Pythagoras Study Guides

Pythagoras Among the most influential figures in history, both in his wise and foolish moments, was the Greek philosopher Pythagoras, whose effect can be seen in both ancient and modern times—my topic for discussion in this chapter. He is the origin of mathematics in the demonstrative deductive reasoning sense, and he is closely associated with a particular kind of mysticism. Since his time, mathematics has had a tremendous and regrettable impact on philosophy, which is partially his fault. Starting with the limited knowledge available about his life. He was born on the island of Samos and lived to a prosperous age in 532 B.C. I'll let the reader decide which of these theories best fits him; some claim he was the son of the deity Apollo, while others claim he was the son of a prominent man named Mnesarchos. During his reign, Samos was governed by the despot Polycrates, an elderly roughheart who amassed enormous wealth and a sizable fleet. Samos was Miletus's commercial adversary; its traders traded as far as Tartessus, a mine-rich town in Spain. Around 535 B.C., Polycrates took over as Samoan tyrant, and he ruled the island until 515 B.C. Moral qualms did not seem to bother him much; he disinherited his two brothers, who had initially supported the tyrant, and he mostly utilized his navy for piracy. The fact that Miletus had just succumbed to Persia worked to his advantage. He joined himself with Amasis, the King of Egypt, to prevent the Persians from moving any farther west. However, upon seeing that Cambyses, the King of Persia, was going to succeed, Polycrates switched sides and focused all of his efforts on conquering Egypt. His political adversaries made up the fleet he dispatched to invade Egypt, but the sailors rebelled and came back to assault him from Samos. Despite defeating them, he ultimately succumbed to a cunning plea to his greed. Polycrates went to the mainland for an interview, was caught and crucified, and the Persian satrap at Sardes indicated that he wanted to rebel against the Great King and would pay large sums for the support of Polycrates. Polycrates was a patron of the arts and left beautiful public works that enhanced Samos. His court poet was named Anacreon. But Pythagoras despised his administration, and for that reason he departed Samos. Although it is possible that Pythagoras traveled to Egypt and picked up much of his knowledge there, it is likely that he finally settled in Croton, southern Italy. The Greek cities of southern Italy, like as Miletus and Samos, were wealthy and prosperous, and they were also safe from Persian attack. * Sybaris and Croton were the two best. Sybaris has continued to be associated with luxury; Diodorus claims that at the city's heyday, there were 300,000 people living there, but this is undoubtedly inflated. Croton was almost the same size as Sybaris. The economy of both cities depended on the importation of Ionian goods into Italy, which were intended for both domestic use and resale to Gaul and Spain via the western coast. The Greek towns of Italy were at war with one another; Croton had recently fallen to Locri when Pythagoras came. But in a battle against Sybaris soon after his arrival, Croton emerged triumphant and the city was entirely devastated (510 B.C.). Miletus and Sybaris maintained a close commercial relationship. Croton was well-known for its medical expertise; one of its citizens, Democedes, later served as Polycrates' and Darius' physician. Pythagoras established a discipleship organization in Croton, which had some influence in the city for a while. However, the populace eventually turned against him, so he relocated to Metapontion, which is also in southern Italy, where he passed away. He founded a school of mathematicians in addition to being a legendary character who was attributed with magic abilities and marvels. Thus, his memory was contested by two conflicting traditions, and it is difficult to separate fact from fiction. Among the most fascinating and perplexing individuals in history is Pythagoras. The legends around him not only include an almost unbreakable blend of fact and fiction, but even in their most basic and undisputed form, they reveal an intriguing psychology to us. In a nutshell, he's kind of like Einstein meets Mamma Eddy. He established a religion whose major beliefs were that eating beans is evil and that souls transmigrate. His faith was represented in a religious organization that occasionally took over the government and instituted a saints' kingdom. However, the unregenerate had a strong need for beans and eventually revolted. Among the Pythagorean order's rules were: 1. should avoid eating beans. 2. not to gather up the debris. 3. A white cock should not be touched. 4. not to shatter bread. 5. to avoid stepping on a crossbar. 6. not to use iron to fan the flames. 7. not to consume the entire bread. 8. Not to cut a wreath. 9. To not linger over a quarter measure. 10. To not consume the heart. 11. Avoid strolling on freeways. 12. to avoid living with swallows on one's roof. 13. Stir the ashes together after removing the saucepan from the heat, making sure not to leave any trace of it behind. 14. Never glance at a mirror that is next to a light. 15. Roll your bedsheets together and smooth out the body impression as soon as you get out of bed. * All of these rules stem from archaic tabu ideas. According to Cornford (From Religion to Philosophy), Parmenides— whom he refers to as "the discoverer of logic"—is "an offshoot of Pythagoreanism, and Plato himself as finding in the Italian philosophy the chief source of his inspiration." Pythagoreanism, he says, was a movement of reform in Orphism, and Orphism was a movement of reform in the worship of Dionysus. Cornford believes that "The School of Pythagoras represents the main current of that mystical tradition which we have set in contrast with the scientific tendency." The Greeks are the first to record the historical contrast between the Olympic gods and other less civilized gods who were more akin to the primordial beliefs studied by anthropologists. This opposition between the logical and the mystical has persisted throughout history. Pythagoras supported mysticism in this debate, although his mysticism was distinctly cerebral in nature. Dikaiarchos claims that Pythagoras taught "first, that the soul is an immortal thing, and that it is transformed into other kinds of living things; further, that whatever comes into existence is born again in the revolutions of a certain cycle, nothing being absolutely new; and that all things that are born with life in them ought to be treated as kindred." * It is said that Pythagoras, like all the other gods, was semidivine. Cornford claims that all the systems that he inspired "tend to be otherworldly, putting all value in the unseen unity of God, and condemning the visible world as false and illusive, a turbid medium in which the rays of heavenly light are broken and obscured, in mist and darkness.” Men and women were accepted on equal footing, property was owned in common, and there was a shared way of life in the community he built. Even after Pythagoras's death, scientific and mathematical accomplishments were still credited to him in a collective and mystical sense. This law was broken by Hippasos of Metapontion, who was shipwrecked by the wrath of God for his impiety. However, how is any of this related to mathematics? It is linked by an ethic that extols the virtues of contemplative living. Burnet summarizes this moral code as follows: "We must not strive to escape by selfmurder; for we are God's chattels, his herdsman, and we have no right to make our escape without his word. We are aliens in this world, and the body is the tomb of the soul. Just like there are three types of individuals that attend the Olympic Games, there are three types of men in this world. Those who come to purchase and sell make up the lowest class; those who compete are the next in line. The people who just come to observe are the best, though. * Word meaning changes are frequently quite informative. Disinterested knowledge is, then, the greatest purification of all, and the individual who commits himself to that, the real philosopher, has most effectively freed himself from the "wheel of birth." Having discussed the word "orgy" previously, I now want to discuss the word "theory." Originally an Orphic word, Cornford defines it as "passionate sympathetic contemplation," in which "the spectator is identified with the suffering God, dies in his death, and rises again in his new birth." Pythagoras, however, believed that this type of contemplation was intellectual in nature and derived from knowledge of mathematics. In this way, Pythagoreanism helped "theory" eventually take on its current definition, but it nevertheless had a sense of exhilarating discovery for everyone who was influenced by Pythagoras. The Pythagorean view, though incorrect, may seem entirely natural to those who have loved mathematics and have occasionally experienced the exhilarating delight of sudden understanding. However, this may seem strange to those who have reluctantly learned a little mathematics in school. While the pure mathematician, like the musician, is a free creator of his universe of ordered beauty, the empirical philosopher may appear to be a slave to his material. It's fascinating to see how Burnet's explanation of the Pythagorean morality contrasts with contemporary ideals. When it comes to a football game, modern men consider the players to be more important than the onlookers. In the same way with reference to the State, people value politicians who participate in the game more highly than those who only observe it. The warrior, the gentleman, the plutocrat, and the dictator, each with his own standard of what is real and good, are the symbols of this shifting social structure. The reason for the gentleman's lengthy career in philosophical theory is his association with the Greek genius, the religious validation of contemplation's virtue, and the academic life's dignity brought by the ideal of disinterested truth. A gentleman is a member of an equal society who subsists, if not entirely on the labor of slaves, then at least on the labor of men whose inferiority is accepted. It should be noted that inasmuch as the lives of the saint and the sage are contemplative rather than active, these individuals are included in this description. Industrialism, not aristocracy, is the source of modern notions of truth, such as pragmatism and instrumentalism, which are practical rather than contemplative. Whatever one's opinion of a social structure that permits slavery, we owe pure mathematics to gentlemen in the aforementioned meaning. Due to the contemplative ideal's contribution to the development of pure mathematics, it gained prominence and achieved success in the fields of philosophy, theology, and ethics that it may not have otherwise had. Describes Pythagoras in great detail in terms of his two roles as a religious prophet and a pure mathematician. He had enormous influence in both areas, and the two were not as unlike as they may appear to the modern eye. The majority of sciences were first associated with some sort of myth, which gave them an artificial significance. Alchemy and chemistry were linked to astronomy and astrology, respectively. A more sophisticated kind of mistake was linked to mathematics. It seemed that mathematical knowledge could be acquired only by thinking, without the need for observation, and that it was also precise, definite, and useful in the actual world. As a result, it was believed to provide an ideal that ordinary empirical knowledge could not match. Mathematical reasoning led to the assumption that mind is superior to sensation and intuition to observation. The world of sense will suffer greatly if mathematics does not match the world of sensation. Many approaches were explored to get closer to the ideal of the mathematician, and the ideas that emerged were largely incorrect in the fields of metaphysics and theory of knowledge. The founder of this school of philosophy is Pythagoras. As is well known, Pythagoras asserted that "all things are numbers." While a modern interpretation of this claim renders it illogical, Pythagoras' meaning was not quite absurd. He saw numbers as forms, just as they appear on dice or playing cards, and he recognized the significance of numbers in music. His relationship between music and arithmetic is preserved in the mathematical words "harmonic mean" and "harmonic progression." We still refer to numbers as squares and cubes, which are his inventions. In addition, he discussed numbers that were pyramidal, triangular, oblong, and so on. These were the quantities of pebbles—or, maybe a more appropriate term—necessary to create the desired forms. Presumably, he believed that everything was atomic and that molecules, which are made up of atoms organized in different ways, comprise bodies. He wanted to establish mathematics as the foundational subject in physics, just like it is in aesthetics. The idea of right-angled triangles—that the square on the side opposite the right angle is equal to the square on the remaining side, or the hypotenuse—was the most important discovery made by Pythagoras or his direct pupils. Egyptians had previously discovered that a triangle with sides of 3, 4, and 5 forms a right angle, but it appears that Greeks were the first to notice that 3 2 + 4 2 = 5 2, and they were the ones who followed up on this observation by finding a demonstration of the general idea. Unfortunately for Pythagoras, the discovery of incommensurables immediately followed his theorem, seemingly refuting his entire worldview. The hypotenuse square in a right-angled isosceles triangle is double the size of the squares on each side. Assuming one inch for each side, what would be the hypotenuse's length? Assume for the moment that it is m/n inches long. Consequently, m2 / n 2 = 2. Divide it out if m and n share a factor; if so, one of them must be odd. Considering that m2 = 2n 2, m2 is even, m is even, and n is odd. Assume m = 2p. Hence, 4p 2 = 2n 2, which implies that n 2 = 2p 2, and n is even, contra hyp. Thus, the hypotenuse cannot be measured by any fraction m/n. The aforementioned proof is essentially taken from Book X of Euclid. * This argument demonstrated that there exist lengths that, regardless of the unit of measurement we choose, have no exact numerical relationship to it. That is to say, there are no two numbers, m and n, such that m times the length in issue equals n times the unit. The Greek mathematicians were persuaded by this that geometry needed to be established apart from arithmetic. There are instances in Plato's dialogues that demonstrate how independent geometry study was already well underway in his day; Euclid is credited with perfecting it. In Book II, Euclid provides geometrical proofs for some concepts that algebra should easily demonstrate, including (a + b)2 = a 2 + 2ab + b 2. He believed that this training was essential due to the complexity of incommensurables. The same holds true for his handling of proportion in Books V and VI. The entire method makes sense and foreshadows the meticulousness of mathematicians in the nineteenth century. In the absence of a suitable arithmetical theory of incommensurables, Euclid's technique was the best available in geometry. Descartes believed there could be a solution to the incommensurables problem when he created coordinate geometry, once more elevating arithmetic to the top. However, at the time, no such answer had been discovered. Geometry has had a significant impact on both philosophy and the scientific method. The Greeks developed geometry, which begins with axioms that are (or are thought to be) self-evident and moves through deductive reasoning to theorems that are far from self-evident. It is believed that the axioms and theorems hold true for actual space, which is a given in experience. Thus, it seemed plausible to make inferences about the real world by first observing what is obvious and then applying that knowledge. This kind of view affected most of the middle-class thinkers as well as Plato and Kant. The Declaration of Independence takes its cue from Euclid when it states, "We hold these truths to be self-evident." Seeking political Euclidean principles, the theory of natural rights emerged in the seventeenth century. * Despite its information that is undoubtedly empirical, Euclid completely dominates the form of Newton's Principia. Theology borrows its style from the same source in its same scholastic forms. Theology comes from mathematics, personal religion from ecstasy, and Pythagoras has both. I think that the main basis of the belief in an accurate and everlasting truth, as well as in a reality that is supersensible and comprehensible, is mathematics. No tangible object is perfectly round, despite the fact that geometry deals with accurate circles; faults and irregularities will always exist. It is natural to go farther and contend that mind is nobler than sense and that the things of thought are more real than those of sense-perception. This implies the perspective that all perfect reasoning applies to ideal rather than sensual objects. Pure mathematics also supports mystic notions regarding the relationship between eternity and time since, if they are real at all, mathematical things like numbers are eternal and not temporally oriented. One can imagine such everlasting things to be God's thoughts. That is why Sir James Jeans thinks God is addicted to math and Plato believes that God is a geometer. Since Pythagoras and especially since Plato, mathematics and mathematical technique have dominated rationalistic religion as opposed to apocalyptic religion. Beginning with Pythagoras, the integration of religion and mathematics shaped religious thought in Greece, the Middle Ages, and modern times, all the way to Kant. Prior to Pythagoras, orphism was comparable to Asian mystery religions. But the close blending of religion and reason, of moral aspiration with logical admiration of what is timeless, which derives from Pythagoras, separates the more straightforward mysticism of Asia from the more intellectualized theology of Europe in the works of Plato, Saint Augustine, Thomas Aquinas, Descartes, Spinoza, and Kant. It has only been feasible to pinpoint the precise areas in which Pythagoras was incorrect in recent years. There has never been a guy in the history of thinking as significant as he was. I say this because analysis reveals that what at first glance appears to be Platonism is really Pythagoreanism. He is the source of the entire idea of an everlasting universe that is disclosed to the mind but not the senses. However, he would not have believed that Christ was the Word, nor would theologians have looked for rational evidence of God's existence and immortality. However, all of this is still implied in him. You'll see how it got clear.