芝诺Zeno of Elea

关于芝诺的生平所知甚少。我们确实知道他是哲学家，据说他是Teleutagoras的儿子。我们对芝诺的了解主要来自柏拉图所写的对话巴门尼德。

芝诺是哲学家巴门尼德的学生和朋友，并在芝诺与他一起学习。埃利亚学派是希腊哲学中领先的前苏格拉底学派之一，由巴门尼德在意大利南部的芝诺创立。他的一元论哲学声称，看似存在的许多事物仅仅是一个永恒的现实，他称之为存在。他的原则是“一切是一”，变化或非存在是不可能的。当然，芝诺深受巴门尼德论证的影响，柏拉图告诉我们这两位哲学家在公元前450年左右一起访问了雅典。

尽管柏拉图描述了芝诺和巴门尼德访问雅典的情况，但这次访问是否真的发生过远未得到普遍接受。然而，柏拉图告诉我们，当时还年轻的苏格拉底在他们访问雅典时会见了芝诺和巴门尼德，并与他们讨论了哲学。根据对这三位哲学家出生日期的最佳估计，苏格拉底当时大约20岁，芝诺大约40岁，巴门尼德大约65岁，所以柏拉图的说法当然是可能的。

芝诺在访问雅典之前就已经写了一部哲学著作，柏拉图报告说，芝诺的书意味着他在访问雅典之前就已经在雅典获得了一定的名声。不幸的是，芝诺的著作没有留存下来，但几乎没有证据表明他写了不止一本书。芝诺在访问雅典之前写的书是他的著名著作，根据普罗克洛，其中包含四十个关于连续统的悖论。其中四个悖论，我们将在下面详细讨论，对数学的发展产生了深远的影响。

第欧根尼·拉尔修 [10]给出了芝诺生平的更多细节，这些细节通常被认为不可靠。芝诺在访问雅典后回到了芝诺，第欧根尼·拉尔修声称他在试图从芝诺城驱逐暴君的英勇行动中丧生。关于他的英雄事迹和在暴君手中遭受酷刑的故事很可能纯属虚构。第欧根尼·拉尔修还写到了芝诺的宇宙论，同样没有关于此事的佐证，但我们将在下面给出一些细节的说明。

根据柏拉图 [8]，芝诺关于四十个悖论的书是：-

……这是一部年轻时的作品，被人窃取了，因此作者没有机会考虑是否出版它。其目的是通过攻击关于事物的通常观念来捍卫巴门尼德的体系。

普罗克洛也描述了这部作品，并确认1：-

……芝诺在假设存在多样性和运动的前提下，精心构造了四十个不同的悖论，所有这些悖论显然都基于对连续统进行分析时所产生的困难。

在反对世界包含不止一个事物的观点的论证中，芝诺从这样一个假设出发推导出他的悖论：如果一个量可以被分割，那么它就可以被无限次地分割。芝诺还假设，一个没有量的事物不可能存在。西里西亚的辛普利修斯，即雅典柏拉图学园的最后一位首领，保存了包括巴门尼德和芝诺在内的许多早期作者的残篇。他在六世纪上半叶写作时，解释了芝诺关于为什么没有量的事物不可能存在的论证[1]：-

因为如果它被加到别的事物上，它不会使那个事物变大；如果它被减去，它也不会使那个事物变小。但是，如果它被加到某物上时不能使该物变大，从某物中减去时也不能使该物变小，那么显然，被加上或减去的乃是无。

尽管芝诺的论证并不完全令人信服，但至少正如Makin在[25]中所写的那样：-

芝诺对简单多元论的挑战是成功的，因为他迫使反巴门尼德主义者超越了常识。

芝诺关于运动的悖论更加令人困惑。亚里士多德在其著作Physics中给出了芝诺的四个论证：二分法、阿基里斯、飞矢和运动场。对于二分法，亚里士多德描述了芝诺的论证（在托马斯·利特尔·希思的译本[8]中）：-

没有运动，因为被移动的东西必须先到达其路程的中点，然后才能到达终点。

为了穿过一条线段，必须到达它的中点。要做到这一点，必须到达$\large\frac{1}{4}\normalsize$点，要做到这一点，必须到达$\large\frac{1}{8}\normalsize$点，如此等等ad infinitum。因此运动永远无法开始。这里的论证并没有被众所周知的无穷和所回答

$\large\frac{1}{2}\normalsize + \large\frac{1}{4}\normalsize + \large\frac{1}{8}\normalsize + ... = 1$

一方面，芝诺可以论证和$\large\frac{1}{2}\normalsize + \large\frac{1}{4}\normalsize + \large\frac{1}{8}\normalsize + ...$实际上永远达不到1，但更令人困惑的是试图反向求和$\large\frac{1}{2}\normalsize + \large\frac{1}{4}\normalsize + \large\frac{1}{8}\normalsize + ...$。在穿过单位距离之前，我们必须到达中点，但在到达中点之前，我们必须到达$\large\frac{1}{4}\normalsize$处，但在我们到达$\large\frac{1}{4}\normalsize$处之前，我们必须到达$\large\frac{1}{8}\normalsize$处，等等。这个论证使我们意识到我们永远无法开始，因为我们试图从“错误”的一端来构建这个无穷和。确实，这是一个巧妙的论证，至今仍困扰着人类的心灵。

芝诺将二分法悖论和对简单多元论的攻击都建立在这样一个事实上：一旦一个事物是可分的，那么它就是无限可分的。人们可以通过假设一个原子论来反驳他的悖论，其中物质由许多小的不可分割的元素组成。然而，芝诺给出的其他悖论之所以造成问题，恰恰是因为在这些情况下，他认为看似连续的量是由不可分割的元素组成的。这样的悖论就是“飞矢”，我们再次给出亚里士多德对芝诺论证的描述（在托马斯·利特尔·希思的译本[8]中）：-

如果，芝诺说，当一切事物占据与自身相等的空间时，要么静止要么运动，而被移动的对象处于瞬间之中，那么移动的箭就是不动的。

这一论证基于这样一个事实：如果在不可分割的时间瞬间中箭移动了，那么这个时间瞬间就确实是可分割的（例如，在更小的‘瞬间’中，箭会移动一半的距离）。亚里士多德通过以下主张来反驳这一悖论：-

……因为时间并非由不可分割的‘现在’组成，任何其他量度也是如此。

然而，一些人认为这与芝诺的论证无关。此外，否认‘现在’作为划分过去与未来的瞬间而存在，似乎也违背直觉。当然，如果瞬间‘现在’不存在，那么箭就从未占据任何特定位置，这似乎也不对。再者，芝诺提出了一个深刻的问题，尽管经过数百年努力解决，似乎仍缺乏真正令人满意的解答。正如Frankel在[20]中所写：-

人类心智在试图准确描述运动时，发现自己面对现象的两个方面。两者都不可避免，但同时相互排斥。要么我们着眼于运动的连续流动；那么我们就无法思考对象处于任何特定位置。要么我们把对象视为占据其路径所引导它经过的任何位置；而当我们的思想固定在那特定位置时，我们不禁要固定对象本身，并让它在一个短暂瞬间内静止。

Vlastos（见[32]）指出，如果我们使用速度的标准数学公式，我们有$v = \large\frac{s}{t}\normalsize$，其中$s$是行进的距离，$t$是所花的时间。如果我们看某一瞬间的速度，我们得到$v = \large\frac{0}{0}\normalsize$，这是无意义的。因此可以说，芝诺在这里指出了一个数学困难，直到极限和微分学被研究并置于适当基础之上，这个困难才得到妥善处理。

从上述讨论可以看出，芝诺的悖论在无穷小概念的发展中很重要。事实上，一些作者声称芝诺的悖论是针对那些引入无穷小的人而提出的。阿那克萨戈拉和毕达哥拉斯的追随者，随着他们对incommensurables的发展，也被一些人认为是芝诺论证的目标（例如见[10]）。当然，柏拉图给出的理由，即捍卫Parmenides的哲学立场，似乎不太可能是芝诺撰写其关于悖论的著名作品的全部解释。

在芝诺的论证中，最著名的无疑是阿基里斯。托马斯·利特尔·希思从亚里士多德的Physics中的翻译是：-

……跑得慢的永远不会被跑得快的追上；因为追赶者必须先到达逃跑者出发的那一点，所以跑得慢的必然总是领先一段距离。

大多数作者，从亚里士多德开始，认为这个悖论本质上与二分法相同。例如Makin [25]写道：-

……只要二分法能够解决，阿基里斯就能解决。解决方案将是平行的。

与关于芝诺悖论的大多数陈述一样，对于任何特定立场都没有完全一致的意见。例如Toth [29]对这两个悖论的相似性提出异议，声称亚里士多德的评论有很多不足之处，并暗示这两个论证具有完全不同的结构。

柏拉图和亚里士多德都没有充分认识到芝诺论证的重要性。正如托马斯·利特尔·希思所说[8]：-

亚里士多德称它们为“谬误”，却无法反驳它们。

伯特兰·罗素在[13]中写道时，肯定没有低估芝诺的重要性：-

在这个反复无常的世界里，没有什么比死后的名声更反复无常了。后世缺乏判断力的最显著受害者之一就是埃利亚的芝诺。他发明了四个无比精妙和深刻的论证，但后来哲学家的粗俗却宣称他只是一个巧妙的魔术师，他的论证全都是诡辩。经过两千年的不断反驳，这些诡辩被恢复，并成为数学复兴的基础……

这里伯特兰·罗素想到的是格奥尔格·康托尔、戈特洛布·弗雷格和他自己关于无限的工作，特别是卡尔·魏尔斯特拉斯关于微积分的工作。在[2]中也讨论了悖论与数学的关系，作者得出了与上述引文中Frankel相似的结论：-

尽管它们常常被视为逻辑上的无稽之谈而被摒弃，但也有许多尝试通过数学定理来消除它们，例如收敛级数理论或集合论。然而，最终，他的论证中固有的困难总是变本加厉地卷土重来，因为人类心智的构造使得它能够以两种并不完全协调的方式来看待连续统。

很难确切说出芝诺的悖论对希腊数学的发展产生了什么影响。B L 巴特尔·伦德特·范德瓦尔登（见[31]）认为，公元前五世纪下半叶发展起来的数学理论表明芝诺的工作影响甚微。然而托马斯·利特尔·希思似乎察觉到了更大的影响[8]：-

然而，数学家们……意识到芝诺的论证对无穷小是致命的，他们看到，只有一劳永逸地将无穷——甚至潜无穷——的观念从他们的科学中彻底驱逐出去，才能避免与之相关的困难；因此，从此以后，他们不再使用无限增大或减小的量，而是满足于可以随意取任意大或任意小的有限量。

我们在上文评论过，芝诺在10中描述了一种他认为出自芝诺的宇宙论。根据他的描述，芝诺提出了一种由若干世界组成的宇宙，这些世界由“热”与“冷”、“干”与“湿”构成，但没有虚空或空的空间。由于这似乎与他的悖论毫无共同之处，通常认为芝诺弄错了。然而，有一些证据表明，这种类型的信念在公元前五世纪就已存在，尤其与医学理论有关，而且它很可能就是芝诺对爱利亚学派所持信念的表述。

Very little is known of the life of Zeno of Elea. We certainly know that he was a philosopher, and he is said to have been the son of Teleutagoras. The main source of our knowledge of Zeno comes from the dialogue Parmenides written by Plato.

Zeno was a pupil and friend of the philosopher Parmenides and studied with him in Elea. The Eleatic School, one of the leading pre-Socratic schools of Greek philosophy, had been founded by Parmenides in Elea in southern Italy. His philosophy of monism claimed that the many things which appear to exist are merely a single eternal reality which he called Being. His principle was that "all is one" and that change or non-Being are impossible. Certainly Zeno was greatly influenced by the arguments of Parmenides and Plato tells us that the two philosophers visited Athens together in around 450 BC.

Despite Plato's description of the visit of Zeno and Parmenides to Athens, it is far from universally accepted that the visit did indeed take place. However, Plato tells us that Socrates, who was then young, met Zeno and Parmenides on their visit to Athens and discussed philosophy with them. Given the best estimates of the dates of birth of these three philosophers, Socrates would be about 20, Zeno about 40, and Parmenides about 65 years of age at the time, so Plato's claim is certainly possible.

Zeno had already written a work on philosophy before his visit to Athens and Plato reports that Zeno's book meant that he had achieved a certain fame in Athens before his visit there. Unfortunately no work by Zeno has survived, but there is very little evidence to suggest that he wrote more than one book. The book Zeno wrote before his visit to Athens was his famous work which, according to Proclus, contained forty paradoxes concerning the continuum. Four of the paradoxes, which we shall discuss in detail below, were to have a profound influence on the development of mathematics.

Diogenes Laertius [10] gives further details of Zeno's life which are generally thought to be unreliable. Zeno returned to Elea after the visit to Athens and Diogenes Laertius claims that he met his death in a heroic attempt to remove a tyrant from the city of Elea. The stories of his heroic deeds and torture at the hands of the tyrant may well be pure inventions. Diogenes Laertius also writes about Zeno's cosmology and again there is no supporting evidence regarding this, but we shall give some indication below of the details.

Zeno's book of forty paradoxes was, according to Plato [8]:-

... a youthful effort, and it was stolen by someone, so that the author had no opportunity of considering whether to publish it or not. Its object was to defend the system of Parmenides by attacking the common conceptions of things.

Proclus also described the work and confirms that [1]:-

... Zeno elaborated forty different paradoxes following from the assumption of plurality and motion, all of them apparently based on the difficulties deriving from an analysis of the continuum.

In his arguments against the idea that the world contains more than one thing, Zeno derived his paradoxes from the assumption that if a magnitude can be divided then it can be divided infinitely often. Zeno also assumes that a thing which has no magnitude cannot exist. Simplicius, the last head of Plato's Academy in Athens, preserved many fragments of earlier authors including Parmenides and Zeno. Writing in the first half of the sixth century he explained Zeno's argument why something without magnitude could not exist [1]:-

For if it is added to something else, it will not make it bigger, and if it is subtracted, it will not make it smaller. But if it does not make a thing bigger when added to it nor smaller when subtracted from it, then it appears obvious that what was added or subtracted was nothing.

Although Zeno's argument is not totally convincing at least, as Makin writes in [25]:-

Zeno's challenge to simple pluralism is successful, in that he forces anti-Parmenideans to go beyond common sense.

The paradoxes that Zeno gave regarding motion are more perplexing. Aristotle, in his work Physics, gives four of Zeno's arguments, The Dichotomy, The Achilles, The Arrow, and The Stadium. For the dichotomy, Aristotle describes Zeno's argument (in Heath's translation [8]):-

There is no motion because that which is moved must arrive at the middle of its course before it arrives at the end.

In order the traverse a line segment it is necessary to reach its midpoint. To do this one must reach the $\large\frac{1}{4}\normalsize$ point, to do this one must reach the $\large\frac{1}{8}\normalsize$ point and so on ad infinitum. Hence motion can never begin. The argument here is not answered by the well known infinite sum

$\large\frac{1}{2}\normalsize + \large\frac{1}{4}\normalsize + \large\frac{1}{8}\normalsize + ... = 1$

On the one hand Zeno can argue that the sum $\large\frac{1}{2}\normalsize + \large\frac{1}{4}\normalsize + \large\frac{1}{8}\normalsize + ...$ never actually reaches 1, but more perplexing to the human mind is the attempts to sum $\large\frac{1}{2}\normalsize + \large\frac{1}{4}\normalsize + \large\frac{1}{8}\normalsize + ...$ backwards. Before traversing a unit distance we must get to the middle, but before getting to the middle we must get $\large\frac{1}{4}\normalsize$ of the way, but before we get $\large\frac{1}{4}\normalsize$ of the way we must reach $\large\frac{1}{8}\normalsize$ of the way etc. This argument makes us realise that we can never get started since we are trying to build up this infinite sum from the "wrong" end. Indeed this is a clever argument which still puzzles the human mind today.

Zeno bases both the dichotomy paradox and the attack on simple pluralism on the fact that once a thing is divisible, then it is infinitely divisible. One could counter his paradoxes by postulating an atomic theory in which matter was composed of many small indivisible elements. However other paradoxes given by Zeno cause problems precisely because in these cases he considers that seemingly continuous magnitudes are made up of indivisible elements. Such a paradox is 'The Arrow' and again we give Aristotle's description of Zeno's argument (in Heath's translation [8]):-

If, says Zeno, everything is either at rest or moving when it occupies a space equal to itself, while the object moved is in the instant, the moving arrow is unmoved.

The argument rests on the fact that if in an indivisible instant of time the arrow moved, then indeed this instant of time would be divisible (for example in a smaller 'instant' of time the arrow would have moved half the distance). Aristotle argues against the paradox by claiming:-

... for time is not composed of indivisible 'nows', no more than is any other magnitude.

However, this is considered by some to be irrelevant to Zeno's argument. Moreover to deny that 'now' exists as an instant which divides the past from the future seems also to go against intuition. Of course if the instant 'now' does not exist then the arrow never occupies any particular position and this does not seem right either. Again Zeno has presented a deep problem which, despite centuries of efforts to resolve it, still seems to lack a truly satisfactory solution. As Frankel writes in [20]:-

The human mind, when trying to give itself an accurate account of motion, finds itself confronted with two aspects of the phenomenon. Both are inevitable but at the same time they are mutually exclusive. Either we look at the continuous flow of motion; then it will be impossible for us to think of the object in any particular position. Or we think of the object as occupying any of the positions through which its course is leading it; and while fixing our thought on that particular position we cannot help fixing the object itself and putting it at rest for one short instant.

Vlastos (see [32]) points out that if we use the standard mathematical formula for velocity we have $v = \large\frac{s}{t}\normalsize$, where $s$ is the distance travelled and $t$ is the time taken. If we look at the velocity at an instant we obtain $v = \large\frac{0}{0}\normalsize$, which is meaningless. So it is fair to say that Zeno here is pointing out a mathematical difficulty which would not be tackled properly until limits and the differential calculus were studied and put on a proper footing.

As can be seen from the above discussion, Zeno's paradoxes are important in the development of the notion of infinitesimals. In fact some authors claim that Zeno directed his paradoxes against those who were introducing infinitesimals. Anaxagoras and the followers of Pythagoras, with their development of incommensurables, are also thought by some to be the targets of Zeno's arguments (see for example [10]). Certainly it appears unlikely that the reason given by Plato, namely to defend Parmenides' philosophical position, is the whole explanation of why Zeno wrote his famous work on paradoxes.

The most famous of Zeno's arguments is undoubtedly the Achilles. Heath's translation from Aristotle's Physics is:-

... the slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead.

Most authors, starting with Aristotle, see this paradox to be essentially the same as the Dichotomy. For example Makin [25] writes:-

... as long as the Dichotomy can be resolved, the Achilles can be resolved. The resolutions will be parallel.

As with most statements about Zeno's paradoxes, there is not complete agreement about any particular position. For example Toth [29] disputes the similarity of the two paradoxes, claiming that Aristotle's remarks leave much to be desired and suggests that the two arguments have entirely different structures.

Both Plato and Aristotle did not fully appreciate the significance of Zeno's arguments. As Heath says [8]:-

Aristotle called them 'fallacies', without being able to refute them.

Russell certainly did not underrate Zeno's significance when he wrote in [13]:-

In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance ....

Here Russell is thinking of the work of Cantor, Frege and himself on the infinite and particularly of Weierstrass on the calculus. In [2] the relation of the paradoxes to mathematics is also discussed, and the author comes to a conclusion similar to Frankel in the above quote:-

Although they have often been dismissed as logical nonsense, many attempts have also been made to dispose of them by means of mathematical theorems, such as the theory of convergent series or the theory of sets. In the end, however, the difficulties inherent in his arguments have always come back with a vengeance, for the human mind is so constructed that it can look at a continuum in two ways that are not quite reconcilable.

It is difficult to tell precisely what effect the paradoxes of Zeno had on the development of Greek mathematics. B L van der Waerden (see [31]) argues that the mathematical theories which were developed in the second half of the fifth century BC suggest that Zeno's work had little influence. Heath however seems to detect a greater influence [8]:-

Mathematicians, however, ... realising that Zeno's arguments were fatal to infinitesimals, saw that they could only avoid the difficulties connected with them by once and for all banishing the idea of the infinite, even the potentially infinite, altogether from their science; thenceforth, therefore, they made no use of magnitudes increasing or decreasing ad infinitum, but contented themselves with finite magnitudes that can be made as great or as small as we please.

We commented above that Diogenes Laertius in [10] describes a cosmology that he believes is due to Zeno. According to his description, Zeno proposed a universe consisting of several worlds, composed of "warm" and "cold, "dry" and "wet" but no void or empty space. Because this appears to have nothing in common with his paradoxes, it is usual to take the line that Diogenes Laertius is in error. However, there is some evidence that this type of belief was around in the fifth century BC, particularly associated with medical theory, and it could easily have been Zeno's version of a belief held by the Eleatic School.