MATHEMATICS AS HISTORY.
W. W. Sawyer.
In describing the development of any subject, a choice has to be made; can we reestrict ourselves to discussion of the subject itself, or is it necessary to consider it in the context of the time and place where it grew? For instance, in the history of music, is it sufficient to describe and analyze the differences between Mozart and Beethoven, or is it necessary to consider such external influences as the French Revolution and the rise of romantic nationalism ? A similar question arises in relation to mathematics.
Of the writers who have discerned some pattern in history the three best known are Toynbee, seeing civilisations in terms of challenge and response; Marx, often misinterpreted as ascribing all human actions to the economic interest of classes; and the most spectacular of all, Spengler.
To the best of my knowledge Toynbee does not deal with mathematics. Marx maintains that mathematics develops in two ways; one is related to the needs of society, the other to the natural, internal development of the subject itself. A good example of this internal development would be Gauss' construction of the regular 17-sided polygon by ruler and compasses. Surely no society will ever have a practical need for the regular 17-gon, and if for some strange reason they did,they would not resort to a lengthy construction with ruler and compasses.
What led Gauss to his theorem was purely intellectual curiosity. The Greeks had succeeded in constructing the regular pentagon - in itself quite a remarkable achievement. Gauss asked himself,"What property does the number 5 have, not possessed by 7,11 and 13 , that makes this construction possible?" He succeeded in answering this question, and his work has since led to a whole series of studies of what can be done with a specified type of mathematical operations.
The 17th century gives an excellent example of the influence of society on mathematical development. The mathematical needs of society were then particularly intense. This can be seen in the dedication of the Royal Society in 1660. It expresses the hope that improvements in navigation will "Deliver the Anxious Seamen from the Fatal Accidents that frequently attend their Mistaken Longitude."
To-day longitude is determined by comparing the time, based on local observations, with the Greenwich time-signal on radio. In the 17th century help was sought in two ways - perfecting clocks that would keep accurate time in a storm-tossed ship, and the improvement of astronomy, so that the Greenwich time could be deduced by observing, for instance, the time at which a particular star passed behind the moon. Newton's Principia goes into immense detail on the movement of the moon, something that would certainly be absent to-day in a publication announcing the law of universal gravity. The theory of dynamics was involved in both approaches.
A young Russian, later murdered by Stalin, wrote an interesting and instructive account of the background to 17th century mathematics.  Mechanics and geometry are both involved in the design of ships. How far this work had gone was shown dramatically when Peter the Great visited England in 1698. Peter had some very unpleasant traits, but I find one aspect of his character very attractive, and most unusual in a king. When he wanted to build a Russia navy, he decided to get a thorough understanding by himself working as a shipwright, first in Holland and then in England. As a result of this experience, when he left England he took 500 Englishmen with him, to play a decisive part in directing and carrying out the construction of his navy. For, he said, the Dutch simply copied what had worked in the past. In Holland he had learned little more than ship's carpentry. "Peter wanted to grasp the basic secrets of ship design ... He wanted blueprints made scientifically, controlled by mathematics"  Incidentally, I do not know the details of what scientific principles for ship design were available at that time, and would be very grateful to anyone who can inform me of these.
HISTORY AS DESTINY.
The most extreme view to be considered is that of Oswald Spengler, an incredibly learned German
schoolteacher, who claimed to have discovered a universal pattern in world history. He supported his views by reference
to the history of civilizations in Europe, Russia, Egypt, the Middle East, India and China throughout the centuries.  With this wealth of material, it is not surprising that his
book, quite apart from its general theme, contains many interesting pieces of information. For instance, early
in the book he scoffs at those who take a superficial view of history, such as the Jacobin clubs in the French
Revolution, who had a cult admiring Brutus as a revolutionary. Spengler points out that Brutus was in fact "a millionaire and an exploiter who, as a supporter of the oligarchic regime, stabbed the
man of the democracy amid the plaudits of the aristocratic Senate."  I never realized this, when I did Julius Caesar at school.
According to Spengler, there was a close analogy between the history of a culture and the life of a human being.
A culture was born; there followed a vigorous youth and a period of maturity; finally came an age of decadence,
in which old beliefs and loyalties faded and money dominated everything. He gave the years A.D. 1800 to 2000 for
this stage of our society. Without admitting the truth of his general theory, one must admit that this has considerable
resemblance to the
He maintained this progression, from vigour to decadence, was as inevitable as the successive changes in a person who survives from infancy to old age. Moreover, at each stage, every subject, every activity reflected both the surrounding culture and the stage that had been reached. His example of a subject depending on the civilization in which it develops -, we think of our geometry as being the same as that of the Greeks, but in fact it is quite different.It is impossible for us to understand the classical view that everything is inside the sphere of the fixed stars. A child to-day, shown the classical picture of the universe, would immediately ask, "What is outside the largest sphere?" Spengler says classical man had neither the word nor the concept of empty space. Greek geometry deals with the sizes and shapes of material objects.
Modern geometry is entirely different. It may start, "Suppose there are three kinds of objects, called points, lines and planes, with the property that any two points determine a line...." and so on. It is a study of logical relationships, quite distinct from Euclid, in which you can pick up the triangle ABC and put it down on the triangle DEF. The Greek fear of the distant. Spengler says the Greeks had an acute fear of anything distant in time or space. He relates this to the existence of the Polis. the city-state in which Greeks lived. "Home for classical man was what he could see from the citadel of his state. Anything beyond this was strange, indeed hostile."  This feeling of dread came to attach to distance in any form. Here we may begin to think Spengler is letting his imagination run away with him. But in another place we read, "In the last years of Pericles a law was passed in Athens that threatened with the severe punishment of impeachment anyone who propagated astronomical theories."  Such theories seem perfectly harmless to us, but evidently upset the Greeks very much. Spengler's explanation seems as good as any.
Spengler asserts that we are in a late stage of a dying civilization. Now this is a view that could be held quite rationally without any reference to Spengler. We can apply Toynbee's condition; a civilization disappears when a challenge arises to which its governing circles fail to respond. To-day there are many serious problems, social, economic and international, on which no European government - and for that matter no opposition party - seems to be preparing a determined onslaught.
Unemployment, perhaps, is the most dangerous of these. The 1930s showed the serious consequences it could have. However at present nations do not try to get rid of it, but only to push it onto somebody else. We must admit that we may fail, but what Spengler says is quite different from that. He maintains we are in the grip of an implacable destiny, which makes it certain we shall fail. Fortunately we can prove quite rigorously that Spengler's reasoning is unsound. An essential point of Spengler's theory, on which he insists many times, is that every activity reflects the stage its civilization has reached, so it is quite impossible for any subject to flourish triumphantly in an age of total decadence. If even one subject does this, it means the whole theory has to be rejected. And in fact there is one subject that has reached supreme heights in our century, a subject for which Spengler prophesied a future which turned out to be the exact opposite of what actually happened. That subject is mathematics.
The reason for this is not that Spengler was ignorant of mathematics. In his book Spengler displays a grasp of what was happening in mathematics that would have been quite creditable for a professional mathematician of his day. What caused his mistake was his belief that, in an age of decay, mathematics must necessarily share that decline. It was this belief - not any ignorance of mathematics - that led him to make a forecast of the future of mathematics that could not have been more untrue. He wrote
In reality, while Spengler was putting his material together, two epoch-making events in mathematics took place.These were two doctoral dissertations;- in 1902 Lebesgue's announcement of Lebesgue integration, which gave analysts an entirely new and extremely powerful weapon; in 1906 Frechet's thesis on abstract spaces, an idea that has permeated nearly all mathematical papers since that time.  Spengler's reference to "work on details" is particularly inappropriate to this latter topic, abstract spaces, for these are characterized by sweeping generalizations. Indeed some of the more conservative mathematicians have complained that, in the pursuit of such generalizations, modern research workers have not paid any attention to certain detailed traditional problems, the solution of which would be of great value.
 B.Hessen. The social and economic roots of Newton's Principia. Translated with introduction by R.S.Cohen. New York. 1971.
 Peter the Great. R.K.Massie. Gollancz. 1980. p.201.
 Oswald Spengler. The Decline of the West. The manuscript was ready in 1914; owing to the war, published only in 1918. The pages quoted below are from Der Untergang des Abendlandes. C.H.Beck (Munich). 1969 reprint.
 Spengler, p.6.
 Spengler, p.113. .
 Spengler, p.123. At the end of the section on numbers.
 An outline of abstract spaces was given in my paper in The Mathematical Gazette for March 1996.
Copyright © W. W. Sawyer & Mark Alder 2000
Version: 22nd March 2001