Zeno of Sidon

Zeno of Sidon出生在of Sidon城，位于今天黎巴嫩的地中海沿岸。Sidon是最古老的腓尼基城市之一，自公元前3千年建城以来，曾被许多不同的民族统治：亚述、巴比伦、波斯、亚历山大大帝、叙利亚的塞琉古王朝、埃及的托勒密王朝和罗马人。

要理解芝诺的哲学，我们需要对哲学家伊壁鸠鲁作一些评论，他创立了芝诺后来所属的Epicurean School。伊壁鸠鲁生活在公元前341年至公元前270年，他基于自己的学说创立了自己的哲学学派。这些学说旨在指明一种生活方式，既旨在保证幸福，也旨在提供找到幸福的方法。伊壁鸠鲁对科学本身没有兴趣，并且是数学的严厉批评者。关于科学，他写道：-

如果我们不为对天空现象和死亡的疑虑所困扰，也不为未能把握痛苦和欲望的限度所困扰，我们就不需要自然科学。

他对数学的批评非常肤浅，无关紧要，因为他显然对这门学科了解甚少。公元前306年，他在雅典自家花园里创立了自己的学派。顺理成章地，这个学派被称为“花园”。

阿波罗多罗斯，写了400多本书，是伊壁鸠鲁的著名追随者，生活于公元前2世纪。Zeno of Sidon是Apollodorus的学生，他在雅典的花园中学习，后来也在那里任教。西塞罗于公元前79年在那里听到他讲课。

Zeno是一位学识渊博的人，其著述涉及极为广泛的主题。据信，在他所研究的领域中，他对逻辑学、原子论、生物学、伦理学、文学风格、修辞学、诗歌、知识论以及数学都有贡献。除了最后提到的两个主题外，我们对他所作贡献的了解甚少。这里我们将讨论Zeno有贡献且其贡献细节为我们所熟知的仅有的两个领域，即知识论和数学。

尽管Epicurus——Zeno所属学派的创始人——并无真正的数学才能，并站在无知的立场上批评这门学科，但Zeno的情况远非如此，他对这门学科有深刻的理解。Zeno对欧几里得在The Elements中提出的公理作了深刻的批评。例如，他声称欧几里得的第一命题假定两条直线最多只能相交于一点，但欧几里得并未将此作为公理，也无法从其他公理推出。

Zeno还抨击了欧几里得关于直角相等的证明，理由是它预设了直角的存在。普罗克洛也说，一位Epicurus学派的人（几乎可以肯定是Zeno，但普罗克洛未点其名）声称欧几里得假定每条曲线都可无限分割，但这同样无法从公理推出。

一些现代作者提出，这些主张使Zeno of Sidon有几分理由被认为是最早考虑non-Euclidean geometry可能性的人。这有些牵强，尤其是因为Zeno的目的肯定不是这个。相反，他的目的是给出实质性的论证来反对数学，以支持Epicurus的反数学信念。

托马斯·利特尔·希思在[2]中关于普罗克洛对芝诺的评论写道：-

芝诺一般地论证说，即使我们承认几何学的基本原理，从这些原理推出的结论也无法得到证明，除非同时承认某些未被包含在上述原理中的其他东西，他打算通过这些批评来摧毁整个几何学。

数学家们当然是为自己的学科辩护，而不是试图理解芝诺深刻而有根据的评论。正如von Fritz在[1]中写道：-

然而，芝诺对欧几里得的批评是切中要害的，如果任何试图反驳这些批评的古代哲学家和数学家能够把握其全部含义，数学的发展可能会走上不同的道路。

许多人在历史上获得重要地位，或者未能获得这样的地位，都是运气使然。假如在芝诺之后有一位数学家能够继续发展他的思想，那么我们今天可能会把芝诺视为一位主要人物，他的数学天才的闪光改变了数学的进程。然而，这并未发生，芝诺思想的辉煌许多世纪以来都未得到赏识。

我们通过Zeno的一位学生、Gadara的Philodemus对Zeno有了更多了解。Philodemus在雅典师从Zeno，然后于公元前75年移居罗马，为罗马贵族Lucius Calpurnius Piso工作。Philodemus随后前往那不勒斯附近Herculaneum的Lucius别墅居住，带去了他相当可观的纸草书图书馆。

公元79年维苏威火山爆发时，赫库兰尼姆与庞贝和斯塔比伊一同被摧毁。赫库兰尼姆被大约16米厚的致密物质掩埋，这使该城得以保存，直到18世纪开始发掘。地面湿度的特殊条件保存了木材、布料、食物，尤其是⟦N1⟧的纸草卷。

纸草文献中包含了菲洛德穆所写的非凡信息，描述了他的老师芝诺与斯多亚派的争论。尽管芝诺追求快乐的伊壁鸠鲁哲学似乎与斯多葛派的义务伦理直接对立，但它们对生活方式的影响却相当相似。菲洛德穆所描述的争论涉及知识的基础。冯·弗里茨在[1]中写道：-

在这场争论中，芝诺捍卫了古老的伊壁鸠鲁学说，即所有人类知识都完全源自经验。然而，使其有趣的是，他的辩护基于一种理论……这本质上是对弗瑞兹·约翰斯图尔特·密尔的归纳理论的预期。……芝诺坚持认为，所有知识从根本上都是通过从大量案例中推断到所有案例而无需观察到反例来获得的。

当然，数学中有许多例子表明，芝诺对许多事例的观察实际上会暗示一个错误的结果。

Zeno of Sidon was born in the city of Sidon on the Mediterranean coast of what today is Lebanon. Sidon was one of the oldest Phoenician cities and, from its founding in the 3rd millennium BC, was ruled by many different peoples: Assyria, Babylonia, Persia, Alexander the Great, the Seleucids of Syria, the Ptolemys of Egypt, and the Romans.

To understand the philosophy of Zeno we need to make some comments about the philosopher Epicurus who founded the Epicurean School to which Zeno later belonged. Epicurus, who lived from 341 BC to 270 BC, founded his own School of philosophy based on his teachings. These teachings were designed to indicate a means of living ones life, and they aimed both to guarantee happiness and to provide a means to find it. Epicurus had no interest in science for its own sake and he was a severe critic of mathematics. On science he wrote:-

If we were not troubled by our suspicions of the phenomena of the sky and about death, and also by our failure to grasp the limits of pain and desires, we should have no need of natural science.

His criticisms of mathematics were very superficial of little importance since he clearly had very little understanding of the subject. In 306 BC he founded his School in Athens in the garden of his house. Reasonably enough the School became known as The Garden.

Apollodorus, the writer of more than 400 books, was a prominent follower of Epicurus who lived in the 2nd century BC. Zeno of Sidon was a student of Apollodorus and he studied, and later taught, in the Garden in Athens. Cicero heard him teaching there in 79 BC.

Zeno was a man of great learning who wrote on a very wide range of topics. It is believed that, among the areas he studied, he contributed to logic, atomic theory, biology, ethics, literary style, oratory, poetry, the theory of knowledge, and to mathematics. Except for the last mentioned two topics, we know very little about the contributions which he made. Here we shall discuss the only two areas to which Zeno contributed where details of his contributions are quite well known to us, namely the theory of knowledge and to mathematics.

Although Epicurus, the founder of the School to which Zeno belonged, had no real mathematical abilities and criticised the subject from a position of ignorance, this is far from true of Zeno who had a deep understanding of the subject. Zeno made deep criticisms of the axioms that Euclid set out in The Elements. For example he claimed that Euclid's first proposition assumes that two straight lines can intersect in at most one point but Euclid does not have this as an axiom, nor can it be deduced from the other axioms.

Zeno also attacked Euclid's proof of the equality of right angles on the grounds that it presupposes the existence of a right angle. Proclus also says that an Epicurean (almost certainly Zeno but Proclus does not name him) claimed that Euclid assumes that every curve is infinitely divisible, but again this cannot be deduced from the axioms.

Some modern authors have suggested that these claims give Zeno of Sidon some justification to be considered as having been the first person to consider the possibility of non-Euclidean geometry. This is a little far fetched particularly since Zeno's aim was certainly not this. Rather his aim was to give substantial arguments against mathematics supporting the anti-mathematical beliefs of Epicurus.

Heath writes in [2] regarding comments by Proclus concerning Zeno:-

Zeno argued generally that, even if we admit the fundamental principles of geometry, the deductions from them cannot be proved without the admission of something else as well which has not been included in the said principles, and he intended by means of these criticisms to destroy the whole of geometry.

Mathematicians of course, came to the defence of their subject, rather than to try to understand the deep and justified comments of Zeno. As von Fritz writes in [1]:-

Zeno's criticisms of Euclid are pertinent, however, and if any of the ancient philosophers and mathematicians who tried to refute them had been able to grasp their full implications, the development of mathematics might have taken a different turn.

Many people gain an important position in history, or fail to gain such a position, as a result of luck. Had there been a mathematician following Zeno who could have continued to develop his ideas then we might know Zeno today as a major figure whose flash of mathematical genius changed the course of mathematics. This was not to be, however, and the brilliance of Zeno's ideas were not appreciated for many centuries.

We know more of Zeno through one of his students Philodemus of Gadara. Philodemus studied under Zeno in Athens and then moved to Rome in 75 BC to work for the Roman aristocrat Lucius Calpurnius Piso. Philodemus then went to live in Lucius's villa at Herculaneum, near Naples, taking with him his considerable library of papyri.

When Vesuvius erupted in 79 AD, Herculaneum together with Pompeii and Stabiae, was destroyed. Herculaneum was buried by a compact mass of material about 16 metres deep which preserved the city until excavations began in the 18th century. Special conditions of humidity of the ground conserved wood, cloth, food, and in particular Philodemus's papyri.

The papyri contain remarkable information written by Philodemus describing the arguments of his teacher Zeno with the Stoics. Although Zeno's Epicurean philosophy of the desire for pleasure seems the direct opposite of the Stoic's ethic of duty, the consequences on how they lived their lives were quite similar. The arguments described by Philodemus concerned the foundations of knowledge. Von Fritz writes in [1]:-

In this dispute Zeno defended the old Epicurean doctrine that all human knowledge is derived exclusively from experience. What make it interesting, however, is that he bases his defence on a theory ... that is essentially an anticipation of John Stuart Mill's theory of induction. ... Zeno insisted that all knowledge is fundamentally derived by inference to all cases from a great many cases without observed counter-instance.

Of course there are many examples in mathematics where Zeno's observations of many cases would actually suggest a false result.