Chapter 6 Chapter 3 Pythagoras

The influence of Pythagoras, ancient and modern, is the subject of my chapter; Pythagoras, as far as he was wise or unwitted, thought One of the most important figures.Mathematics, in the sense of demonstrative-deductive inferences, began with him; and mathematics in his thinking was intimately associated with a special form of mysticism.Since his time, and partly because of him, the influence of mathematics on philosophy has been both profound and unfortunate. Let's start with some of the few facts that are known about his life.He is a native of Satsuma and flourished around 523 BC.Some say that he was the son of a wealthy citizen called Mnesark, others that he was the son of the god Apollo; I leave the reader to choose one of the two.In his day, Summer was ruled by the tyrant Polycrates, an old rascal who had made a fortune and had a large navy.

Summer was a commercial competitor of Miletus; its merchants traveled as far as Tartesus, Spain, famous for its minerals.Polycrates became tyrant of Samoa around 535 BC and ruled until 515 BC.He was not much troubled by moral censure; he expelled his two brothers, who had joined him in the tyranny, and his navy was largely devoted to piracy.The fact that Miletus had surrendered to Persia not long ago was very favorable to him.In order to prevent the Persians from continuing to expand westward, he allied with the Egyptian king Amasis.But when the Persian king Cambyses concentrated all his efforts on conquering Egypt, Polycrates realized he was going to win, and changed sides.He sent a fleet of his political enemies against Egypt; but the sailors rebelled and returned to Samos to attack him.Although he defeated them, in the end he was caught in a conspiracy to exploit his greed for money and fell.The Persian satrap at Sardis pretended to betray the great king of Persia, and offered a large sum of money in return for Polycrates' assistance; was crucified.

Polycrates was a patron of the arts and beautified Samoa with many marvelous buildings.Anacreon was his court poet.However, Pythagoras did not like his government, so he left Samo.It is said—and it is not improbable—that Pythagoras visited Egypt, where he acquired most of his wisdom; whatever the circumstances, it is certain that he finally settled at Croton in southern Italy. . The Greek cities of southern Italy were as rich and prosperous as Samo and Miletus; moreover, they were not threatened by the Persians.The two largest cities are Sybaris and Croton.The splendor of Sybaris is still celebrated; its population at its height, according to Diodorus, amounted to three hundred thousand, though no doubt this is an exaggeration.Croton is about the same size as Sebaris.Both cities lived by importing Ionian goods into Italy, some for consumption in Italy and some for re-export from the western coast to Gaul and Spain.Many Greek cities in Italy fought fiercely against one another; Croton had just been defeated by Laucre when Pythagoras reached it.Shortly after Pythagoras' arrival, however, Croton won a complete victory over Sybaris, who was completely destroyed (510 BC).Sybaris had always been closely connected commercially with Miletus.Croton was famous for his medicine; one of Croton's men, Demosides, was once the physician to Polycrates, and later to Darius.Pythagoras and his disciples established a community at Croton, which was for a time very influential in the city.But in the end the citizens turned against him, and he moved to Metapontion (also in southern Italy), where he died.He soon became a mythical figure endowed with miracles and powers, but he was also the founder of a school of mathematicians.Thus two opposing legends contended about his deeds, and the truth was difficult to ascertain.

Pythagoras is one of the most interesting and difficult to understand figures in history.Not only are the legends about him an almost inextricable mixture of truth and fable, but even in their simplest and least disputed form they offer us a most curious psychology.In short, he could be described as a sort of Einstein-Mrs. Eddy.He founded a religion whose main teachings were the reincarnation of the soul and the sinfulness of eating beans.His religion manifested itself in a religious order which everywhere gained control of the state and established a regime of saints.But the unreformed man craves beans, and sooner or later he rebels.

Some rules of Pythagoreanism are: 1.Do not eat beans. 2.If you drop something, don't pick it up. 3.Don't touch the white cock. 4.Do not break bread. 5.Don't step over the latch. 6.Do not poke fire with iron. 7.Don't eat the whole loaf. 8.No garlands. 9.Don't sit on the bucket. 10.Don't eat your heart. 11.Don't walk on the road. 12.No swallows are allowed in the room. 13.When the pot is removed from the heat, do not leave the mark of the pot on the ashes, but wipe it off. 14.Don't look in the mirror next to the light. 15.When you take off your pajamas, roll them up to smooth out any marks on your body.

All these commandments belong to the primitive idea of ​​taboo. Comfortford ("From Religion to Philosophy") says that, in his view, "Pythagoras represents the dominant current of that mystical tradition that we regard as opposed to the scientific tendencies." He sees Parmenides— He called it "the discoverer of logic" - "a branch of Pythagoras, while Plato himself derived his chief source of inspiration from Italian philosophy".He said that Pythagoreanism was a reform movement within Orphism, which in turn was a reform movement within the cult of Dionysus.The opposition of the rational to the mystical, perpetuated throughout history, first appeared among the Greeks as an opposition between the Olympian gods and other less civilized gods, which were closer to those studied by anthropologists. Primitive belief.On this distinction Pythagoras was on the side of mysticism, although his mysticism had a peculiarly intellectual character.He considered himself to be of a semi-divine nature, and seems to have said, "There are men, there are gods, and there are creatures like Pythagoras." Systems "are all tending toward the otherworldly, placing all values ​​in the invisible unity of God, and dismissing the visible world as illusory, as a cloudy medium in which the rays of heaven destroyed and blinded in color and darkness".

Pythagoras, says Dicheax, taught, "First, that the soul is an immortal being, which can be transformed into other beings; Things are absolutely new; all things that are born with life should be considered relatives." It is said that Pythagoras, like St. Francis, once preached to animals. In the societies he established, men and women could join; property was communal, and there was a common way of life, and even the discoveries of science and mathematics were considered communal, and, in a mystical sense, All thanks to Pythagoras; even after his death.Hibassus of Metapontion, who violated this rule, was shipwrecked and died, as a result of the wrath of the gods at his impiety.

But what does all this have to do with mathematics?They are connected by a morality that celebrates the contemplative life.Burnett sums up this morality as follows: "We are all strangers in this world, and the body is the grave of the soul, yet we must not kill ourselves to escape; for we are God's property, and God is our shepherd, and we have no right to escape without his command There are three kinds of people in this world, just as there are three kinds of people who come to the Olympic Games. Those who come to buy and sell are the lowest class, and those who come above them are those who come to compete. However, The highest kind are those who come only to see. Therefore the greatest purification of all is the science of doing nothing, and only he who devotes himself to this enterprise, that is, the true philosopher, can really Free yourself from the wheel of life." The change in meaning of words is often very instructive.I have already mentioned the word "orgy" above; now I shall speak of the word "theory."The word was originally an Orphic word, which Cornford interpreted as "passionate and moving meditation."In this state, he says, "the observer is one with the suffering God, who died in his death and was raised again in his new birth"; for Pythagoras, this "passionate Moving contemplation" is intellectual and results in mathematical knowledge.Thus, through Pythagoreanism, "theory" has gradually acquired its modern significance; but it has always retained, for all who were inspired by Pythagoras, a kind of intoxicated Ingredient of revelation.This may seem strange to those who have had no choice but to learn some mathematics at school; To many people, the Pythagorean view seems quite natural, even if it is untrue.As if the empirical philosopher were only a slave to his material, and the pure mathematician, like the musician, the free creator of his well-ordered and beautiful world.

The most interesting thing is that we can see from the ethics of Pythagoras described by Burnett that it is contrary to modern values.For example, in a football game, people with modern minds always think that the players are far greater than the spectators.As for nations, the situation is similar: they worship politicians (who are competitors in the game) more than people who are mere spectators.Changes in this value are related to changes in social institutions—warriors, gentlemen, plutocrats, dictators, each with its own standards of goodness and truth.The gentleman had a long reign in philosophical theory, because he was united with the Greek genius, because the virtue of contemplation gained theological assurance, and because the ideal of the truth of inaction enshrined the academy. life.A junzi can be defined as a member of a society of equals who live off the labor of slaves, or at least the undoubtedly lowly working people.It should be noted that saints and sages are also included in this definition, for these too were contemplative rather than active as far as their lives were concerned.

Modern definitions of truth, such as the pragmatist and instrumentalist ones, are practical rather than contemplative, inspired by industrial civilization as opposed to aristocratic regimes. Whatever one may think about the social institutions that allow slavery, it is from the gentleman in that sense that we have pure mathematics.The contemplative ideal, which leads to the creation of pure mathematics, is therefore the source of a useful activity; this adds to its prestige and gives it a status in theology, ethics, and philosophy. A success not enjoyed otherwise. We have explained a great deal about Pythagoras both as a religious prophet and as a pure mathematician.In both respects he had an immeasurable influence, and the two respects were not as separate as modern people imagined.

Most of the sciences have been associated from their inception with some false form of belief, which gives them a illusory value.Astronomy is linked with astrology, chemistry with alchemy.Math incorporates a more refined type of error.Mathematical knowledge appears to be reliable, accurate, and applicable to the real world.Moreover, it is obtained by pure thinking, and does not require observation.For this reason it is supposed to provide an ideal of which the knowledge of everyday experience is powerless.From mathematics it is assumed that thought is superior to senses, and intuition is superior to observation.If the world of the senses doesn't agree with mathematics, the world of the senses is even worse.People seek methods that are closer to the mathematician's ideal in various ways, and the resulting revelations become the source of many errors in metaphysics and epistemology.This form of philosophy also began with Pythagoras. As you all know, Pythagoras said "everything is number".This assertion, interpreted in modern terms, is logically meaningless, but what Pythagoras was referring to was not entirely meaningless.He discovered the importance of numbers in music, and the "harmonic middle term" and "harmonic progression" in mathematical terms still retain the connection established by Pythagoras for music and mathematics.He imagined numbers as shapes like those represented on dice or playing cards.We still speak of numbers squared and cubed, and these terms come from him.He also mentions rectangular numbers, triangular numbers, pyramidal numbers, and so on.These are the number of small blocks (or we should say more naturally, the number of small balls) necessary to form the various shapes mentioned above.He assumed that the world was atomic, and that objects were formed by molecules composed of atoms arranged in various forms.In this way he hoped to make arithmetic a fundamental object of study in physics as well as in aesthetics. The greatest discovery of Pythagoras, or that of his immediate disciples, was the proposition concerning right triangles; that the sum of the squares of the two sides of a right angle is equal to the square of the other side, the square of the chord .The Egyptians already knew that if the side lengths of a triangle are 3, 4, and 5, there must be a right angle.But apparently the Greeks were the first to observe 32 + 42 = 52, and found a proof of this general proposition on the basis of this hint. Unfortunately, however, Pythagoras' theorem immediately led to the discovery of incommensurable divisors (irrational numbers), which seemed to negate his entire philosophy.In an equilateral right triangle, the square of the chord is equal to twice the square of each side.Let's assume each side is an hour long, how long should the string be?Let us assume its length is m/n.Then m2/n2=2.If m and n have a common divisor, we can cancel it, so one of m and n must be an odd number.Now m2=2n2, so m is even, so m is even; therefore n is odd.Suppose m=2p.Then 4p2 = 2n2, therefore n2 = 2p2, and therefore n is even, contrary to the assumption.So there is no m/n fraction that can be used to approximate the chord.The above proof is essentially the proof in the tenth book of Euclid. This argument proves that no matter what unit of length we adopt, there will always be some lengths that cannot have an exact numerical relationship to that unit; that is, there cannot be two integers m, n such that the m times A unit whose length is equal to n times.This made the Greek mathematicians firmly believe that the establishment of geometry must be independent and has nothing to do with mathematics.Several passages in Plato's dialogues attest that geometry was already dealt with independently in his time; geometry was completed in Euclid.Euclid proves geometrically in the second book many things that we would naturally prove algebraically, such as (a+b)2=a2+2ab+b2.It was precisely because of the difficulty of incommensurable divisors that he considered this approach necessary.The same is true when he treats proportions in Books V and VI.The whole system is logically striking and already prefigured the rigor of nineteenth-century mathematicians.As long as no proper mathematical theory of incommensurable denominators exists, Euclid's method is the best possible method in geometry.When Descartes reasserted the supremacy of arithmetic by introducing coordinate geometry (analytic geometry), he had imagined the possibility of a solution to the problem of incommensurable denominators, although such a solution had not yet been discovered by him. The influence of geometry on philosophy and scientific method has been profound.The geometry established by the Greeks proceeds from axioms that are self-evident, or considered to be self-evident, and proceeds by deductive reasoning to theorems that are far from self-evident.Axioms and theorems are supposed to be true of real space, which in turn is all that is experienced.In this way, first noticing the self-evident and then using the deductive method, it seems possible to discover everything in the actual world.This view influenced Plato and Kant and most philosophers in between. The "Declaration of Independence," which says: "We hold these truths to be self-evident," itself grew out of Euclid.The doctrine of natural rights in the eighteenth century was a pursuit of Euclidean axioms in politics.Newton's Principia, although its material was admittedly empirical, its form was completely dominated by Euclid.The genre of the strictly scholastic form of theology derives from the same source.Personal religion is derived from communication, theology from mathematics; and both are to be found in Pythagoras. Mathematics, I believe, is the chief source of our belief in eternal and exact truths, and of a supersensory, knowable world.Geometry deals with exact circles, but no sensible object is exactly circular; no matter how carefully we use our compasses, there will always be some incompleteness and irregularity.This suggests the view that all strict reasoning applies only to ideal objects as opposed to sensible objects; Perceived objects are more real.The mystic doctrine of the relation of time to eternity is also consolidated by pure mathematics; for the objects of mathematics, such as numbers, if they are real, must be eternal and not in time.This eternal object can then be imagined as the mind of God.Thus, Plato's doctrine is that God is a geometer; and Sir James Chance believed that God was fond of arithmetic.The religion of rationalism, as opposed to the religion of revelation, has been completely dominated by mathematics and mathematical method since Pythagoras and especially since Plato. The combination of mathematics and theology began with Pythagoras, which represented the characteristics of Greek, medieval and modern religious philosophy up to Kant.Orphic teachings before Pythagoras were similar to the mystical teachings of Asia.But in Plato, St. Augustine, Thomas Aquinas, Descartes, Spinoza, and Kant there is an intimate interweaving of religion and reasoning, a pursuit of morality and logic of the timeless. The close interweaving of cults; this came from Pythagoras and distinguished the intellectualized theology of Europe from the more straightforward mysticism of Asia.Only in recent times has it been possible to say definitively where Pythagoras was wrong.I know of no one else who has had as much influence on the world of thought as he has.I say this because what is called Platonism, when analyzed, is found to be in essence nothing but Pythagoreanism.The whole idea of ​​an eternal world which can only be revealed to the intellect and not to the senses is derived from Pythagoras.If it were not for him, Christians would not think that Christ is the way; if it were not for him, theologians would not pursue the logical proof of the existence of God and the immortality of the soul.But on him, none of this was obvious.Here's how it all came to be.