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The Marvels of Mathematics
"The Mirror of Civilization”
The Coming Of An Arithmetic Of Social Welfare

(By O. N. Gillespie ).

The great Edmund Burke of whom a cynic raid that “his words were always golden but his logic often brummagem,” uttered a great truth by accident in one of his tremendous onslaughts on the French Revolution. He said this: “The Age of Chivalry is gone. That of sophists, economists, and calculators has succeeded, and the glory of Europe is extinguished for ever.” It was profoundly untrue that the glory of Europe was over, but it was exactly correct to say that the age of economists and calculators had arrived.

The distinction between modern civilisation and the great cultures of the misty past, is that we have learned to use figures. A railway engine might possibly have emerged in Athens or Babylon, but a railway time-table could neither have been planned nor understood by the most scientific mind in Greece or Assyria. In other words, we have lately learned to calculate. This advance is recent, and the use of calculation is daily taking fresh form. The ordinary person has no trouble in finding the answer to questions which baffled the best mathematical minds of ancient times, and the time will come when Einstein will be easily understood by boys and girls of the middle standards. New Zealand has just produced a brilliant example of the fusion of mathematical and literary statement in a survey of social research, and this article will try to trace a little of the history of the science on which, in Julian Huxley's words, “depends the progress of any democratic society.”

The modern science of mathematics has arisen from the needs of everyday life, and has grown up day by day with the everyday life of the world. Intelligent social planning will have to use mathematics more and more, and the study of mathematics will have to be pursued with more and more ardour. In the words of one of the greatest of all English writers, the aid of mathematics is needed “by every intelligent youth from fifty to ninety who is trying to get the hang of things in the universe.”

First of all let me try to explain the definition of mathematics, cleverly devised and reduced to simple terms by the great Professor Hogben. He says that mathematics is the language of size as distinct from the language which describes the sorts of things. The rules of mathematics form its grammar.

The first men had a form of talking; they had to convey their ideas to each other by sounds, and the ideas mostly dealt with the description of things, or emotions.

“The language in which people describe the different sorts of things there are in the world is vastly more primitive and more conservative than the size languages which have been multiplied to cope with the increasing precision of man's control over nature.” Moreover, it is abundantly clear that the language of mathematics has other notable qualities; it is international; it is rationally planned; it has no place for sentiment or national prejudices; it has no social distinctions, and no inheritance of emotion.

It is not an overstatement, then, to predict that it is in the study of mathematics, and the diffusion of the knowledge of pure mathematics, that will lie the solution of the world's countless troubles.

We have a reasonably accurate knowledge of the methods of counting used by several of the older cultures, and the story of how and why men first learned to count, is as fascinating as a good thriller.

Some one said in reply to a scoffing critic who claimed that science did not give a true picture of the world, that “Science is not a picture of anything. It is an ordnance map to direct our efforts in changing the world.”

Obviously, the ordnance map is of no possible use without understanding the figures on it.

The tale of man's first effort to count is an exciting one, but stranger still is the romance which surrounds his first efforts to record the results of his counting.

You will see in our illustration samples of four of the ancient scripts, used in civilisations that had reached lofty-heights of culture. They are easy to understand. After them came the Roman system and the Etruscan variation. This idea involved the use of different letters as symbols. “V” stood for five. “L” for fifty, and so on. This, as you will see, was derived from the number of fingers on one hand. Six is represented by “VI” which is five followed by one. With the use of “X” for ten, “C” for one hundred, “D” for five hundred, and so on, a script for numbers, or rather a writing method for figures had arrived, and was a fairly useful medium.

It is certain that counting commenced when men started to collect flocks and herds. Its next development was caused by the need to estimate days and seasons, and then, of course, came all the necessities arising out of the trading with goods.

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Now there is a fatal defect in all these ancient methods of putting calculations on record. None of them allow of such a statement as 1/1000 or 9.998. Imagine the simplest time-table to Palmerston North expressed in any of the scripts shown in our picture!

The ancient Greeks, with all their superb culture and extraordinary powers of thought, had the whole of their scientific investigations limited by this fact; they had no workable method of division; they had the abacus, the bead frame for doing additions and simple multiplication, but it also refused to go beyond a limit of numbers. Division has insuperable difficulties. All the mental processes of man are limited, naturally, by his social background, and more, by the mechanical aids to calculation which are available to him. One authority says that this difficulty was “the Nemesis of Greek culture.”

It led to the most ludicrous misapprehensions. Having no apparatus or any device to express such a number as 1,000,000,000, Greek thinking stopped at numbers which seem ridiculous to us today. Anaxagoras was thought guilty of blasphemy when he asserted that the sun was probably as large as the mainland of Greece.

The sides of the counting frame, or abacus, actually formed a prison for the shining minds of that great land of thinkers.

Relief was to come from a strange quarter. The Hindus starting far behind the Greeks in cultural standards, had evolved a series of symbols for numbers which could be used without mechanical aids. The most amazing and effectual discovery of all was the symbol “o.” With a method of representing all the numbers up to nine, and a separate symbol for “o,” or zero, all modern arithmetic became possible.

Laplace, the great astronomer, says of this revolution that “it was a profound and important idea which appears to us so simple that we ignore its true merit …. we shall appreciate the true grandeur of this achievement when we remember it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.

When the Mohammedan civilisation swept through Africa and established itself in Spain, this script went with it, took charge of Europe, and modern civilisation, modern science and all the works of man's mind, became possible.

Perhaps the most important effect of all this was to clear away the mystical nonsense about numbers. Pythagoras and a number of successors right on into the Middle Ages erected an imposing priesthood of numbers. Numbers were invested with all sorts of magical significance and even were given sex and profound qualities of good and evil.

Even Plato regarded the study of mathematics as the privilege of the learned few. Now it is the common mental stock of mankind. In other words mathematics has been put to work. Even the Alexandrians realised this; they listened and thrilled under the highbrow erudition of the high priests of Mathematics but they used geometry to build their temples and warehouses. To-day we are on the same road, with an infinitely wider vision.

Cobbett, the great reformer, answering a query as to whether the labour of learning grammar was worth while, pointed out that without the knowledge of the grammar of a language, no effort towards the gain of human freedom could ever be placed on record. Without the grammar of mathematics, further progress towards human liberty and happiness is impossible.

Without further labouring of the subject, it is to be said that the analogy between the grammar of a language and the grammar of mathematics is almost exact.

Modern developments in such branches as algebra, trigonometry, the calculus and other branches of higher mathematics, have given the science, adverbs, verbs, adjectives and propositions for the expression of every finest shade of meaning.

The final glory of the language of numbers is still to be stressed; it is completely international. (a-b) (a+b)=(a2 - b2) is intelligible to Russian or Chilian, the French or the Italian schoolboy.

The language of mathematics represents an emancipating force, freeing intellects and setting free influences which pass over national barriers.

From the study of its eternal truths, from the appreciation of its beneficence of discovery, its inescapable exactness of conclusion, will come the ultimate realisation of happiness for all mankind.

This being so, I am proud to be able to instance a recent New Zealand book which is a notable example of the use of modern statistical research methods. “Littledene” is the study of a small New Zealand community. It combines a personal and human knowledge of the people with an exact set of figure calculations relating to production, social activities, and the general economic pattern of the whole entity. This is the combined method described with so much zest in Bell's great book, “The Search for Truth.”

The author, Mr. H. C. D. Somerset, is a schoolmaster with an equipment of profound scholarship and the ability to write in a way which is denied to most authors of such books as “Littledene.” He lived in the district, worked there and entered fully into the life. His observations are made from inside the sitting room, not peeping in the window of the kitchen, armed with a notebook. Listen to this account of the Jubilee Procession: “It was decided to hold a procession in which every organisation could take part. Most people found that they were eligible to take part on half a dozen counts. There was much preparation by all concerned. When the day arrived the procession was half a mile long; everybody was in it. First came the brass band, playing a march; behind the band the various lodges in full regalia. Then a float, representing Britannia and her colonies. There followed the displays of the Farmers’ Union, the various sports clubs, the Women's Christian Union and so on, and so on. The Salvation Army brought up the rear with its blood and fire banner. But few loyal Littledenians saw the procession; everyone participated so fully that the writer of this survey and a few latecomers were the only ones privileged to see it pass by.”

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Those of us who know our New Zealand, will recognise and appreciate that picture at once.

But the marvel of the book is this: on page 8 there is a graph showing “Animals per average 100 acres, according to Size of Farm.” This is at once understood by any reader, but it would have been completely beyond the grasp of the great Democritus; it could not have been shown, either, under any system of Greek writing.

On page 81, there is a table showing the percentages of occupations adopted by the ex-pupils of the school. This is perfectly clear to anyone who has passed the fifth standard. Plato, however, would not have been successful, nor had the Greeks of his time any method of picturing such a calculation.

As one reads this fascinating survey, “Littledene,” it becomes increasingly clear that mathematics is the base of it. The truth shines clear through its pages because the language of figures which help in its expression, lets in the light.

The farmer at Littledene is measuring his field for ploughing and sowing, perusing his stock company accounts, and reading his daily paper, with a background of mathematical knowledge denied to the most profound thinker of the days of Imperial Rome. His wife uses feats of memory and skill to do her feats of cooking but her recipes are based on exact mathematical calculation, and the watch or clock that she uses for timing would represent an amazing and unintelligible piece of mechanical wizardry to Archimedes.

By the way, I must digress to quote from the book on the subject of cooking.

“It is impossible to take one sunset, some skill, and abundant leisure to make a picture. The farm wife takes a pint of cream, six eggs, and the spur of the moment while the meat and potatoes are cooking, and lo! a cream sponge-cake six inches high, with two inches of whipped cream in its depths. The cookery section of the Littledene Agricultural and Pastoral Show is like a confectioner's heaven. And experts agree that the cookery book produced by the Women's Division of the New Zealand Farmers’ Union would do credit to a Paris chef.

It is as well to remind ourselves that in the leisurely days of the ancient civilisations, there was not a device for counting time which measured smaller units than the time needed to cook some object. Hours were known, but not seconds.

It is the harnessing of mathematics to human needs, its removal from the sphere of lofty and mystic realms of thought, that makes human progress possible.

“The sellers of cars, radios, electric light and telephones can revolutionise the work and play of a community in a way that the philosopher with his reasoning can never hope to rival.”

But all those instruments of human enlightenment and recreation were only possible of creation through the evolution of mathematics.

But there is something more important than the mere bringing of these instruments of human culture within the reach of mankind in general.

It is at once obvious that the statement that 7 plus 5 is a truth which is different in essence from countless other assertions which claim to be final truths. It is in this pure quality of mathematical truth that its helpfulness finally resides. An English scientist has pointed out that no word is so loosely used as the word “law,” particularly in such phrases as “laws of Nature,” “laws of economics,” and so on. One use of “law” is to describe “observed regularities in nature.” In other words things which have happened millions of times at regular intervals may be predicted to go on happening. When the word is lifted and applied to developments of human society and human motives, it is a definite misuse. It wrecks in the opinion of many the value of much of the writing by economists. Professor Hogben says that, if the word “law” in this connection, must be retained we had better call them “Bye-laws,” implying our right to repeal them.

This confusion is impossible in the realm of pure mathematics. The fact that the language of mathematics is a planned and rationalised language, means that each term has its permanent and distinct significance. It is safe to predict therefore that to the mathematicians can be entrusted the task of where we are heading and of finding us the direction chart to show where we ought to go.

But there is a step to be taken. In the words of Bertrand Russell, “we must remove mathematics from the remote regions of apparent uselessness.” In other words we have each and all of us to learn the Arithmetic of Social Progress. We have to realise that learning about science or the art of living means mainly learning more about mathematics. Put shortly, mathematics has to be democratised. Out of that diffusion will spring the men of genius who will collate and put in order the thought of the community. Robert Burns would not have produced his magic poetry unless he lived in a community of poets, of people who worshipped and practised the art of verse making. The work of Isaac Newton, Lord Rutherford, Madame Curie and Einstein was possible because they dwelt in a world emancipated from the intellectual prisons of the earlier centuries. It is not absurd to suppose that as the centuries go by, the mathematicians and scientists of the future will bear the same relation to Lord Rutherford and Einstein as those two giants of achievement bear to Pythagoras or Archimedes.

As Cobbett said, Prynne would not have been able to impeach Archbishop Laud if his command of grammar had not been sufficient to make himself understood. When the common men of the world have universally a command of the grammar of mathematics, there will arise from them greater men still, more profound thinkers, who will conduct the impeachment of the evils that infest our world.

It is a vision of comfort and of glory.