Evolution Of Mathematics

A Story by MartinDickens

An essay on the history of mathematics.

Some people will tell you that mathematics did not begin until there was a class with leisure to play with figures and numbers. However, mathematics has advanced when there has been real work for the mathematicians to do and has stagnated whenever it has become a plaything of a class which is isolated from the common life of mankind.

The Greeks, who were the foremost writers of mathematics felt a need for it due to the surrounding circumstances. They lived in a world in which they saw people measuring the angles between the stars, building temples aided by diagrams traced on the sand, reckoning heights by the length of shadows, designing figures in clay, and making tiles. The men who first wrote books on mathematics lived in a world in which the priestly architecture of the Pyramids, magical games with numbers, Cyprian vases ornamented with geometrical patterns, walls and floors covered with mosaic tiles, were familiar.

By the time, when people first began to write books on mathematics, mankind had found ways of answering several sorts of questions to which the answer is a number. The need for mathematics arose due to questions somewhat like the following:

HOW MANY INDIVIDUALS MAKE UP A GROUP ?

The extinct Tasmanian aborigines, did not count beyond four. We may assume that the need for counting large numbers was not felt until men began to keep flocks and herds. The shepherds and herdsmen must count their flocks. In our number system we group objects for enumeration in tens, tens of tens(hundreds) and tens of tens of tens(thousands). This is what is meant when we say that ten is the base of our number system.

Some multiple of five recurs as the base of nearly all number systems. This is due to the fact that primitive man uses his fingers as a tally to check off the things he counts. There are traces of hand and foot counting in our own language as seen in the frequent use of the word “score” in the Old Testament. An even earlier way of grouping numbers in twos and fours is seen in the base two of the Syriac numerals.

HOW LONG WAS IT ?

When primitive man learned to sow grains and keep beasts which bear at certain periods of the year, he had to take stock of the seasons. He notices that the moon rises a little later and sets a little later each night between one full moon and the next. The Egyptians had already fixed the length of the year to three-sixty-five days before four thousand BC, by counting the days that intervened between two successive occasions when the dog-star, Sirius, was just visible at its rising immediately before sunrise.

Civilised mankind developed written symbols for numbers long before the need for rapid and simple means of calculation arose. In fashioning their number script men had no prevision of the requirements of a script with which simple arithmetical performances could be carried out. When man got out of the stage of relying entirely upon tally sticks, representing numbers by notches, he hit on the practice of using pebbles or sticks which could be rapidly used or discarded again.

So came the counting frame. At first it was probably a series of grooves on a flat surface. Then it was a set of upright sticks on which pierced stones, shells or beads could be placed. The counting frame named abacus was a very early achievement. It follows the megalithic culture routes all around the world. The Mexicans and Peruvians were using the abacus when the Spaniards got to America. The Chinese and Egyptians already possessed the abacus several millennia before the Christian era. The Romans took it from the Etruscans. Till about the beginning of the Christian era, this fixed frame remained the only instrument for calculation that mankind possessed.

To us numbers are symbols with which sums can be done. This conception of number was completely foreign to the most advanced mathematicians of ancient Greece. The ancient scripts were merely labels to record the result of doing work with an abacus instead of doing work with a pencil and pen. In the whole history of mathematics there has been no more revolutionary step than the one which the Hindus made when they invented the sign “0” to stand for the empty column of the counting frame.

The new number vocabulary of Hindus allows you to add on paper in the same way as you add on the abacus. How the invention was made and how it affected the after history of mathematics must be left for the present. The important thing to realize is that the mathematicians of classical antiquity inherited a social culture which was equipped with a number script before the need for laborious calculations was felt. So they were completely dependent upon mechanical aids which have now been banished to the nursery.

The need for accurate measurement grew naturally from the practice of time-keeping, which was an essential prerequisite of settled city life. It is fairly certain that people began to measure angles long before they troubled themselves very much with measuring lengths. The number of days in a year had been fixed by the heliacal rising of Sirius at the very beginning of settled life along the Nile.

For many millennia men were content to use crude anatomical units of length for most practical purposes. The Semitic peoples used the cubit or distance from the tip of the middle finger to the elbow, as farmers still use their legs to pace out a field in “feet” or yards. For ordinary purposes they were content with a unit of length which varied from one individual to another. Temple architecture demanded a far higher standard of precision, and was based on the long-lost art of shadow reckoning in the sunnier climates where civilization began. Heights were reckoned by the length of the shadow and the angle of the sun above the horizon, and reckoning heights in this way depends on certain simple truths about the relation between the lengths of the sides of a triangle. The earliest mathematical discoveries belong to this class of problems.

We have to make “laws” of discourse to regulate the communication of laws which are made without our help. The latter are the real laws of nature, and they do not change because Einstein is a better mathematician than Newton was. Indeed, when Einstein applied a different sort of mathematics to the mechanical problems for which Newton’s system was devised a party of scientists had to trek half-way across the earth to decide whether it really made any difference.

Einstein’s robust common sense has not prevented his disciples from erecting his teaching into a new system of theological apologetics. The followers of Mach, who anticipated some of Einstein’s criticisms of Newton’s mechanics, had already anticipated their exploits. The sequel was a comic controversy about whether science is a true picture of the world. Science is not a picture of anything. It is an ordnance map to direct our efforts in changing the world. The world view of science is not a by-product of human cerebration. The mountains and the valleys remain, when we use a new color scale to paint in the altitudes. The ether does not remain. It is the ink used in the nineteenth century edition of the map. Both parties were right, and both were wrong. One argued as if mathematical symbols were eternal, and the other, as if regularities of nature did not exist before Pithecanthropus became extinct. Controversies of this sort will continue, until we make the language of science part of the language of mankind, and realize that the future of the human reason lies with those who are prepared to face the task of rationally planning the instruments of communication. Many people talk as if we had reached the limit of the educability of mankind. In Bacon’s words, they prefer to extol the powers of their own minds instead of seeking “the true helps” by which intelligent citizenship can be encouraged. If we ask what true helps exist we need not look far afield for new materials. When the educational powers of the cinema have been adopted to visualizing the use of mathematical symbols anything which can be achieved by books and blackboards will seem trivial by comparison.

( With reference to “Mathematics For The Million” by

“Lancelot Hogben” )