Pythagoras

 

The Mathematical Basis of All Things

Across a span of water from Miletus, located in the Aegean Sea, was a small island of Samos, which was the birthplace of a truly extraordinary and wise man, Pythagoras. From the scrapes of information we have about him and those who were his followers, an incomplete but still fascinating picture of his new philosophical reflections emerges. Pythagoras migrated to Southern Italy and settled there in the prosperous Greek city of Crotone, where his active philosophic life is usually dated from about 525 to 500 B.C.E. We are told by Aristotle that "the Pythagoreans … devoted themselves to mathematics, they were the first to advance the study, and having been brought up in it they thought its principles were the principles of all things …" In contrast to the Milesians, the Pythagoreans said that things consist of numbers. Although it is quite strange to say that everything consists of numbers, the strangeness, as well as the difficulty, of this doctrine is greatly overcome when we consider why Pythagoras became interested in numbers and what his conception of numbers was.

Pythagoras became interested in mathematics for what appear to be religious reasons. His originality could be said to consist in his conviction that the study of mathematics is the best purifier of the soul. He is, therefore, referred to as the founder of both a religious sect and at the same time a school of mathematics. What gave rise to the Pythagorean sect was people’s yearning for a deeply spiritual religion that could provide the means for purifying the soul and for guaranteeing its immortality. The Homeric gods were not gods in the theological sense, since they were as immortal as human beings and as such could be neither the objects of worship nor the source of any spiritual power to overcome the pervading sense of moral uncleanliness and the anxiety that people had over the shortness of life and the finality of death.

The Pythagoreans were clearly concerned with the mystical problems of purification and immortality, and it was for this reason that they turned to science and mathematics, the study of which they considered the best purge for the soul. Thought and reflection represent a clear contrast to the life of trade and competition for various honors. It was Pythagoras who first distinguished three different kinds of lives, and by implication the three divisions of the soul, by saying that there are three different kinds of people who go to the Olympian Games. The lowest class is made up of those who go there to buy and sell, to make a profit. Next are those who go there to compete, to gain honors. Best of all, he thought, are those who come as spectators who reflect upon and analyze what is happening. Of these three, the spectators illustrate the thinkers, whose activity as philosophers liberates them from the involvements of daily life and its imperfections. To "look on" is one of the meanings of the Greek word "theory." Theoretical thinking, or pure science and pure mathematics, was considered by the Pythagoreans as a purifier of the soul, particularly as mathematical thought could liberate people from thinking about particular things and lead their thought, instead, to the permanent and ordered world of numbers. The final mystical triumph of the Pythagorean is liberation from "the wheel of birth," from the migration of the soul to animal and other forms in the constant process of death and birth, for thus the spectator achieves a unity with god and shares his immortality.

We are here concerned only with the cosmological element in this body of doctrine and how Pythagoras himself may have dealt with the problem of nature. The Pythagorean cosmography, or picture of the world, suggests that Pythagoras in this respect remained a true disciple of the Ionian School. Like Anaximenes, he pictured the world as suspended in a boundless three-dimensional ocean of vapor and inhaling nourishment from it. Like both Anaximenes and Anaximander, he thought of it as a rotating nucleus in this vapor, having the earth at its center; the rotary movement serving to generate and segregate opposites. A new discovery of his own seems to have been that the earth is spherical in shape. In his cosmology or theoretical commentary on this picture, Pythagoras broke new ground, with momentous consequences, according to Collingwood. So definite was the break on this point between Pythagoras and his predecessors that we can guess how his thought actually moved.

Pythagoras suggested that the qualitative differences of nature were based on differences of geometrical structure. The point of the new theory is that we need not bother to ask what the primitive matter is like; that makes no difference; we need not ascribe to it any character differing from that of space itself: all we must ascribe to it is the power of being shaped geometrically. The nature of things, that by virtue of which they severally and collectively are what they are, is geometrical structure or form. What must have facilitated the development of the doctrine that all things are numbers was the Pythagorean practice in counting and their way of writing numbers. Apparently they built numbers out of individual units, using pebbles to count. The number "one" was therefore a single pebble and all other numbers were created by the addition of pebbles, somewhat like the practice of representing numbers on dice by the use of dots. But the significant point is that the Pythagoreans discovered a relation between arithmetic and geometry. A single pebble, as a point is "one," but "two" is made up of two pebbles or two points, and these two points make a line. Three points, as in the corners of a triangle, create a plane or area and four points can represent a solid. This suggested to the Pythagoreans a close relationship between number and magnitude, and Pythagoras is credited with discovering that the square of the hypotenuse is equal the squares of the other two sides of a right-angled triangle. This correlation between numbers and magnitude provided immense consolation to those who were seeking evidence of a principle of structure and order in the universe.

Figures

The importance of the relation between number and magnitude was that numbers, for the Pythagoreans, meant certain "figures" such as a triangle, square, rectangle and so forth. The individual points were "boundary stones" which marked out "fields." Moreover, these "triangular numbers," "square numbers," "rectangular numbers," and "spherical numbers" were differentiated by the Pythagoreans as being "odd" and "even," thereby giving them a new way of treating the phenomena of the conflict of opposites. In all these "forms," numbers were, therefore, far more than abstractions – they were specific kinds of entities. To say, then, as the Pythagoreans did, that all things are numbers meant for them that there is a numerical basis for all things which possess shape and size; thus they moved from arithmetic to geometry and then to the structure of reality. All things had numbers, and their odd and even values explained such opposites in things as one and many, square and oblong, straight and curved, rest and motion. Even light and dark are numerical opposites as are male and female, good and evil.

The Concept of Forms

This way of understanding numbers lead the Pythagoreans to formulate their most important philosophical notion, their most significant contribution to philosophy; namely, the concept of "forms." The Ionians had conceived the idea of a primary "matter" or stuff out of which everything was constituted, but they had no coherent concept of how specific things are differentiated from this single matter. They all spoke of an unlimited stuff, whether it be water, air, or the indeterminate boundless, by which they all meant some primary "matter." It was the Pythagoreans who now came forth with the conception of "form." For them form meant limit, and limit is understandable especially in numeric terms. It is no wonder that the two arts in which the Pythagoreans saw the concept of limit best exemplified were music and medicine, for in both of these arts the central fact is harmony, and harmony is achieved by taking into account proportions and limits. In music there is a numeric ratio by which different notes must be separated in order to achieve concordant intervals. Harmony is the form that the limiting structure of numerical ratio imposes upon the unlimited possibilities for sound possessed by the strings of the musical instrument. In medicine, the Pythagoreans saw the same principle at work, health being the harmony or balance or proper ratio of certain opposites such as hot and cold, wet and dry, and the volumetric balance of various specific elements later known as biochemicals. The Pythagoreans looked upon the body as they would a musical instrument, saying that health is achieved when the body is "in tune" and that disease is a consequence of undue tensions or the loss of proper tuning of the strings. The concept of number was frequently used when translated to mean "figure," in connection with health and disease in the literature of early medicine especially pertaining to the constitution of the human body. The "true" number, or figure, therefore, refers to the proper balance of all the elements and functions of the body. Number, then, represents the application of "limit" (form) to the "unlimited" (matter), and the Pythagoreans referred to music and medicine only as vivid illustrations of their larger concept, namely, that all things are numbers. 

What the problem of physics needed for its solution was to be approached from the standpoint of mathematics. The principle of which physics stood in need, identified with something unintelligible, namely matter, was now identified with something supremely intelligible, namely mathematical truth. Once people had learned to think mathematically (and the Greeks had learned from the Ionians) it was obvious that mathematics provided a field in which the human mind was completely at home; a field in which clear and certain knowledge was attainable than in any other: far more than in the astronomical predictions or cosmological speculations of Ionia. This peculiarly clear and certain kind of knowledge was put by the Pythagoreans in quite a new but instantaneously convincing position on the map, as knowledge of the essence of things: not only of shapes which things may assume but of what gives them their peculiar properties and their difference from one another. Incidentally this gave a most powerful stimulus to mathematical studies; but its philosophical importance was still greater, as a declaration that the essence of things, what makes them what they are, is supremely intelligible.

Hence, when Socrates claimed that ethical concepts were even more intelligible than mathematical, and when he or his pupil, Plato, identified the ultimate nature of things with the concept of the good, the new movement of thought, though to some extent it diverted attention from mathematics, was philosophically no change at all, and that is why Aristotle looking back over the history of Greek thought, could describe Plato as a Pythagorean. For if form is essentially something that differentiates itself into a hierarchy of forms, it is not necessary to suppose that mathematical forms, infinite though they are in their own diversity, exhausts the whole of this hierarchy: there may be non-mathematical forms as well.

The brilliance of Pythagoras and his followers is measured to some extent by the enormous influence they had upon later philosophers and particularly upon Plato. There is much in Plato that first came to life in the teachings of Pythagoras, including the importance of the soul and its threefold division and the importance of mathematics as related to the concept of form and the Forms.