Chapter IX. THE LANGUAGE OF MATHEMATICS

YOUR grandfather leaves $5,000 to you and your sister. The will provides that she is to receive $650 more than you receive. How will you tell her how much she is to get? In ordinary language you can shuffle the figures around and after a time find the answer. But by using mathematical language you can communicate the news much more quickly and accurately. Let x be your share. Then x + 650 is her share. Both shares equal $5,000. Making an equation or a mathematical sentence of these statements:

x + (x + 650) = 5000 or ix + 650 = 5000

or 2x= 5000 — 650 = 4350

or #=2175 your share

x + 650 = 2825 her share

Adding the shares for proof 5000

Mathematics has been called the language of science. This is not quite accurate. Each branch of science has also an argot of its own, and as we have seen, even physicists often use ordinary language like the rest of us. Some scientific concepts, however, cannot be communicated except in mathematical terms. This is the Case with the central concept of relativity. You must know some calculus to grasp it and make it your own. Many books have been advertised as reducing Einstein to simple terms which “any intelligent layman can understand.” Strictly speaking, the blurb if not the book is fraudulent. One does not understand a story in Russian just because it is written in words of one syllable.

A similar situation holds for quantum theory. Ordinary language is not adapted to describe processes within the atom. It is adapted to deal with everyday processes involving exceedingly large numbers of atoms. To talk about what is happening inside one atoni, a special language is required.

Must we all turn mathematicians, then, to understand our world? No. But two important observations are in order. For some of the more complicated aspects of nature, mathematics provides the only key; for everyday activities in the Power Age, mathematics provides a very useful aid to clear thinking. Even if one does not master higher mathematics, a knowledge of what this language is about—how it developed, and the ability to handle a little algebra and geometry, to plot a few simple graphs—is worth having. It helps to solve many problems of communication and meaning.

One of the pleasantest ways I know to obtain this knowledge is to read Lancelot Hogben’s Mathematics for the Million. It is not guaranteed painless, but the fact that it has been an outstanding best-seller both here and in England is evidence of its human and practical value. Mathematics, he says, began in the nomadic age to fill a need. It was necessary to count herds and flocks to keep them straight. When agriculture was developed, it became essential to measure crop lands. We have already noted how the Nile washed out boundary marks every spring and encouraged a science of land measure, or geometry. Recurring seasons for planting, harvesting, high-water periods, demanded an accurate calendar, and for this astronomical measurements had to be taken. Many of the first writings were calendar notations. I have seen beautiful examples of such stone writings on Maya stelae in Mexican jungles. As cities grew, timekeeping became essential, and mathematics was broadened to count the hours. The building of temples, especially pyramids, required careful measurements and a geometry of solids. How much stone must be quarried for a truncated pyramid of such and such size? When corn and wine were bartered or sold, standard measurements were essential, so that neither party would be defrauded. Presently galleys and ships began to take journeys beyond sight of land, and navigation was demanded. How human this is. Some Heavy Thinker of ancient times did not begin by sitting in his portico and evolving numbers and planes and truncated pyramids out of his head, to plague children in schoolrooms forever-more. The numbers and planes came out of the need of shepherds, farmers, traders, builders—out of the day-by-day life of the people. They came without mystery,

but not without a kind of mental revolution. Bertrand Russell has observed that it must have taken many ages to discover that a brace of partridges and a couple of days are both instances of the number 2.

Certain Greek philosophers took this useful tool and made a dull fetish out of it. They lifted it from the market place and put it in the cloister. They believed bedrock had been reached when they had isolated a point, a line, an angle—something changeless, timeless, eternal. From these absolutes, truth could be reared by reason. The intellectuals of Athens and Alexandria rarely examined the sort of things about which these words can be intelligently used. They dealt with pure theory, not with a living world. In Euclid the analysis of flatness reached its climax, so perfect and often so unreal that it has been a major educational subject ever since. No wonder so many schoolboys are bored by geometry; it connects with nothing in their experience, and no meaning comes through. Euclid was a great man, and his geometry is useful in limited contexts. The objection raised by Hogben is to the worship of Euclid’s valuable findings as truth, good everywhere, for everything, at all times. The word “worship” is used advisedly. A noted divine once wrote a book proving to his own satisfaction that if you destroy Euclid, you necessarily destroy the revealed word of God. Nobody can destroy Euclid. All that can be done is to put his work in the place where it functions and keep it out of places where it does not function. If Einstein had stuck to Euclidian geometry, relativity would never have been heard of.

Pythagoras formulated some excellent mathematics more than a century before Euclid. He also made a major contribution to the technique of human knowledge by working out the concept of “proof.” He insisted that assumptions or postulates must first be set down clearly. No extraneous matter must be subsequently introduced. Proof is arrived at by applying close deductive reasoning to the postulates. Having thus immortalized himself, Pythagoras went off into the blind alley of magic numbers, and founded a whole school which stood in awe of the portents and omens of 7’s and n’s. “Bless us, divine Number, thou who generatest gods and men.” Some people today still cower before the number 13.

The abacus or counting frame was invented to do sums in addition, subtraction, multiplication, and division. One can still see this device in active use in Russia or at a Chinese laundry. It consists of little balls on wires which one pushes around, “carrying over” from one line to the next. The ancients did not know how to do sums on paper; fractions were avoided, decimal points unheard of. One counted on one’s ringers or used the abacus. It was tedious work. Not until long after the fall of the Roman Empire did Western countries adopt the Arabic zero, the sign for the “empty” column of numbers, together with the rudiments of algebra. The zero and Arabic numerals produced a revolution more important than that of the printing press. They liberated mankind from the prison bars of the abacus. The revolution was not, however, unopposed. An edict of 1259 forbade the bankers of Florence to use the infidel symbols. The bankers must still write four characters for 8 (vin), six characters for 48 (xlviii), fifteen for 3,888 (mmmdccclxxxviii). But by using pencil, paper, and the decimal point which the zero permitted, merchants could solve in minutes sums which used to take hours. The electric adding machine is a great improvement over pencil and paper, but it is as nothing compared to the improvement in mental machinery provided by the zero. Observe also the increased efficiency of algebraic symbols:

Before algebraic symbols: 3 census et 6 demptis 5 rebus

aequatur zero After: 3X 2 — $x -f 6 = o

Observe also that the algebra is direct shorthand translation of longhand talk, a tidier, defter language, not something incomprehensible out of the sky. Slang sometimes performs a similar service: rather than “Disperse yourselves as rapidly as possible,” the American policeman remarks laconically, “Scram!”

When early observers could not readily express their measurements in everyday language, they were driven to experiment with symbols. If further observations were inexpressible in the symbols available, new symbols were sought. Thus Newton was stumped to tell the world or

even himself what he had discovered about the movement of celestial bodies until he had perfected the differential calculus, which is an admirable language for accurately expressing movement. Thus Gauss was forced to perfect co-ordinates and the integral calculus. So Lobatchevsky in 1826 invented symbols to express non-Euclidian geometry; so Einstein applied and improved the calculus of tensors—not to drive us crazy, but to meet a genuine need. We noted earlier how everyday language has developed by a process of filling the gaps, supplying a new word to take the place of a long, clumsy description. Mathematics has followed a similar course. Furthermore, as Bell points out: One significant fact stares us in the face. Mathematics is the inexhaustible matrix of new development in the art of thinking. When it declines, close reasoning petrifies into stereotyped and unimaginative repetition of the classics.

The Middle Ages was such a period of petrifaction.

Hogben calls ordinary speech a language of “sorts,” and mathematics a language of “size.” The writing of sort language was once a mystery closely monopolized by priests. The time has come, he says, for another Reformation like that of five hundred years ago when the priestly monopoly was broken, and the mass of the people were permitted to read the Bible and learn to write themselves. Most people today can neither read nor write size language, yet the world they live in depends upon it. Without mathematics there would be no elec

trie power, no steel bridges, no public statistics, no railroads, automobiles or telephones. Without the theory of analytic functions we could not study temperatures or the flow of electricity, and so control them. Without such numbers as V —1 in vector algebra we could not have learned how to build radios or to send telegrams. Without multidimensional geometry we could not construct automobile engines or deal with gases under pressure.

We need to know at least the rudiments of mathematics in sheer self-defense. No society is safe in the hands of priests. Think of the mathematical accompaniment of our daily life: timetables, unemployment figures, insurance based on actuarial computations, taxes, debts, interest, wage rates, pensions, old-age security legislation, bond yields, speed limits, betting odds, baseball averages, football gaining and scoring, calories, weights, temperatures, rainfall, meter readings, radio wave lengths, tire pressures, freight charges, calculation of flood crests, birth rates, death rates. . . . Little of this was essential in Athens, Alexandria, or Rome. With greater urgency than ever before, the mathematician and the plain man need to understand each other. Without a knowledge of the grammar of size and order, we cannot hope to plan an age of plenty. Priests and pundits will prove that it cannot be done and we shall have to submit, unless we know the hocus-pocus in the proof. Modern engineering is possible because of the similarity in structure between mathematics and the outside world. With confidence we rely upon the structural abstractions which engineers employ to build skyscrapers, bridges, motorcars, and airplanes. A large part of modern behavior, many social institutions, are dependent upon the engineers’ ability to predict what will happen when steel and stone and chemicals are combined thus and so. Without such sureness, bridges might collapse, Boulder Dams might fail. Upon predictability of this nature modern “civilization” has been built.

Predictability, observes Dr. D. G. Campbell, depends upon the discovery of structure, the representation of that structure by a language with similar structural characteristics, and then the manipulation of the symbols of that language to determine what will happen under the conditions of such structural arrangements in the future. It is like a miniature stage used by a stage designer to study lights and color, like a wind tunnel for testing airplane design, like the tank which Starling Burgess uses for testing models of cup-defender yachts. A mathematician, for instance, predicts torsion stresses in a steel bar by measuring stresses in a soap film in which he finds characteristics of similar structure. Relationships are similar and may be represented by mathematical symbols of relationship. Laborious methods of trial and error become unnecessary. From a few measurements, structure in the soap film is discovered; a language cor-

responding to the structure is utilized; predictability of the behavior of steel bars is made possible.

But no man alive predicted the great depression of 1929 with any structural knowledge to support him, and no man knows surely when the next collapse will come.

Let us follow Hogben in a few simple exercises in translating mathematics into ordinary language:

Area equals length times breadth. In the language of mathematics, this sentence reads:

A = lxb

X and = are the verbs in this expression, while the nouns are A, I and b. Comparing the two languages:

Mathematical Ordinary language language The length /

must be multiplied by X

the breadth b

to get a measure of =.

the area A

Observe the saving in time and space. Observe further that in algebraic symbols of this kind no actual objects, no referents in the world outside, are included. Mathematics is a language of action and relation. But it is easy to supply referents for the sentence above by measuring your kitchen floo? in square units. The equation gives orders and relations which must be obeyed. You are not to multiply area by length to get breadth; you follow

the rules. Then it is possible to substitute yards or feet or kilometers to apply to oblongs anywhere (within the confines of this “length” concept) and get an answer for what you want to know.

You must be careful, however, to deal in similar units, always yards, or always kilometers, in this equation. You must not multiply yards by kilometers, must not add yards to gallons, or you will create a mathematical monster. Before you know it, you will be preparing indices of wholesale prices for the guidance of economists and statesmen.

Everyday language contains gerunds, or noun and verb combined in one word, as in “working.” Gerunds are also found in mathematics in such symbols as —3, or V—1, which are numbers with direction attached to them by convention. We find conjunctions: .”. (therefore) and ‘•’ (because). The verbs + and — were originally chalked on bales in warehouses to show surplus or short weight. In using these verbs with referents attached we must again be careful not to add or subtract dissimilar things. Two boys + 10 green apples do not foot up to anything—except possibly a couple of stomachaches. This warning has been called the “rule of quantitative similarity.” There are collective nouns in mathematics, but nothing corresponding to the high-order abstractions in everyday language. Capital letters of the alphabet are used, like A or M, to symbolize whole families of numbers having something definite in common. Area, or

A, must translate into square units—inches, or feet, or kilometers. This makes it difficult to create fictional entities without observable referents. Mathematics is a powerful corrective for the spook-making of ordinary language. The term “elegant” is frequently applied to mathematical style. It means that rotundities have been removed by the process of elimination. “In the international language of mathematics, we sacrifice everything to making the statement as clear as possible.”

Suppose we translate into mathematics the famous problem with which Zeno baffled the Greek logicians. Zeno said that if Achilles allowed a tortoise a head start in a race, no matter how much faster Achilles ran, he could never overtake the tortoise. Why? Because he must first reach the place where the tortoise was when Achilles started. By the time he reaches it, the tortoise, however slow, has made some progress. So Achilles must reach this second place. But by the time he gets there, the tortoise has moved to a third place still ahead, and so on ad infinitum. The distance between them ever narrows, but Achilles can never overcome it. When last seen, to paraphrase Bell, the tortoise was .000005 of an inch ahead and Achilles’ tongue was hanging out half a yard.

Dealing in words alone, the logic is unimpeachable. Let it whirl around your cortex from reference to symbol and the chances are that you will be unable to discover anything wrong with it. You may settle yourself in an armchair and think until kingdom come, or until you go mad, and you cannot get around it. But the moment you begin to look for referents, to perform an operation, to place an actual turtle here, and a young athlete there, and start them off, the mental blockage dissolves. When I hear a problem of this nature, my

Drawing by J. F. Horrabin. Reprinted from Mathematics for the Million, by Lancelot Hogben by permission of W. W. Norton & Company, Inc.

impulse is to reach for a pencil and paper, and undoubtedly you share this impulse. It is a sign of semantic progress. The ancients had no scribbling paper and no adequate symbols for attacking such problems. They knew of course that Achilles could lick the tar out of the tortoise, but how were they to prove it? Here are two simple methods of translation unknown to the Greeks. In this mathematical language of graphs we draw the rate at which the tortoise moves and the rate of Achilles, and where the two lines meet the tortoise is overtaken. Assuming that the tortoise can run a yard a second, that Achilles runs ten times as fast and gives the tortoise a start of ioo seconds, they will meet ni.ii yards down the track. It does not make any difference what rates are taken so long as the tortoise starts first and Achilles runs faster. The slope of the lines will change, but the meeting-point will always appear.

With the same assumptions, let us translate the problem into simple algebra. Let r be the rate of the tortoise. Then Achilles’ rate will be ior. Let x be the time in seconds taken by the tortoise before they meet. We know that the distance traveled by the tortoise equals the distance traveled by Achilles. The distance a body travels is its rate of travel multiplied by the time traveled. Using this formula:

Tortoise’s distance =r X x Achilles’s distance = ior X (x — ioo) or rx = lorx — iooor

or qx = iooo

Therefore x = 111.11 seconds

They will meet 111.11 seconds after the tortoise starts, or 111.11 yards down the course, as in the graph. Incidentally, most algebra can be translated into graphs, with curved lines for higher powers of x. Engineers are very partial to graph language. The Greeks had no algebra, no graphical methods, while the geometry of

Euclid which they did possess dealt only in spaces and made no allowance for times. Motion, rates of motion, velocity, could not be handled. So you see in what a predicament the logicians found themselves when Zeno, perhaps with an ironical smile, put the problem before them.

At this point a little journey through the fourth dimension with Bell may prove enlightening. Suppose you want to identify and label all the men in Middletown. For each man you ask (i) his age in years, (2) his height in inches, (3) his weight in pounds, (4) dollars in his pocket or bank. You allot a symbol for each characteristic: A for age, H for height, W for weight, D for dollars, and rigorously maintain the order. Having got your facts together, any man can be accurately and quickly identified as follows:

A H W D William Black 35 60 160 2 Arthur White 42 68 135 10,000

Black, then, is 35 years old, 60 inches tall, weighs 160, and has $2. White is 42, 68 inches tall, weighs 135, and has $10,000. The set of labels A, H, W, D is a simple kind of four-dimensional manifold—a term which has long terrorized the nonmathematical. We can make it five-dimensional by adding S for size of shoe, and six-dimensional by adding C for number of children, and so on.

Moving this idea over into the field of mechanics, we can set up ordered symbols for three distance meas-

urements and one time measurement. With this four-dimensional manifold the position of any particular object can be fixed at any particular instant. Consider a fly in a room. Let E be the east wall, N the north wall, F the floor, T the time after 12 o’clock. The units are inches and seconds.

E N F T

12 2 3 5

means that the fly was 12 inches from the east wall, 2 inches from the north wall, 3 inches from the floor, at 5 seconds after 12 o’clock. The next label might be 60, 100, 36, 6—which you can translate yourself. “By refining our observations to the limit of endurance, we could fit labels enough to describe the erratic flight of the fly for an hour with sufficient accuracy for all human purposes.” This hour’s history is a four-dimensional manifold by definition, and gives us a useful method of describing the order of certain happenings in the outside world. 1

It is meaningless to talk of the “the fourth dimension.” We can construct as many dimensions in manifolds of this kind as we wish. Yet no sooner did relativity become news than a lady with a piercing eye undertook to tell the good people of Pasadena—for a fee—exactly how “the fourth dimension” would enable them to re-

1 Bassett Jones points out that such a manifold is not homogeneous, having both space units and a time unit, and holds it to be more a conversational device than a mathematical one.

 capture their virility, their dividends, their faith in God, and their straying husbands or wives. Remember that fly, and do not cringe before the fourth dimension again.The relatively young science of agrobiology is an excellent example of the usefulness of mathematics to farmers, to gardeners, and to the public in general. Observe the progression. First Liebig determined, by growing plants in earthenware pots, the various chemical substances essential to plant life—phosphorus, potash, nitrogen, sulphur, magnesium, and the rest. Then Mitsch-erlich carried on the experiments to show the specific effect of each chemical on plant yields. For a pot of oats, no nitrogen resulted in no yield; .35 grams of nitrogen gave 80 grams of dried oat plant; .7 grams of nitrogen gave 120 grams of plant; 1.4 grams gave 150, and 3.5 grams 160. This was the end. No matter how much more nitrogen was added to the pot, the yield could not be raised above 160 grams.

Now when these figures are neatly tabulated and checked by scores of experiments, mathematics enters, and a curve is plotted. Curves are similarly prepared for the effect on oats of potassium, phosphorus, and the other chemicals. Measured quantities of water are applied, and curves are prepared for that. When oats are finished, corn, wheat, roses, and other plants are grown in pots and their curves in turn are plotted. Presently a law is derived: “When we take as the unit of a growth factor that quantity of it that will produce 50 per cent of the total yield, then each cumulative unit is only

half as effective as the unit that went before.” The more fertilizer you add, the greater the yield—this side of the limit—but at a diminished rate. It was once thought that plant growth went up in a straight line as fertilizer was added.

Thus the agrobiologists discovered with the aid of mathematics a practical law of the utmost importance. They have done more. They have calculated the possible maximum yield of many plants, and are prepared to do it for any plant that grows except the fungi, which follow different rules. The maximum yield of corn, for instance, is 225 bushels per acre. This quantity of corn has actually been raised. Agrobiology is already revolutionizing the art of growing things, and the future effects, economic, political, and international, promise to be epoch-making. It employs the language of mathematics to a great extent, and indeed would be nonexistent without mathematics. Note carefully, however, that this machine does not run on air; it runs on pots of oats and corn. 1

Mathematical language is also susceptible of abuse. It is not so widespread as the abuse of ordinary language, but it is serious enough. It takes various forms. Eve* since numbers were invented people have become intoxicated with their possible combinations and have gone off on magnificent ghost chases after mystical numbers. Numbers, of course, are nothing but useful symbols to

1 For a fascinating account of this science see The ABC of Agrobiology, by O. W. Willcox. W. W. Norton, 1937.

fill gaps in meaning and communication. They originated in the human cortex and are unreported anywhere in nature. We have objectified them, as we have done with so many other symbols, into puissant forces in the world outside. Consider the vast amount of tosh erected around the number 7: the Seven Candlesticks, the Seven Deadly Sins, the Seven Planets. In the year that Piazzi discovered Ceres—or planet Number 8—Hegel wrote upbraiding the scientists for their neglect of philosophy. Philosophy, he said, had established seven as the only possible number of planets. Why waste time looking for more?

Again, many men, including mathematicians, have failed to realize the limitations of the language. The technique is wonderfully useful for establishing relations and orders, but what are the things between which we desire to establish the relations? Mathematics is purely abstract, and says nothing about that. Just to whirl relations about in the head may be an amusing method of killing time, but no knowledge is gained until concrete things are hitched to the symbols. These objects must be carefully selected. A distinguished professor recently sent me a monograph in which calculus was solemnly applied to various kinds of consumers’ goods, such as potatoes and automobiles, including, if you please, “subjective wants.” Try to count anything in the real world corresponding to “subjective wants”! The result of all this fine mathematics, of course, was blab.

Mathematics has been likened to a sausage machine. Feed it proper raw materials and turn the crank. Something useful, if not edible, comes out. Feed it nothing and turn the crank. There is much grinding of gears, but nothing comes out. Feed it scrap iron mixed with broken glass and the machine refuses to work. A good deal of what passes for pure mathematics consists in whirring the works with nothing edible inside. Meanwhile the makers of various kinds of economic index numbers are feeding the sausage machine scrap iron and broken glass. Bertrand Russell has characterized pure mathematics as “that science in which we neither know what we are talking about, nor whether what we say is true.”

Always examine the data assumed. If the assumptions are without tangible validity, the mathematical theories deduced from them may scintillate with dazzling plausibilities, but they will be worthless. A dangerous abuse of mathematics appears in the practice of extrapolation-described earlier as riding a trend curve to Cloudcuckoo-land. Here deductions are made from facts, but there are not enough -facts. Example: The earth will maintain vegetation for the next 5,000,000 years. A wild guess. No operations are available except some crude data on the rate of the earth’s cooling. Another example: The New York Metropolitan Area will have a population of 2 r,-000,000 by 1965. This is a more careful guess, based on actual population trends prior to 1920. The fact of birth control, among others, was neglected, and it now looks

as though the New York metropolitan population in 1965 would be far less than the original estimate.

Be exceedingly chary of large generalizations under the caption “Science Says That Universe Is Running Down,” or “Science Says That Universe Is Blowing Up,” or “Science Says That in Ten Thousand Years the Human Race Will Have Lost Its Teeth.” Some professor is probably making an extrapolating ass of himself. Bell furnishes a list of famous extrapolations about the age of the earth.

Bishop Ussher 5,938 years

Lord Kelvin 20,000,000 to 40,000,000

Helmholtz 22,000,000

G. H. Darwin 57,000,000

J. Joly 80,000,000 to 90,000,000

Joly and Clarke 100,000,000

Assorted geologists … 2,000,000,000

Assorted astronomers .. 2,000,000,000 to 8,000,000,000

Step right up, ladies and gentlemen, and pick your winner! Here is another choice item of extrapolation paraphrased from a book by Stewart and Tait, physicists of fifty years ago. Matter is made up of molecules (size A) which are vortex rings composed of luminiferous ether. The ether is made up of much smaller molecules (size B), vortex rings in the subether. This is the Unseen Universe. Here the human Soul exists. It is made up of B molecules. It permeates the body like a gas. Thought is vibratory motion in the A molecules, but part of the vibration, following the law of the conservation of energy, will be absorbed by the

B molecules, the Soul. Therefore the Soul has memory. When the body dies, the Soul keeps memory intact, and becomes a free agent in the subether. The physical possibility of the immortality of the Soul is thus demonstrated.

The volume in which this charming balderdash appeared was widely read in the 1870’s and i88o’s.

Mathematics can do no more, explain no more, than the tangible things to which its symbols are hitched permit. Beyond this limit, it goes off the deep end and has no meaning. In the language of mathematics no less than in ordinary language we must find the referent for the symbols. There is no truth in the machinery of mathematics as such, only an endless series of tautologies. Two plus 2, we say, is 4, and with glittering eye challenge the world to get around this great truth. Bartenders get around it every day, for 2 quarts of alcohol plus 2 quarts of water do not make 4 quarts of highball, but something less. A chemical change shrinks the volume of the mixture. “Two plus 2 is 4” is a statement which may be true in one context and untrue in another. Find the referent. Two what? Where? When? Upward reports a primitive tribe whose language unmasks this absolute even more effectively. The tribe has a word for “one,” a word for “two,” but no word for “four.” The word for “two” is “burla.” So when the chief intones the great truth, he says, “Burla and burla is always and forever burla-burla!”

Mathematics, as Korzybski presents it, is a language with structural similarity to the human nervous system and to the world outside. If the cortex exercises its switchboards with mathematics, the man inside can improve his grasp of the world without. Witness the case of applying mathematics to Achilles and the tortoise. But if he neglects facts from the world without, or makes false assumptions about them, he can strangle meaning as effectively” with mathematics as with other languages. The process is even more dangerous, for it is widely supposed that figures speak with extrahuman authority. In its field, mathematics is good human talk, just as music is. It developed, as we have seen, to meet urgent human needs. Combined with the operational approach of modern physics, it has extended knowledge into unprecedented areas, and the extension goes steadily forward. It has been taught us badly, and we shy away from its symbols. This is our misfortune, for mathematics might be a shield and buckler against verbal confusions.

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