In spite of its mystical character, the Pythagorean doctrine marked a step forward in the development of philosophy. There is nothing strange about this. In the evolution of human thought, there are many instances in which the pursuit of irrational and unscientific goals nevertheless have furthered the cause of science. For centuries, the alchemists exerted themselves fruitlessly in an attempt to discover the "philosopher’s stone." This ended in failure. But in the process, they made extremely important discoveries in the field of experiment which provided the basis upon which modern science, especially chemistry, later developed.
The basic tendency of Ionian philosophy was an attempt to generalise from the experience of the real world. Pythagoras and his followers attempted to arrive at an understanding of the nature of things by a different route. Schwegler puts it thus:
"We have the same abstraction, but on a higher stage, when the sensuous concretion of matter in general is looked away from; when attention is turned no longer to the qualitative aspect of matter, as water, air, etc., but to its quantitative measure and relations; when reflection is directed, not to the material, but to the form and order of things as they exist in space." (Schwegler, History of Philosophy, p. 11.)
The progress of human thought in general is closely linked to the capacity to make abstractions from reality, to be able to draw general conclusions from a host of particulars. Since reality is many-sided, it is possible to interpret it in many different ways, reflecting this or that element of the truth. This we see many times in the history of philosophy, where great thinkers laid hold of one aspect of reality, and held it up as an absolute and final truth, only to be swept away by the next generation of thinkers, who in turn repeat the process. Yet the rise and fall of great philosophical schools and scientific theories represents the development and enrichment of human thought by a process of endless successive approximations.
The Pythagoreans approached the world from the standpoint of number and quantity relations. For Pythagoras, "all things are numbers." This idea was linked to the search for the underlying harmony of the universe. They believed that number was the element out of which all things developed. Despite the mystical element, they made important discoveries which greatly stimulated the development of mathematics, especially geometry. They invented the terms odd and even numbers, odd numbers being male and even ones female. Since no women were allowed in the community, they naturally declared odd numbers to be divine and even ones earthly! Likewise, our terms squares and cubes of numbers come from the Pythagoreans, who also discovered harmonic progression in the musical scale, linking the length of a string and the pitch of its vibrating note.
These important discoveries were not put to any practical use by the Pythagoreans, who were interested in geometry purely from an abstract mystical point of view. Yet they had a determining influence on subsequent thought. The mystique of mathematics as an esoteric subject, inaccessible to ordinary mortals, has persisted down to the present day. It was transmitted through the idealist philosophy of Plato, who placed over the entrance of his school the inscription: "Let no man destitute of geometry enter my doors."
"The cosmology of the Pythagoreans," writes professor Farrington, "is very curious and very important. They did not, like the Ionians, try to describe the universe in terms of the behaviour of certain material elements and physical processes. They described it exclusively in terms of number. Aristotle said long afterwards that they took number to be the matter as well as the form of the universe. Numbers constituted the actual stuff of which their world was made. They called a point One, a line Two, a surface Three, a solid Four, according to the minimum number of points necessary to define each of these dimensions." (Farrington, op. cit., p. 47.) They attached magical significance to particular numbers—three, four, seven. Of particular significance was the number ten which is the sum of 1, 2, 3, and 4. These superstitions still persist in the Holy Trinity, the four horsemen of the Apocalypse, the seven deadly sins, and the like. "It is also apparent," adds Bernal, "in modern mathematical physics whenever its adepts try to make god the supreme mathematician." (Bernal, op. cit., p. 124.)
The history of science is characterised by the most fierce partisanship, at times bordering on fanaticism, in defence of particular schools of thought, which put themselves forward as the protagonists of an absolute truth, and who do in fact embody the maximum point reached by human knowledge at a given point in time. Only the further development of science itself reveals the limitations and inner contradictions of a given theory, which is then negated by its opposite, which is itself negated, and so on ad infinitum. This process is precisely the dialectic of the history of science, which for centuries proceeded in tandem with the history of philosophy, and initially was virtually indistinguishable from it.
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