In the materialist view, then, to understand what mathematics means it is necessary to look at its history and see how it fits in with the general development of society and, in particular, what it was used for. There have been a number of worthwhile attempts to map out such a history [#!Hog68!#,#!Kli54!#,#!Str59!#] not just as a history of ideas (as it is usually presented) but as one aspect of the history of class society. Here restrictions on space allow only some brief remarks about three of the more significant factors in that history: the rise of trade, problems of technology and contradictions internal to mathematics. As should be clear already the history of mathematics is a highly complex process and it is in no way suggested that these are the sole determining factors.
Hunter-gatherer societies may have only counted as high as three (though the evidence is not conclusive), and we can assume that this was sufficient to allow them to perform the very simple tasks they faced. The first class societies, based on farming, needed to count their livestock and to measure the lengths of the seasons and so they could count much higher. But it may be that the rise of trade, right through history, has given the most significant impetus to the development of arithmetic, geometry and algebra. This is for three main reasons:
So it is not a coincidence that the great trading societies all made contributions to the advance of mathematics: the Sumerians and Phoenicians who started counting in groups; the Hebrews, Romans and Greeks who used abstract symbols for numbers; the Chinese who probably invented the abacus and may have discovered Pythagoras' theorem before him; the Mayans who invented the numeral 0; the Babylonians who had a place value system; the Hindus who invented fractions and the modern number system; the great Arabic trading empire of the medieval period that developed algebra (algebra first occurs in the works of al-Khwarizmi (c. 780-c. 850)) and algorithms for multiplication, division, etc.; Europe since the 17'th Century where, amongst other things, differential calculus was invented. In the twentieth century the enormous expansion of trade has meant that large companies require huge banks of the most up to date computers, expert statisticians, mathematicians and even artificial intelligence.
The drawing of scaled diagrams for the construction of buildings was clearly important for the early development of geometry. The Egyptian are known to have used Pythagorean triangles and a relatively advanced number system for the building of the pyramids, but the ideological function of the priesthood meant that this was kept secret and very little written evidence remains. By contrast the later Greek philosophers scorned all concern with practical problems, and so Platonism was quite a natural consequence of this. Hogben argues, perhaps taking the point too far, that it was because of this separation of theory and practice that Greek mathematics stagnated [#!Hog68!#]. Interestingly, Kline [#!Kli54!#] argues the opposite: it was the abstract nature of Greek mathematics that gave it such generality and power. Certainly it is accepted that the Greek contribution to mathematics was considerable, but when faced with paradoxes, like Zeno's paradox or the problem of irrational numbers mentioned earlier, instead of pushing mathematics forward, they retreated into philosophy and failed to solve the problems. It was the Alexandrians, with their practical interest in technology (like Archimedes pump, various war machines involving cog-wheels, catapults, etc.) who were able to make real advances in geometry (the circular function sine and cosine, an estimation of correct to two decimal places) and arithmetic (where they solved Zeno's paradox and could sum infinite series). It seems the interplay between theoretic and practical considerations typical in mathematics, is more subtle that either of the two cited books acknowledges.
Hessen [#!Hes71!#] convincingly demonstrates how Newtonian mechanics and calculus were required to solve very concrete technical problems. Mining, water transport, arms manufacture and the metal industry were crucial.
In the twentieth century the effect of technology on mathematics can be seen clearly in subjects as diverse as number theory, logic, complexity and chaos theory all of which have been closely connected to the development of computer technology. More generally science under capitalism (which was dependent on the knowledge of other societies, notably Hindu mathematics and the influx of Arabic science into Spain, Italy and Southern France from about the 12'th Century) has made an accelerating progress reflecting the development of technology and industry.
As has already been said, mathematics does not simply deal with the practical problems facing society, it also learns from other sciences and other parts of mathematics. Examples of very fruitful developments caused by internal contradictions include the following: the formalisation of logic in ancient Greece caused by the discovery of irrational numbers; Descartes' synthesis of algebraic geometry (co-ordinate geometry) out of the previously unrelated subjects of algebra and geometry; axiomatic set theory following the discovery of paradoxes in set theory; Gödel's synthesis of arithmetic and logic etc. etc.