Through a Glass Darkly


IT is to be feared that many readers of this magazine will pass this article by—the modest as too learned, the learned as too ignorant; the light as too heavy, and the heavy as too light. All will be mistaken. What is here said is very real truth — even if softened now and again with an attempt to alleviate it by what is very real fun — and represents a very real and important issue.

The clergy, in their similar despairing call for attention at the opening of a sermon, take a text. Let me, therefore, as the prelude to this essay, take mine from the pages of one of the most eminent and conspicuous works in economics that have appeared in the past few years. The writer of it is the holder of one of the most respected chairs in England, in a university of which I dare not breathe the name. If I did, I should be crushed flat at once under the dead weight of prestige and authority. In any case it would seem invidious and personal to say that in my opinion the particular book of a particular author is tommyrot, when what I really mean is that a hundred recent books by a hundred recent authors are tommyrot.

The author in the case before us is undertaking a discussion of what he calls the ‘size of real incomes.’ That is about as near to a plain intelligible phrase as a trained economist can get. An ordinary person would prefer to say ‘what people get for their money,’ but that would be just a little too easy to understand. The writer goes on to say that if, conceivably, all people used and consumed one and the same thing, and only one, then we could compare what each got with what every other got by the mere quantity or number. But in reality people consume all kinds of different things with all kinds of preferences.

So far the sky is clear. There has been no warning of mathematics. The readers are as unsuspecting as the crowd in the Paris streets before Napoleon Bonaparte turned on the grapeshots of Vendémiaire.

Now comes the volley: —

It may perhaps be thought that the difficulty can be overcome by comparing real incomes, not in themselves, but in respect of their values. It is, of course, always possible, with a pricing system, to value each of two real incomes in terms of any commodity that we choose, and to set the values so reached over against one another. This is frequently done in terms of money. Unfortunately, however, the two valuations will, in general, be related to one another in different ways according to what commodity is taken as the measure of value. Thus, suppose that we have two incomes each comprising items of three sorts —A, B, C; that in the first income the quantities of these items are a, b, c, with money prices pa, pb, pc; and in the second α, β, γ, with money prices πα, πβ, πγ.The money value of the first income divided by that of the second is then apa + bpb + cpc απα + βπβ + γπγ

Call this m. The value of the first income divided by that of the second in terms of

commodity A is πα/Pa in terms of commodity

B, αβ/Pb in terms of commodity C, πγ/Pc.

These quantities are obviously, in general, different. There is nothing to prevent one of them being greater, while another is less, than unity. Thus the result of comparisons depends on the choice we make of the commodity in terms of which valuations are to be made; and this is purely arbitrary. Nothing useful, therefore, can be accomplished on this plan.

As the last echo of the paragraph dies away, the readers are seen to lie as thickly mown down as the casualties of Vendémiaire. The volley has done its work. There will be no further resistance to the argument on the part of the general public. Theirs not to reason why, theirs but to do and die. They will learn to surrender their economic thought to the dictation of the élite. They are not to question where they do not understand.

The last sentence of the paragraph, the final shot, is not without humor. ‘Nothing useful,’ it says, ‘can be accomplished on this plan.’ No, indeed, nothing much, except getting rid of the readers. For the whole of the ‘plan’ and its pretentious mathematics, when interpreted into plain talk, amounts to something so insignificant and so self-evident that it is within reach of the simplest peasant who ever lived in Bœotia, or failed at Cambridge. It only means that different people with the same money would buy different things; one might buy roses, one cigars, and another concert tickets; and you could n’t very well compare them because the weight would n’t mean anything, and the color would n’t, nor the number. As to what you pay for them in money and why you paid — well, that is the very thing we want to find out.

Or shall I state the same thing like this: ‘It is hard to compare Janie’s doll with Johnny’s dog.’ Or let us put it into rural Yorkshire: ‘There’s a mowt of folks i’ counthry; happen one loike this aw t’other chap that; dang me if I know ’oo gets best on it.’ Or in Cree Indian (Fort Chipewyan, H. B. Post, Athabaska Lake): ‘Hole-in-the-Sky take four guns, two blanket; squaw take one looking glass, one hymnbook.’

What the problem means is that he can’t really compare what Hole-in-theSky got and what the squaw got. That’s all.


What has been just said is not meant as fun: it is meant in earnest. If the mathematical statement helped the thought, — either in presentation or in power of deduction, — it would be worth while. But it does n’t. It impedes it. It merely helps to turn economics into an esoteric science, known only to the few. The mathematician is beckoning economics toward the seclusion of the dusty chamber of death, in the pyramid of scholasticism, He stands at the door that he has opened, his keys in his hand. It is dark within and silent. In the darkness lie the mummified bodies of the learnings that were, that perished one by one in the dead mephitic air of scholasticism; of learning that had turned to formalism and lost its meaning, to body and lost its soul, to formula and lost its living force. Here lie, centuries old, the Scholarship of China, the Learning of Heliopolis, the Medicine that the Middle Ages killed, and the Reason that fell asleep as Formal Logic.

All are wrapped in a sanctity that still imposes. They sleep in all the symbols of honor, with a whisper of legend still about them. But the work they would not do, the task they could not fulfill, is left still to the fresh bright ignorance of an inquiring world.

Put without prolixity: Any wellestablished dignified branch of knowledge, finding its problems still unsolved, turns to formalism, authority, symbolism, the inner system of a set of devotees, excluding the world; philosophy becomes scholasticism, science turns to thaumaturgy, religion to dogma, language to rhetoric, and art to symbolism.

Modern economics and philosophy and psychology have so far utterly failed to solve their main problems. So they are beginning to ‘dig in’ as scholasticism. For economics, mathematical symbolism is the means adopted.


So few people are accustomed to use mathematical symbols that it is hard to discuss them in an essay of this sort without incurring the very danger here denounced and ‘sidetracking’ the reader. But something of their nature everybody knows. Very often a mathematical symbol or expression does convey an idea very quickly and clearly. Thus the simple and self-evident little charts and graphs used in newspapers to show the rise and fall of production and trade, the elementary index numbers used to show the movement of prices — these things are immensely useful. But they are only a method of presentation of what is known, not a method of finding out what is not known.

Very often we use simple mathematical expressions as a vehicle of common language, as when we say ‘fifty-fifty,’ or ‘a hundred per cent American,’ or ‘half-soused,’ or ‘three-quarters silly.’ We could go further if we liked, and instead of saying ‘more and more’ we could say A+n+n. . . . We could express a lot of our ordinary dialogue in mathematical form. Thus: —

‘How is your grandmother’s health?’ , it depends a good deal on the weather and her digestion, but I am afraid she always fusses about herself: to-day she about fifty-fifty.’

Mathematically this is a function of two variables and a constant, and reads: —

f(W.D.+fuss) = ½

The result is, in all seriousness, just as illuminating and just as valuable as the mathematics quoted above.

We could even go further and express a lot of our best poetry in mathematical form: —


Half a league, half a league,
Half a league onward . . .
Then they rode back, but not,
Not the six hundred.

The mathematician would prefer: —

½+½+½/600 = 600 — N

Or, try this as an improvement on Byron: —


Did ye not hear it? — No; ’t was but the wind,
Or the ear rattling o’er the stony street;
On with the dance!

d+d+d+d . . . d (n)

Let joy be unconfined;

j+j+j+ . . . infinity

No sleep till morn, when Youth and Pleasure meet.

M-S = Y+P

Or, to quote a verse of ‘Lord Ullin’s Daughter’ (done as mathematics), in which I once depicted the desperate efforts of the highland boatman: —

The angry water gains apace
Both of his sides and half his base,
Till as he sits he seems to lose
The square of his hypotenuse.

Or, to go a little deeper, by venturing into Descartes’s brilliant method of indicating space and motion by means of two or more coördinates as a frame of reference, we can make the opening of Gray’s ‘Elegy’ a little more exact.

The lowing herd winds slowly o’er the lea.

We can indicate the exact path by a series of points at successive moments of time (p —p1 —P2 —P3 • • . pn), and by dropping perpendiculars from each of these to the coördinates we can indicate the area swept by the lowing herd, or rather the area which it ought to sweep but does n’t.


Let me explain here that in this essay I do not wish in any way to deny the marvelous effectiveness of mathematical symbols in their proper field. I have for mathematics that lowly respect and that infinite admiration felt by those of us who never could get beyond such trifles as plane trigonometry and logarithms, and were stopped by a nolle prosequi from the penetration of its higher mysteries. Mathematical symbols permit of calculation otherwise beyond our powers and of quantitative expression that otherwise would require an infinity of time. It is no exaggeration to say that mathematical symbols are second only to the alphabet as an instrument of human progress. Think what is entailed by the lack of them. Imagine a Roman trying to multiply LXXYI by CLX. The Roman, indeed, could make use of an abacus, — the beads on wires of the Chinese, the familiar nursery toy, — but multiplication with beads only, and without written symbols on a decimal or ascending place-plan, is a poor and limited matter. See who will in this connection the mediaeval work called Accomptyng by Counters—A.D. 1510.

Contrast with these feeble expedients

the power of expression and computation that symbols give us! The Hebrew psalmist used to ask with awe who could number the sands of the sea! Well, I can! Put them, let us say, at (100)100and we’ve only used six figures and two crooked lines! And if that is not enough use three more figures: —


Where are the sands of the sea now? Gone to mud! Light moves fast and space is large, but symbols can shoot past them at a walk. Take the symbol for a ’light-year’ and cube it! You can see it all there in half a dozen strokes, and its meaning is as exact as the change out of a dollar.

Consider this. There is a famous old Persian story, known to everybody, of the grateful king who asked the physician who had saved his life to name his own reward. The physician merely asked that a penny — or an obol or something — be placed on the first square of a chessboard, two on the next, and then four, and so on, till all the sixty-four spaces were filled! The shah protested at the man’s modesty and said he must at least take a horse as well. Then they counted the money, and presumably the shah fell back dead! The mathematical formula that killed him was the series (1 + 2 + 4 . . . N), where N = 64: the sum of a geometrical progression — and, at that, the simplest and slowest one known to whole numbers.

As a matter of fact, if the king and the physician had started counting out the pennies at the rate of five thousand an hour and had kept it up for a sevenhour day, with Sundays off, it would have taken them a month to count a million. At the end of a year they’d be only on square No. 20 out of the sixty-four; granting that the king and the physician were each sixty-two years old (they’d have to be that to have got so far in politics), their expectation of life would be fifteen years, and they’d both be dead before they got to the thirtieth square; and the last square alone would call for 10,000,000,000,000,000,000 pennies. In other words, they are both alive now and counting.1


But all of this wonder and power and mystery is of no aid in calculating the incalculable. You cannot express the warmth of emotions in calories, the pressure on the market in horsepower, and the buoyancy of credit in specific gravity! Yet this is exactly what the pseudo-mathematicians try to do when they invade the social sciences. The conceptions dealt with in politics and economics and psychology — the ideas of valuation, preference, willingness and unwillingness, antipathy, desire, and so forth — cannot be put into quantitative terms.

It would not so much matter if this vast and ill-placed mess of mathematical symbolism could be set aside and left to itself while the real work of economics went on. Thus, for example, is left aside by the real modern physicists, such as Rutherford and Soddy, the whole mass of the Einstein geometry — which from their point of view is neither here nor there. (Many people don’t know that.) But in the case of economic theory these practitioners undertake to draw deductions; to dive into a cloud of mathematics and come out again holding a theory, a precept, an order, as it were, in regard to the why of the depression, or a remedy for unemployment, or an explanation of the nature of saving and investment. They are like — or want to be like — a physician prescribing a dose for the docile and confiding patient. He writes on a piece of paper, ‘σΔρo/o,’ and says, ‘Take that.’ Thus one of the latest and otherwise most deservedly famous of the mathematical economists advises us in a new book, heralded as the book of the year, that our salvation lies in the proper adjustment of investment and demand. Once get this right and all the rest is easy. As a first aid the great economist undertakes to explain the relation of investment and demand in a preliminary, simple fashion as follows: —

More generally the proportionate change in total demand to the proportionate change in investment equals

ΔY/Y ΔI/I = ΔY/Y.Y-C/ΔY-ΔC=1-c/y/dc1-dc/dy

To 99.9 per cent of the world’s readers this spells good-bye. If economics can only be made intelligible in that form, then it moves into the class of atomic physics. The great mass of us are outside of it. We can judge it only by its accomplishments; and, as economics so far has accomplished nothing, the outlook is dark.

Now I do not know what all that Delta and Y stuff just quoted means, but I am certain that if I did I could write it out just as plainly and simply as the wonderful theorem up above about different people spending their money on different things. In other words, mathematical economics is what is called in criminal circles ‘a racket.’

  1. If any reader doubts these calculations I refer him to my colleague, Professor Charles Sullivan of McGill University, and if he doubts Professor Sullivan I refer him so far that he will never get back. — AUTHOR