The Modularity of
Knowing

The paradox is old enough. The ridicule is plain, and yet that ambitious professor of Lagado was logically without flaw. More than that, the extension of his method has convinced us today that, just as all our prose might indeed appear from his mindless but modular frame, so our pictures, fabrics, devices, formulae--all knowledge--share the modular nature. The whole even of our world--radiant energy and protean matter, crystals and cells, stars and atoms--all is built of modules, whose identity and simplicity belie the unmatched diversity of the works of man and nature. The world is atomic, which is to say modular; our knowledge is modular as well. All can be counted and listed; our very analysis implies atoms of knowing, as the material itself is atomic. The prodigality of the world is only a prodigality of combination, a richness beyond human grasp contained in the interacting multiplicity of a few modules, but modules which nature has made in very hosts.

The explosive growth of sheer number out of the apparent poverty of combination is hard to grasp. Here is the seat of the paradox. School algebra will quickly show that the letter-frame would probably display one meaningful three-letter word in English--if not Lagadoan--for every few-score-randomly-selected squares of wood. Yet to expect to find any single meaningful sentence of the length, say, of the proverb "a stitch in time saves nine" would require not a mere folio or two of the professor's but a library of folios, well-printed, stored in shelves filling a large university building like the great libraries of Harvard or Berkeley. And the land masses of the earth would have to be covered with such buildings cheek by jowl before you would be sure--barely sure--of reading this one particular sentence. Of course you would find all shorter statements, Himalayas of chaff and gibberish for each pithy E=mc2. Everything would be there, everything brief enough. A vast desert of meaninglessness would surround the oasis of a sentence or two of sense, while a very few longer remarks would be found, in all the languages of Babel. You could not reasonably hope for a meaningful paragraph. The lottery of meaning is an unfair game. [p. 1]

The philosopher Gottfried Wilhelm von Leibnitz himself, before Swift, gave up atoms and modularity. Even he could not believe the computations he knew how to make. For he argued against NewtonÍs corpuscles that if indeed the world were modular, some repetition must logically occur, and no one--not even a diligent friend of his who searched a whole afternoon over the lawn of a German princess--had ever found two blades of grass which were absolutely identical. He was logically correct, but absurd before the combinatorial facts. The palace lawn was worlds too small. The storyteller Joge Luis Borges alone among imaginative writers has fully grasped the paradox in his Library of Babylon, a universe filled to the ends of space with random books, which house men still lost for want of meaning. The parable is as keen for us as was Lagado for the Royal Society, but it is more soundly reasoned. The world is modular, yet it never repeats, nor does it supply meaning randomly. The possibility of the typewriting apes and the script of Hamlet is no more than an arithmetical joke, a game with logic. [p. 3] [Fig. of "The book"-writing frame of the Grand Academy of Lagado, in the shadow of the flying island of Laputa. Random throws of the cranks would rearrange the characters of this exotic language and allow the logical hope of eventually spelling out all knowledge. From Jonathan Swift, Gulliver's Travels, 1726.

Our Latin alphabet, and the construction of every expressed meaning out of its few dozen modular bits is the most natural of introductions to the atomic world. For the claim is made that modular atoms build all of matter. But is this enough? Can we extend that claim to all languages, to all knowledge? The Chinese calligrapher might appear to defeat the analysis. His characters do not much repeat; he spins a book, perhaps out of many thousands of distinct characters. The atomic model, the modular device, arises much less readily out of such a literary tradition. Indeed, there has not been a lack of bold speculators to base, on this difference in our alphabets, the strength of atomic ideas in the West, in contrast with the vitality of an organic wholeness in Chinese thought. But the matter is not so simple. The Chinese scholar himself consults a dictionary of radicals, arranged by the number of strokes in the character. The strokes are his alphabet. The objection still persists, for the variety of placement and turn of the stroke of the brush far exceeds the number of letters in the richest alphabet. Yet it is limited. A few widths of stroke, a few turns at the ends, a few lengths of stroke, and the rest is placement. But placement itself is limited. In principle, one imagines there is an infinite choice for where to begin or to slant the stroke within the imaginary square that bounds each character. Of course this is not so; choice among a dozen positions vertically or horizontally will more than suffice. The tiny variations which lie between such a rough net of possibilities represent changes, not in meaning, but in calligraphic style. Our handwriting, too, presents letter variability. So in fact even the character is modular, though the strokes of which it is built, and the pictorial composition they enter, are far more complex than our letters. Chinese text can be rendered in English, or paraphrased by using a numbered list of characters, or redrawn by a division into small squares which remain blank or become blackened, as one copies a map. No, Chinese cannot escape the analysis of the frame, no more than does our Western cursive manuscript hand. [p. 5] [Fig. - from: Chiang Yee. Chinese Calligraphy, Cambridge, Mass: Harvard University Press, 1954.]

We have reached the stronghold of the position. All written knowledge can be modularly expressed. Consider now the image made by the painter of the photographer. In a land of the TV set, who does not know that somehow an image too can be read off in the simplicity of electrical pulses? The illustrations shown here *display the process in detail. A picture--to be sure, a rather coarse dramatic image--is spelled out in ordinary letters and digits. In orbit TIROS flashed to the ground by radio what its cameras saw. The spelled-out picture is recombined into an image. To improve the images by adding the subtlest of visual detail, to represent the most delicate of palettes as well, to present not merely a flat image, but a sculpture in the round, are all merely to demand a numerically greater list of letters from our machine. Since every letter is translatable into a choice of one among a few dozen options, and since a few dozen options can be described in a handful of true-false, on-off, or yes-no choices, it is not hard to prove that every image and piece of knowledge is expressible via the [p. 7] intermediation of a tediously long, bland series of mere on-off decisions. This is the basis of the theory of information on which a point of view towards knowledge more unifying than any other mode of analysis has in the last decades been built. Knowledge itself, and science of course within it, is seen to be modular at base.

How close this is to the structure of eye and brain, the material foundation of sight and of insight, is not yet clear. Certainly the tangled knot of the brain and its ravelling out into hand and eye are woven out of a numerous skein of rather similar neurons. But their complex geometry and their subtle function are still a good deal beyond contemporary understanding. The general view of the information theorist predisposes one to the conviction that not even in the brain, the most complex of known physical systems, will the fundamental simplicity of nature be found wanting. It has been known for years that the behavior of simpler beings, say of the army ant which has a complex enough way of life, is in fact woven again out of a simple modular set of cliché responses, whose "words" are combined of the exigencies of the environment and the stereotyped alphabet of the insect's response. A troop of such ants set moving on a smooth floor can sometimes be caused to form a circle, each ant following the scent of his predecessors in line. The circle will move aimlessly in fatal procession toward nowhere until exhaustion. The situation spells march; in the natural environment, the external world varies the command enough, by blurring the signals between ants, to allow a message of behavior more serviceable than a suicidal marathon.

The work of men too has many obvious modularities. The masonry wall and the woven fabric are two of the oldest. How varied are the world's masonry buildings, how diverse the patters of woven stuffs! yet in each case the modular idea gave the craftsman a small alphabet from which he chose and chose again to yield the mosaic pillars of Erech or the rich, figured cloth of the Peruvian coast. Here it is not knowledge but a material structure which has been synthesized from simple recombining in parts. Illustrated on the following pages is a modern and beautiful example of the same idea, far from the craftsmanship of the past, yet remaining in its full debt. This up-to-date example is a style of building computers themselves just adopted by the largest manufacturer of such devices. By its designers, the machine is "written' as a book might be. For letters, they use the individual circuit elements of the electronics trade, transistors, resistors, capacitors, and the like. A major distinction from the literary metaphor is that, unlike our letters, the electronic "letters" have not reached a stable standardization. They are, on the contrary, very much in the process of change, for after all they function for [p. 8] the fast-evolving machine, while the printer's letters must appeal to human eyes, which remain constant. These evolving small components are placed onto the receptive surfaces of small ceramic chips or cards with connections printed in conducting inks to form the words of the text. These cards, of which a large and growing font exists in the factory, are then themselves chosen to be set into racks, in an order and choice which parallels the assembly of sentences. The complete statement of many related sentences is the finished computer. [p. 9] [All illustrations in the text refer to the IBM System/360 computers/photos courtesy International Business Machines Corporation:

4a. Here is a set of the building blocks, individual identical transistors, in the course of their manufacture. They are not used as a block pattern, but rather one by one. In metaphor, this is the type-founder's shop producing a block of lead type, all one single letter. Each square transistor is about the width of a typewritten period.

4b. A collection of "letters" broken apart and ready for use. This is like a font of "a's" of a typesetter. An ordinary transistor, like many found in the familiar portable radio, is shown for comparision beside a thimbleful of the miniature modules.

4c. The assembly of "letters" into a meaningful "word." The ceramic tile, about one half inch square, is receiving the proper transistors from an automatic machine. It already bears a screen-printed copper-and-solder circuit pattern. The small thickish square indicated by an arrow is a transistor; the larger dark rectangles are printed resistance elements.

4d. A "word" is sealed into a metal shell, labeled, and inserted into a functional "sentence." the modules are meaningfully interconnected behind the circuit card. Other cards bear six, twelve, twenty-four . . . modules.

4e. The sentences are arranged into a statement of power. The circuit cards are plugged into a connection board to provide the logical operations for a computer's arithmetic processing unit, its memory, or an input-output translating device.

Such a computer spins much sense, mathematical and verbal, out of pulses which forever flow in its modules of small parts. This has seemed strange to many persons, who have only now in seeing it happen come to realize how great a change in quality can stem from a sufficient quantity of intrinsically elementary decisions. It is still passing strange to watch the computer output so well approximate some forms of thought. But the idea that a mere machine could embody repeated meaningful choice is very old, arising not first in the more abstract modes, but in fact in the weaving of cloth, which not merely in metaphor is based on the self-same analytical idea. The next illustration [Instructions for the making of cross-stitch embroidery, to be followed by a human craftsman] serves as a reminder that the weaver and maker of fabrics was the artificer who first stored information, structure, to be read, not by humans, but by a machine. The simple pattern of cross-stitch instructions suitable for a human embroiderer to read is worth presentation; it seems likely that older versions of such instructions, pricked out on squared paper for the operator of a draw loom making figured cloth, were the direct ancestral strain of the familiar punched card machine-language. Its line of descent runs from the human reader to card-reading looms of the eighteenth-century French silk textile industry, to the very successful Jacquand figure-weaving machine of about 1800, to Babage's difference engine, to the Hollerith card machines of the United States Census of 1890. It was always easiest for machines to finger and not to gaze at the messages sent them; they still read mainly braille, for machines were until very lately blind. [The machine which reads the strange figures on your bank check has a magnetic sense of touch, more subtle than that of the machines which read punched cards only; it recognizes the digits by detecting the iron content in the special inks that form them.]

So for artifact, machine and wall, cloth and picture, book and drawing, all the expression of human hand and mind, the modular idea proves itself both as the ground plan of much of our product--of everything expressed graphically--and as an irresistible analystic scheme, capable of measuring knowledge itself as the sum of simpler choices. This point of view, this power of analysis is not more than a generation old. It arose out of an even bolder analysis, now perhaps as old as the century: the quantum physics. Built upon the modular idea, it has won sway over the whole of the universe of physics, save only gravitation.

Let us enter it step by step. Jean-Baptiste Fourier, great mathematical physicist of a hundred and fifty years ago, felt it plausible to found his whole treatise on an axiom, that the infinitely repeated subdivision of a pure substance, say of water, would not change its intrinsic properties. This is of course the perceptual continuity we see and touch and taste in this world in which we live. Exactly as Leibnitz [p. 13] is refuted by mere number, so is Fourier. his was the error of size. There are atoms, but they are so sm all that we cannot expect our senses to reveal them; to our coarse inborn instrumentation, water remains watery to the tiniest drop. But in fact the history of the study of matter is largely a history of the refutation of that percept by reasoning from the properties of matter to the existence of atoms, and supporting the inference by even more powerful means of perception. Today the battle is well worn; atoms are as real as chairs, and rather better understood. No one would argue that woven cloth, for example, would retain its properties in squares one millimeter on an edge. The thread is easily seen, and indeed the weaver will testify that the properties of his yarn are not the properties of the fruit of his loom, though the yarn may foreshadow some of them. Nowadays we can very nearly weave materials from atoms, and we observe the same results: matter has properties which depend on the assembly of many modular constituents, tediously repeated, in a small alphabet of relationships, but spelling out in the end new and unexpected messages. Cloth is more than yarn, and yet it is only yarn modularly repeated.

The first lexicon of matter was the roster of chemical elements, the next, the modularity of crystals, then the color patterns of visible light which are the spectra. By now the chain of argument is so intricate and so complete, and yet still so fruitful, that it is an unsuitable task to try to sketch it here. Rather look at two examples which display the modules to our eyes, mediated to be sure through complicated apparatus, but with results so rich in connotation and detail that we can bury skepticism, recalling only that every photograph can reveal its object only in some single light; yet every real object contains more than one image for photography.

Reproduced here [Fig. 7. An electron micrograph showing a crystalline heap of tiny identical modules. The whole heap is about one-tenth the diameter of a hair . . . Such modules-tiny for us, giant for atoms - are typical of the fine structure of living beings. Photo courtesy Dr. Ralph Wyckoff.] is a small bit of a remarkable crystal. It is very plainly assembled out of identical, rather simple modules, near-spheres by this light. They are piled in a regular and repetitive three-dimensional display, which is the crystal. But these modules are by no means atoms. Each one is a proto-organism, a virus particle, capable of inducing in a suitable living host, the tobacco plant, a characteristic disease, leaf necrosis. [The pathology is inessential to the argument, and reflects only the investigator's stake in choosing this particular virus for study.] What turns out to happen is that each of the modules is capable of entering a particular host cell, and once within, of subverting the metabolic machinery of the cell to turn out replicates of the module, at the expense of the normal growth and function of the leaf cell. Hence the disease. In this illustration one sees a photograph made by using the geometrical principles of the microscope, with lenses not shaped of glass to bend light, but rather made of well-shaped magnetic fields to guide rays of electrons to form the photographic image. The [p. 14] while crystal fragment shown is about one-tenth the diameter of a human hair. These modules hold some five or ten million atoms. The existence of these intermediate structures, modular, but far greater than atomic size, is one of the hallmarks of life. No metal or mineral shows such great modules in its assembly to crystalline form. The domain of these modules--can we call them giant molecules?--is the domain of life. How cunningly they are marshalled to lend life its mobility, to allow it growth, to endow it with heredity, are stories told in another place. Suffice it here to say that modularity, on a scale below the microscope but far above the atom, is the scheme of life. The intimacy of relationship between the metaphors of computers, with the theory of modular information on the one hand, and modern genetics and developmental embryology on the other is the burden of the most dazzling scientific tale of the last twenty years. We must pass it by.

Not only the biological giant molecules, but even individual atomic modules can be displayed. We can see them in the illustration on the opposite page. [Fig. 8. Another kind of electrical super-microscope view. Here we see the glowing screen image of the individual platinum atoms in a tiny tip of platinum metal. The whole picture is as far across as one five-hundredth of a human hair, and could be spanned by one of the puffy modules of fig. 7--photo courtesy Prof. Erwin W. Müller.] This is a photograph of a fluorescent screen of a vacuum tube like that of a TV tube, with an image painted out of it, not this time by electrons, but by the heavier charged particles called ions of helium. These ions were made out of the gas of the tube, produced very close to a tiny metal tip, which is a perfectly smooth near-hemispherical crystal of platinum. They are electrically repelled by the voltage on the tip, shoot out, strike the screen, and set it aglow. They image the atomic arrangement of the crystal; nearly a thousand facets of the complex surface of intersection between the regular hexagonal crystal lattice and over-all hemispherical shape of the tip appear. A near-ideal regularity is displayed, usually by small groups of atoms blurring into a single image, once in a while even by individual atoms. Some central facets appear blank because their atomic surfaces are too flat; the ion formation favors those locations where individual atoms tend to protrude beyond their neighbors, because they lie at edges or corners of the atomic latticework. The whole pictured tip is about one five-hundredth part of a hair's breadth. [p. 16]

One impression cannot escape us; whatever else we may see, the modular construction of the metal crystal is plain. No continuity, no smooth ground stuff of malleable metal appears to our eyes; our most powerful, almost magical magnification has yielded the discrete muster of atomic parts, multiplied in a pattern austere and elegant. Hidden within the luster of metal worked by hammer and roll, there always lie the patterns of the snowflake or of the Alhambra's tiled walls, patterns conforming in most details to the severe mathematical canons for the uniform assembly of identical modules. Exactly this did the crystallographers long ago infer from the well-developed forms of crystals; it has remained for our time to display the arrays of atoms themselves.

What is most striking of all is the absolute identity of all the hordes of atoms, members of one species. Indeed, we now have criteria for identity so stringent as to imply that no surprises can arise which will cause us to subdivide certain classes whose members we have recognized as one and the same! The atom is not the uncuttable, as the Greeks would have it. Far from it; we part atoms in each match flame, or in rubbing lucite against wool. Even the core of the atom, the nucleus, a million-fold more refractory than the outer atomic shell, we part easily enough, if more dramatically. But what is in essence atomic is identity. The variety of our world which lies so richly at hand is a modular variety: matter is modular. Its modules fulfill the most precise of stereotypes, free from the interesting variations of the craftsman's hand or the furnace heat. Their number alone, and the motions and interactions of those myriads, weaves the intricacies of reality.

Those uncuttable particles, the atoms of the philosophers, could well be identical. They were made so, it was put forward, in the dawn of time. But how can objects with structure, with subsystems, like electrons and nuclei, sensitive to environment under by no means unrealizable circumstances, hold their immutable identity of form? This is the puzzle whose solution is the theory of the quantum. For it turns out that motion, too, the kind of orbit traced out by a planetary electron in the solar system of the atom, is modular. The concept of freely assigned orbit, as in the familiar mechanics of our human scale or larger, turns out to fall into contradiction before the theory of the mechanics of the quantum. Only certain states of motion are possible, modules of energy or momentum, if you wish, and a system has a set of modules which define its nature. True, systems of many atoms may combine these so deftly that they approximate the smoothly continuous motions of Newton and of common experience, They do this much in the manner that the modular atomic lattices allow the sculptor to make any shape he chooses by the rough judgment of the eye, though in reality his materials on the atomic scale lie in rigidly controlled pattern. [p. 18]

If such abstract measures of material motion as energy are affected with the brick-like modularity which is the natural order, it may not become as wholly strange that radiation, the light we see, shares the same quality. Indeed, we nowadays regard all the world as built out of certain subatomic fundamental modules, or particles, whose classes we are still enumerating. Why they are precisely identical by classes we know not. They seem to occur in families, as the tiles of a mosaic, grouped not by color but by sets of other intrinsic properties. They interact with each other in ways which are complex, but which themselves reflect a complex set of essentially discrete rules. Out of all this tangled skein time has woven the fabric of the world. Continuity, and its strange child, the randomness of chaos and formless motion, are present only in the gaseous state and in the gravitational orbits which hold the planets and the stars in their courses. In time and space we still see no modularity; these categories are less real to us still, more tinged with metaphysics, than the matter which they somehow contain, the matter which somehow lends them features. But here we grope at the edge of knowledge.

The world is both richly strange and deeply simple. That is the truth spelled out in the graininess of reality; that is the consequence of modularity. Neither gods nor men mold clay freely; rather they form bricks. If it were not so, order and diversity would be no allies, but eternally at war.

NOTE: The illustrations described here are not available here at present.
* Fig. 3a. On September 22, 1962, a camera flying in orbit in the weather satellite TIROS 5, about seven hundred kilometers above the surface of the eastern Mediterranean, and a hundred kilometers northwest of Alexandria, took this photograph. Some time later the electronic translation of the image was flashed to the ground at Point Mugu, California. This illustration shows the face of a TV screen bearing the result. One can see dark croplands of the Nile delta, the irrigated course of the Nile, the oasis of El Fairõm, the Suez Canal, and the Dead Sea. The south direction is uppermost in this illustration.

* Fig. 3b. This illustration shows the printed output of the computer which spells out exactly the picture seen in Fig. 3a., now translated into numbers and letters. The digits and letters represent thirty-six levels of grayness at each point of the array which forms the picture, 0,1,2, . . . A,B, . . . Z, with 0 the darkest and Z the brightest level reproduced. Actually only the upper right quarter of Fig. 3a is reproduced here. The dark Nile can be found flowing numerically onto the print about the seventeenth line from the left [Photos courtesy Dr. Albert Arking, of the Institute of Space Studies, NASA Goddard Space Flight Center, NY]