Pythagoras,ancient GreekMathematicianphilosopher。Pythagoras was born inaegean seainSamoa(Today's small island in the east of Greece), a noble family, who was smart and eager to learn from childhood, studied under a famous teachergeometry, natural science and philosophy.
Pythagoras is a little more advanced than some scholars who started teaching in the same era;Because he allowed women (of course, aristocratic women rather than slave maidservants) to attend classes.He believes that women have the same right to seek knowledge as men, so there are more than ten women scholars in his school.This is a phenomenon not found in other schools.
Pythagoras was the first to propose“Heliocentric theory”One of the people.[8]
The first person who pays attention to "number" Pythagoras theorem（Pythagorean theorem） It is proved that the number of regular polyhedrons Many influential associations have been established Pythagorean schoolopen
Key achievements
The first person who pays attention to "number" Pythagoras theorem（Pythagorean theorem） It is proved that the number of regular polyhedrons Many influential associations have been established Pythagorean school findgolden section(Golden ratio)
In 580 BC, Pythagoras was born inmiletus NearbySamoa(TodayGreeceEast Island)——Ionian IslandsAt that time, the islands were in their heyday, and were far ahead of the city states in Greece in terms of economy, culture and other aspects.
Pythagoras' father was a rich businessman who was sent by his father when he was nine years oldTyr , onSemiticSyriaScholars study there, where he came into contact with the Eastern religion and culture.Later, he went on business trips with his father many timesAsia Minor。
In 551 BC, Pythagoras camemiletus , Delos and other places, visited mathematicians and astronomersThales、AnaximanderAnd Phil Kurdes, and became their students.Before that, he had beenSamosThe poet Clefelos studied poetry and music there.
In 550 BC, 30 year old Pythagoras caused local people's aversion because of his propaganda of rational theology, wearing oriental clothes and wearing hair. Since then, Samos have always had prejudice against Pythagoras, believing that he was innovative and advocated heresy.Pythagoras was forced to leave his home in 535 BCEgyptHe was in Phoenicia on the waycoastal cityStop to learn local mythology and religion, and meditate in the Temple of Tyre.
After arriving in Egypt, King Amasis recommended him to study in the temple.During the ten years from 535 BC to 525 BC, Pythagoras learnedHieroglyphicsandEgyptian mythology, history and religion, and publicityGreek philosophyHe was respected by many Greeks, and many people went to his school.
At the age of 49, Pythagoras returned to his hometown of Samoa and began to give lectures and open schools, but the results did not meet his expectations.
Around 520 BC, in order to get rid of the tyranny of the monarch at that time, he left Samoa with his mother and the only disciple and moved tosicily, later settled in Crotone.There he recruited many disciples and established a religious, political and academic group.
His speech attracted many people from all walks of lifeUpper class societyTo attend the lecture.According to the custom at that time, women were forbidden to attend public meetings. Pythagoras broke thisConventionalAnd allow them to listen.Among the enthusiastic listeners was his later wife, Sienna, who was young and beautiful. She once wrote a biography of him, but unfortunately it was lost.
Statue of Pythagoras
Pythagoras founded aSecret associationThere are both men and women in this society, all of them have equal status, and all of their property belongs to the public.The organization discipline of the community is very strict, even with a strong religious color.Each student should reach a certain academic level, and go through a series of mysterious rituals to join the organization in order to achieve "spiritual purification".
They should receive long-term training and assessment, abide by many norms and precepts, and swear never to reveal the secrets and doctrines of the school.They believe that relying on mathematics can sublimate the soul. Through mathematics, they can pry into God's thought. All things contain numbers, even all things are numbers. Numbers are the truth behind the ever-changing world.This isPythagorean schoolThe main difference from other sects.
The members of the school have common philosophical beliefs and political ideals. They eat simple food and carry out strict training.The teachings of the school encourage people to control themselvesAbstinencePure and obedient.They started inGreat Greece(Today's southern part of Italy) has won a high reputation, had a considerable impact, and therefore aroused the envy of the enemy.Later, they were impacted by the democratic movement, and the community's venue in Crotone was severely damaged.Pythagoras was forced to move to Tallington (todayItalysouthTaranto）, and died in 500 BC at the age of 80.
Many of the disciples fled back to Greece and reestablished their stronghold in Phrios. Others went to Taranto to continuePhilosophy of mathematicsResearch, as well as political activities, until the middle of the 4th century BC.Pythagorean schoolIt has been prosperous for two centuries.
It is said that he is a very excellent teacher. He thinks that everyone should know something about geometry.Once he saw a diligent poor man who wanted to teach him geometry, so he suggested to him that if he could learn a theorem, he should give him three silver coins.The man learned geometry from him for the sake of money, but after a period of time, the student became very interested in geometry and asked Pythagoras to teach it faster. He suggested that if the teacher taught more theorems, he would give a coin.It didn't take much time. Pythagoras took back all the money he had given to the student.
Key achievements
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Research on Number Theory
Under the guidance of the mathematical philosophy that number is the origin of all things, Pythagoras devoted himself to mathematics and gave many special meanings to number. Pythagoras believed that the hexahedron contains the mystery of the earth, the pyramid contains the meaning of fire, and the dodecahedron contains the mystery of the universe.It also divides numbers into intelligent, stupid, contented and unsatisfied, and gives the concepts of average number, affinity number and perfect number.[2]
At the same time, Pythagoras also divided natural numbers into odd, even, triangular and pentagonal numbers.Pythagoras believes that these different abstract numbers are the foundation of all things in the universe. The numbers in different situations form or constitute different objects in the universe according to the number or the different proportional relationship between them, which forms the unity of many and one objects in the universe.[2]
Pythagoras also abstracted numbers from concrete things.He separated the number of "three" from three tables, three trees, three chairs, three people, three elephants and other specific things, which seemed very common, but for later mathematical philosophy, its significance was very great, making people's vision of mathematical research from name to constant,It makes people study the origin of everything from concrete things to numbers that are separated from concrete things, which has played an important role in promoting the development of mathematics and philosophy in later generations.[2]
Pythagoras theorem (Pythagorean theorem)
It is said that there is such a record in the history of the discovery of Pythagoras' theorem that Pythagoras visited a friend's house and everyone was bored waiting for dinner. Pythagoras used this time to carefully observe the beautiful square porcelain floor tiles of his friend's house. Driven by Pythagoras' thought of "all things are counted", he must have thought of what connection they have with "number".So he took out a pen and squatted on the ground, drawing a square with the diagonal of one of the tiles as the edge, and found that the area of the square was exactly equal to the sum of the areas of the two tiles.[2]
Although many people have explored it before Pythagoras, Pythagoras was the first person to generalize the Pythagorean theorem (from the isosceles right triangle to the general right triangle).The Pythagorean theorem (Pythagorean theorem), which reflects the trilateral relationship of right triangles, is quantified, abstracted and logically proved strictly.[2]
golden section
golden section
Pythagoras, under the guidance of the mathematical philosophy of "all things are equal", carefully studied the structure of numbers inside things and the proportional relationship of numbers, and discovered the golden section in the 6th century BC, which some people call the "sacred section".
In essence, the golden section divides a line segment into two parts of different length. The ratio of the longer part to the shorter part is equal to the ratio of the whole part to the longer part, and the ratio is 1:0.618 or 1.618:1. From this, the golden figure of 0.618 or 1.618 is derived.[2]
astronomy
Pythagoras introduced mathematics into astronomy.They have built a complete universe system.Pythagoras believed that ten was a perfect number."People study the activity and essence of 'number' according to the ability existing in 'ten', because its' ten 'is great, perfect and omnipotent." There are ten celestial bodies in circular motion.Or the center of these ten celestial bodies, the earth is not the center.Copernicus acknowledged that Pythagoras' theory of the universe was the pioneer of his hypothesis.[5]
music
Pythagoras was not only a famous philosopher in ancient Greece, but also the first music theorist in the history of western music.Pythagoras believed that the origin of all things is "number", and all things can be deduced by mathematical principles.In music, he linked "number" with music and proposed that "number" is the root of music, that is, "music is number".This point of view is to think and study music from the perspective of "mathematics", so that music has a rational consciousness, thus broadening the field of vision of music.Pythagoras permeated the precise concept of "mathematical" logic in music, linked music and science, and exceeded the perceptual aesthetic experience of music.Since Bidagras put forward the idea of "mathematics and music", the study of "mathematics and music" has never stopped in the history of western music.[4]
Character ideology
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Idealism
Pythagoras is an idealist philosopher. He uses numbers to explain human nature, social morality and everything. He believes that the whole universe is generated by numbers."Ten" is the most harmonious.He believed that all things in the whole universe were established orderly according to the harmonious relationship of numbers.Human beings are a big universe, and everyone is a small universe and should be harmonious.[3]
Pythagoras
Pythagoras believed that it was the harmony of various opposite beauty systems embodied by odd and even numbers.He summed up these antagonistic relationships into ten basic relationships, namely odd and even, some many, finite and infinite, left and right, yang and yin, movement and static, straight and curved, light and dark, good and evil, square and long.Among these ten antagonistic relationships, the most fundamental is one and many.The number starts from the beginning. The essence and foundation of all things is one. Like lines and planes, the body starts from the point. The opposition between one gui and two is the opposition between odd numbers and even numbers. Two can be divided into many. Therefore, the opposition between one and two and the number after it is the opposition between one and many, finite and infinite.The harmony of all these opposites constitutes the order of all things.
They sometimes say that two is opinion, four is justice, five is marriage, seven is death, and eight is friendship and love.He also said that the holy numbers were four and ten.Because there are three, two and four in four, 4+3+2+1=10.Of course, this farfetched comparison is childish and ridiculous, which is simply "the little nine to grind women". However, in their various absurd explanations, they also implement their basic orientation, including a kind of morality. The core of this morality is to emphasize "harmony" and "order".They believe that "the whole sky is a harmony, a number".
Morality comes from the soul. Man is a harmonious whole composed of soul and body. Human nature, morality and happiness all depend on the soul.The soul is the number that drives itself, and the soul is divided into divinity. From this, it can be concluded that the soul does not die, and only people have a soul. After the death of the body, the soul will be implanted in animals and plants. The soul does not die. Knowing the four incarnations 2700 years ago, the soul attributes human nature and moral roots to mysterious numbers, which is idealistic and negates the relationship between human nature and morality and the outside world,In essence, it denies the necessity for people to pursue realistic material interests, represents the interests of slave owners and nobles, and negates the necessity and rationality of the new slave owners' democrats and civilians fighting for realistic interests.[3]
Pythagoras did not study numbers for practical application, but to explore the eternal truth of the universe by revealing the mysteries of numbers.Believing in all things selling numbers, recognizing the mathematical nature of music, and using the music of celestial bodies to promote this concept to the digital harmony of all things in the universe.[3]
Aesthetic outlook
The Pythagoras School, from the perspective of the harmony of music and the relationship between man and art, believes that people can appreciate art and beauty because of the "simultaneous response" between man's internal harmony and external harmony, and then emphasizes the "purification" role of music in aesthetic education.Pythagoras and his school put forward the idea that music purifies the soul, which can be said to be the earliest aesthetic education thought in the West.Pythagoras School believes that number is the origin, foundation and principle of all things: "All other things, in terms of their entire nature, take number as an example. Pythagoras not only applies the principle of number to natural things, the movement of the universe (the order of the universe is the harmony of numbers), but also to social things and human activities.They associate the principle of number with beauty, emphasizing that "order and symmetry are both beautiful and useful, while disorder and asymmetry are ugly and useless". "The beauty of body really lies in the proportional symmetry between parts." Pythagoras School not only links beauty with number, proportion and harmony, but also introduces harmony and order into the fields of life and society,It is related to the "purification" of human soul and the acquisition of eternal life.Therefore, he not only sought the invariable order and principle for the universe and nature, but also took the "purification" and "harmony" of the soul and the pursuit of an idealized and eternal order construction as the theme of philosophy.It was from then on that the rational thinking of Greek philosophy on life and society had a truly profound humanistic character and aesthetic character."The most powerful part of people is the soul. The soul is both good and evil. People are happy when they have a good soul. They never stop. Their life is a change of the house.".4） The purification of the soul, the pursuit of life ideals, and the pursuit of truth are not only the pursuit of wisdom, but also the way of life.[1]
They expanded the proportional relationship of beauty, put forward the concept of two universes, regarded man as a small universe and the world as a big universe. They believed that the human body, like the universe and celestial bodies, was governed by the principle of harmony of numbers. The internal harmony of man was affected by the external harmony. Once the internal harmony of man and the external harmony "corresponded", man entered the state of art appreciation,So as to purify people's mind.[1]
virtue
Pythagoras believed that virtue is a kind of harmony, just like health and God.Friendship is also a kind of harmonious equality.Therefore, in Jin Yan, he proposed to respect God and parents, to choose friends with moral standards, to be modest and patient with friends: to be fair in words and deeds, and to yield modestly even when people say bad things.These thoughts of his seem to be moral advice put forward in a narrow life circle, and are only self serving experiences and purposes.However, he tried to extract a universal moral requirement from the range of groups and social life he experienced, which is still positive for adding a moral rule to actual life.Pythagoras, as the "first scholar who tried to speak morally", his "Golden Words" should be said to be the embryonic form of the ancient Greek moral code system.[3]
Educational outlook
The philosophical and scientific achievements of Pythagoras School form the theoretical basis of its educational thought.The most important part of their philosophy is the theory about people, human soul and the relationship between soul and everything in the universe.Pythagoras School believes that the soul is immortal and reincarnated.They especially emphasized that the purification and education of the soul is an important way to achieve the purification of the soul.Zeller believes that "the spirit, principles and behavior of the Pythagorean group are rooted in the theory of transmigration." On the moral issues of life, the Pythagorean school advocates "harmony" and "order".They believe that the arrangement of numbers is harmonious and orderly, so society should also have a harmonious order.According to the law of equality of geometric proportions, the rights of individuals should correspond to their contributions, so everyone has a designated position in the country.The state should rely on the law to regulate and maintain this order and should communicate among all levels through love.They think virtue is a kind of harmony.People should respect God and parents, be humble to friends, and be patient and fair to others.In fact, this theory is realized by the function of education guidance and restriction.Therefore, the philosophy of Pythagoras School requires education to ensure its realization and education obtains development space by virtue of its philosophy.[5]
Soul transmigration
Pythagoras accepted the religious superstition of the immortal soul of the Egyptian priests and believed that the human soul was divided into three parts, namely, appearance, spirit and vitality.Animals have only appearance and vitality without soul, while human beings have three.The position of the soul is from the heart to the brain of the human body.The part in the heart is anger, and the part in the brain is appearance and mind, namely reason.He believes that the rational part of the soul is immortal.Therefore, he advocated a kind of soul transmigration theory, advocating that the soul should be transferred from one organism to another according to the destiny after death, as Engels pointed out: "In the Pythagorean school, the soul is already immortal and immovable, and the body is purely accidental for it."It is self-evident that, according to the destiny, the soul of a person can also be transformed into the soul of a dog after death. It is said that one day when Pythagoras saw someone beating a dog, he pitifully said to that person," Don't hit it, because I heard its voice. It was the spirit of a friend who attached to it. Pythagoras was the first to introduce the theory of reincarnation of this soul into Greece.This made him play a very bad role in the history of the development of philosophy and ethics in ancient Greece, and this role was particularly highlighted in Plato's philosophy and ethics.[3]
Personage school
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sketch
Pythagorean schoolAlso known as the "Southern Italian School", it is an organization integrating politics, academia and religionAncient Greek philosopherFounded by Pythagoras.The school came into being at the end of the 6th century BC and was forced to dissolve in the 5th century BC. Most of its members were mathematicians, astronomers and musicians.It is the first school in the history of western aesthetics to explore the essence of beauty.
Pythagoras once lived in Egypt, and later traveled around. During his travels, he was influenced by the local culture and learned many mysterious thingsreligious ritesThey are also familiar with the relationship between them and the knowledge of numbers and geometric rules.After the trip, he returned to his hometown of Samos.For political reasons.He later moved to Crone, a Greek port in southern Italy.A group studying philosophy, mathematics and natural science was founded here, and later developed into a religious school organization with secret rituals and strict disciplines.
The Bishop School believes that meditation on the relationship between geometric forms and numbers can achieve spiritual liberation, while music is regarded as a means of purifying the soul to achieve liberation.
Pythagorean schoolThe contemporary research theme ofAnthropologistZhang Ronghuan returns politics, academia and religion to the first proposition of human destiny, that is, the rise of human personality and its ecologystaySocialLevel and self level.
viewpoint
1. Human happiness can only be the product of personality society;It is a new personality and the result of new ecological harmony.
2. Happiness is not the abstinence experience of sectarian theology, nor the hedonic feeling of Confucian ethics, nor the unlimited desire for money and status, butbeliefAnd the personality satisfaction of yearning for realization.
3. AttentionHuman value, requiring improvementThinking abilityAnd creative potential to encourage positiveLife attitude, advocating the spirit of active exploration;
4. Everyone can believe in the truth.
5. Reflect the high wisdom of people in modern society with sincere democracysocial existence。
6. Improve people's intelligence ability in public affairs of the groupthe United NationsIn the system, let the relationship between people develop harmoniously.
7. ReformationThe RenaissanceAnd the ultimate goal of legal revival is the rectification of people.
13. When the pot is taken off the fire, do not leave the mark of the pot on the ash, but wipe it off.
14. Don't look in the mirror beside the light.
15. When you take off your pajamas, roll them up and smooth the marks on your body.
All these commandments belong to primitive taboo concepts.
Irrational number
Legendary Pythagorean studentsHippersos(Hippasus) foundIrrational number。When he studied a square with one side length, he found that its diagonal could not be expressed by the ratio of integers.This discovery broke the creed of Pythagoras and his followers.It is said that Hippasus was thrown into the sea and killed by the school because he publicized the incident.However, the discovery of irrational numbers has made an important contribution to the development of mathematics[6]。
Character evaluation
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Comfort(From Religion to Philosophy) said that, in his view, "Pythagoras represents the mainstream of the mysterious tradition that we think is opposite to the scientific trend." He believed thatParmenides——He called it "the discoverer of logic" - "a branch of Pythagoras, andPlatoI got his main source of inspiration from Italian philosophy.
He saidPythagorasismyesOrpheusAn internal reform movement, and Orpheus isDionysusThe reform movement in worship.The opposition between rational things and mysterious things runs through the whole history. It was initially shown among the Greeks as the opposition between the Olympic gods and other relatively uncivilized gods, the latter being closer to what anthropologists have studiedPrimitive belief。In this division, Pythagoras standsMysticismAlthough his mysticism has a special intellectual nature.He thinks he has a semi divine nature, and seems to have said, "There are people, gods, and creatures like Pythagoras.",The various systems inspired by him "tend to be born, place all values in the invisible unity of God, and reject the visible world as illusory, saying that it is a turbid medium, in which the light of heaven is damaged and obscured in fog and darkness".
Bernett put thisMoralityThe summary is as follows: "We are all foreigners in this world, and the body is the tomb of the soul, but we can never commit suicide to escape; because we are the property of God, and God is our shepherd, without his command, we have no right to escape. There are three kinds of people in this life, just like toOlympic GamesThere are three kinds of people coming up.Those who come to do business belong to the lowest class, and those who come to compete are higher than them.However, the highest one is those who only come to watch.Therefore, the greatest purification of all is the science of doing nothing. Only those who devote themselves to this cause, that is, true philosophers, can really free themselves from the 'wheel of life'. "
Changes in the meaning of words are often very instructive.I mentioned the word "orgy" above;Now I want to talk about the word "theory".This word was originally a word of the Orpheus sect, and Conford explained it as "passionate and moving meditation".He said that in this state, "the observer and the suffering God become one, die in his death, and rise again in his new life";For Pythagoras, this "passionate and moving meditation" is rational, and the result is mathematical knowledge.
Thus, through Pythagorean doctrine, "theory" gradually gained its modern significance;However, for all the people inspired by Pythagoras, it has always kept an element of revelations of ecstasy.This seems strange to those who have learned some mathematics helplessly in school;However, for those who always experience being intoxicated by the sudden mastery of mathematics, and for those who love mathematics, Pythagoras's view seems very natural, even if it is not true.It seems that the experienced philosopher is only the slave of materials, while the pure mathematician, just like the musician, is the free creator of his orderly and beautiful world.
The most interesting thing is that we can see the concept contrary to modern values from the Pythagorean ethics described by Bernett.For example, in a football match, people with modern minds always think that football players are much greater than the audience.As for the country, the situation is similar: they treat politicians (politicians are in competitioncompetitor）We worship more than those who are just onlookers.The change of this value andsocial systemRelated to the change of soldiers, gentlemen, chaeboldictatorEach has its own standards of goodness and truth.A gentleman once had a long period of power in philosophical theory, because he was combined with Greek genius, because the virtue of meditation was guaranteed by theology, and because the ideal of the truth of doing nothing dignified the life of the college.A gentleman can be defined as a member of the society of equal people. They live by slave labor, or at least by those who undoubtedly have low statusworking peopleAnd live.It should be noted that this definition also includes saints and sages, because as far as their lives are concerned, they are also contemplative rather than active.
Modern definitions of truth, such aspragmatismAnd ofInstrumentalismIts definition of truth is practical rather than contemplative, because it is opposed to the aristocratic regimeindustrial civilization Aroused.
No matter what people think of toleranceSlaveryWhat is the idea of the existing social system, but it is from the gentleman in the above sense that we have pure mathematics.The ideal of meditation can lead people to create pure mathematics, so it is the source of a useful activity;This has increased its prestige and enabled it to achieve a success in theology, ethics and philosophy that it would not otherwise enjoy.
We have explained a lot about Pythagoras as a religious prophet and as a pure mathematician.In these two aspects, he had an inestimable influence, and these two aspects were not separated as modern people imagined at that time.
Most of the sciences are connected with some wrong forms of belief from the very beginning, which makes them have an illusory value.Astronomy andAstrologyThey are linked together, chemistry and alchemy are linked together.Mathematics combines a more sophisticated type of error.Mathematical knowledge seems to be reliable, accurate, and can be applied to the real world.In addition, it is obtained from pure thinking and does not require observation.Therefore, people think that it provides an ideal that the knowledge of daily experience cannot do.According to mathematics, people assume that thought is higher than sense and intuition is higher than observation.If the sensory world is not consistent with mathematics, then the sensory world is even worse.People seek methods that are closer to the ideal of mathematicians in various ways, and the revelations from the results becomemetaphysicsAndEpistemologyThe source of many mistakes in.This form of philosophy also started from Pythagoras.
As we all know, Pythagoras said, "All things are numbers.".This assertion, if interpreted in a modern way, is logically meaningless, but what Pythagoras meant is not completely meaningless.
He discovered the importance of numbers in music,Mathematical nounIn“Harmonic middle term”And“Harmonic series”The connection between music and mathematics established by Pythagoras is still preserved.He imagines numbers as shapes that appear on dice or cards.We still talk about the square and cube of numbers, and these nouns come from him.
He also mentioned the number of rectanglestriangleNumber, pyramid number, etc.These are the necessary number of small blocks (or we should say more natural number of small balls) to form the above shapes.He imagined the world as atoms and objects as molecules formed by atoms arranged in various forms.In this way, he hopes to make mathematics the fundamental research object of physics and aesthetics.
The greatest discovery of Pythagoras, or hisA qualified discipleThe greatest discovery ofright triangleProposition of;That is, two right anglesHemmingThe sum of the squares of is equal to the square of the other side, that is, the square of the chord.Egyptians have known that if the side length of a triangle is 3, 4, or 5, there must be a right angle.But apparently the Greeks were the first to observe that 3 ²+4 ²=5 ², and found the proof of this general proposition according to this hint.Unfortunately, Pythagoras' theorem immediately became impossibleCommon divisor(Irrational number）This seems to deny all his philosophy.In an equilateral right triangle, the square of the chord is twice the square of each side.Let's assume that each side is one time long. How long should the string be?Let's assume its length is m/n.Then m ²/n ²=2.
If m and n have a common divisor, we can eliminate it, so one of m and n must be odd.Now m ²=2n ², so if n is even, then m is also even;So n is an odd number.Assume m=2p.So 4p ²=2n ², so n ²=2p ², so n is even, contrary to the assumption.So there is no fraction of m/n that can approximate the end chord.The above proof is actuallyEuclidProof in Part 10 ①.
This argument proves that no matter what kind ofLength unitThere will always be some lengths that cannot have an exact numerical relationship with that unit;In other words, there cannot be two integers m and n, so that the length of m times in the problem is equal to the unit of n times.This made Greek mathematicians firmly believe that,geometryThe establishment of must be independent andArithmeticsirrelevant.
Dialogues of Plato There are several sections in it that can prove that someone had dealt with geometry independently at that time;Geometry was completed in Euclid.In Part II, Euclid geometrically proved many things that we would naturally use algebra to prove, such as (a+b) ²=a ²+b ²+2ab.It was precisely because of the difficulty of incommensurability that he thought this approach was necessary.This is also the case when he discusses proportion in Parts V and VI.The whole system is logically striking, and already indicates the preciseness of mathematicians in the 19th century.As long as there is no proper computational theory about incommensurable numbers, Euclid's method is the best possible method in geometry.WhenDescartesIntroducedCoordinate geometryWhen learning (analytic geometry) again established the supremacy of mathematics, he imagined that there was a possibility to solve problems that could not be common divisors, although he had not found such a solution at that time.
Geometry for philosophy andscientific methodIts influence has been far-reaching.The geometry established by the Greeks started from the axioms that are self - evident, or are considered self - evident, and proceeded according to deductive reasoning to reach the theorems that are far from self - evident.Axioms and theorems are believed to be true for real space, which is everything in experience.In this way, first pay attention to what is self-evident and then apply itDeductive method, as if it is possible to discover everything in the real world.This view has influencedPlatoandKant And most of the philosophers between them.The "Declaration of Independence", "We believe that these truths are self-evident", was born of Euclid itself.eighteenth centuryNatural human rightsHis theory is a kind of axiom that pursues Euclidean style in politics.NewtonAlthough its material is generally recognized as empirical, its form is completely dominated by Euclid.The strict scholastic form of theology derives from the same source.Individual religion derives from the understanding of heaven and man, while theology derives from mathematics;Both can be found in Pythagoras.
I believe that mathematics is the main source of our belief in eternal and strict truth, and also the main source of our belief in a super sensible and knowable world.Geometry discusses strict circles, but no perceptible object is strictly circular;No matter how carefully we use ourcompassesThere will always be some incompleteness and irregularity.This suggests that all strict reasoning can only be applied to the ideal object opposite to the perceptible object;Naturally, it can be further demonstrated that thought is more noble than sense and the object of thought is more real than the object of sense perception.The theory of mysticism about the relationship between time and eternity is alsoPure mathematicsConsolidated;Because mathematical objects, such as numbers, if they are real, must be eternal and not in time.This eternal object can be imagined as the thought of God.Therefore, Plato's theory is: God is a geometer;Sir James Jens also believed that God was fond of mathematics.The religion of rationalism, as opposed to the religion of revelation, has been completely integrated by mathematics andMathematical methodDominated.
The combination of mathematics and theology began in Pythagoras, which represents the Greek, medieval and modernReligious philosophyCharacteristics of.The doctrine of Orpheus before Pythagoras was similar toAsiaThe mysticism of.But inPlato、Saint Augustine、thomas aquinas 、Descartes 、SpinozaandKant There is a close interweaving of religion and reasoning, a close interweaving of moral pursuit and worship of the logic of things without time;This is from Pythagoras and distinguishes Europe's intellectualized theology from Asia's more straightforward mysticism.Only in the recent period can people clearly tell where Pythagoras was wrong.I don't know what other people thinkthe thought circleHe had such a great influence.I say this because of the so-calledPlatonismIf we analyze it, we can find that it is only Pythagorean in essence.There is an eternal world that can only be shown in reason but not in the senses. All this idea is derived from Pythagoras.If it were not for him, Christians would not think that Christ is the Tao;If it were not for him, theologians would not pursue God's existence andImmortality of soulLogical proof of.But in his body, all this is not obvious.
Author: YingBertrand Russell Source:《History of Western Philosophy》Pythagoras' influence on ancient and modern times is the theme of my chapter;Pythagoras is one of the most important figures in thought since he was born, whether in terms of his cleverness or not.
Mathematics, in the sense of deductive inference of proof, starts from him;Moreover, mathematics is closely combined with a special form of mysticism in his thought.Since then, and partly because of him, the influence of mathematics on philosophy has been profound and unfortunate.
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Let's start with a few known facts about his life.He isSamoThe people of the island flourished in about 523 BC.Some people say that he is the son of a wealthy citizen called Munisalke, while others say that he isApolloSon of God;I invite readers to choose one of the two statements.In his time, Samo was ruled by the tyrant Polucardi, an old rascal who made a lot of money and had a huge navy.
Samo is a commercial competitor of Milidou;Its merchants have traveled as far as Talesus, Spain, which is famous for its minerals.Polucardi became the tyrant of Samo in about 535 BC and ruled until 515 BC.He is unconcerned with moral censure;He threw out his two brothers. They were working with himTyrant politicsHis navy was mostly used for plundering at sea.Not so long ago, Millie surrenderedPersiaThis matter is very beneficial to him.
In order to prevent the Persians from expanding westward, he allied himself with King Amasis of Egypt.But when the Persian king, like the Sith, concentrated on conquering Egypt, Polucardi realized that he would win, so he changed his position.He sent a fleet of his political enemies to attack Egypt;But the sailors mutinied and returned to Samo Island to attack him.Although he defeated them, he finally fell into a conspiracy to take advantage of his greed.The Persian governor in Sardis pretended to betray the Persian king, and was willing to pay a large sum of money in return for the assistance of Polucardi to him;When he went to the mainland to meet with the Persian governor, he was captured and crucified.
Pollucrates was a patron of art and had beautified Samo with many great buildings.Anacreon was his court poet.However, Pythagoras did not like his government, so he left Samo Island.It is said - and not impossible - that Pythagoras had been to Egypt, where he learned most of his wisdom;Whatever the circumstances, it is certain that he finally settled in Croton in southern Italy.The Greek cities in southern Italy, like Samo and Mili, are rich and prosperous;In addition, they are not threatened by the Persians.The two largest cities are West Barres and Croton.The luxury of Sibaris is still popular today;according toTheodorusIt is said that its population reached as many as 300000 in its heyday, although this is undoubtedly an exaggeration.Croton is roughly the same size as West Barres.Both cities make a living by importing goods from Ionia to Italy, some of which are used as consumer goods in Italy, and some of which are re exported from the western coast to ItalyGaulAnd Spain.
Many Greek cities in Italy fought fiercely against each other;When Pythagoras arrived at Croton, Croton had just been defeated by Laurie.However, shortly after Pythagoras arrived, Croton won the war against Sibaris, and Sibaris was completely destroyed (510 BC).Sibaris andmiletus There has always been a close connection in business.Croton is famous for medicine;There was a man in Croton, Democydides, who was once the imperial physician of Polycrates and laterDariusThe imperial physician of.
Pythagoras and his disciples established a group in Croton, which was very influential in the city for a time.But in the end, the citizens opposed him, so he moved to Meda Pontion (also in southern Italy) and died here.Soon he became a mythical figure, endowed with various miracles and supernatural powers, but he was also a founder of the school of mathematicians ②.In this way, there are two opposite legends arguing about his deeds, and the truth is difficult to understand.
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Pythagoras is one of the most interesting and difficult characters in history.Not only are his legends almost a mixture of inextricable truth and absurdity, but even in the simplest and least controversial form of these legends, they also provide us with the most peculiar psychology.
To put it simply, he can be described as a combination of Einstein and Lady Eddie.He established a religion whose main doctrines were the transmigration of the soul and the sin of eating beans.His religion is embodied as a religious group, which has made great contributions to the nationalcontrol powerAnd establish a set of sage's rule.But people who have not been reformed are eager to eat beans, so sooner or later they rebel.
