Perhaps the best known of the pre-Socratic philosophers, Pythagoras was a near-mythical figure who established a cultlike community devoted to the pursuit of science, mathematics, and mysticism.
在前苏格拉底时期的哲学家中，毕达哥拉斯或许是最为人所熟知的一位。他堪称一个近乎神话般的人物，创立了一个颇具教派色彩的团体，致力于钻研科学、数学以及神秘主义。
A cosmos governed by numbers
由数字主宰的宇宙
Pythagoras (c.570–c.495 bce) is remembered as the mathematician who gave his name to the theorem of right-angled triangles—that the square of the hypotenuse is equal to the sum of the squares of the other two sides. However, in his own time, he was better known for his belief in the transmigration (rebirth) of the soul. Little is known of what he actually thought, since he left no written legacy and many of the ideas ascribed to him may very well be those of others. However, it is certain that he set up a community in southern Italy and trained his followers in philosophical and scientific inquiry. The “so-called Pythagoreans,” as Aristotle later described them, studied astronomy and geometry and examined the link between numbers, mathematics, and the natural world. For example, the Pythagoreans— notably Philolaus—discovered that musical harmony is based on mathematical ratios using the first four whole numbers (see below).
毕达哥拉斯（约公元前570年—约公元前495年）作为数学家而名垂青史，著名的勾股定理（即直角三角形斜边的平方等于另外两条直角边的平方和）便以他的名字命名。然而，在他所处的那个时代，他其实更因坚信灵魂转世（重生）之说而广为人知。 由于他并未留下任何书面著作，我们对他的真实思想了解甚少，许多被归到他名下的观点很可能实际上出自他人之手。不过，可以确定的是，他在意大利南部建立了一个社群，并教导他的追随者们进行哲学与科学探究。 亚里士多德后来将这些人称作“所谓的毕达哥拉斯学派成员”。他们钻研天文学与几何学，探寻数字、数学与自然世界之间的内在联系。例如，毕达哥拉斯学派的成员——尤其是菲洛劳斯——发现，音乐的和谐源自数学比例，而且运用到了最初的四个自然数。
Pythagoras is believed to have learned geometry from Thales (see pp.16–17). However, he was also familiar with the cosmological theories of the Milesian school, and Anaximander in particular, whose chief thesis was that the cosmos is formed from “the Boundless”—an inexhaustible, unobservable, lifegiving substance. Pythagoras reasoned that the cosmos must have an underlying structure determined by the laws of mathematics, which imposes limits on the Boundless, giving form to the universe. For the Pythagoreans, the cosmos—and everything in it—is governed by numbers, so numbers have an almost divine significance.
人们认为毕达哥拉斯曾向泰勒斯学习几何学。不过，他也熟知米利都学派的宇宙论，尤其是阿那克西曼德的理论。阿那克西曼德的核心观点是，宇宙由“无定”（阿派朗）构成——那是一种无穷无尽、无法观测却孕育生命的物质。毕达哥拉斯推断，宇宙必然存在一种由数学法则决定的底层结构，这些法则对“无定”（阿派朗）加以约束，赋予了宇宙以具体的形态。对于毕达哥拉斯学派而言，宇宙以及其中的一切都受数字支配，因此数字几乎具有神圣的意义。
Sacred numbers
Numbers took on a mystical significance for the Pythagoreans as they made links between mathematics and the natural world. The first four integers (whole numbers) were especially important: 1) the fundamental number associated with the origin of everything; 2) the material derived from it; 3) the beginning, middle, and end; and 4) the number of the elements. Together, they add up to 10—the “perfect number.”
神圣数字
对于毕达哥拉斯学派的成员而言，数字具有神秘的意义，因为他们发现了数学与自然世界之间的关联。最初的四个自然数尤为关键：1）代表着与万物起源相关的基本数字；2）由1衍生出的物质；3）象征着开始、中间和结束；4）代表着元素的数量。这四个数字相加等于10——这个被视为“完美数字”的数值。
GEOMETRIC OBJECTS
Pythagoreans revered the number 1, from which they believed all numbers derive. For example, geometric figures can be created from a single point: connecting two points creates a line; connecting parallel lines forms a square; and connecting parallel squares creates a cube.
几何图形
毕达哥拉斯学派尊崇数字1，他们认为所有数字皆源于1。例如，几何图形可以从一个点开始构建：连接两个点形成一条线段；连接相互平行的线段形成一个正方形；连接相互平行的正方形则形成一个立方体。
THE OCTAVE
Pythagoras also discovered that musical intervals that sound harmonious when played together correspond to the mathematical ratios of 1:2, 2:3, and 3:4. This means that if a string sounds the note A, a string half of its length will sound the A an octave higher (an eighth above), a string two-thirds its length will sound the note E (a fifth above), and a string three-quarters its length will sound the note D (a fourth above). For Pythagoras, it was no coincidence that these ratios only involve the first four integers, which add up to the perfect number, 10.
八度音程
毕达哥拉斯还发现，当同时演奏时听起来和谐悦耳的音乐音程，与1:2、2:3和3:4这些数学比例相对应。这意味着，如果一根弦发出的音高为A，那么长度为其一半的弦会发出高一个八度（高八度音）的A音；长度为其三分之二的弦会发出E音（高五度音）；长度为其四分之三的弦会发出D音（高四度音）。对毕达哥拉斯来说，这些比例仅涉及最初的四个自然数，而它们相加又等于完美数字10，这绝非偶然。
“The Pythagoreans ... fancied that the principles of mathematics were the principles of all things.”
Aristotle, Metaphysics (4th century bce)
“毕达哥拉斯学派……认为数学原理是万物的原理。”
——亚里士多德，《形而上学》（公元前4世纪）
THE COSMOS
Philolaus, Pythagoras’s student, is credited with the idea that all the heavenly bodies—including the Earth and a “Counter-Earth”—orbit a central fire called the Hearth. The distances of the stars and planets from the center correspond to the ratios of the consonant musical intervals, creating what the Pythagoreans referred to as the “harmony of the spheres.”
宇宙
毕达哥拉斯的学生菲洛劳斯被认为提出了这样一种观点：所有的天体——包括地球和一个“对地”——都围绕着一个被称为“中心火”的中央火源运转。恒星和行星与中心的距离，与和谐的音乐音程比例相对应，从而形成了毕达哥拉斯学派所称的“天体的和谐”。
THE TETRACTYS
The tetractys—a triangle composed of 10 dots—had great symbolic significance for the Pythagoreans. Its rows of one, two, three, and four add up to the perfect number 10, and its central dot is comparable to the Hearth at the center of the cosmos.
四元体
四元体——一个由10个点组成的三角形——对毕达哥拉斯学派而言具有重大的象征意义。它的四行点，分别为1个、2个、3个和4个，相加等于完美数字10，而且其中心的点可类比为宇宙中心的“中心火”。
