Pre Socratic Philosophers

1. Pre Socratic Philosophers Presented at Central University of Finance and Economics 中央财经大学 Beijing by 卜若柏 Robert Blohm Chinese Economics and Management Academy 中国经济与管理研究院 http://www.blohm.cnc.net March 16 & 21, 2008 2008年3月16日和21日

2. What is philosophy? • Clarification of very basic concepts used for knowledge and actions • Basic rules of thought, of procedure for knowledge, or of action • According to Russell: it is located between science and dogma (belief, faith)

3. History of Philosophy • Begins in the 6th century BC both in Greece and in China. Declines in the West with the Roman Empire (end of 5th Century) and in China with the Han Dynasty (3rd century) • Second Great period is the later Middle Ages (11th to 14th centuries) under the Catholic Church in Europe, and Neo-Confucianism in China (10th to the 13th century) under the Sung Dynasty • The interim period was dominated by the rise of Christianity on the ruins of the Roman Empire (6th to 10 century) and the rise of Buddhism during the Tang Dynasty (7th to 9th century).

4. History of Philosophy (cont.d) • 15th & 16th centuries was the interim period of the Protestant Revolt and the Renaissance in the Europe. In China it was the age of the Universal Mind school of Neo-Confucianism under the Ming Dynasty which was exposed to European developments through Jesuits like Matteo Ricci.. • The third great period is the modern age of science and economy from the 17th century. During most of that time China fell under non-Chinese Manchu rule which brought great prosperity but little philosophical development.

5. Two Opposite Greek City States • Sparta (斯巴达). • Citizen’s duty to the City. • Gave rise eventually to Roman administration, Roman armies, Roman law.

6. Two Opposite Greek City States(cont.d) • Athens (雅典). • Individual freedom. • Subsequent occupation by Alexander and the Romans gave rise to the non-political Stoicism of communion of the individual soul with God, that ultimately was manifested in Christianity.

7. Two Opposite Greek City States(cont.d) • Athens (cont.d) • Religious allegiance ultimately become important in Northern Europe under various barbarian conquerors, while the Roman Emperors eventually became Christian. This evolved into conflict between Church and King, between a unified Mediterranean Europe and the fragmented Northern Europe of decentralized principalities. The Church dominated through education and through maintenance of order desired by the mercantile class, until the Reformation destroyed the unity of the Church and the Renaissance destroyed the unity of the universe and science. Individual interpretation of the Bible, individual sense perception, inward meditation gave rise to the modern philosophy of Descartes for whom self-awareness was the fundamental knowledge from which all other knowledge could be derived. That introduced a danger of subjectivism and naturalism, countered by the liberalism of Locke, and the absolutism of Hobbes, Rousseau and Hegel.

8. Two Opposite Greek City States(cont.d) • Sparta and Athens represent two opposite dangers: • too much discipline and reverence of tradition • dissolution and subjection to foreign conquest. Important civilizations start with a rigid and superstiutious system, gradually relaxed, and leading to a period of brilliant genius.

9. Presocratic (第一篇 前苏格拉底)Philosophers

10. Egypt and Babylon埃及和巴比伦 • Egypt and Mesopotamia (美索不达米亚) were river and land-based agricultural societies. They had developed arithmetic and some geometry as rules of thumb. Babylonians discovered a cycle in eclipses. Greeks developed deductive reasoning from general premises as a way to derive, discover, and organize arithmetic and geometry.

11. Egypt and Babylon (cont.d) • Egyptian and Babylonian religion were fertility cults. Earth female and sun male. • Egyptian religion is death-oriented: soul ultimately rejoins the body. • Babylon was warlike and its religion was oriented to prosperity in this world rather than happiness in the next. Calvinism (加尔文派教义). Human laws come from God.

12. Crete (克里特) • Minoan (米诺) culture. 2500 - 1400 BC. Commerce, especially with Egypt. • Cheerfulness versus terrifying gloom of Egyptian temples. • Like Egyptian religion: reward or retribution after death. • Spread to Greek mainland as Mycenaean (迈锡尼) civilization 1600-900 BC, as Greek civilization had spread to Roman. Mycenaeans settled Greek islands and Sicily (西西里).

13. Greece • Geography: fertile coastal valleys separated by mountains. • Founded colonies in Asia Minor (小亚细亚) & Sicily. • Government. From monarchy, to aristocracy, to non-hereditary leader (“tyrant”, usually gold or silver mine owners after coinage invented in 700 BC), to democracy. • Greek religion • Fate played a big role: basis for natural law. • Gods were the gods of a conquering aristocracy, not the fertility gods of farmers.

14. Greece (cont.d) • Greek religion (cont.d) • Dionysius 狄奥尼索斯 (Bacchus 巴库斯), god of wine and drunkenness. • Originally the fertility god of Thracean 色雷斯 (northern) farmers. • Surrounding mysticism influenced Greek philosophers and later Christianity • Forethought, enduring pain for future pleasure, is characteristic of agriculture, versus hunting which is impulsive. Develops into laws and customs enforcing the purposes of the community on the individual, sacrificing the present for the future. Olympic gods versus less civilized gods.

15. Greece (cont.d) • Bacchic ritual compensated for prudence by producing “enthusiasm” which means “having the god enter the worshipper” who became one with the god. “Much of what is greatest in human achievement involves some element of intoxication, sweeping away of prudence by passion”. Sober science by itself is not satisfying. Men also need passion, religion & art.

16. Greece (cont.d) • Orpheus (奥尔弗斯) spiritualized, reformed Dionysian worship. Orphic doctrines originated in Egypt through Crete. • Transmigration of souls to heaven or hell. • Ceremonies of purification • Man may become one with Bacchus who was twice born • Ritual tearing of a wild animal and eating its flesh reenacted the mythical tearing of Dionysus into pieces before Dionysus was born again. Those who ate Dionysus’ flesh became divine.

17. Greece (cont.d) • Orphics were an ascetic sect: wine was a symbol, as in the Christian sacrament. Symbolic intoxication was a way of acquiring mystic knowledge not available by ordinary means. Pythagoras (毕达哥拉斯) introduced Orphism into philosophy and, through Plato (柏拉图), into later religious philosophy. Orphism espoused: • feminism. Plato: complete political equality of women. • violent emotion (“catharsis”, purification). Greek tragedy is rooted in rites of Dionysus.

18. Greece (cont.d) • According to Orphism life is pain and weariness and only by purification, renunciation and an ascetic life can we escape it and attain to the ecstasy which is union with God. • 2 tendencies in Greece: one passionate, religious, mystical, the other cheerful, empirical, rationalistic & interested in acquiring knowledge. • 6th century: Religious revival, while Greek science, philosophy & mathematics began and prevented Greek religion from reaching the stage attained by religions in the East. Greek religion had no priesthood to preserve dogma.

19. Greece (cont.d) • Orphic-like beliefs were present in India at the same time, but no contact. For Orphics, “orgy” meant “sacrament” • Orphics organized into churches, religious communities open through initiation. Origin of the conception of philosophy as a way of life.

20. http://upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Map_Greco-Persian_Wars-en.svg/750px-Map_Greco-Persian_Wars-en.svg.pnghttp://upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Map_Greco-Persian_Wars-en.svg/750px-Map_Greco-Persian_Wars-en.svg.png

21. Milesian (米利都) School • Miletus (米利都) supressed by Persian Emperor Darius in the 4th century BC. • No Orphic mystical influence. • Brought the Greek mind into contact with Babylonia and Egypt. • Scientific hypotheses. No anthropomorphic desires or moral ideas.

22. Thales (泰勒斯) • First Greek philosopher, from Miletus, flourishing city in Asia Minor allied with Lydia (吕底亚) which had cultural relations with Babylonia. Babylonian astronomers discovered that eclipses occur in 19-year cycles. They could predict lunar eclipses, but not solar which occur in different locations. Thales was the first Greek to predict an eclipse. • Water is the original substance. Everything is made of water. In early 20th century physics the received view was that everything is made of hydrogen, which is 2/3rds of water (H2O).

23. Thales (cont.d) • Thales brought Egyptian geometric rules of thumb to the Greeks. • Discovered how to determine distance on shore to a remote object on water by taking observations from two locations, and how to estimate the height of a pyramid from its shadow.

24. Anaximander (阿那克西曼德) • Milesian school, like Thales • All things come from a single unspecified infinite, eternal, ageless primal substance into which all others are transformed. • Our world is only one of many. • There is eternal motion, in the course of which the worlds originated. Worlds were not created but evolved. • Precursor of the scientific concept of evolution.

25. Anaximander (cont.d) • Living creatures arose from the moist element as it was evaporated by the sun. Man and all other animals must be derived from fishes. • The profound Greek concept of cosmic “justice”. Necessity or natural law maintains a balance between the different substances earth, fire and water and their tendencies to enlarge their role. Where there has been fire, there are ashes which is earth. This is a mix of prescriptive and descriptive law. • First man who made a map, and estimated the size of the sun as a multiple of the earth’s size.

26. Anaximines (阿那克西美尼) • Milesian school, like Thales and Anaximander. • The fundamental substance is air. The soul is air. Fire is rarefied air. When condensed air becomes first water, then earth if further condensed, and finally stone. • His theory has the merit of making the differences between substances entirely quantitative, in terms entirely of degree of condensation. • Believed earth was a disc. That view was upheld by Atomists (原子论者). • Influenced the Pythagoreans (毕达哥拉斯学派) who discovered the earth is spherical.

27. Pythagoras (毕达哥拉斯) • Native of Samos (萨摩), an island commercial rival of Miletus. Migrated to southern Italy. • He founded a reformed Orphic religion. Members of his sect lived in a collective. • Rational mysticism. Escape through science. The body is the tomb of the soul. The best purification (from buying, selling, and competing) is disinterested science. • “Theory”, like “orgy”, is an Orphic term for passionate, sympathetic contemplation, which Pythagoras found in mathematics, in the intoxicating delight of sudden understanding.

28. Pythagoras (cont.d) • Gave theological endorsement to the virtue of contemplation. The well-supported “gentleman” in pursuit of disinterested truth. Aristocratic approach to truth, versus pragmatic and instrumentalist concepts of truth borne of industrialism. • Mathematical knowledge is exact and applicable to the world but obtainable by mere thinking. • Thought is superior to sense, intuition, observation.

29. Pythagoras (cont.d) • “All things are numbers”. • He connected music and arithmetic in the concepts of harmonic mean and harmonic progression. • Galileo's father, Vincenzo Galilei, a lutenist and music theorist, performed experiments establishing perhaps the oldest known non-linear relation in physics: for a stretched string, the pitch varies as the square root of the tension. These observations lay within the framework of the Pythagorean tradition of music, well-known to instrument makers, which included the fact that subdividing a string by a whole number produces a harmonious scale. Thus, a limited amount of mathematics had long related music and physical science. http://en.wikipedia.org/wiki/Galileo_Galilei • He thought of numbers as shapes. Whence “squares” and “cubes” of numbers. • Atomism (原子论). “Oblong numbers”, “pyramidal numbers” were the number of pebbles (shot) needed to make the shapes. Like the world of bodies composed of molecules composed of atoms arranged in various shapes. • Arithmetic as the fundamental study in physics like mathematics.

30. IEEE History of Computing, until 1500 AD http://pages.cpsc.ucalgary.ca/~williams/History_web_lsite/timeline%203000BCE_1500CE/time_3000_1500.html At the pictorial stage a number would designate particular things: a picture of ten tents would represent ten family groups, for example. At the symbolic stage of number, a picture of a cow would be followed by five strokes to represent five cattle. At this stage, strokes have become adjectives rather than nouns. The notation system would be based on a hierarchy of group sizes, and on counting a number of items into a group located at the next level of the hierarchy before starting a new group at that immediately lower level of the hierarchy. The Egyptian number system was strictly based on ten, but was additive, rather than positional in nature.. The following seven basic signs were all that were necessary to represent a number: Numbers were written with the highest valued symbol(s) always preceding the lower one(s). The Chinese used a positional decimal number system, rather than an additive number system like the Egyptians, from about 1300 BCE, possibly even earlier. Early integers 1 to 5 were represented by a corresponding number of vertical strokes, and the integers 6 to 9 by a horizontal stroke, representing 5, underneath which additional vertical strokes were added until the desired value was attained. In each decimal place, the same rules applied. Of course, the problem was that for a number like 24, a representation of of 6 vertical strokes could be somewhat ambiguous. So, to fix this problems, the strokes in every second decimal place were turned 90 degrees. Without a representation for zero, the orientation of the numerals was still vital to their understanding. A representation for zero was added to the system, probably from contact with India, around 800 CE.

31. An Egyptian scribe wishing to find the answer to "How much is 23 taken 13 times" would proceed as follows: * Write down the first number (23) in the first column * Write down 1 in the second column * In the next row, double each number in the previous row * Repeat the previous step until the number in the right column is about to become larger than the second number (13) * Check off the numbers in the right column that add up to make the second number (13) * Total the numbers in the left column that correspond to the numbers that have been checked off in the right column - this is your answer (299)

32. Doubling was the basic operation involved in this method of doing multiplication and, because the Greeks adopted the technique for use in their schools, this mechanism spread into Europe. Until the introduction of Hindu-Arabic numerals, Europeans used Egyptian doubling (duplation) methods for multiplication and a similar halving (mediation) technique for division. Of course, division led to the complication of fractions being needed to represent some of the results, which in turn led to the development of a rather strange and complex system of fractional notation. One system of fractions (the Horus Eye or Corn Measure fraction system) is labeled below with the respective fractions. It is based on a mythological tale in which the eye of the god Horus was torn into pieces, each of which came to represent a fraction. Further discussion of Egyptian fractions is beyond the scope of this site.

33. The Greeks eventually developed two separate forms of numerical notation which were both in use at the same time. In the "Attic" system the numerals appear to have been derived from the first letter of the names of the numbers, except in the case of 1, which was represented in the Egyptian fashion by a simple line. This system was strictly decimal with the exception of having a separate symbol for 5. This may well be a holdover from an early Greek number system based on 5 instead of 10. The Attic symbols were combined in an additive system, and no symbol would be repeated more than 4 times. The second set of numerals, usually called the alphabetic numerals, eventually replaced the Attic system, apparently some time after 100 BCE partially because of the concise nature of number representation in the alphabetic system, making it more usable on coins and other places where space is limited. It is usually thought that the Greeks obtained their alphabet from the Phoenicians and then changed some of the consonants to represent vowels, which were not written in Phoenician. Then they assigned values to the various letters. The Greeks may have got the idea to use the alphabet to represent numerical quantities through trade contact with other groups, such as the Hebrews, early Arabs, Hindus, and many others,

34. The oldest surviving counting board or accounting tablet is the Salamis tablet from the Island of that name, and used by the Babylonians circa 300 B.C.The lines scribed into the stone could be used in conjunction with pebbles or beads to accomplish addition, subtraction, multiplication, or division. The Salamis counting board is a marble tablet with its grooves already inscribed. With as few as four grooves, it was possible to add and subtract to 10,000. Counting tables were used throughout the middle East and Europe until the middle of the 19th century. Called "counters" even after nobody counted on them, counting tables became the counters in department stores and our kitchens. http://imrl.usu.edu/OSLO/technology_writing/001_002.htm

35. Many of our modern mathematical and commercial terms can be traced to the early use of the table abacus. For example, the Romans used small limestone pebbles, called calculi, for their abacus counters; from this we take our modern words calculate and calculus. By the thirteenth century the European table abacus had been standardized into some variant of a simple table, sometimes covered by a cloth, upon which a number of lines were drawn in chalk or ink. The lines indicated the place value of the counters. Thus the bottom line represented units, and each line above increased by ten times the value of the line below; also each space between the lines counted for five times that of the line below it. No more than four counters could be placed on a line and no more than one in any space. As soon as five counters appeared on a line, they were removed and one placed in the next higher space; if two appeared in a space, they were removed and one placed on the next higher line. By the thirteenth century the counters had changed from the simple pebbles used in earlier days into specially minted coin-like objects. They first appeared about 1200 in Italy but, because it was there that the use of Hindu-Arabic numerals first replaced the abacus, the majority of the counters now known come from north of the Alps. These coin-like counters were cast, thrown, or pushed on the abacus table; thus they were generally known by some name associated with this action. In France they were called jetons from the French verb jeter (to throw). A new "nest" of between 20 and 100 jetons was considered a very suitable gift to celebrate New Year's Day. The Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of the previous Babylonian abacus. It was the first portable calculating device for engineers, merchants and presumably tax collectors. It greatly reduced the time needed to perform the basic operations of Roman arithmetic using Roman numerals. For more extensive and complicated calculations, such as those involved in Roman land surveys, there was, in addition to the hand abacus, a true reckoning board with unattached counters or pebbles. Some appreciation of the power of the abacus can be gained by noting that, on November 12, 1946, Kiyoshi Matsuzake of the Japanese Savings Bureau of the Ministry of Postal Administration used a soroban (the Japanese version of the abacus) to beat Private Thomas Nathan Wood of the U.S. Army of Occupation (240th Finance Disbursing Section of General MacArthur's headquarters), who had won a contest as the most expert operator of the most modern electrically driven mechanical calculating machine in the U.S. Army, in a contest of speed and accuracy in calculation. The contest consisted of addition and subtraction problems (adding up long columns of multiple digit numbers) multiplication of five to twelve digit integers, division problems containing five to twelve digit integers, and a complex problem consisting of a mixture of all these operations on 45 different numbers. Matsuzake clearly won in four out of the five contests held, being only just beaten out by the electrically driven calculator when doing the multiplication problems.

36. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |O| |O| |O| |O| |O| |O| |O| |O| MM CM XM M C X I 0 ~3 --- --- --- --- --- --- --- --- --- | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ) |O| |O| |O| |O| |O| |O| |O| |O| | | |O| |O| |O| |O| |O| |O| |O| |O| | | |O| |O| |O| |O| |O| |O| |O| |O| | | |O| |O| |O| |O| |O| |O| |O| |O| |O| 2 |O| |O| Roman Hand Abacus http://en.wikipedia.org/wiki/Roman_abacus The Late Roman hand abacus shown here contained seven longer and seven shorter grooves used for whole number counting, the former having up to four beads in each, and the latter having just one. The rightmost two grooves were for fractional counting. The lower groove marked I indicates units, X tens, and so on up to millions. The beads in the upper shorter grooves denote fives—five units, five tens, etc., essentially in a bi-quinary coded decimal system. Computations are made by means of beads which would probably have been slid up and down the grooves to indicate the value of each column. The upper slots contained a single bead while the lower slots contained four beads, the only exceptions being the two rightmost columns, marked 0 and ~3. These latter two slots are apparently for mixed-base math. The longer slot with five beads below the 0 position allowed for the counting of 1/12th of a whole unit, making the abacus useful for Roman measures and Roman currency. Many measures were aggregated by twelfths. Thus the Roman pound ('libra'), consisted of 12 ounces (unciae) (1 uncia = 28 grams). A measure of volume, congius, consisted of 12 heminae (1 hemina = 0.273 litres). The Roman foot (pes), was 12 inches (unciae) (1 uncia = 2.43 cm). The actus, the standard furrow length when plowing, was 120 pedes. There were however other measures in common use - for example the sextarius was two heminae. The as, the principal copper coin in Roman currency, was also divided into 12 unciae. Again, the abacus was ideally suited for counting currency. It is most likely that the rightmost slot or slots were used to enumerate fractions of an uncia and these were from top to bottom, 1/2 s , 1/4 s and 1/12 s of an uncia. The upper character in this slot (or the top slot where the righmost column is three separate slots) is the character most closely resembling that used to denote a Semuncia or 1/24. The name Semuncia denotes 1/2 of an uncia or 1/24 of the base unit, the As. Likewise the next character is that used to indicate a Sicilius or 1/48 th of an As which is 1/4 of an uncia. Finally, the last or lower character is most similar but not identical to the character denoting 1/144 of an As, the Dimidio Sextula which is the same as 1/12 of an uncia.

37. It is thought that early Christians brought the abacus to the East (note that both the suan-pan and the Roman hand-abacus have a vertical orientation). Aspects of Roman culture could have been introduced to China as early as 166 A.D, during the Han Dynasty, as Roman emperor Antoninus Pius' embassies to China spread along the Silk Road. The abacus as we know it today, appeared (was chronicled) circa 1200 A.D. in China. http://www.ee.ryerson.ca/~elf/abacus/history.html IEEE History of Computing, until 1500 AD (cont.d) http://pages.cpsc.ucalgary.ca/~williams/History_web_lsite/timeline%203000BCE_1500CE/time_3000_1500.html The origin of our current positional number system, and the 10 digits on which it relies, remains unclear. It certainly came into Europe from the Arabs, and it is quite clear that they in turn obtained it from the people on the Indian subcontinent. The earliest undoubted occurence of a zero in a written inscription in India was in 876 CE with the numbers 50 and 270 being written in the local version of Indian digits. It may have been an indigenous invention, it may have come from further east in Indo-China, or it may have developed out of the Babylonian use of an "empty" column symbol. Early in the ninth century, an Arab scholar called Al-Khowarizmi wrote textbooks on both arithmetic and algebra and calculated astronomical tables for use in Baghdad. In his book, Arithmetic, he distinctly indicates that the systems used came to him from the Indians. Arithmetic begins with the subject of numeration, then discusses the digits, and the use of zero at great length, and follows this with a systematic discussion of the fundamental operations upon integers. Al-Khowarizmi's work was very influential, and the use of the Hindu numerals quickly spread throughout the Arab empire. The first great attempt to introduce the Hindu-Arabic system was made by Fibonacci, when he published a work titled Liber Abaci, (The Book of the Abacus), in 1202. after finding various current European systems woefully inadequate. The book introduces Arabic notation, fundamental operations on integers, and various applications, as well as methods of calculation involving series, proportion, square and cube roots, and a short discussion of geometry and algebra. Unfortunately, his book was not the influence it could have been; it was too long to be easily reproduced in an era before printing, and much of the subject matter was for a more scholarly readership than your average European citizen. The Italians, quick to see the usefulness of the system in commerce, led Europe in the general adoption of the place value system. It was known throughout Europe by about 1400, but more conservative merchants kept records in Roman numerals until about 1550, and many monasteries and colleges kept them until the middle of the seventeenth century.

38. (hypotenuse) c b 90o a Pythagoras (cont.d) • Pythagorean theorem • The Egyptians knew numerical • combinations that would make the • sides of a right triangle, like 3,4,5. • Pythagoras discovered the formula c2 = a2 + b2 . • Proof: Area: (a+b)2 = 4(1/2ab) + c2. Area: (a+b)2 = 4(1/2ab)+ a2 + b2 = http://jwilson.coe.uga.edu/EMT668/emt668.student.folders/HeadAngela/essay1/Pythagorean.html

39. m/n 1 90o 1 Pythagoras (cont.d) • Pythagorean theorem (cont.d) • Pythagorean theorem led to discovery of “incommensurables” (“irrational” numbers, like ) which seemed to disprove his philosophy. • If each side of a right triangle is 1 inch long, how long is • the hypotenuse supposing it’s length is m/n, where m & n are integers. Then m2/n2 = 2. After canceling any common factor, either m or n must be odd because, if they were both even, then 2 would be a common factor. Since m2 = 2n2 , m2 is even, so m is even (because if 2 is a factor of m2, it must also be a factor of m) and, therefore, n is odd. Suppose m = 2p. Then 4p2 = 2n2. Therefore n2 = 2p2 and therefore n is even, contrary to hypothesis. Therefore no fraction will measure the hypotenuse. In other words, there are no two integers m,n such that n times the length m/n in question is m times the unit.*This convinced Greek mathematicians that geometry must be established independently of arithmetic. • *Rational numbers are the ratio of two integers • **Blue phrases are mine. Russell’s book apparently misprints m for • the red n, and n for the red m, in that sentence.

40. Pythagoras (cont.d) • Pythagorean theorem (cont.d) • Pythagorean theorem led to discovery of “incommensurables” 不可公约数 (“irrational” numbers 无理数, like ) which seemed to disprove his philosophy. (cont.d) • Geometry seemed to accord with the senses. It’s pictorial. • Arithmetic is number theory and deals with integers (vs the full set of real numbers ) • Descartes prematurely restored the equivalence of numerical mathematics with geometry but no solution had been found yet to the problem of incommensurables • Axiomatic method • Geometry’s lasting contribution to philosophy and scientific method lies in the axiomatic method of theory construction & the resulting method of proof.

41. Pythagoras (cont.d) • Axiomatic method (cont.d) • Geometry, because it had to conform to the physical world, seemed a way to discover physical truths from “self-evident” axioms. • But axioms are assumptions. • Non-Euclidean, observed curved-space geometry of (非欧几里几何学) relativity-theory (where parallel lines intersect at infinity) • Method of Newton’s (牛顿的) mathematical physics (“natural philosophy”) • Through Plato, Descartes (笛卡尔), and Kant (康德), the 18th century doctrine of “natural rights” tried to introduce the axiomatic method into political philosophy and law.

42. Pythagoras (cont.d) • Axiomatic method (cont.d) • As a method of • discovering conclusions (theorems) • testing assumptions (axioms) by testing conclusions: • Propositional calculus (T: true proposition; F: false proposition; : implies; : or) false conclusion means assumption is false.

43. Pythagoras (cont.d) • Mathematics as the guide to knowledge • Theology derived from mathematics is like Pythagorean personal religion derived from ecstasy • Mathematics as the chief source of belief in eternal and exact truth • Actually no real objects correspond to pure geometric objects, like circles • Thought approximates the real world or vice versa. • Mathematics provides hints, conclusions. If they correspond to reality, that’s evidence that the assumptions hold. By testing for the applicability of the conclusions, we prove the correspondence of the assumptions. • In this methodological order, thought seems nobler than sense.

44. Pythagoras (cont.d) • Mathematics as suggestive of the divine • Mathematics reinforces mystical doctrines about the relation of time to eternity. Numbers are • timeless, eternal • conceived as God’s thoughts • Plato: God is a geometer. • Mathematics dominates rationalistic (versus apocalyptic) religion • Before Pythagoras, Orphism was like Asian mystery religions (versus the intellectualized theology of Europe) • Rationalistic religion combines the moral aspiration of religion with logical admiration for the timeless

45. Pythagoras (cont.d) • Originated the conception of the external world as revealed to the intellect rather than to the senses: • Core of Platonism • But for Pythagoras, • Christ would not be “the word”, the alpha and omega • No logical proofs of God or immortality • Laid the basis for Parmenides (巴门尼德) and Plato.

46. Xenophanes (色诺芬尼) • Ionian 伊奥尼亚人 (Asia Minor) • All things are made from earth and water • One unknowable God • Opposed mystical tendencies of Pythagoras

47. Heraclitus (赫拉克利特) • Ionian (Asia Minor). Un-Milesian school. Mystic. • Fire is the fundamental substance. Perpetual flux: • like flame in a fire, everything is born of the death of something else. • the motion is a harmony • you cannot step twice in the same river • nothing ever is but is becoming • Dialectic, strife. • The unity in the world is formed by diversity, the mingling of opposites. • All things come into being and pass away through strife which is justice. • War is the father of all. • Greater deaths win greater portions.

48. Heraclitus (cont.d) • Primacy of unity: • all things are an exchange for fire and fire for all things • many come from one and vice versa but the many have less reality than the one • the one is God • good and ill are one • cosmic justice, as for Anaximander, prevents a complete victory of either • progress itself is permanent

49. Heraclitus (cont.d) • Primacy of unity (cont.d): • Eternity • existence outside the temporal process. Not endless duration. Ever-living fire. • Change as the rearrangement of persistent elements. • Atoms are indestructible (since disproved by radio-activity). • Energy has replaced matter as what is permanent • The wise soul as the vehicle for ambition • The soul with the most fire is driest and wisest • The moist soul has too much pleasure and passion. Hostility to Bacchic religion • Power through self-mastery, asceticism.

50. Parmenides (巴门尼德) • S. Italy where more inclined to mysticism and religion • Interviewed by Socrates • Invented metaphysics deduced from language. • Nothing changes. The only one true being is “the One” which is • infinite & indivisible • not God, but material & extended everywhere, so cannot be divided. • a sphere.