安提丰Antiphon

安提丰是一位演说家和政治家，他以修辞学为职业。他是智者，也是苏格拉底的同时代人。然而，这些明确的论断受到一些历史学家的质疑。问题似乎围绕着究竟是否有一位名叫安提丰的安提丰哲学家生活在这一时期，还是存在两位，或者如一些专家所声称的，三位不同的安提丰。

在下文中，我们将假设至少那位名叫安提丰的演说家与取得数学进展的安提丰是同一个人。这与[1]中所采取的立场相同，而在[2]中只讨论了作为演说家的安提丰，没有提及哲学或数学著作。在[7]中讨论了安提丰是一个还是几个不同的人的假说，但没有偏向任何一种明确的观点。

安提丰所写的一些演说词被保存了下来。其中三篇是安提丰在谋杀案审判中作为公诉人所作的真实演说。十二篇是安提丰为在案件中教授学生起诉和辩护技巧而写的示范演说。这些演说分为三组，每组四篇；三个不同案件各包括两篇起诉演说和两篇辩护演说。

安提丰出版了一些哲学著作，这些著作已经失传，只有少数残篇被发现，还有一些其他作者著作中对这些作品的引用。这些著作包括On Truth, On Concord, The Statesman和On Interpretation of Dreams。著作On Truth是为了支持巴门尼德的观点而写的，后者认为存在一个唯一的实在，而众多事物的表象世界是不真实的。在这部著作中，安提丰捍卫了芝诺用其悖论所支持的相同哲学思想。

在On Concord 安提丰 [1]中：-

……捍卫共同体的权威，以此作为防止无政府状态的保障，并推荐共同体内部以及个人灵魂内部的和谐与自我克制的理想。最可能的是，他仅仅关注于通过询问一个城邦的法律是否满足个人的“自然”需求来批评这些法律。

Hobbs在[7]中指出：-

……有些人怀疑同一个人是否可能既写了《论真理》，又写了《论和谐》中那些传统的格言式言论。

在[7]中给出了三个理由，至少支持这两部哲学著作出自同一位作者：-

(1) "On Truth" is not as radical as it appears, but simply a plea for legal reform;

(2) its doctrines, although radical, are not endorsed by Antiphon;

(3) Antiphon changed his mind。

最后，在讨论哪些著作是由安提丰所写时，值得指出的是，一些历史学家否认安提丰写了归于他名下的另外两部著作The Statesman和On Interpretation of Dreams。

安提丰对数学做出了早期而重要的贡献，他尝试化圆为方。在此过程中，他成为第一个提出穷竭法的人，尽管他对自己提议的理解程度并不完全清楚。他提出将圆内接正多边形内接的边数连续加倍，使得面积之差最终被穷竭。

我们通过亚里士多德及其评注者了解他的工作。亚里士多德声称，几何学家只需证明基于几何学的错误论证是错的，否则可以忽略它们。亚里士多德在其Physics中写道（例如见[4]）：-

……因此，用线段来反驳求积是几何学家的事，但反驳安提丰的则不是他的事。

如果读者想知道亚里士多德所说的“用线段求积”指的是什么，那么几乎可以肯定他指的是lunes的希俄斯的希波克拉底方法。

然而西里西亚的辛普利修斯未能正确理解安提丰在做什么。他认为安提丰声称化圆为方。他写道（托马斯·利特尔·希思的译文见[4]）：-

安提丰认为，用这种方法圆的面积会被穷尽，我们终将得到一个内接于圆的多边形，其边由于极短而与圆周重合。既然我们可以作一个等于任意多边形的正方形……我们就能作一个等于圆的正方形。

然而，根据托马斯·利特尔·希思的说法，这并非安提丰所主张的[4]：-

因此，安提丰在几何学史中理应享有光荣的地位，因为他首创了用边数不断增加的内接正多边形来穷尽一个区域的思想，……欧多克索斯正是基于这一思想创立了他划时代的穷竭法。

1中的Kerferd提出，安提丰可能把圆视为一个边数众多的多边形：-

在现代，人们常常认为安提丰在几何学上犯了一个严重的错误，即认为任何近似都永远不可能达到一个无论有多少条边的多边形与一个连续弯曲的圆的圆周之间的重合。……这或许不是正确的看法。安提丰似乎相信，通过他的方法可以实现完全的重合……这可能意味着安提丰把圆视为一个边数极多（或可能无穷多）的多边形。

安提丰卷入了一场失败的反民主革命。修昔底德在其著名的History中认为安提丰是这场革命的领袖[2]：-

[安提丰]构思了整个事件以及实现它的手段。

尽管他的职业是辩护词撰写人，他那被修昔底德描述为如下的精彩演说：-

……一个人为生命受审时所作的最伟大的演说……

在他因叛国罪受审时未能救他，他被处决了。

Antiphon was an orator and statesman who took up rhetoric as a profession. He was a Sophist and a contemporary of Socrates. These definite assertions are, however, disputed by some historians. The problem seems to revolve round whether there was one Sophist philosopher named Antiphon who lived around this time or whether there are two, or as some experts claim, three distinct Antiphons.

In what follows we shall assume that at least the orator named Antiphon was the same person as the Sophist who made the mathematical advances. This is the same line as taken in [1] while in [2] only Antiphon as an orator is discussed without reference to the philosophical or mathematical works. In [7] the hypothesis that Antiphon is one, or several different men is discussed without any definite view being preferred either way.

A number of speeches which were written by Antiphon have been preserved. Three of these speeches were real speeches made by Antiphon as the prosecutor in murder trials. Twelve speeches are specimen speeches written by Antiphon for use in teaching students the skills of prosecuting and defending clients in cases. The speeches come as three collections of four; two prosecution speeches and two defence speeches for each of three different cases.

Antiphon published a number of works on philosophy which have been lost except for a small number of fragments which have been discovered together with some quotations from the works in the writings of other authors. These works include On Truth, On Concord, The Statesman, and On Interpretation of Dreams. The work On Truth is written to support the views of Parmenides who believed that there was a single sole reality and that the apparent world of many things was unreal. In this work Antiphon is defending the same philosophical ideas which Zeno of Elea supported with his paradoxes.

In On Concord Antiphon [1]:-

... defends the authority of the community as a safeguard against anarchy and recommends the ideals of concord and self-restraint both within communities and within the individual soul. Most probably he was only concerned to criticise the laws of a city by asking whether or not they satisfy the "natural" needs of the individual.

Hobbs in [7] notes that:-

... some have doubted whether the same man could have written "On Truth" and the conventional gnomic utterances of "On Concord".

In [7] three reasons are given to support at least the same author for these two philosophical works:-

(1) "On Truth" is not as radical as it appears, but simply a plea for legal reform;

(2) its doctrines, although radical, are not endorsed by Antiphon;

(3) Antiphon changed his mind.

Finally while discussing which works were written by Antiphon it is worth remarking that some historians reject the idea that Antiphon wrote the two other works attributed to him The Statesman and On Interpretation of Dreams.

Antiphon made an early and important contribution to mathematics when he made an attempt to square the circle. In doing so he became the first to propose a method of exhaustion although it is not entirely clear how well he understood his own proposal. He proposed successively doubling the number of sides of a regular polygon inscribed in a circle so that the difference in areas would eventually become exhausted.

We know of his work via Aristotle and his commentators. Aristotle claims that a geometer only needs to show that false arguments are false if they are based on geometry, otherwise he can ignore them. Aristotle writes in his Physics (see for example [4]):-

... thus it is the geometer's business to refute the quadrature by means of segments, but it is not his business to refute that of Antiphon.

In case the reader is wondering what Aristotle refers to with his phrase 'quadrature by means of segments' then it is almost certain that he means the method of lunes of Hippocrates.

However Simplicius failed to properly understand what Antiphon was doing. He thought that Antiphon was claiming to have squared the circle. He wrote (translation by Heath given in [4]):-

Antiphon thought that in this way the area of the circle would be used up, and we should some time have a polygon inscribed in the circle the sides of which, owing to their smallness, coincide with the circumference of the circle. And as we can make a square equal to any polygon ... we shall be in a position to make a square equal to a circle.

However, according to Heath, this was not what Antiphon claimed [4]:-

Antiphon therefore deserves an honourable place in the history of geometry as having originated the idea of exhausting an area by means of inscribed regular polygons with an ever increasing number of sides, an idea upon which ... Eudoxus founded his epoch-making method of exhaustion.

Kerferd in [1] suggests that Antiphon may have regarded a circle as a polygon with a large number of sides:-

In modern times it has often been supposed that Antiphon was simply making a bad mistake in geometry by supposing that any approximation could ever amount to coincidence between a polygon with however many sides and a continuously curved circumference of a curved circle. ... This may not be the right view to take. Antiphon appears to have believed that complete coincidence could be achieved by his method ... This may mean that Antiphon regarded the circle as a polygon with a very large (or possibly infinite) number of sides.

Antiphon was involved in an anti-democratic revolution which failed. Thucydides' in his famous History believes that Antiphon was the leader of the revolution [2]:-

[Antiphon] conceived the whole matter and the means by which it was brought to pass.

Despite his profession as a writer of defence speeches, his brilliant speech, described by Thucydides as:-

... the greatest ever made by a man on trial for his life...

failed to save him when he was tried for treason and he was executed.