What Lies Behind Pythagoras’ Theorem

Pythagoras (570-500 BC) was born on the island of Samos in what is now Greece. On the advice of Thales of Miletus, he travelled to Memphis in Egypt where he came into contact with priests renowned for their wisdom. At the age of 40, he fled the tyranny of Polycrates to Croton in Southern Italy, where he established a philosophical and religious community. Those who entered the community’s inner circle were governed by a strict set of ascetic and ethical rules, forsaking personal possessions, assuming a mainly vegetarian diet, and—since words are so often careless and misrepresentative—observing the strictest silence. Some of the community’s more eccentric rules, such as ‘do not break bread’ or ‘do not poke the fire with a sword’ may have been riddles or allegories that required interpreting. Pythagoras’ brotherly community has been hailed as a prototype for later philosophical institutions such as Plato’s Academy, Aristotle’s Lyceum, and Epicurus’ Garden, and, indeed, for the monastic life and associated early universities.

Music played an important role in Pythagoras’ community. Pythagoreans recited poetry, sang hymns to Apollo, and played on the lyre to cure illnesses of both body and soul. It is said that, one day, Pythagoras passed by some blacksmiths at work, and found that their hammering on anvils produced especially harmonious sounds. He then found that the anvils were simple ratios of one another, one being half the size of the first, another two thirds of the size, and so on. This discovery of a relationship between numerical ratios and musical intervals led Pythagoras to believe that the study of mathematics was the key to understanding the structure and order of the universe. According to his ‘harmony of the spheres’, the heavenly bodies move according to mathematical equations that correspond to musical notes and form part of a grand cosmic symphony.

Pythagoras never separated religion from science and philosophy, which, even in his day, left him open to accusations of mysticism. No doubt under the influence of Orphism, an Ancient Greek mystery religion that arose from pre-Hellenic beliefs and the Thracian cult of Zagreus, he believed in the transmigration of the soul; that is, in the reincarnation of the soul over time into the bodies of human beings, animals, or plants (metempsychosis) until such a time as it became moral. He claimed to have lived four lives and to remember them all in great detail, and once recognized the cry of his dead friend in the yelping of a puppy. After his death, the Pythagoreans deified him, and attributed him with a golden thigh and the gift of bilocation (being in two places at once). But in his own lifetime Pythagoras had always been a paragon of modesty, declining to be called a ‘wise man’ or ‘sophos’, and preferring instead to be called ‘a lover of wisdom’ or ‘philosophos’—thereby coining the term ‘philosopher’.

It is said that Numa Pompilius, the second king of Rome, had been schooled by Pythagoras, whence his great wisdom and piety. This story is referred to and discredited by Plutarch and Livy, not least because the dates do not tally, with Pythagoras having lived from about 570 to 500 BC, and so considerably later than the semi-legendary King Numa. Even though Pythagoras and Numa never met, Pythagoras and the Pythagoreans exerted a strong influence on the Roman mind. In the Tusculan Disputations, Cicero indicates that Pythagoras rose to fame in southern Italy at just the same time that Brutus brought an end to the monarchy, and that many Roman usages derived from the Pythagoreans. Unfortunately, he does not elaborate on the nature of these usages. In his Natural History, Pliny the Elder tells us that, in 343 BC, during the war with the Samnites, the god Apollo ordered the Romans to erect one statue to the wisest and another to the bravest of all Greeks, with their choices falling upon Pythagoras for the former and Alcibiades for the latter. Pliny expresses surprise that they picked Pythagoras over Socrates, whom Apollo himself had called the wisest of all men. But the fact is that the Romans liked to think of the Greek-Italian Pythagoras as their very own philosopher, and spun all sorts of stories, such as the one about Numa, to better appropriate him.

Apart from this, Pythagoras also exerted a strong indirect influence on Roman thinking, and indeed on all philosophy and theology, through the teachings of Plato, the principal architect of the western mind. Aristotle, who was Plato’s pupil of twenty years, claimed that his master’s teachings owed much to those of Pythagoras; so much, in fact, that, in his History of Western Philosophy, Bertrand Russell upheld not Plato but Pythagoras as the most influential of all Western philosophers. Pythagoras’ influence is especially evident in Plato’s mystical approach to the soul and in his emphasis on mathematics and, more generally, abstract thinking, as a secure basis for the practice of philosophy.


Aristotle’s Metaphysics, Book Alpha

Aristotle: The Master of those who know

Outline of Book 1 (of 13)

All men by nature desire to know. Thus, the senses are loved not only for their usefulness but also for themselves. Sight is loved best of all, for, of all the senses, it is the one that brings the most knowledge. Animals are by nature sensing, and from sensation memory is produced in some of them, which are thereby more intelligent and apt at learning than those which cannot remember. Those that have both memory and the sense of hearing can be taught, but the others cannot. Animals other than man live by appearances and memories and have but little of connected experience, but man lives also by art and reasoning. From several memories of the same thing man produces a single experience, and it is through this single experience that come science and art. With a view to action, experience (knowledge of individuals) is not inferior to art (knowledge of universals), and men of experience succeed better than those with theory but no experience, for actions are concerned not with the universal but with the individual. And yet people suppose artists to be wiser than men of experience because artists know the ‘why’ and the cause, and can therefore teach, whereas men of experience cannot teach. Again, none of the senses are regarded as Wisdom because, although they give the most authoritative knowledge of particulars, they do not reveal the ‘why’ of anything. At first all the arts were admired, but as more arts were invented, the recreational arts (those that pertain to Wisdom) were admired more than the practical arts.

What are the causes and principles of Wisdom? As far as possible, the wise man knows all things, even though he may not have detailed knowledge of them, and he can learn things that are difficult and farthest from mere sense perception. He is more exact, more capable of teaching, and more suited to ordering than to obeying. The most exact of the sciences are those that deal most with first principles, for the sciences that involve fewer principles are more exact than those that involve additional principles. First principles are most truly knowledge, and most knowable; from these all other things come to be known, but not vice versa. The science which knows to what end each thing must be done is the most authoritative, and this end is the good of that thing, and in general the supreme good in nature. As the good is one of the causes, this science must be the same as that which investigates the first principles and causes. That it is not a science of production is obvious even from the earliest philosophers, owing to whose wonder men first began to philosophise. A man who wonders and who is puzzled thinks of himself as ignorant, and philosophises to escape ignorance and accede to knowledge, not for the sake of something else but for its own sake. Such a free science only God can have, or God above all others; and God himself is thought to be among the causes of all things and to be a first principle.

Evidently, then, we have to acquire knowledge of the original causes, and causes are spoken of in four senses (see the Physics). In one sense, a cause is the substance or essence, in another the matter or substratum, in a third the source of the change, and in a fourth the purpose or the good that it serves. Of the first philosophers, most think that the principles of matter are the only principles of all things. They argue that that of which all things consist, that from which they come to be, and that into which they are resolved (the substance remaining, but changing its modification) is the element and the principle of things; thus, nothing is either generated or destroyed in the sense that the substratum (or substrata) remains. Yet they do not agree as to the number and nature of these principles. Thales says the principle is water (a view that may have been shared by those who first framed accounts of the gods), Anaximenes and Diogenes that it is air, Hippasus and Heraclitus that it is fire, Empedocles that it is all four of the elements, and Anaxagoras that it is infinite in number. However true it may be that all generation and destruction proceed from some one or several elements, why does this happen and what is the cause? The substratum does not make itself change, bronze does not manufacture a statue, but something else is the cause of the change, and to seek this is to seek the second cause, namely, that from which comes the beginning of movement. Some of the first philosophers who maintain that the substratum is one, as if defeated by the search for the second cause, say that the one and nature as a whole is unchangeable not only in respect of generation and destruction, but also of all other change. Those who allow for more elements are better able to account for the second cause; however, it is unlikely that fire or earth or any one element, or indeed spontaneity and chance, can explain why things manifest goodness and beauty both in their being and in their coming to be. When Anaxagoras and Hermotimus of Clazomenae first suggested that reason is present, as in animals, so throughout nature as the cause of order and movement, they must have seemed like sober men. Perhaps Hesiod is the first to look for such a thing, and Parmenides and some others also think of love or desire as the first principle. Certainly, Empedocles is the first to conceive not only of an aggregative first principle which he calls love or friendship, but also of a contrary segregative first principle which he calls strife. Empedocles is also the first to speak of four material elements, even though he only treats them as two, fire as one kind of thing, and earth, air, and water as another. Leucippus and Democritus say that the full and the empty are the elements, calling the one being and the other non-being, and making these the material causes of things. Those who make the underlying substance one generate all other things by its modifications; similarly, they make differences in the elements (namely, differences in shape, order, and position) the causes of all other qualities. All these thinkers evidently grasp, if only imprecisely, two of the causes which I distinguish in the Physics, namely, the matter and the source of movement.

For the Pythagoreans, all things seem to be modelled on numbers, and so they suppose the elements of numbers to be the elements of all things. Evidently, they also consider that number is the principle as matter for things and as both their modifications and their permanent states. According to them, the elements of number are the even and the odd, from which the One, which is both even and odd, proceeds, and number from the One. Other Pythagoreans say that there are ten principles, which they arrange into two columns of cognates, limit and unlimited, odd and even, one and plurality, right and left, male and female, resting and moving, straight and curved, light and darkness, good and bad, square and oblong. Alcmaeon of Croton also advances that the contraries are the principles of things, but how these principles can be brought together under the causes that I have named neither Alcmaeon nor the Pythagoreans can explain, although they do seem to range the elements under the head of matter. There are also those who speak of the universe as if it were one entity, but since they also maintain that change is impossible, the discussion of them is in no way appropriate to my present investigation of causes. In summary, then, of the earliest philosophers, there are on the one hand those who regard the first principle, whether single or plural, as corporeal, and on the other hand those who posit both this cause and also the source of movement, whether single or dual.

In most respects, Plato follows these thinkers. In this youth, Plato became familiar with Cratylus and with the teachings of Heraclitus that all sensible things are in a state of flux and that there can hence be no knowledge about them. Whereas Socrates seeks out the universal in ethical matters, Plato holds that the problem applies not to sensible things, which are always changing, but to the Ideas or Forms in which sensible things participate. For the Pythagoreans things exist by ‘imitation’ of numbers, whereas for Plato they exist by ‘participation’ in Forms, but what ‘imitation or ‘participation’ involve they do not say. Moreover, Plato maintains that, besides sensible things and Forms, there are the objects of mathematics, which occupy an intermediate position. Since the Forms are the causes of all other things, their elements are the elements of all things. As matter, the great and the small are principles; as essential reality, the One; for from the great and the small, by participation in the One, come the Numbers. Plato agrees with the Pythagoreans that the One is substance and not a predicate of something else, and that Numbers are the causes of the reality of other things. However, he constructs the infinite out of great and small instead of treating it as one, and conceives of the Numbers as existing apart from sensible things.

The essence, that is, the substantial reality, no one expresses distinctly. It is hinted at chiefly by Plato, who does not suppose either that the Forms are the matter of sensible things and the One the matter of the Forms, or that they are the source of movement. Instead, he advances that the Forms are the essence of every other thing, and that the One is the essence of the Forms. When the early philosophers speak of a cause, for instance, reason or friendship, they do not speak as if anything that exists came into being for the sake of it, but as if movements started from it. Thus, they both say and do not say that reason or friendship is a cause, in the sense that it is only an incidental cause.

Those who say that the universe is one and posit one kind of thing as matter, and as corporeal matter, only posit the elements of bodies and not of incorporeal things, though there are also incorporeal things. In giving a physical account of all things, they neglect the cause of movement. Furthermore, they do not posit the substance, that is, the essence, as the cause of anything, and call one of the simple bodies (water, fire, air) the first principle without asking how the simple bodies are produced out of each other, and so without considering their priority and posterity. Empedocles posits that all four bodies are the first principles, but he can be criticised on the same ground and also on grounds that are peculiar to his position. The Pythagoreans do not say how there can be movement if limited and unlimited and odd and even are the only things assumed. It appears that they have nothing to say about perceptible things, for if spatial magnitude does indeed consist of these elements, how, for instance, could some bodies be light and others heavy? Moreover, is the number that is each abstraction the same number that is exhibited in the material universe, or is it another than this? According to Plato, both bodies and their causes are numbers, but intelligible numbers are causes whereas the others are sensibles.

Unfortunately, to posit the Ideas as causes is, so to speak, to introduce an equal number of causes to the causes. Besides which, there is no convincing proof for the existence of the Forms: from some proofs no inference necessarily follows, and from other proofs there arise Forms even of things which are not thought of as having Forms. Of the more accurate arguments, some lead to Ideas of relations, and others introduce the ‘third man’. There are also other objections to the Ideas. Above all, one might ask what on earth the Forms contribute to sensible things, whether eternal or perishable, if they cause neither movement nor change in them.

Adapted from Aristotle’s Universe (NYP)

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