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.

Mechanism in biology

William Bechtel, Discovering Cell Mechanisms: The Creation of Modern Cell Biology (2006)

William Bechtel, “Mechanism and Biological Explanation,” Philosophy of Science (2011)

William Bechtel, “Biological Mechanisms: organized to maintain autonomy,” in Systems Biology: Philosophical Foundations (2007)

Carl Craver and James Tabery, “Mechanisms in Science,” Stanford Encyclopedia of Philosophy (2015)

Carl F. Craver and Lindley Darden, In Search of Mechanisms: Discoveries Across the Life Sciences (2013)

Margaret Gardel, “Moving beyond molecular mechanisms,” Journal of Cell Biology (2015)

Daniel J. Nicholson, “The concept of mechanism in biology,” Studies in History and Philosophy of Biological and Biomedical Sciences (2012)

Rob Phillips, “Musings on mechanism: quest for a quark theory of proteins?” Journal of the Federation of American Societies for Experimental Biology (2017)

James Tabery, Monika Piotrowska, and Lindley Daren, “Molecular Biology,” Stanford Encyclopedia of Philosophy (2015)

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.

Breath, blood, sex, and lightning

Bertrand Russell, “The Uses of Language,” in Human Knowledge: Its Scope and Limits (1948):

Language, like other things of mysterious importance, such as breath, blood, sex and lightning, has been viewed superstitiously ever since men were capable of recording their thoughts. Savages fear to disclose their true name to an enemy, lest he should work evil magic by means of it. Origen assures us that pagan sorcerers could achieve more by using the sacred name Jehovah than by means of the names Zeus, Osiris or Brahma. Familiarity makes us blind to the linguistic emphasis in the Commandment: ‘Thou shalt not take the nameof the Lord in vain.’ The habit of viewing language superstitiously is not yet extinct. ‘In the beginning was the Word’, says our version of St John’s Gospel, and in reading some logical positivists I am tempted to think that their view is represented by this mistranslated text.

Philosophers, being bookish and theoretical folk, have been interested in language chiefly as a means of making statements and conveying information, but this is only one of its purposes, and perhaps not the most primitive. What is the purpose of language to a sergeant-major? On the one hand there is the language of words of command, designed to cause identical simultaneous bodily movements in a number of hearers; on the other hand there is bad language, designed to cause humility in those in whom the expected bodily movements have not been caused. In neither case are words used, except incidentally, to state facts or convey information.

What it has become

Adorno, Aesthetic Theory (1970), translated by Robert Hullot-Kentor:

Posed from on high, the question whether something such as film is or is no longer art leads nowhere. Because art is what it has become, its concept refers to what it does not contain. The tension between what motivates art and art’s past circumscribes the so-called questions of aes­ thetic constitution. Art can be understood only by its laws of movement, not ac­ cording to any set of invariants. It is defined by its relation to what it is not. The specifically artistic in art must be derived concretely from its other; that alone would fulfill the demands of a materialistic-dialectical aesthetics. Art acquires its specificity by separating itself from what it developed out of; its law of movement is its law of form. It exists only in relation to its other; it is the process that tran­ spires with its other. Nietzsche’s late insight, honed in opposition to traditional philosophy, that even what has become can be true, is axiomatic for a reoriented aesthetic. The traditional view, which he demolished, is to be turned on its head: Truth exists exclusively as that which has become.

You can get some way towards the secret

Coleridge’s “Metrical Feet: Lessons for a Boy” (1834), written for his sons:

Trochee trips from long to short;
From long to long in solemn sort
Slow Spondee stalks; strong foot! yet ill able
Ever to come up with Dactyl’s trisyllable.
Iambics march from short to long;—
With a leap and a bound the swift Anapests throng.
One syllable long, with one short at each side,
Amphibrachys hastes with a stately stride;—
First and last being long, middle short, Amphimacer
Strikes his thundering hoofs like a proud high-bred racer.


A taxonomy of metrical feet from George Sainsbury’s A History of English Prose Rhythm (1912):


From Sainsbury’s preface, a little too tortured by classical learning, in the manner of Shaw:

As I approach, contemplating it still from whatever distance, the end of these studies of metre and rhythm which I may never reach, that sense of the unending endless quest,” which I suppose all but very self-satisfied and self-sufficient persons feel, impresses itself more and more upon me. An, I suppose, youthful reviewer of some different but kindred work of mine not very long ago, reproached me with ignorance or neglect of the fact that he and his generation had quite given up positive deliverances in criticism. They regarded it (I think he said) as hopeless and wrong and to “pin” something or other “to the rainbow beauty of what was really a miracle of incrustation.” The proceeding appeared to me to be difficult, if not impossible, and the phrase to be really a miracle of galimatias. But, as a fact, I hope that almost all who have read me will acquit me of the impudence or the folly of thinking that I could say even an interim last word on the secrets of rhythmical charm, whether in the slightly more tangible form of verse, or the far more intangible one of prose. Here, as everywhere, and almost more than anywhere, beauty incipit in mysterio as well as exit in mysterium. Here, and almost more also, it is as when you see a face and say to it with Browning—

Lie back; could thought of mine improve you?

and decide that, if improvement is possible, the interpretation of the actual charm is equally so. You can get some way towards the secret. The spring of the wing of the nostril; the plunge into the clear pool of the eyes, with its impenetrable background of agate or lapis lazuli, of chrysoprase or avanturine; the sweep of the cheek-edge from ear to chin; the straight descent, or curved and recurved wave, of the profile; the azure net-work of the closed eyelids; “the fringed curtains” at their juncture; the infinite intricacies of the mouth and hair,—ask yourself about any one of these, and you cannot tell why it is beautiful, why the combination of the whole makes a beautiful face. But you can, to some extent, fix for yourself the character of those parts and the composition of that whole, and, so far at least, you are ahead of the mere gaper who stares and “likes grossly.”

So it is with literature. You can never get at the final entelechy which differentiates Shelley and Shakespeare from the average versifier, Cluvienus and myself from Pater or from Browne. But you can attend to the feature-composition of the beautiful face, to the quality of the beautiful features, in each of these masters, and so you can dignify and intensify your appreciation of them. That this is best to be done in prose, as in verse, by the application of the foot-system—that is to say, by studying the combinations of the two great sound-qualities which, for my part, I call, as my fathers called them from the beginning, “long” and “short,” but which you may call anything you like, so long as you observe the difference and respect the grouping—I may almost say I know; having observed the utter practical failure of all other systems in verse, and the absence even of any attempt to apply any other to prose.

With this I may leave the present essay to its chances; only repeating my acquaintance with two quotations which I made thirty-six years ago when touching, for the first time, the subject of Prose Style generally. One was Nicholas Breton’s warning “not to talk too much of it, having so little of it,” and the other, Diderot’s epigram on Beccaria’s ouvrage sur le style où il n’y a point de style. These are, of course, “palpable hits” enough. But you may criticise without being able to create, and you may love beauty, and to the possible extent understand it, without being beautiful.


from Paul Fussell’s Poetic Meter and Poetic Form (1965):

Because the concept of the foot is an abstraction, we will never encounter a pure example of any of the standard feet. “For that matter,” as Hugh Kenner says, “you will never encounter a round face, though the term is helpful; and if the idea of a circle had never been defined for you, you might not be clearly aware of how a round face differs from a long one, even though the existence of some sort of difference is evident to the eye. The term ‘iambic foot’ has the same sort of status as the term ’round face.'” Although we will probably never meet a really pure spondee or pyrrhic, in which the two syllables are of exactly the same weight, there would seem to be no need for such over scrupulous formulations as the terms “pseudo-spondee” or “false spondee,” which suggest that our work as scansionists and critics ought to be more objective and accurate than of course it ever can be. The goal of what we are doing is enjoyment: an excessive refinement of terms and categories may impress others but it will probably not help us very much to appreciate English poetic rhythms.


Sainsbury’s remark on “the combinations of the two great sound-qualities” reminds me of Quine, “Universal Library,” in Quiddities (1987):

There is a melancholy fantasy, propounded a century and more ago by the psychologist Theodor Fechner and taken up by Kurt Lassiwitz, Theodor Wolff, Jorge Luis Borges, George Gamow, and Willy Ley, of a complete library. The library is strictly complete, boasting as it does all possible books within certain rather reasonable limits. It admits no books in alien alphabets, nor any beyond the reasonable length say of the one you are now reading, but within those restrictions it boasts all possible books. There are books in all languages, transliterated where necessary. There are coherent books and incoherent, predominantly the latter. The principle of accession is simple, if uneconomical: every combinatorially possible sequence of letters, punctuation, and spaces, up to the prescribed book length, uniformly bound in half calf.

Other writers have sufficiently belabored the numbing combinatorial statistics. At 2,000 characters to the page we get 500,000 to the 250-page volume, so with say eighty capitals and smalls and other marks to choose from we arrive at the 500,000th power of eighty as the number of books in the library. I gather that there is not room in the present phase of our expanding universe, on present estimates, for more than a negligible fraction of the collection. Numbers are cheap.

It is interesting, still, that the collection is finite. The entire and ultimate truth about everything is printed in full in that library, after all, insofar as it can be put in words at all. The limited size of each volume is no restriction, for there is always another volume that takes up the tale—any tale, true or false—where any other volume leaves off. In seeking the truth we have no way of knowing which volume to pick up nor which to follow it with, but it is all right there.

We could narrow down the choice by weeding out the gibberish, which makes up the bulk of the library. We could insist on English, and we could program a computer with English syntax and lexicon to do the scanning and discarding. The residue would be an infinitesimal fraction of the original, but still hyperastronomic.

There is an easier and cheaper way of cutting down. Some of us first learned from Samuel Finley Breese Morse what others of more mathematical bent knew before this time: that a font of two characters, dot and dash, can do all the work of our font of eighty. Morse actually used three characters, namely dot, dash and space; but two will suffice. We could use two dots for the space and then admit no initial or consecutive dots in encoding any of the other old characters.
If we retain the old format and page count for our volumes, this move reduces the size of the library’s collection to the 500,000th power of two. It is still a big number. Written out it would fill a hundred pages in standard digits, or two volumes in dots and dashes. The volumes are skimpier in thought content than before, taken one by one, because our new Morse is more than six times as long-winded as our old eighty-character font of type; but there is no loss in content over all, since for each cliff-hanging volume there is still every conceivable sequel on some shelf or other.

This last reflection—that a diminution in the coverage of each single volume does not affect the cosmic completeness of the collection—points the way to the ultimate economy: a cutback in the size of the volumes. Instead of admitting 500,000 occurrences of characters to each volume, we might settle for say seventeen. We have no longer to do with volumes, but with two-inch strips of text, and no call for half-calf bindings. In our two-character code the number of strips is 2^17, or 131,072. The totality of truth is now reduced to a manageable compass. Getting a substantial account of anything will require extensive concatenation of out two-inch strips, and re-use of strips here and there. But we have everything to work with.

The ultimate absurdity is now staring us in the face: a universal library of two volumes, one containing a single dot and the other a dash. Persistent repetition and alternation of the two is sufficient, we well know, for spelling out any and every truth. The miracle of the finite but universal library is a mere inflation of the miracle of binary notation: everything worth saying, and everything else as well, can be said with two characters. It is a letdown befitting the Wizard of Oz, but it has been a boon to computers.

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 virtue of intolerance

the opening sentences of M. T. McClure’s review in The Journal of Philosophy of W. T. Stace’s A Critical History of Greek Philosophy (1920):

Clarity is often a virtue of intolerance. A man with convictions knows precisely what he believes and is able to measure the worth of ideas as any want of conformity unto or transgression of his standards of belief. Mr. Stace is a man with convictions. He knows exactly what he means by philosophy and writes a “critical” history of Greek thought in the light (or darkness) of this meaning. The style and manner of presentation are extraordinarily simple and clear. There are more monosyllables to the paragraph than in any philosophical treatise with which I am familiar. Lucidity is the chief merit of the book. As a contribution to historical scholarship it is altogether unimportant.

The immediate perceptive epithet

from Bruce Wilshire, “William James’s Pragmatism: A Distinctly Mixed Bag,” in 100 Years of Pragmatism, edited by John Stuhr:

William James is a tragic figure. I will try to fully explain what I mean by that. But right off the bat, we can point out a feature of this tragic stance. It’s fairly widely believed that James is a major philosopher. Yet in no other such philosopher’s work, I believe, are great strengths so vividly mixed with major defects. His famous, often read—too often read, I think—popular lectures, Pragmatism, gaudily illustrate this claim.

What does it take to be a major philosopher? A most difficult question. Wilfrid Sellars’s one-liner statement of what philosophy seeks to discover is hard to better: how things, in the broadest sense, hang together, in the broadest sense.

But how does one start a process of discovery without begging crucial questions that philosophy should endeavor to answer? How does one begin to comprehend the farthest reaches of complexity without prejudging things—or occluding whole horizons of possibilities and viewpoints—stupidly? James’s description in Pragmatism of expertness in philosophy is arresting: “Expertness in philosophy is measured by the definiteness of our summarizing reactions, by the immediate perceptive epithet with which the expert hits such complex objects off” (P, 25). Thee summarizing that emerges through perceptual epithet! A taking in at a glance that delivers the first sketch of the whole lay of the land. Is there any better way to avoid getting lost in the details of some corner of the subject matter, any better way to begin doing philosophy unprejudiciously?