Pythagoras 毕达哥拉斯

Pythagoras 毕达哥拉斯

Birth c. 580 BC – 572 BC

Death c. 500 BC – 490 BC

School/tradition 学派/传统

Pythagoreanism 毕达哥拉斯主义

Main interests 主要兴趣

Metaphysics, Music, Mathematics, Ethics, Politics 形而上学、音乐、数学、伦理学、政治学

Notable ideas 著名思想

Musica universalis, Golden ratio, Pythagorean tuning, Pythagorean Theorem 音乐普遍性、黄金比例、毕达哥拉斯调音、毕达哥拉斯定理

Influenced by 所受影响

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

Influenced 影响

Philolaus, Alcmaeon, Parmenides, Plato, Euclid, Empedocles, Hippasus, Kepler菲洛劳斯，阿尔克梅翁，巴门尼德，柏拉图，欧几里德、恩培多柯勒、希勃索斯、开普勒

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.

萨默斯岛的毕达哥拉斯是一位爱奥尼亚希拉数学家，也是名之曰毕达哥拉斯主义的宗教运动的创始者。他常常被推崇为伟大的数学家、神秘主义者和科学家；可是也有人怀疑他对数学与自然哲学的贡献范围。希罗多德称他为“希腊最有才干的哲学家”。他的名字把他和达尔菲的太阳神联系在一起；阿里斯提普这样解释他的名字：“他道出的真理不亚于皮提亚[希腊神]”，杨布里科斯讲了这样一个故事：皮提亚预言他怀孕的母亲将生出一个无比美貌、智慧并有益于人类的儿子。

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毕达哥拉斯定理而最为知名。他被称为“数字之父”，毕达哥拉斯在公元前6世纪后期对哲学与宗教工作做出了颇有影响的贡献。由于传说与困惑是我们对他的工作还不如对前苏格拉底哲学家更清楚，所以我们谈论他的生活与教义没有多少信心。我们不知道毕达哥拉斯及其门徒相信—一切都与数学联系在一起—数是终极实在—通过数学，什么都可以预测出来，什么都可以有节奏的样式或循坏加以计量。按阿里斯提普的看法，毕达哥拉斯曾经说过“数是形式和思想的统治者也是众神与魔鬼的目标”。

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.

他是自称哲学家或爱智慧者的第一人，他的思想对柏拉图具有深远影响。不幸的是，由于他没有任何作品幸存下来，因此我们对毕达哥拉斯知之甚少。归之于毕达哥拉斯的许多成就实际上有可能是他的同行和继承者作出的。

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

所谓的毕达哥拉斯主义者，是第一批从事数学的人，他们不仅提出了这门学科，而且投身于它。他们猜测数学的原则乃是万物的原则。

--亚里士多德，《形而上学》

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.

此组织在某些方面就像一个学派，在某些方面像兄弟会，在某些方面像修道院。它以毕达哥拉斯的宗教教义为基础，非常保密。起初此学派非常关注社会道德，它要求其成员道德地生活，彼此爱护，共享政治信仰，饯行和平主义并把自己贡献给自然的数学。

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.

毕达哥拉斯的追随者一般被称之为毕达哥拉斯学派。他们被公认为对公理化几何学（此学科在之后的两百年的发展由欧几里德在《几何原本》中加以总结）的创立有影响的哲学数学家。

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.

毕达哥拉斯学派遵守被称之为“echemythia”的沉默规则，违背这一规则就会被处死。这是因为毕达哥拉斯学派认为，人之言语常有疏漏并歪曲自己。他们也认为一个人“如果不知道应该怎么说，它就应该保持沉默”。他们的另一个规则是帮助“负担加重的人，但不是帮助他放下负担，因为鼓励懒惰是一种巨大的罪恶”，他们说“离开你的家园，不要回头，因为复仇女神将成为你的仆人”；这一格言提醒他们：与根本就不学习而学到一点东西相比，还不如学不到什么有关数学、上帝与宇宙的东西好。

Musical theories and investigations 音乐理论与研究

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.

毕达哥拉斯与其追随者都对音乐深感兴趣。毕达哥拉斯学派既是音乐家也是数学家。毕达哥拉斯想改进他所处时代的音乐，因为他认为那时的音乐不和谐、太杂乱。

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."

传说毕达哥拉斯是这样发现音符可以转化成数学方程的：有一天他经过铁匠铺的时候就想，从击打铁砧飞出来的声音很好听也很和谐，他就确定：产生这种现象的科学规律必定是数学的而且可以用到音乐中来。他到铁匠那里通过观察它们的工具了解这是怎么回事儿，他发现：这是因为铁砧之间“彼此之间成简单的比利，一个是另一个的一半大小，另一个是2/3大小，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.

毕达哥拉斯详细叙述数的理论，其准确的含义至今仍在学者中有争议。毕达哥拉斯相信被称之为“天体和谐”的东西。他认为行星与恒星按照数学方程运动，其符合音符因而产生旋律。

Influence  影响

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.

从公元四世纪开始，毕达哥拉斯就被公认为毕达哥拉斯定理的发现者。毕达哥拉斯定理是一个几何定理，说的是直角三角形的斜边的平方（c2）等于另外两边的平方和（a2+b2）——即，a2+b2= c2。

Religion and science 宗教与科学

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.

在毕达哥拉斯看来，他的宗教与科学观是不可分割地交织在一起的。在宗教上，毕达哥拉斯是一个轮回转世的信奉者。他相信轮回转世或者是灵魂的反复再生进入人、动物或植物直到变成精神的。他的转世观点受到了古希腊宗教的影响。他是第一批提出思想过程和灵魂处于大脑而不是心脏的人之一。他自称他已经活了四世，能够记得其细节，能够听到狗叫时已死朋友的哭喊。

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.

毕达哥拉斯一个信念是“存在的本质是数”。所以存在有赖于创世的所有东西的稳定性。像健康这样的事情就有赖于各种成分的稳定比利；一种东西太多或太少都会引起导致不健康的不平衡。毕达哥拉斯认为思考就是使用观念数进行计算。和民间推测结合在一起，哲学就产生了一种这样的信念：存在本质的知识可以在数的形式中发现。如果再往前走一步，我们可以说，由于数学是隐藏的本质，所以存在的本质就是可以通过数学研究来发现的隐藏特征。

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

尽管有很多以毕达哥拉斯名义伪造的作品—至今仍有一些留存下来—在古代流传，但是毕达哥拉的作品没有遗留下来。

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”.

毕达哥拉斯的主要成就是发现音乐基于1到4这些书的比例间隔。他认为数的体系因而宇宙体系，都是基于这些数字和：10。毕达哥拉斯学派用Tetractys 与十而不是神起誓（注：The Tetractys is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row.）偶数女性的，奇数是男性的。这一学派的一个成员西帕索斯也发现了无理数，但这对毕达哥拉斯来说是不可想象的，据说西帕索斯被处死了。毕达哥拉斯或毕达哥拉斯学派也发现了平方数。他们发现，如果取四块小石头；把它们放到一个正方形中，正方形不仅边与边相等，而且当两个边相乘时，其结果等于正方形中石头的数量之和，因而得出“平方根”这种叫法。他也是第一批认为地球是圆形、所有行星都有一个轴而且都围绕中心点运动的人士之一。一开始，他确定这一中心点是地球，但后来他又用“行星都围绕着中心火转动”代替了这种看法，但中心火他从来没有认为是太阳。他也认为月亮是一颗行星，名之曰“对地”。

Pythagoras

Testimonies 证言

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’. 恩培多柯勒对此很有信心，他写道：“在他们之中有个人具有罕见的学识，最通晓各种智慧性工作，此人赢得了最多的智慧财富；因为他总是陷于深思，容易看出人们十辈子、二十辈子经历的所有事情的一切。”

2. Pythagoras wrote nothing… 毕达哥拉斯没有写下任何东西…

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. 最起码下面的观点尽人皆知；第一，他认为灵魂是不朽的，第二，他认为灵魂转变为其他生命；他也认为时间以某种方式轮回，没有绝对新的东西；最后，他认为所有的生命都应被看成同类。毕达哥拉斯看来是第一位把这种观念带给希腊的人。

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. 如果你相信毕达哥拉斯：同样的事情会轮回发生，那么，我要告诉你—再次坐在你现在的地方，我的手中拿着这个打字机，其他别的都和现在一样，那么，推测时间也和现在一样就是合理的。

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. 让我们如此考虑规则：在你出去到教堂之时，先拜神，在路上不说也不做与日常有关的事儿…不穿鞋进行供奉与崇拜…听从神并限制谈论别的…没有灯就莫谈毕达哥拉斯学派的事儿…对神与宗教信仰深信不疑…不被难以压抑的高兴所征服…禁止吃豆子…禁止吃活物。

6. So Pythagoras turned geometrical philosophy into a form of liberal education by seeking its first principles in a higher realm of reality…因此毕达哥拉斯通过在更高的实在王国寻求第一原则把几何哲学转变成自由教育的形式…

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. 他说，生活就像节日；正当有些人在节日中角逐之时，有些人从事商业，但是最好的人士当观众，所以在生活中奴性人追逐的是名或利，而哲学家则追求真理。

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观点，毕达哥拉斯使用药物保持身体的纯洁，使用音乐保持灵魂的净化。

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. 十是数字的本质…毕达哥拉斯也坚持认为，数字10的力量存在于数字4、4元组之中。原因是：如果你用1+2+3+4就会得到数字10…所以毕达哥拉斯学派习惯于把四元组[由1、2、3、4点组成的三角形]在为最有约束力的誓言：而且是他给我们这一代人留下四元组，其中包含不朽自然的源泉与根源。

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”. 与哲学哲学家同时代以及在他们之前的哲学家，所谓的毕达哥拉斯学派，首先从事数学研究，他们不仅开展了这种研究，而且也产生了这样的思想：数学的原则是万物的原则。由于这些原则在性质上是第一位的，而且在数字上他们似乎看到了已存与即存事物的许多类似性，这些类似性比在火、土和水中找到的还要多；由于他们也看到了改变与音阶的比例皆可用数字来表达；---也因为所有其他事物在整个性质上都是由数来展示的，并且数字是整个自然的首要的东西，他们就认为数元素就是万物的元素，整个天体就是音节与数字。所有数与音阶的性质都应该表现为与整个天体安排的性质与组成部分相一致，他们搜集并整合他们的这一图式；如果任何地方有什么空缺，他们就准备加以弥补一边让他们的整个理论保持一贯。例如，由于数字10倍认为是完美的，它包含数字的所有性质，所以他们认为在天空运动的天体是10个，但可见的天体只有9个，他们就发明了第十个“对地”，用于满足这种说法。