The development of the quantitative side of investigating nature was obviously of crucial importance. Without it, science would have remained on the level of mere generalities incapable of further development. However, when such a breakthrough takes place, there is inevitably a tendency to make exaggerated claims on its behalf. This is particularly true in this case, where science is still entangled with religion.

The Pythagoreans saw number—quantitative relations—as the essence of all things. "All things are numbers." Indeed, it is possible to explain many natural phenomena in mathematical terms. Nevertheless, even the most advanced mathematical models are only approximations of the real world. The inadequacy of the purely quantitative approach, however, was evident long ago. G. W. F. Hegel, who, as a convinced idealist and a formidable mathematician, might have been expected to be enthusiastic about the Pythagorean school, however, this was far from the case. Hegel poured scorn on the idea that the world could be reduced to quantitative relations.

From Pythagoras onwards, the most extravagant claims have been made on behalf of mathematics, which has been portrayed as the queen of the sciences, the magic key opening all doors of the universe. Breaking free from all contact with crude material reality, mathematics appeared to soar into the heavens, where it acquired a god-like existence, obeying no rule but its own. Thus, the great mathematician Henri PoincarŽ, in the early years of this century, could claim that the laws of science did not relate to the real world at all, but represented arbitrary conventions destined to promote a more convenient and "useful" description of the corresponding phenomena. Many physicists now openly state that the validity of their mathematical models does not depend upon empirical verification, but on the aesthetic qualities of their equations.

Thus, the theories of mathematics have been, on the one side, the source of tremendous scientific advance, and, on the other, the origin of numerous errors and misconceptions which have had, and are still having profoundly negative consequences. The central error is to attempt to reduce the complex, dynamic and contradictory workings of nature to static, orderly quantitative formulae. Starting with the Pythagoreans, nature is presented in a formalistic manner, as a single-dimentional point, which becomes a line, which becomes a plane, a cube, a sphere, and so on. At first sight, the world of pure mathematics is one of absolute thought, unsullied by contact with material things. But this is far from the truth, as Engels points out. We use the decimal system, not because of logical deduction or "free will," but because we have ten fingers. The word "digital" comes from the Latin word for fingers. And to this day, a schoolboy will secretly count his material fingers beneath a material desk, before arriving at the answer to an abstract mathematical problem. In so doing, the child is unconsciously retracing the way in which early humans learned to count.

The material origins of the abstractions of mathematics were no secret to Aristotle: "The mathematician," he wrote, "investigates abstractions. He eliminates all sensible qualities like weight, density, temperature, etc., leaving only the quantitative and continuous (in one, two or three dimensions) and its essential attributes." (Metaphysics, p. 120.) Elsewhere he says: "Mathematical objects cannot exist apart from sensible (i.e., material) things." (Ibid, p. 251.) And, "We have no experience of anything which consists of lines or planes or points, as we should have if these things were material substances, lines, etc., may be prior in definition to body, but they are not on that account prior in substance." (Ibid, p. 253.)

The development of mathematics is the result of very material human needs. Early man at first had only ten number sounds, precisely because he counted, like a small child, on his fingers. The exception were the Mayas of Central America who had a numerical system based on twenty instead of ten, probably because they counted their toes as well as their fingers. Early man, living in a simple hunter-gatherer society, without money or private property, had no need of large numbers. To convey a number larger than ten, he merely combined some of the ten sounds connected with his fingers. Thus, one more than ten is expressed by "one-ten," (undecim, in Latin, or ein-lifon—"one over"—in early Teutonic, which becomes eleven in modern English). All the other numbers are only combinations of the original ten sounds, with the exception of five additions—hundred, thousand, million, billion and trillion.

The real origin of numbers was already understood by the great English materialist philosopher of the 17th century Thomas Hobbes: "And it seems, there was a time when those names of number were not in use; and men were fayn to apply their fingers of one or both hands, to those things they desired to keep account of; and that thence it proceeded, that now our numerall words are but ten, in any Nation, and in some but five, and then they begin again." (Hobbes, Leviathan, p. 14.)

Alfred Hooper explains: "Just because primitive man invented the same number of number-sounds as he had fingers, our number-scale today is a decimal one, that is, a scale based on ten, and consisting of endless repetitions of the first ten basic number-sounds…Had men been given twelve fingers instead of ten, we should doubtless have a duo-decimal number-scale today, one based on twelve, consisting of endless repetitions of twelve basic number-sounds." (A. Hooper, Makers of Mathematics, p. 4-5.) In fact, a duodecimal system has certain advantages in comparison to the decimal one. Whereas ten can only be exactly divided by two and five, twelve can be divided exactly by two, three, four and six.

The Roman numerals are pictorial representations of fingers. Probably the symbol for five represented the gap between thumb and fingers. The word "calculus" (from which we derive "calculate") means "pebble" in Latin, connected with the method of counting stone beads on an abacus. These, and countless other examples serve to illustrate how mathematics did not arise from the free operation of the human mind, but is the product of a lengthy process of social evolution, trial and error, observation and experiment, which gradually becomes separated out as a body of knowledge of an apparently abstract character.

Similarly, our present systems of weights and measures have been derived from material objects. The origin of the English unit of measurement, the foot, is self-evident, as is the Spanish word for an inch, "pulgada," which means a thumb. The origin of the most basic mathematical symbols + and - has nothing to do with mathematics. They were the signs used in the Middle Ages by the merchants to calculate excess or deficiency of quantities of goods in warehouses.

The need to build dwellings to protect themselves from the elements forced early man to find the best and most practical way of cutting wood so that their ends fitted closely together. This meant the discovery of the right angle and the carpenters’ square. The need to build a house on level ground led to the invention of the kind of levelling instrument depicted in Egyptian and Roman tombs, consisting of three pieces of wood joined together in an isosceles triangle, with a cord fastened at the apex. Such simple practical tools were used in the construction of the pyramids. The Egyptian priests accumulated a huge body of mathematical knowledge derived ultimately from such practical activity.

The very word "geometry" betrays its practical origins. It means simply "earth-measurement." The virtue of the Greeks was to give a finished theoretical expression to these discoveries. However, in presenting their theorems as the pure product of logical deduction, they were misleading themselves and future generations. Ultimately, mathematics derives from material reality, and, indeed, could have no application if this were not the case. Even the famous theorem of Pythagoras, known to every school pupil, that a square drawn on the longest side of a right triangle is equal to the sum of the squares drawn on the other two sides, had been already worked out in practice by the Egyptians.

The Pythagoreans, breaking with the Ionian materialist tradition which attempted to generalise on the basis of experience of the real world, asserted that the higher truths of mathematics could not be derived from the world of sensuous experience, but only from the workings of pure reason, by deduction. Beginning with certain first principles, which have to be taken as true, the philosopher argues them through a series of logical stages until he arrives at a conclusion, using only facts that are agreed first principles, or are derived from such. This was known as a priori reasoning, from the Latin phrase denoting "from what comes first."

Using deduction and a priori reasoning, the Pythagoreans attempted to establish a model of the universe based on perfect forms and governed by divine harmony. The problem is that the forms of the real world are anything but perfect. For instance, they thought that the heavenly bodies were perfect spheres moving in perfect circles. This was a revolutionary advance for its time, but neither of these assertions is really true. The attempt to impose a perfect harmony on the universe, to free it from contradiction, soon broke down even in mathematical terms. Internal contradictions began to surface which led to a crisis of the Pythagorean school.

About the middle of the 5th century, Hippius of Metapontum discovered that the quantitative relations between the side and the diagonal of simple figures like the square and the regular pentagon are incommensurable, that is, they cannot be expressed as a ratio of whole numbers, no matter how great. The square root of two cannot be expressed by any number. It is, in fact, what mathematicians call an "irrational" number. This discovery threw the whole theory into confusion. Hitherto, the Pythagoreans had taught that the world was constructed out of points with magnitude. While it might not be possible to say how many points there were on a given line, still they were assumed to be finite in number. Now if the diagonal and the side of a square are incommensurable, it follows that lines are infinitely divisible, and that the little points from which the universe was built do not exist.

From that time on, the Pythagorean school entered into decline. It split into two rival factions, one of which buried itself in ever more abstruse mathematical speculation, while the other attempted to overcome the contradiction by means of ingenious mathematical innovations which laid the basis for the development of the quantitative sciences.

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