Pythagoras 畢達哥拉斯

Birth c. 580 BC – 572 BC

Death c. 500 BC – 490 BC

School/tradition 學派/傳統(tǒng)

Pythagoreanism 畢達哥拉斯主義

Main interests 主要興趣

Metaphysics, Music, Mathematics, Ethics, Politics 形而上學船侧、音樂捷雕、數(shù)學手趣、倫理學临燃、政治學

Notable ideas 著名思想

Musica universalis, Golden ratio, Pythagorean tuning, Pythagorean Theorem 音樂普遍性淋袖、黃金比例屯伞、畢達哥拉斯調(diào)音芜辕、畢達哥拉斯定理

Influenced by 所受影響

Thales, Anaximander, Pherecydes 泰勒斯律杠、阿那克西曼德、斐瑞居德斯

Influenced 影響

Philolaus, Alcmaeon, Parmenides, Plato, Euclid, Empedocles, Hippasus, Kepler菲洛勞斯难菌，阿爾克梅翁，巴門尼德蔑滓，柏拉圖郊酒，歐幾里德、恩培多柯勒键袱、希勃索斯燎窘、開普勒

Pythagoras of Samos was an Ionian Greek mathematician and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician, mystic and scientist; however some have questioned the scope of his contributions to mathematics and natural philosophy. Herodotus referred to him as "the most able philosopher among the Greeks". His name led him to be associated with Pythian Apollo; Aristippus explained his name by saying, "He spoke (agor-) the truth no less than did the Pythian (Pyth-)," and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and beneficial to humankind.

薩默斯島的畢達哥拉斯是一位愛奧尼亞希拉數(shù)學家，也是名之曰畢達哥拉斯主義的宗教運動的創(chuàng)始者蹄咖。他常常被推崇為偉大的數(shù)學家褐健、神秘主義者和科學家；可是也有人懷疑他對數(shù)學與自然哲學的貢獻范圍澜汤。希羅多德稱他為“希臘最有才干的哲學家”蚜迅。他的名字把他和達爾菲的太陽神聯(lián)系在一起；阿里斯提普這樣解釋他的名字：“他道出的真理不亞于皮提亞[希臘神]”俊抵，楊布里科斯講了這樣一個故事：皮提亞預言他懷孕的母親將生出一個無比美貌谁不、智慧并有益于人類的兒子。

He is best known for the Pythagorean theorem, which bears his name. Known as "the father of numbers", Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. Because legend and obfuscation cloud his work even more than with the other pre-Socratics, one can say little with confidence about his life and teachings. We do know that Pythagoras and his students believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured in rhythmic patterns or cycles. According to Iamblichus, Pythagoras once said that "number is the ruler of forms and ideas and the cause of gods and demons."

他因以自己名字命名的畢達哥拉斯定理而最為知名徽诲。他被稱為“數(shù)字之父”刹帕，畢達哥拉斯在公元前6世紀后期對哲學與宗教工作做出了頗有影響的貢獻。由于傳說與困惑是我們對他的工作還不如對前蘇格拉底哲學家更清楚谎替，所以我們談論他的生活與教義沒有多少信心偷溺。我們不知道畢達哥拉斯及其門徒相信—一切都與數(shù)學聯(lián)系在一起—數(shù)是終極實在—通過數(shù)學，什么都可以預測出來钱贯，什么都可以有節(jié)奏的樣式或循壞加以計量挫掏。按阿里斯提普的看法，畢達哥拉斯曾經(jīng)說過“數(shù)是形式和思想的統(tǒng)治者也是眾神與魔鬼的目標”喷舀。

He was the first man to call himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato. Unfortunately, very little is known about Pythagoras because none of his writings have survived. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors.

他是自稱哲學家或愛智慧者的第一人砍濒，他的思想對柏拉圖具有深遠影響。不幸的是硫麻，由于他沒有任何作品幸存下來爸邢，因此我們對畢達哥拉斯知之甚少。歸之于畢達哥拉斯的許多成就實際上有可能是他的同行和繼承者作出的拿愧。

The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.

—Aristotle, Metaphysics

所謂的畢達哥拉斯主義者杠河，是第一批從事數(shù)學的人，他們不僅提出了這門學科，而且投身于它券敌。他們猜測數(shù)學的原則乃是萬物的原則唾戚。

--亞里士多德，《形而上學》

The organization was in some ways a school, in some ways a brotherhood, and in some ways a monastery. It was based upon the religious teachings of Pythagoras and was very secretive. At first, the school was highly concerned with the morality of society. Members were required to live ethically, love one another, share political beliefs, practice pacifism, and devote themselves to the mathematics of nature.

此組織在某些方面就像一個學派待诅，在某些方面像兄弟會叹坦，在某些方面像修道院。它以畢達哥拉斯的宗教教義為基礎卑雁，非常保密募书。起初此學派非常關注社會道德，它要求其成員道德地生活测蹲，彼此愛護莹捡，共享政治信仰，餞行和平主義并把自己貢獻給自然的數(shù)學扣甲。

Pythagoras's followers were commonly called "Pythagoreans". They are generally accepted as philosophical mathematicians who had an influence on the beginning of axiomatic geometry, which after two hundred years of development was written down by Euclid in The Elements.

畢達哥拉斯的追隨者一般被稱之為畢達哥拉斯學派篮赢。他們被公認為對公理化幾何學（此學科在之后的兩百年的發(fā)展由歐幾里德在《幾何原本》中加以總結）的創(chuàng)立有影響的哲學數(shù)學家。

The Pythagoreans observed a rule of silence called echemythia, the breaking of which was punishable by death. This was because the Pythagoreans believed that a man's words were usually careless and misrepresented him and that when someone was "in doubt as to what he should say, he should always remain silent". Another rule that they had was to help a man "in raising a burden, but do not assist him in laying it down, for it is a great sin to encourage indolence", and they said "departing from your house, turn not back, for the furies will be your attendants"; this axiom reminded them that it was better to learn none of the truth about mathematics, God, and the universe at all than to learn a little without learning all.

畢達哥拉斯學派遵守被稱之為“echemythia”的沉默規(guī)則琉挖，違背這一規(guī)則就會被處死启泣。這是因為畢達哥拉斯學派認為，人之言語常有疏漏并歪曲自己粹排。他們也認為一個人“如果不知道應該怎么說种远，它就應該保持沉默”。他們的另一個規(guī)則是幫助“負擔加重的人顽耳，但不是幫助他放下負擔坠敷，因為鼓勵懶惰是一種巨大的罪惡”，他們說“離開你的家園射富，不要回頭膝迎，因為復仇女神將成為你的仆人”；這一格言提醒他們：與根本就不學習而學到一點東西相比胰耗，還不如學不到什么有關數(shù)學限次、上帝與宇宙的東西好。

Musical theories and investigations 音樂理論與研究

Pythagoras was very interested in music, and so were his followers. The Pythagoreans were musicians as well as mathematicians. Pythagoras wanted to improve the music of his day, which he believed was not harmonious enough and was too hectic.

畢達哥拉斯與其追隨者都對音樂深感興趣柴灯。畢達哥拉斯學派既是音樂家也是數(shù)學家卖漫。畢達哥拉斯想改進他所處時代的音樂，因為他認為那時的音樂不和諧赠群、太雜亂羊始。

According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was when one day he passed blacksmiths at work, and thought that the sounds emanating from their anvils being hit were beautiful and harmonious and decided that whatever scientific law caused this to happen must be mathematical and could be applied to music. He went to the blacksmiths to learn how this had happened by looking at their tools, he discovered that it was because the anvils were "simple ratios of each other, one was half the size of the first, another was 2/3 the size, and so on."

傳說畢達哥拉斯是這樣發(fā)現(xiàn)音符可以轉(zhuǎn)化成數(shù)學方程的：有一天他經(jīng)過鐵匠鋪的時候就想，從擊打鐵砧飛出來的聲音很好聽也很和諧查描，他就確定：產(chǎn)生這種現(xiàn)象的科學規(guī)律必定是數(shù)學的而且可以用到音樂中來突委。他到鐵匠那里通過觀察它們的工具了解這是怎么回事兒柏卤，他發(fā)現(xiàn)：這是因為鐵砧之間“彼此之間成簡單的比利，一個是另一個的一半大小匀油，另一個是2/3大小缘缚，等等”。

The Pythagoreans elaborated on a theory of numbers, the exact meaning of which is still debated among scholars. Pythagoras believed in something called the "harmony of the spheres." He believed that the planets and stars moved according to mathematical equations, which corresponded to musical notes and thus produced a symphony.

畢達哥拉斯詳細敘述數(shù)的理論敌蚜，其準確的含義至今仍在學者中有爭議桥滨。畢達哥拉斯相信被稱之為“天體和諧”的東西。他認為行星與恒星按照數(shù)學方程運動弛车，其符合音符因而產(chǎn)生旋律该园。

Influence? 影響

Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle), c, is equal to the sum of the squares of the other two sides, b and a—that is, a2 + b2 = c2.

從公元四世紀開始，畢達哥拉斯就被公認為畢達哥拉斯定理的發(fā)現(xiàn)者帅韧。畢達哥拉斯定理是一個幾何定理，說的是直角三角形的斜邊的平方（c2）等于另外兩邊的平方和（a2+b2）——即啃勉，a2+b2= c2忽舟。

Religion and science 宗教與科學

Pythagoras’ religious and scientific views were, in his opinion, inseparably interconnected. Religiously, Pythagoras was a believer of metempsychosis. He believed in transmigration, or the reincarnation of the soul again and again into the bodies of humans, animals, or vegetables until it became moral. His ideas of reincarnation were influenced by ancient Greek religion. He was one of the first to propose that the thought processes and the soul were located in the brain and not the heart. He himself claimed to have lived four lives that he could remember in detail, and heard the cry of his dead friend in the bark of a dog.

在畢達哥拉斯看來，他的宗教與科學觀是不可分割地交織在一起的淮阐。在宗教上叮阅，畢達哥拉斯是一個輪回轉(zhuǎn)世的信奉者。他相信輪回轉(zhuǎn)世或者是靈魂的反復再生進入人泣特、動物或植物直到變成精神的浩姥。他的轉(zhuǎn)世觀點受到了古希臘宗教的影響。他是第一批提出思想過程和靈魂處于大腦而不是心臟的人之一状您。他自稱他已經(jīng)活了四世勒叠，能夠記得其細節(jié)，能夠聽到狗叫時已死朋友的哭喊膏孟。

One of Pythagoras' beliefs was that the essence of being is number. Thus, being relies on stability of all things that create the universe. Things like health relied on a stable proportion of elements; too much or too little of one thing causes an imbalance that makes a being unhealthy. Pythagoras viewed thinking as the calculating with the idea numbers. When combined with the Folk theories, the philosophy evolves into a belief that Knowledge of the essence of being can be found in the form of numbers. If this is taken a step further, one can say that because mathematics is an unseen essence, the essence of being is an unseen characteristic that can be encountered by the study of mathematics.

畢達哥拉斯一個信念是“存在的本質(zhì)是數(shù)”眯分。所以存在有賴于創(chuàng)世的所有東西的穩(wěn)定性。像健康這樣的事情就有賴于各種成分的穩(wěn)定比利柒桑；一種東西太多或太少都會引起導致不健康的不平衡弊决。畢達哥拉斯認為思考就是使用觀念數(shù)進行計算。和民間推測結合在一起魁淳，哲學就產(chǎn)生了一種這樣的信念：存在本質(zhì)的知識可以在數(shù)的形式中發(fā)現(xiàn)飘诗。如果再往前走一步，我們可以說界逛，由于數(shù)學是隱藏的本質(zhì)昆稿，所以存在的本質(zhì)就是可以通過數(shù)學研究來發(fā)現(xiàn)的隱藏特征。

No texts by Pythagoras survive, although forgeries under his name — a few of which remain extant — did circulate in antiquity.

盡管有很多以畢達哥拉斯名義偽造的作品—至今仍有一些留存下來—在古代流傳仇奶，但是畢達哥拉的作品沒有遺留下來貌嫡。

One of Pythagoras's major accomplishments was the discovery that music was based on proportional intervals of the numbers one through four. He believed that the number system, and therefore the universe system, was based on the sum of these numbers: ten. Pythagoreans swore by the Tetrachtys of the Decad, or ten, rather than by the gods. Odd numbers were masculine and even were feminine. One member of his order, Hippasos, also discovered irrational numbers, but the idea was unthinkable to Pythagoras, and according to legend, Hippasos was executed. Pythagoras (or the Pythagoreans) also discovered square numbers. They found that if one took, for example, four small stones and arranged them into a square, each side of the square was not only equivalent to the other, but that when the two sides were multiplied together, they equaled the sum total of stones in the square arrangement, hence the name "Square Root". He was one of the first to think that the earth was round, that all planets have an axis, and that all the planets travel around one central point. He originally identified that point as Earth, but later renounced it for the idea that the planets revolve around a central “fire” that he never identified as the sun. He also believed that the moon was another planet that he called a “counter-Earth”.

畢達哥拉斯的主要成就是發(fā)現(xiàn)音樂基于1到4這些書的比例間隔比驻。他認為數(shù)的體系因而宇宙體系，都是基于這些數(shù)字和：10岛抄。畢達哥拉斯學派用Tetractys 與十而不是神起誓（注：The Tetractys is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row.）偶數(shù)女性的别惦，奇數(shù)是男性的遥皂。這一學派的一個成員西帕索斯也發(fā)現(xiàn)了無理數(shù)亚享，但這對畢達哥拉斯來說是不可想象的，據(jù)說西帕索斯被處死了纪蜒。畢達哥拉斯或畢達哥拉斯學派也發(fā)現(xiàn)了平方數(shù)蹭秋。他們發(fā)現(xiàn)扰付，如果取四塊小石頭；把它們放到一個正方形中仁讨，正方形不僅邊與邊相等羽莺，而且當兩個邊相乘時，其結果等于正方形中石頭的數(shù)量之和洞豁，因而得出“平方根”這種叫法盐固。他也是第一批認為地球是圓形、所有行星都有一個軸而且都圍繞中心點運動的人士之一丈挟。一開始刁卜，他確定這一中心點是地球，但后來他又用“行星都圍繞著中心火轉(zhuǎn)動”代替了這種看法曙咽，但中心火他從來沒有認為是太陽蛔趴。他也認為月亮是一顆行星，名之曰“對地”例朱。

Pythagoras

Testimonies 證言

1. Empedocles too bears witness to this, writing of him: ‘And there was among them a man of rare knowledge, most skilled in all manner of wise works, a man who had won the utmost wealth of wisdom; for whensoever he strained with all his mind, he easily saw everything of all the things that are, in ten, twenty lifetimes of men’. 恩培多柯勒對此很有信心孝情，他寫道：“在他們之中有個人具有罕見的學識，最通曉各種智慧性工作茉继，此人贏得了最多的智慧財富咧叭；因為他總是陷于深思，容易看出人們十輩子烁竭、二十輩子經(jīng)歷的所有事情的一切菲茬。”

2. Pythagoras wrote nothing… 畢達哥拉斯沒有寫下任何東西…

3. None the less the following became universally known; first, that he maintains that the soul is immortal’ next, that it changes into other kinds of living things; also that events recur in certain cycles, and that nothing is ever absolutely new; and finally, that all living things should be regarded as akin. Pythagoras seems to have been the first to bring these beliefs into Greece. 最起碼下面的觀點盡人皆知派撕；第一婉弹，他認為靈魂是不朽的，第二终吼，他認為靈魂轉(zhuǎn)變?yōu)槠渌贫模凰舱J為時間以某種方式輪回，沒有絕對新的東西际跪；最后商佛，他認為所有的生命都應被看成同類喉钢。畢達哥拉斯看來是第一位把這種觀念帶給希臘的人。

4. If one were to believe the Pythagoreans, with the result that the same individual things will recur, then I shall be talking to you again sitting as you are now, with this printer in my hand, and everything else will be just as it is now, and it is reasonable to suppose that the time then is the same as now. 如果你相信畢達哥拉斯：同樣的事情會輪回發(fā)生良姆，那么肠虽，我要告訴你—再次坐在你現(xiàn)在的地方，我的手中拿著這個打字機玛追，其他別的都和現(xiàn)在一樣税课，那么，推測時間也和現(xiàn)在一樣就是合理的痊剖。

5. Let the rules to be pondered be these: When you are going out to a temple, worship first, and on your way neither say nor do anything else connected with your daily life…Sacrifice and worship without shoes on…Follow the gods and restrain your tongue above all else…Speak not of Pythagorean matters without light…Disbelieve nothing strange about the gods or about religious beliefs . . . Be not possessed by irrepressible mirth…Abstain from beans…Abstain from living things. 讓我們?nèi)绱丝紤]規(guī)則：在你出去到教堂之時韩玩，先拜神，在路上不說也不做與日常有關的事兒…不穿鞋進行供奉與崇拜…聽從神并限制談論別的…沒有燈就莫談畢達哥拉斯學派的事兒…對神與宗教信仰深信不疑…不被難以壓抑的高興所征服…禁止吃豆子…禁止吃活物陆馁。

6. So Pythagoras turned geometrical philosophy into a form of liberal education by seeking its first principles in a higher realm of reality…因此畢達哥拉斯通過在更高的實在王國尋求第一原則把幾何哲學轉(zhuǎn)變成自由教育的形式…

7. Life, he said, is like a festival; just as some come to the festival to compete, some to ply their trade, but the best people come as spectators, so in life the slavish men go hunting for fame or gain, the philosophers for the truth. 他說找颓，生活就像節(jié)日；正當有些人在節(jié)日中角逐之時叮贩，有些人從事商業(yè)叮雳，但是最好的人士當觀眾，所以在生活中奴性人追逐的是名或利妇汗，而哲學家則追求真理。

8. The Pythagoreans, according to Aristoxenus, practised the purification of the body by medicine, that of the soul by music. 按照阿里斯托克塞努斯的觀點说莫，畢達哥拉斯使用藥物保持身體的純潔杨箭，使用音樂保持靈魂的凈化。

9. Ten is the very nature of number…And again, Pythagoras maintains, the power of the number ten lies in the number four, the tetrad. This is the reason: if one starts at the unit and adds the successive numbers up to four, one will make up the number ten…If. that is, one takes the unit, adds two, then three and then four, one will make up the number ten…And so the Pythagoreans used to invoke the tetrad as their most binding oath: 'Nay, by him that gave to our generation the tetractys, which contains the fount and root of eternal nature. 十是數(shù)字的本質(zhì)…畢達哥拉斯也堅持認為储狭，數(shù)字10的力量存在于數(shù)字4互婿、4元組之中。原因是：如果你用1+2+3+4就會得到數(shù)字10…所以畢達哥拉斯學派習慣于把四元組[由1辽狈、2慈参、3、4點組成的三角形]在為最有約束力的誓言：而且是他給我們這一代人留下四元組刮萌，其中包含不朽自然的源泉與根源驮配。

10. Contemporaneously with these philosophers and before them, the so-called Pythagoreans, who were the first to take up mathematics, not only advanced this study, but also having been brought up in it they thought its principles were the principles of all things. Since of these principles numbers are by nature the first, and in numbers they seemed to see many resemblances to the things that exist and come into being—more than in fire and earth and water; since, again, they saw that the modifications and the ratios of the musical scales were expressible in numbers;--since, then, all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangement of the heavens, they collected and fitted into their scheme; and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent. E.g. as the number 10 is thought to be perfect and to comprise the whole nature of numbers, they say that the bodies which move through the heavens are ten, but as the visible bodies are only nine, to meet this they invent a tenth—the “counter-earth”. 與哲學哲學家同時代以及在他們之前的哲學家，所謂的畢達哥拉斯學派着茸，首先從事數(shù)學研究壮锻，他們不僅開展了這種研究，而且也產(chǎn)生了這樣的思想：數(shù)學的原則是萬物的原則涮阔。由于這些原則在性質(zhì)上是第一位的猜绣，而且在數(shù)字上他們似乎看到了已存與即存事物的許多類似性，這些類似性比在火敬特、土和水中找到的還要多掰邢；由于他們也看到了改變與音階的比例皆可用數(shù)字來表達牺陶；---也因為所有其他事物在整個性質(zhì)上都是由數(shù)來展示的，并且數(shù)字是整個自然的首要的東西辣之，他們就認為數(shù)元素就是萬物的元素掰伸，整個天體就是音節(jié)與數(shù)字。所有數(shù)與音階的性質(zhì)都應該表現(xiàn)為與整個天體安排的性質(zhì)與組成部分相一致召烂，他們搜集并整合他們的這一圖式碱工；如果任何地方有什么空缺，他們就準備加以彌補一邊讓他們的整個理論保持一貫奏夫。例如怕篷，由于數(shù)字10倍認為是完美的，它包含數(shù)字的所有性質(zhì)酗昼，所以他們認為在天空運動的天體是10個廊谓，但可見的天體只有9個，他們就發(fā)明了第十個“對地”麻削，用于滿足這種說法蒸痹。