who was not involved in the research. “You want to find something that is druggable. For the typical biologist, it hasn’t been meaningful to find these kinds of links without knowing why the inputs and outputs are linked.”

Where earth is forgotten

The opening of Thomas de Quincey’s review of The Works of Alexander Pope, Esquire in the North British Review (August 1848):

Every great classic in our native language should from time to time be reviewed anew; and especially if he belongs in any considerable extent to that section of the literature which connects itself with manners; and if his reputation originally, or his style of composition, is likely to have been much influenced by the transient fashions of his own age. The withdrawal, for instance, from a dramatic poet, or a satirist, of any false luster which he has owed to his momentary connection with what we may call the personalities of a fleeting generation, or of any undue shelter to his errors which may have gathered round them from political bias, or from intellectual infirmities amongst his partisans, will sometimes seriously modify, after a century or so, the fairest original appreciation of a fine writer. A window composed of Claude Lorraine glasses spreads over the landscape outside a disturbing effect, which not the most practiced eye can evade. The eidola theatri affect us all. No man escapes the contagion from his contemporary bystanders.

As books multiply to an unmanageable excess, selection becomes more and more a necessity for readers, and the power of selection more and more a desperate problem for the busy part of readers. The possibility of selecting wisely is becoming continually more hopeless as the necessity for selection is becoming continually more pressing. Exactly as the growing weight of books overlays and stifles the power of comparison, pari passu is the call for comparison the more clamorous; and thus arises a duty correspondingly more urgent of searching and revising until everything spurious has been weeded out from amongst the Flora of our highest literature, and until the waste of time for those who have so little at their command is reduced to a minimum. For, where the good cannot be read in its twentieth part, the more requisite it is that no part of the bad should steal an hour of the available time; and it is not to be endured that people without a minute to spare should be obliged first of all to read a book before they can ascertain whether in fact it is worth reading. The public cannot read by proxy as regards the good which it is to appropriate, but it can as regards the poison which it is to escape. And thus, as literature expands, becoming continually more of a household necessity, the duty resting upon critics (who are the vicarious readers for the public) becomes continually more urgent — of reviewing all works that may be supposed to have benefited too much or too indiscriminately by the superstition of a name. The praegustatores should have tasted of every cup, and reported its quality, before the public call for it; and, above all, they should have done this in all cases of the higher literature — that is, of literature properly so called.

What is it that we mean by literature? Popularly, and amongst the thoughtless, it is held to include everything that is printed in a book. Little logic is required to disturb that definition. The most thoughtless person is easily made aware that in the idea of literature one essential element is some relation to a general and common interest of man — so that what applies only to a local, or professional, or merely personal interest, even though presenting itself in the shape of a book, will not belong to literature. So far the definition is easily narrowed; and it is as easily expanded. For not only is much that takes a station in books not literature; but inversely, much that really is literature never reaches a station in books. The weekly sermons of Christendom, that vast pulpit literature which acts so extensively upon the popular mind — to warn, to uphold, to renew, to comfort, to alarm — does not attain the sanctuary of libraries in the ten-thousandth part of its extent. The drama again — as, for instance, the finest of Shakespeare’s plays in England, and all leading Athenian plays in the noontide of the Attic stage — operated as a literature on the public mind, and were (according to the strictest letter of that term) published through the audiences that witnessed their representation some time before they were published as things to be read; and they were published in this scenical mode of publication with much more effect than they could have had as books during ages of costly copying or of costly printing.

Books, therefore, do not suggest an idea coextensive and interchangeable with the idea of literature; since much literature, scenic, forensic, or didactic (as from lecturers and public orators) , may never come into books, and much that does come into books may connect itself with no literary interest. But a far more important correction, applicable to the common vague idea of literature is to be sought not so much in a better definition of literature as in a sharper distinction of the two functions which it fulfils. In that great social organ which, collectively, we call literature, there may be distinguished two separate offices that may blend and often do so, but capable, severally, of a severe insulation, and naturally fitted for reciprocal repulsion. There is, first, the Literature of Kowledge; and, secondly, the Literature of Power. The function of the first is — to teach; the function of the second is — to move: the first is a rudder; the second, an oar or a sail. The first speaks to the mere discursive understanding; the second speaks ultimately, it may happen, to the higher understanding or reason, but always through affections of pleasure and sympathy. Remotely, it may travel towards an object seated in what Lord Bacon calls dry light; but, proximately, it does and must operate — else it ceases to be a Literature of Power — on and through that humid light which clothes itself in the mists and glittering iris of human passions, desires, and genial emotions. Men have so little reflected on the higher functions of literature as to find it a paradox if one should describe it as a mean or subordinate purpose of books to give information. But this is a paradox only in the sense which makes it honorable to be paradoxical. Whenever we talk in ordinary language of seeking information or gaining knowledge, we understand the words as connected with something of absolute novelty. But it is the grandeur of all truth which can occupy a very high place in human interests that it is never absolutely novel to the meanest of minds: it exists eternally by way of germ or latent principle in the lowest as in the highest, needing to be developed, but never to be planted. To be capable of transplantation is the immediate criterion of a truth that ranges on a lower scale. Besides which, there is a rarer thing than truth — namely, power, or deep sympathy with truth. What is the effect, for instance, upon society, of children? By the pity, by the tenderness, and by the peculiar modes of admiration, which connect themselves with the helplessness, with the innocence, and with the simplicity of children, not only are the primal affections strengthened and continually renewed, but the qualities which are dearest in the sight of heaven — the frailty, for instance, which appeals to forbearance, the innocence which symbolizes the heavenly, and the simplicity which is most alien from the worldly — are kept up in perpetual remembrance, and their ideals are continually refreshed. A purpose of the same nature is answered by the higher literature, viz. the Literature of Power. What do you learn from Paradise Lost? Nothing at all. What do you learn from a cookery-book? Something new, something that you did not know before, in every paragraph. But would you therefore put the wretched cookery-book on a higher level of estimation than the divine poem? What you owe to Milton is not any knowledge, of which a million separate items are still but a million of advancing steps on the same earthly level; what you owe is power — that is, exercise and expansion to your own latent capacity of sympathy with the infinite, where every pulse and each separate influx is a step upwards, a step ascending as upon a Jacob’s ladder from earth to mysterious altitudes above the earth. All the steps of knowledge, from first to last, carry you further on the same plane, but could never raise you one foot above your ancient level of earth: whereas the very first step in power is a flight — is an ascending movement into another element where earth is forgotten.

Were it not that human sensibilities are ventilated and continually called out into exercise by the great phenomena of infancy, or of real life as it moves through chance and change, or of literature as it recombines these elements in the mimicries of poetry, romance, &c., it is certain that, like any animal power or muscular energy falling into disuse, all such sensibilities would gradually droop and dwindle. It is in relation to these great moral capacities of man that the Literature of Power, as contradistinguished from that of knowledge, lives and has its field of action. It is concerned with what is highest in man; for the Scriptures themselves never condescended to deal by suggestion or cooperation with the mere discursive understanding: when speaking of man in his intellectual capacity, the Scriptures speak not of the understanding, but of “the understanding heart” — making the heart, i.e. the great intuitive (or non-discursive) organ, to be the interchangeable formula for man in his highest state of capacity for the infinite. Tragedy, romance, fairy tale, or epopee, all alike restore to man’s mind the ideals of justice, of hope, of truth, of mercy, of retribution, which else (left to the support of daily life in its realities) would languish for want of sufficient illustration. What is meant, for instance, by poetic justice? — It does not mean a justice that differs by its object from the ordinary justice of human jurisprudence; for then it must be confessedly a very bad kind of justice; but it means a justice that differs from common forensic justice by the degree in which it attains its object, a justice that is more omnipotent over its own ends, as dealing — not with the refractory elements of earthly life, but with the elements of its own creation, and with materials flexible to its own purest preconceptions. It is certain that, were it not for the Literature of Power, these ideals would often remain amongst us as mere arid notional forms; whereas, by the creative forces of man put forth in literature, they gain a vernal life of restoration, and germinate into vital activities. The commonest novel, by moving in alliance with human fears and hopes, with human instincts of wrong and right, sustains and quickens those affections. Calling them into action, it rescues them from torpor. And hence the pre-eminency over all authors that merely teach of the meanest that moves, or that teaches, if at all, indirectly by moving. The very highest work that has ever existed in the Literature of Knowledge is but a provisional work: a book upon trial and sufferance, and quamdiu bene se gesserit. Let its teaching be even partially revised, let it be but expanded — nay, even let its teaching be but placed in a better order — and instantly it is superseded. Whereas the feeblest works in the Literature of Power, surviving at all, survive as finished and unalterable amongst men. For instance, the Principia of Sir Isaac Newton was a book militant on earth from the first. In all stages of its progress it would have to fight for its existence: first, as regards absolute truth; secondly, when that combat was over, as regards its form or mode of presenting the truth. And as soon as a Laplace, or anybody else, builds higher upon the foundations laid by this book, effectually he throws it out of the sunshine into decay and darkness; by weapons won from this book he superannuates and destroys this book, so that soon the name of Newton remains as a mere nominis umbra, but his book, as a living power, has transmigrated into other forms. Now, on the contrary, the Iliad, the Prometheus of Aeschylus, the Othello or King Lear, the Hamlet or Macbeth, and the Paradise Lost, are not militant, but triumphant for ever as long as the languages exist in which they speak or can be taught to speak. They never can transmigrate into new incarnations. To reproduce these in new forms, or variations, even if in some things they should be improved, would be to plagiarize. A good steam-engine is properly superseded by a better. But one lovely pastoral valley is not superseded by another, nor a statue of Praxiteles by a statue of Michelangelo. These things are separated not by imparity, but by disparity. They are not thought of as unequal under the same standard, but as different in kind, and, if otherwise equal, as equal under a different standard. Human works of immortal beauty and works of nature in one respect stand on the same footing; they never absolutely repeat each other, never approach so near as not to differ; and they differ not as better and worse, or simply by more and less: they differ by undecipherable and incommunicable differences, that cannot be caught by mimicries, that cannot be reflected in the mirror of copies, that cannot become ponderable in the scales of vulgar comparison.

All works in this class, as opposed to those in the Literature of Knowledge, first, work by far deeper agencies, and, secondly, are more permanent; in the strictest sense they are χτηματα εσ αει: and what evil they do, or what good they do, is commensurate with the national language, sometimes long after the nation has departed. At this hour, five hundred years since their creation, the tales of Chaucer, never equaled on this earth for their tenderness, and for life of picturesqueness, are read familiarly by many in the charming language of their natal day, and by others in the modernizations of Dryden, of Pope, and Wordsworth. At this hour, one thousand eight hundred years since their creation, the Pagan tales of Ovid, never equaled on this earth for the gaiety of their movement and the capricious graces of their narrative, are read by all Christendom. This man’s people and their monuments are dust; but he is alive: he has survived them, as he told us that he had it in his commission to do, by a thousand years; “and shall a thousand more.”

All the Literature of Knowledge builds only ground nests, that are swept away by floods, or confounded by the plough; but the Literature of Power builds nests in aerial altitudes of temples sacred from violation, or of forests inaccessible to fraud. This is a great prerogative of the power literature; and it is a greater which lies in the mode of its influence. The knowledge literature, like the fashion of this world, passeth away. An encyclopedia is its abstract; and, in this respect, it may be taken for its speaking symbol — that before one generation has passed an encyclopedia is superannuated; for it speaks through the dead memory and unimpassioned understanding, which have not the repose of higher faculties, but are continually enlarging and varying their phylacteries. But all literature properly so called — literature χατ εξοχην — for the very same reason that it is so much more durable than the Literature of Knowledge, is (and by the very same proportion it is) more intense and electrically searching in its impressions. The directions in which the tragedy of this planet has trained our human feelings to play, and the combinations into which the poetry of this planet has thrown our human passions of love and hatred, of admiration and contempt, exercise a power for bad or good over human life that cannot be contemplated, when stretching through many generations, without a sentiment allied to awe. And of this let every one be assured — that he owes to the impassioned books which he has read many a thousand more of emotions than he can consciously trace back to them. Dim by their origination, these emotions yet arise in him, and mould him through life, like forgotten incidents of his childhood.

The tree of knowledge

The frontispiece of the first of Pierre Mouchon’s two-volume Table (1780), or index, to the 28 volumes and 5 supplementary volumes of the Encyclopédie originally edited by Diderot and D’Alembert (first edition 1751), engraved by Robert Bénard (whose name appears in the lower righthand corner), with epigraph

Au lecteur

Da veniam Scriptis, quorum non gloria
nobis causa, sed utilitas, officiumque fuit.

Ovid Trist. III. 10.

(which gives an incorrect citation of Ovid! it should be Ex Ponto III.55) and this inscription:

Selon l’Explication détaillée du Système
des Connaissances Humaines dans le Discours
préliminaire des Editeurs de l’Encyclopédie
publiée par M. Diderot et M. d’Alembert,
À Paris en 1751
Reduit en cette forme pour découvrir la connaissance
Humaine d’un coup d’oeil.
Par Chrétien Frederic Guillaume Roth,
À Weimar, 1769

The “tree” of knowledge in the first edition of 1751

came after the “Preliminary Discourse” (translated into English by Richard N. Schwab), with these remarks:

After reviewing the different parts of our knowledge and the characteristics that distinguish them, it remains for us only to make a genealogical or encyclopedic tree which will gather the various branches of knowledge together under a single point of view and will serve to indicate their origin and their relationships to one another. We will explain in a moment the use to which that tree may be put according to our claims, but the execution itself is not without difficulty. Although the philosophical history we have just given of the origins of our ideas is very useful in facilitating such a work, it should not be thought that the encyclopedic tree ought to be, or even can be, slavishly subject to that history. The general system of the sciences and the arts is a sort of labyrinth, a tortuous road which the intellect enters without quite knowing what direction to take. Impelled, first of all, by its needs and by those of the body to which it is united, the intelligence studies the first objects that present themselves to it. It delves as far as it can into the knowledge of these objects, soon meets difficulties that obstruct it, and whether through hope or even through despair of surmounting them, plunges on to a new route; now it retraces its footsteps, sometimes crosses the first barriers only to meet new ones; and passing rapidly from one object to another, it carries through a sequence of operations on each of them at different intervals, as if by jumps. The discontinuity of these operations is a necessary effect of the very generation of ideas. However philosophic this disorder may be on the part of the soul, an encyclopedic tree which attempted to portray it would be disfigured, indeed utterly destroyed.

Moreover, as we have already shown in the subject of Logic, most of the sciences which we regard as including the basic principles of all the others, and which ought for this reason to occupy the first places in the encyclopedic arrangement, do not have the first places in the genealogical arrangements of ideas because they were not invented first. Indeed, in the beginning we [human beings] of necessity studied individual things. It is only after having considered their particular and palpable properties that we envisaged their general and common properties and created Metaphysics and Geometry by intellectual abstraction. Only after the long usage of the first signs have we perfected the art of these signs to the point of making a science of them. And it is only after a long sequence of operations on the objects of our ideas that, through reflection, we have at length given rules to these operations themselves.

Finally, the system of our knowledge is composed of different branches, several of which have a common point of union. Since it is not possible, starting out from this point, to begin following all the routes simultaneously, it is the nature of the different minds that determines which route is chosen. Rarely does a single mind travel along a large number of these routes at the same time. In the study of Nature, men at first applied themselves, as if in concert, to satisfying the most pressing needs. But when they came to less absolutely necessary knowledge, they were obliged to divide it among themselves, and each one moved forward in almost equal step with the others. Thus several sciences have been contemporaneous, so to speak. But when tracing in historical order the progress of the mind, one can only embrace them successively.

It is not the same with the encyclopedic arrangement of our knowledge. This consists of collecting knowledge into the smallest area possible and of placing the philosopher at a vantage point, so to speak, high above this vast labyrinth, whence he can perceive the principal sciences and the arts simultaneously. From there he can see at a glance the objects of their speculations and the operations which can be made on these objects; he can discern the general branches of human knowledge, the points that separate or unite them; and sometimes he can even glimpse the secrets that relate them to one another. It is a kind of world map which is to show the principal countries, their position and their mutual dependence, the road that leads directly from one to the other. This road is often cut by a thousand obstacles, which are known in each country only to the inhabitants or to travelers, and which cannot be represented except in individual, highly detailed maps. These individual maps will be the different articles of the Encyclopedia and the Tree or Systematic Chart will be its world map

But as, in the case of the general maps of the globe we inhabit, objects will be near or far and will have different appearances according to the vantage point at which the eye is placed by the geographer constructing the map, likewise the form of the encyclopedic tree will depend on the vantage point one assumes in viewing the universe of letters. Thus one can create as many different systems of human knowledge as there are world maps having different projections, and each one of these systems might even have some particular advantage possessed by none of the others. There are hardly any scholars who do not readily assume that their own science is at the center of all the rest, somewhat in the way that the first men placed themselves at the center of the world, persuaded that the universe was made for them. Viewed with a philosophical eye, the claim of several of these scholars could perhaps be justified by rather good reasons, quite aside from self-esteem.

In any case, of all the encyclopedic trees the one that offered the largest number of connections and relationships among the sciences would doubtless deserve preference. But can one flatter oneself into thinking it has been found? We cannot repeat too often that nature is composed merely of individual things which are the primary object of our sensations and direct perceptions. To be sure, we note in these individual things common properties by which we compare them and dissimilar properties by which we differentiate them. And these properties, designated by abstract names, have led us to form different classes in which these objects have been placed. But often such an object, which because of one or several of its properties has been placed in one class, belongs to another class by virtue of other properties and might have been placed accordingly. Thus, the general division remains of necessity somewhat arbitrary. The most natural arrangement would be the one in which the objects followed one another by imperceptible shadings which serve simultaneously to separate them and to unite them. But the small number of beings known to us does not permit us to indicate these shadings. The universe is but a vast ocean, on the surface of which we perceive a few islands of various sizes, whose connection with the continent is hidden from us.

One could construct the tree of our knowledge by dividing it into natural and revealed knowledge, or useful and pleasing knowledge, or speculative and practical knowledge, or evident, certain, probable, and sensitive knowledge, or knowledge of things and knowledge of signs, and so on into infinity. We have chosen a division which has appeared to us most nearly satisfactory for the encyclopedic arrangement of our knowledge and, at the same time, for its genealogical arrangement. We owe this division to a celebrated author [Bacon] of whom we will speak later in this preface. To be sure, we have thought it necessary to make some changes in his division, of which we will render an account; but we are too aware of the arbitrariness which will always prevail in such a division to believe that our system is the only one or the best. It will be sufficient for us if our work is not entirely disapproved of by men of intelligence. We do not wish to resemble that multitude of naturalists (censured with such good reason by a modern philosopher) whose energies have been ceaselessly devoted to dividing the productions of Nature into genera and species, consuming an amount of time in this labor which would have been employed to much better purpose in the study of those productions themselves. What would be said of an architect, who, having to build an immense edifice, passed his whole life in drawing the plans for it? Or likewise what would we say of an inquisitive person who, proposing to inspect an enormous palace, spent all his time in observing the entryway?

On the Encyclopédie and the encyclopedic tradition, see:

Phillip Blom, Enlightening the World: Encyclopédie, The Book That Changed the Course of History (2005)

Robert Darnton, The Business of Enlightenment: A Publishing History of the Encyclopédie, 1775–1800 (1987) and W. Paul Vogt’s review, “Paideia for Profit” (1982)

Clorinda Donato and Robert M. Maniquis, The Encyclopédie and the Age of Revolution (1992)

John Lough, The Encylcopédie (1971)

Alan Rauch, Useful Knowledge: The Victorians, Morality, and the March of Intellect (2001)

Robert Yeo, Encyclopaedic Visions: Scientific Dictionaries and Enlightenment Culture (2001)

On the tree and other controlling metaphors for organizing knowledge and information, see:

Tore Frängsmyr, ed., The Structure of Knowledge: Classifications of Science and Learning Since the Renaissance (2001)

D.R. Kelley and R.H. Popkin, eds., The Shapes of Knowledge from the Renaissance to the Enlightenment (1991)

Manuel Lima, The Book of Trees: Visualizing Branches of Knowledge (2014)

Manuel Lima, The Book of Circles: Visualizing Spheres of Knowledge (2017)

A refuge of absolute technique

Glenn Gould’s introductory remarks before playing the last movement of Hindemith’s third piano sonata, in The Anatomy of Fugue, broadcast by Canadian Broadcasting Corporation on March 4, 1963:

In more recent times one often notices that the most prolific fugue writers are the composers who have the greatest difficulty in being direct and lucid in a freer compositional style. One thinks of composers like Max Reger or the extraordinary Russian Nikolai Myaskovsky. Perhaps this is due to the fact that the artistic climate of the early years of this century was more hospitable to the concept of the stargazing composer who received lightning bolts of inspiration than to the image of the hardworking academic burning the midnight oil over his fugues and canons. And in such times those composers who for one reason or another find the philosophical liberties of free form frightening or embarrassing tend to use the fugue as a refuge of absolute technique: one can prove one’s mastery of the material more readily in a fugue than one can prove it in a less self-conscious form. There’s less room for arguments to whether or not one has been successful. One need only suggest certain fundamental propositions of aesthetic right and wrong, and one’s colleagues will have to agree that a particular fugal inspiration has been a happy one, or a successful one, or maybe a slightly too risky one. And so in generations in which fewer marks are given for ingenuity, for contrapuntal dexterity, than for vivid imagination, for dramatic flair, the invention of fugue has often attracted composers temperamentally unsuited to larger forms—composers perhaps of a particularly logical mind, who find it difficult to believe unquestioningly in their own subconscious impulses.

In our century, composers have continued writing fugues, particularly those whose style shows a neoclassic influence. But almost every idiom has seen its share of fugues. The Canadian composer Harry Summers has composed fugues in a style which uses the controversial twelve-tone technique. The brilliant American Lukas Foss has taught some of his colleagues to improvise on some fugues out of chart patterns which he draws on paper. And various jazz groups have attempted to improvise fugues, too. But most of these, because of one basic omission, have negated the main source of discipline inherent in the fugue. The harmonic criteria of most of the contemporary musical language shies away from tonality, and they’ve been unable to develop their linear designs into a system which would produce real demands of chord tension and relaxation. And because of this, fugal technique in their work is little more than a respectful bow to the past, really—it’s, it’s not a living tradition.

Paul Hindemith is one of the few composers of our own time who can undeniably be called a fuguist to the manner born. Hindemith has developed a very special language of his own, a language which is contemporary in the best sense of the word but which in its attempt to provide harmonic logic uses what you might call a substitute tonality. It neatly sidesteps the basic confrontations of tonic and dominant chords of conventional tonality, but it has nonetheless a very strong sense of relative tension. And so since he uses a language in which this structure of fugue can be helped along by the exchange of subject and answer at clearly related harmonic levels, Hindemith already has the edge on fugue writers in the tonal idiom. Besides which, he’s quite at home in an idiom which employs a minimum of what you could call textural irregularities: he’s quite able to continue a structure in three or four or five real voices over a period of some minutes’ duration. In fact for most of his career he’s been writing in a style which rather makes us think of an early Renaissance contrapuntal jamboree. The fugue I’m going to play now by Hindemith is actually the concluding movement of a piano sonata, and it doesn’t pretend to the highly romantic connection of Beethoven’s finale, where the concept of fugue is welded into the structure of the sonata, but it happily provides an assurance that the magic of fugue, however rare it may be nowadays, isn’t yet forgotten.

The dream of an inaccessible eloquence

from Julie K. Ellison’s Emerson’s Romantic Style (1984):

Our exploration of the motives of Emerson’s development starts with his youthful journals, roughly from 1820 to 1824. These documents exhibit a severe case of literary over influence. Emerson would later say, quite accurately, “I have served my apprenticeship of bows & blushes, of fears & references, of excessive admiration” (JMN.IV.278). Awed by the glory of classical and English literature, he expressed his own literary ambitions mimetically. “What we ardently love we learn to imitate,” he writes in the well-known “robe of eloquence” passage (JMN.II.239; April 18, 1824). At the same time, he treats his imitations as proof of his inability to match his models. The intensity of his fantasies of identification with great authors of the past is directly proportionate to his contempt for himself as their critic. His gloomy meditations on history and historical awareness express the Romantic sense that self-consciousness is a belated, sentimental condition. His judgments about history, religion, and literature are manifestations of his first vocational crisis, precipitated by the conflict between the dream of an inaccessible eloquence and the habit of criticism. In his late teens and early twenties, he is plagued with uneasiness that leads eventually to the discovery of self-delighting powers.

Some definite knowledge to be obtained

from the preface to Bertrand Russell’s Dictionary of Mind, Matter, and Morals (1952):

I feel considerably honoured that my philosophy should have been thought worthy to be alphabetically anatomized in this dictionary. I have been accused of a habit of changing my opinions in philosophy and, in so far as this is true, the dictionary will enable readers to find it out. I am not myself in any degree ashamed of having changed my opinions. What physicist who was already active in 1900 would dream of boasting that his opinions had not changed during the last half century? In science men change their opinions when new knowledge becomes available, but philosophy in the minds of many is assimilated rather to theology than to science. A theological proclaims eternal truths, the creeds remain unchanged since the Council of Nicaea. Where nobody knows anything, there is no point in changing your mind. But the kind of philosophy that I value and have endeavoured to pursue is scientific in the sense that there is some definite knowledge to be obtained and that new discoveries can make the admission of former error inevitable to any candid mind.

The bestiary of the imagination

Foucault’s preface to The Order of Things (1966), translated by Routledge (anonymously!), referring to Borges’s story “The Analytical Language of John Wilkins”:

This book first arose out of a passage in Borges, out of the laughter that shattered, as I read the passage, all the familiar landmarks of my thought — our thought, the thought that bears the stamp of our age and our geography — breaking up all the ordered surfaces and all the planes with which we are accustomed to tame the wild profusion of existing things, and continuing long afterwards to disturb and threaten with collapse our age-old distinction between the Same and the Other. This passage quotes a certain Chinese encyclopedia” in which it is written that animals are divided into: (a) belonging to the Emperor, (b) embalmed, (c) tame, (d) sucking pigs, (e) sirens, (f) fabulous, (g) stray dogs, (h) included in the present classification, (i) frenzied, (j) innumerable, (k) drawn with a very fine camelhair brush, (l) et cetera, (m) having just broken the water pitcher, (n) that from a long way off look like flies”. In the wonderment of this taxonomy, the thing we apprehend in one great leap, the thing that, by means of the fable, is demonstrated as the exotic charm of another system of thought, is the limitation of our own, the stark impossibility of thinking that.

But what is it impossible to think, and what kind of impossibility are we faced with here? Each of these strange categories can be assigned a precise meaning and a demonstrable content; some of them do certainly involve fantastic entities — fabulous animals or sirens — but, precisely because it puts them into categories of their own, the Chinese encyclopedia localizes their powers of contagion; it distinguishes carefully between the very real animals (those that are frenzied or have just broken the water pitcher) and those that reside solely in the realm of imagination. The possibility of dangerous mixtures has been exorcized, heraldry and fable have been relegated to their own exalted peaks: no inconceivable amphibious maidens, no clawed wings, no disgusting, squamous epidermis, none of those polymorphous and demoniacal faces, no creatures breathing fire. The quality of monstrosity here does not affect any real body, nor does it produce modifications of any kind in the bestiary of the imagination; it does not lurk in the depths of any strange power. It would not even be present at all in this classification had it not insinuated itself into the empty space, the interstitial blanks separating all these entities from one another. It is not the fabulous” animals that are impossible, since they are designated as such, but the narrowness of the distance separating them from (and juxtaposing them to) the stray dogs, or the animals that from a long way off look like flies. What transgresses the boundaries of all imagination, of all possible thought, is simply that alphabetical series (a, b, c, d) which links each of those categories to all the others.