亚里士多德Aristotle

亚里士多德主要不是数学家，但通过系统化演绎逻辑做出了重要贡献。他写过物理学主题的著作：他的Analytica posteriora某些部分显示出对数学方法的非凡把握。然而，首要的是，他在所有知识的发展中都很重要，因为正如[2]的作者们所写：-

亚里士多德比任何其他思想家都更决定了西方思想史的方向和内容。他是一种哲学和科学体系的作者，这一体系历经数世纪成为中世纪基督教和伊斯兰经院思想的支撑和载体：直到17世纪末，西方文化都是亚里士多德式的。而且，即使在随后数世纪的思想革命之后，亚里士多德式的概念和观念仍然深深植根于西方思维之中。

亚里士多德出生在希腊北部哈尔基迪基半岛的斯塔基鲁斯，或称斯塔基拉，或称斯塔盖鲁斯。他的父亲是尼各马科，一名医生，而他的母亲名叫菲斯蒂斯。亚里士多德出生时，尼各马科肯定住在哈尔基迪基，他很可能就出生在该地区。亚里士多德的母亲菲斯蒂斯来自优卑亚岛的哈尔基斯，她的家族在那里拥有财产。

毫无疑问，尼各马科本打算让亚里士多德成为一名医生，因为传统是医学技能保密并由父传子。这不是一个人们去看医生的社会，而是医生们周游全国照料病人。虽然我们对亚里士多德的早年一无所知，但他很可能在旅行中陪伴父亲。我们确实知道，尼各马科发现哈尔基迪基的条件不如邻近的马其顿国令人满意，他开始在那里工作并取得了如此大的成功，以至于他很快被任命为马其顿国王阿明塔斯三世的私人医生。

没有记录表明亚里士多德是否与他的父亲一起住在马其顿首都佩拉，而尼各马科则在宫廷照料国王阿明塔斯。然而，亚里士多德在几年后肯定与国王阿明塔斯的儿子菲利普友好，而且似乎可以合理地假设，这两个年龄几乎完全相同的人，在佩拉还是幼童时就已成为朋友。

亚里士多德大约十岁时，他的父亲去世了。这肯定意味着亚里士多德现在无法继承父亲的医生职业，而且由于他的母亲似乎也早逝，亚里士多德由监护人阿塔尔纽斯的普罗克西努斯抚养长大，后者是他的叔叔（或者如一些作者所暗示的，可能是家庭朋友）。普罗克西努斯教亚里士多德希腊语、修辞学和诗歌，这补充了尼各马科给亚里士多德的生物学教导，作为训练儿子学医的一部分。由于亚里士多德在后来的生活中写出了优美的希腊散文，这也必定是他早期教育的一部分。

公元前367年，亚里士多德十七岁时成为雅典柏拉图学园的学生。当亚里士多德加入科学院时，它已经运作了二十年。柏拉图不在雅典，而是在他第一次访问叙拉古。我们不应把柏拉图学园看作一个只对抽象思想感兴趣的非政治组织。Academy深深卷入了当时的政治，事实上柏拉图访问西西里是出于政治原因，而科学院以及整个地区的政治将在影响亚里士多德一生历程方面发挥重要作用。

亚里士多德抵达雅典时，Academy在柏拉图缺席期间由Cnidos的欧多克索斯管理。斯彪西波，即柏拉图的侄子，也在科学院任教，Chalcedon的色诺克拉底也是如此。在当了一段时间的学生后，亚里士多德很快成为科学院的教师，并将在那里待二十年。关于亚里士多德在科学院教了什么，我们知之甚少。在[10]中，公元二世纪的第欧根尼·拉尔修写道，亚里士多德教授修辞学和辩证法。当然，亚里士多德此时写了关于修辞学的著作，发表了Gryllus，攻击了Isocrates的修辞学观点，后者在雅典经营着另一所重要的教育机构。亚里士多德此时的所有著作都强烈支持柏拉图的观点以及科学院的观点。

在亚里士多德于科学院的二十年接近尾声时，由于当时的政治事件，他的处境变得困难。马其顿国王Amyntas大约在公元前369年去世，也就是亚里士多德前往雅典加入科学院的前几年。Amyntas的两个儿子Alexander II和Perdiccas III先后统治马其顿一段时间，但王国既受内部纷争之苦，又受外部战争之扰。公元前359年，Amyntas的第三个儿子Philip II在Perdiccas抵抗Illyrian入侵时阵亡后登上王位。Philip运用巧妙的军事和政治策略，使马其顿获得了一段内部和平时期，在此期间他们通过对周边地区的胜利而扩张。

Philip在公元前348年占领了Olynthus并吞并了Chalcidice。亚里士多德出生的城镇Stagirus抵抗了一段时间，但也被Philip击败。雅典担心马其顿强大的威胁力量，然而亚里士多德曾在马其顿宫廷长大，可能一直保持着与Philip的友谊。事件的实际顺序现在有些不确定。柏拉图于公元前347年去世，Speusippus接任了科学院的领导。亚里士多德当然反对Speusippus的观点，他可能在柏拉图去世后离开科学院，原因可能是学术上的，也可能是因为他未能被任命为科学院的领袖。然而，一些资料表明，他可能在柏拉图去世前出于政治原因离开，因为他的马其顿关系使他不受欢迎。

亚里士多德从雅典前往Assos，该地面向Lesbos岛。他不是唯一离开科学院的人，Chalcedon的色诺克拉底也与他同行。在Assos，亚里士多德受到Atarneus的统治者Hermias的热烈欢迎。很可能亚里士多德当时是作为Philip的使者，而Hermias确实以此相待。亚里士多德娶了Pythias，即Hermias的侄女和养女，他们有一个孩子，一个也叫Pythias的女儿。然而，亚里士多德的妻子在结婚约10年后去世。据认为她比亚里士多德年轻得多，结婚时可能大约18岁。

亚里士多德成为赫尔米亚斯在那里聚集的哲学家团体的领袖。色诺克拉底可能也曾一度是该团体的成员。亚里士多德对解剖学和一般生物的结构有浓厚的兴趣，这种兴趣是他父亲在他早年培养起来的，这帮助他发展出了非凡的观察才能。亚里士多德和他的团体成员在阿索斯开始收集观察资料，特别是在动物学和生物学方面。巴恩斯在[6]中写道，亚里士多德的：-

……对动物的研究奠定了生物科学的基础；直到他去世两千多年后，这些研究才被取代。那些伟大著作所基于的探究，可能主要是在阿索斯和莱斯博斯进行的。

亚里士多德可能在阿索斯开始了他的著作Politics以及现已失传的On Kingship。他开始发展出一种与柏拉图不同的哲学，后者曾说国王应该是哲学家，哲学家应该是国王。在On Kingship中，亚里士多德写道，这是：-

……对国王来说，成为哲学家不仅没有必要，甚至是一种劣势。相反，国王应该听取真正哲学家的建议。这样他就会用善行而非善言来充实他的统治。

亚里士多德在阿索斯的时光因政治事件而结束。波斯人袭击了该镇，赫尔米亚斯被俘并被处死。亚里士多德逃脱，在前往马其顿的途中停留在莱斯博斯岛。这不仅仅是一次路过访问，因为他在那里待了大约一年，而且必定带着来自阿索斯的科学家团队，因为他们继续在那里进行生物学研究。

马其顿此时与雅典和平相处，因为腓力在公元前346年缔结了一项条约。公元前343年，亚里士多德到达马其顿宫廷，并在那里待了七年。经常被引用的故事说他成为腓力的儿子、年轻的亚历山大大帝的导师，几乎肯定是后来的虚构，正如Jaeger所指出的，见[16]。Grayeff在[12]中提出，腓力在亚里士多德身上看到了未来雅典科学院的领袖。这当然会非常适合腓力，因为当时的科学院领袖斯彪西波强烈反对腓力，并大力鼓励雅典反对马其顿的崛起。

雅典与马其顿之间的条约在公元前340年开始瓦解，战争的准备工作开始了。第二年斯彪西波去世，但亚里士多德虽然被提名为科学院的领袖，却没有当选。该职位落到了色诺克拉底手中，腓力对支持亚里士多德失去了兴趣。他搬回了斯塔吉拉的家，并带走了他的哲学家和科学家圈子。

亚里士多德在妻子去世后没有再婚，但他与来自家乡斯塔吉拉的Herpyllis建立了关系。他们何时初次见面尚不清楚，但他们共同育有一个儿子尼各马科，以亚里士多德的父亲命名。

腓力此时正处于权力的巅峰，但正如经常发生的那样，这被证明是内部纷争的时期。亚里士多德支持腓力的儿子亚历山大，后者很快成为国王。亚历山大决定采取与其父亲相似的对雅典政策，并寻求通过和平手段来确立自己的权力。亚历山大保护了科学院，并鼓励它继续工作。然而，与此同时，他派亚里士多德到雅典建立了一个竞争机构。

公元前335年，亚里士多德在雅典创办了自己的学校——吕克昂。他带着助手来到这座城市为学校配备人员，并带来了他在马其顿期间收集的大量教学材料；书籍、地图和其他教学材料，这些材料很可能在某一阶段是为了支持亚里士多德争取成为科学院的负责人。学园的旨趣一直较为狭窄，但亚里士多德领导下的吕克昂追求更广泛的学科。亚里士多德突出了对自然的详细研究，在这一学科以及他所研究的所有其他学科中[6]：-

他自己的研究是与人合作进行的，他将自己的想法传达给朋友和学生，从未想过将它们保留为私人宝藏。事实上，他认为一个人除非能够将自己的知识传授给他人，否则不能声称了解一个学科，并且他将教学视为知识的恰当表现。

流传给我们的著作是出自亚里士多德之手还是他学派后来的成员之手，在19世纪末受到许多学者的质疑。Jaeger[16]讨论了这些原因，但在这部著作中，Jaeger认为亚里士多德在不同著作中方法上的明显差异可以用他的思想在多年间的发展来解释。Grayeff[6]详细考察了某些文本，并再次声称它们代表了亚里士多德学派在他去世很久之后的思想发展。他写道：-

根据在他去世约二百五十年后兴起的一个传统，这个传统后来成为主流，甚至今天几乎无人质疑，亚里士多德在这些年里讲授——不是一次，而是两三次，几乎涉及每一门学科——逻辑学、物理学、天文学、气象学、动物学、形而上学、神学、心理学、政治学、经济学、伦理学、修辞学、诗学；并且他写下了这些讲稿，多次扩充和修改，直到达到我们现在读到的阶段。然而，更令人惊讶的是，这些学科中的大多数在他之前并不作为这样的学科存在，因此他将是第一个构想并建立它们作为系统学科的人。

公元前323年亚历山大大帝去世后，雅典的反马其顿情绪使亚里士多德退隐到哈尔基斯，他住在曾属于他母亲的房子里，这所房子仍由家族保留。次年，他因胃病去世，享年62岁。

几乎不可能确切地描绘亚里士多德的个性，但[2]的作者写道：-

关于他的轶事表明他是一个和蔼可亲、充满深情的人，几乎看不出某些学者认为能在其著作中察觉到的自负痕迹。保存下来的遗嘱也显示出同样和善的特点；他提到自己幸福的家庭生活，并对子女和仆人都关怀备至。

Barnes [6]写道：-

他有点花花公子气，手指上戴着戒指，头发剪得很时髦。他消化不良，据说还是细长腿。他善于言辞，讲课清晰，谈话有说服力；而且言辞犀利。他的敌人很多，把他描绘成傲慢专横。……作为一个人，我猜想，他令人钦佩而非和蔼可亲。

我们在上文评论了现代学者之间关于亚里士多德是否写了现在归于他名下的那些论文的争论。我们确实知道，他的作品分为两个截然不同的部分，即他生前发表而现已失传的作品（尽管有些片段通过他人著作中的引用而留存下来），以及流传给我们、并非由亚里士多德在其生前发表的文集。我们可以肯定地说，亚里士多德从未打算让这30部作品——印出来超过2000页——出版。它们肯定是吕克昂学园所授课程的讲义，要么如大多数学者所认为的那样是亚里士多德的作品，要么是后来讲师的。当然，它们也完全可能是亚里士多德最初讲授的课程笔记，但在亚里士多德去世后由其他讲师增补。

这些作品最早由吕克昂学园最后一任园长罗得岛的安德罗尼柯于约公元前60年出版。当然，[2]：-

今天所研究的亚里士多德著作的形式、标题和顺序，是由安德罗尼柯在哲学家去世近三个世纪后赋予的，而对它们漫长的评注史正是从这一阶段开始的。

这些著作包含什么内容？其中有关于逻辑学的重要著作。亚里士多德认为逻辑学不是一门科学，而是必须在研究每一门知识分支之前加以处理。亚里士多德给逻辑学起的名字是“分析学”，而“逻辑学”这一术语是由在科学院工作的色诺克拉底引入的。亚里士多德认为逻辑学必须应用于各门科学[6]：-

各门科学——至少理论科学——应当被公理化。那么，它们的公理应当是什么？一个命题要算作公理，必须满足什么条件？再者，每门科学内部的推导将采取什么形式？定理将依据什么规则从公理中推出？这些都是亚里士多德在其逻辑学著作中，特别是在被称为《前分析篇》和《后分析篇》的著作中提出的问题。

事实上，在Prior Analytics中，亚里士多德提出了如今著名的亚里士多德三段论，这是一种由两个前提和一个结论组成的论证形式。他的例子是：-

(i) Every Greek is a person。
(ii) Every person is mortal。
(iii) Every Greek is mortal。

亚里士多德并不是第一个提出公理系统的人。柏拉图曾大胆提出，可能存在一个囊括所有知识的单一公理系统。亚里士多德则倾向于一个更可行的建议，即为每门科学建立一个公理系统。请注意，欧几里得及其几何学公理系统出现在亚里士多德之后。

亚里士多德作出重大贡献的另一个主题是自然哲学，或者用今天的术语说，是物理学。（我[EFR]如果指出20世纪60年代我的学位课程中包括通过“普通自然哲学”考试，就暴露了我的年龄和圣安德鲁斯大学的传统性质。）亚里士多德考察物质、变化、运动、空间、位置和时间。他还对天文学研究作出贡献，尤其研究了彗星；对地理学作出贡献，考察了河流等特征；对化学作出贡献，对燃烧等过程感兴趣；此外还有气象学和彩虹研究。

除了在动物学和心理学方面的重要著作外，亚里士多德还写了他著名的形而上学著作。根据亚里士多德，这部著作研究[2]：-

……现实的最一般或最抽象的特征以及具有普遍有效性的原理。……形而上学研究的是，一切存在的事物仅就其存在而言必然具有的性质，[并且]它研究任何存在的事物都必须满足的一般条件。

亚里士多德似乎没有在数学上做出任何新发现，但他在数学的发展中很重要。正如托马斯·利特尔·希思在[14]中所解释的：-

正确理解亚里士多德的数学的重要性主要在于，他对科学方法的说明大多取自数学。

亚里士多德对初等数学有透彻的掌握，并相信数学作为三门理论科学之一具有极大的重要性。然而，可以公平地说，他不同意柏拉图，后者将数学提升到如此突出的研究地位，以至于亚里士多德所研究的一系列科学几乎没有空间。亚里士多德声称，另外两门理论科学是（用现代术语来说）哲学和理论物理学。

托马斯·利特尔·希思在[14]的导言中指出了亚里士多德在其著作中提到的一些数学内容：-

... 亚里士多德 知晓 欧多克索斯 的重要发现，这些发现深刻影响了 欧几里得 对《几何原本》的阐述。有一处暗示清楚地表明 亚里士多德 知道 欧多克索斯 伟大的比例理论，该理论由 欧几里得 在其第五卷中阐述，并认识到其重要性。另一段话回顾了 欧多克索斯 据以建立其测量面积和体积的‘穷竭法’的基本假设；当然，亚里士多德 熟悉 同心球 体系，欧多克索斯 和 卡里普斯 通过该体系从理论上解释了太阳、月亮和行星的独立运动。...

incommensurable 被一再提及，但所提及的情况是正方形的对角线与边的关系；没有暗示 西奥多罗斯 和 泰阿泰德 将该理论推广到其他情况...

托马斯·利特尔·希思 [14] 还提到了 亚里士多德 也许令人惊讶地没有提及的数学。有：-

... 没有提及 conic sections、doubling of the cube 或 trisection of an angle。squaring the circle 的问题在 安提丰、Bryson 和 希俄斯的希波克拉底 试图解决它的背景下被提及；但关于 希庇亚 的曲线却只字未提 ...

虽然 托马斯·利特尔·希思 [14] 讨论了 亚里士多德 中的许多数学引用，但该书 [5] 试图构建（或重构）一部关于 亚里士多德 数学哲学观点的著作。正如 Apostle 在 [5] 中所写：-

... 关于数学的众多段落分布在我们所拥有的著作中，并指示出一种明确的数学哲学，因此尝试以相当高的准确度构建或重构该哲学是可能的。

我们以亚里士多德关于数学中“连续”与“无限”的思想的例证来结束讨论。托马斯·利特尔·希思[15]解释了亚里士多德关于“连续”的观点：-

……不可能由不可分的部分组成；连续就是其中两个相邻部分之间的边界或界限，在它们相接之处，是同一个……

至于无限，亚里士多德认为它并非实际存在，而只是潜在存在。亚里士多德在Physics中写道（例如见[15]）：-

但我的论证无论如何并未剥夺数学家的研究，尽管它否认了实际存在意义上的无限，即某种增大到无法穷尽的程度的东西的存在；因为实际上，他们并不需要无限或使用无限，而只要求有限直线可以随他们的意愿任意长。……因此，对于证明的目的而言，这对他们不会有任何区别。

Aristotle was not primarily a mathematician but made important contributions by systematising deductive logic. He wrote on physical subjects: some parts of his Analytica posteriora show an unusual grasp of the mathematical method. Primarily, however, he is important in the development of all knowledge for, as the authors of [2] write:-

Aristotle, more than any other thinker, determined the orientation and the content of Western intellectual history. He was the author of a philosophical and scientific system that through the centuries became the support and vehicle for both medieval Christian and Islamic scholastic thought: until the end of the 17th century, Western culture was Aristotelian. And, even after the intellectual revolutions of centuries to follow, Aristotelian concepts and ideas remained embedded in Western thinking.

Aristotle was born in Stagirus, or Stagira, or Stageirus, on the Chalcidic peninsula of northern Greece. His father was Nicomachus, a medical doctor, while his mother was named Phaestis. Nicomachus was certainly living in Chalcidice when Aristotle was born and he had probably been born in that region. Aristotle's mother, Phaestis, came from Chalcis in Euboea and her family owned property there.

There is little doubt that Nicomachus would have intended Aristotle to become a doctor, for the tradition was that medical skills were kept secret and handed down from father to son. It was not a society where people visited a doctor but rather it was the doctors who travelled round the country tending to the sick. Although we know nothing of Aristotle's early years it is highly likely that he would have accompanied his father in his travels. We do know that Nicomachus found the conditions in Chalcidice less satisfactory than in the neighbouring state of Macedonia and he began to work there with so much success that he was soon appointed as the personal physician to Amyntas III, king of Macedonia.

There is no record to indicate whether Aristotle lived with his father in Pella, the capital of Macedonia, while Nicomachus attended to king Amyntas at the court there. However, Aristotle was certainly friendly with Philip, king Amyntas's son, some years later and it seems reasonable to assume that the two, who were almost exactly the same age, had become friendly in Pella as young children.

When Aristotle was about ten years old his father died. This certainly meant that Aristotle could not now follow in his father's profession of doctor and, since his mother seems also to have died young, Aristotle was brought up by a guardian, Proxenus of Atarneus, who was his uncle (or possibly a family friend as is suggested by some authors). Proxenus taught Aristotle Greek, rhetoric, and poetry which complemented the biological teachings that Nicomachus had given Aristotle as part of training his son in medicine. Since in latter life Aristotle wrote fine Greek prose, this too must have been part of his early education.

In 367 BC Aristotle, at the age of seventeen, became a student at Plato's Academy in Athens. At the time that Aristotle joined the Academy it had been operating for twenty years. Plato was not in Athens, but rather he was on his first visit to Syracuse. We should not think of Plato's Academy as a non-political organisation only interested in abstract ideas. The Academy was highly involved in the politics of the time, in fact Plato's visit to Sicily was for political reasons, and the politics of the Academy and of the whole region would play a major role in influencing the course of Aristotle's life.

When Aristotle arrived in Athens, the Academy was being run by Eudoxus of Cnidos in Plato's absence. Speusippus, Plato's nephew, was also teaching at the Academy as was Xenocrates of Chalcedon. After being a student, Aristotle soon became a teacher at the Academy and he was to remain there for twenty years. We know little regarding what Aristotle taught at the Academy. In [10] Diogenes Laertius, writing in the second century AD, says that Aristotle taught rhetoric and dialectic. Certainly Aristotle wrote on rhetoric at this time, issuing Gryllus which attacked the views on rhetoric of Isocrates, who ran another major educational establishment in Athens. All Aristotle's writings of this time strongly support Plato's views and those of the Academy.

Towards the end of Aristotle's twenty years at the Academy his position became difficult due to the political events of the time. Amyntas, the king of Macedonia, died around 369 BC, a couple of years before Aristotle went to Athens to join the Academy. Two of Amyntas's sons, Alexander II and Perdiccas III, each reigned Macedonia for a time but the kingdom suffered from both internal disputes and external wars. In 359 BC Amyntas's third son, Philip II came to the throne when Perdiccas was killed fighting off an Illyrian invasion. Philip used skilful tactics, both military and political, to allow Macedonia a period of internal peace in which they expanded by victories over the surrounding areas.

Philip captured Olynthus and annexed Chalcidice in 348 BC. Stagirus, the town of Aristotle's birth, held out for a while but was also defeated by Philip. Athens worried about the powerful threatening forces of Macedonia, and yet Aristotle had been brought up at the Court of Macedonia and had probably retained his friendship with Philip. The actual order of events is now a little uncertain. Plato died in 347 BC and Speusippus assumed the leadership of the Academy. Aristotle was certainly opposed to the views of Speusippus and he may have left the Academy following Plato's death for academic reasons or because he failed to be named head of the Academy himself. Some sources, however, suggest that he may have left for political reasons before Plato died because of his unpopularity due to his Macedonian links.

Aristotle travelled from Athens to Assos which faces the island of Lesbos. He was not alone in leaving the Academy for Xenocrates of Chalcedon left with him. In Assos Aristotle was received by the ruler Hermias of Atarneus with much acclaim. It is likely that Aristotle was acting as an ambassador for Philip and he certainly was treated as such by Hermias. Aristotle married Pythias, the niece and adopted daughter of Hermias, and they had one child, a daughter also called Pythias. However, Aristotle's wife died about 10 years after their marriage. It is thought that she was much younger than Aristotle, being probably of age of about 18 when they married.

On Assos, Aristotle became the leader of the group of philosophers which Hermias had gathered there. It is possible that Xenocrates was also a member of the group for a time. Aristotle had a strong interest in anatomy and the structure of living things in general, an interest which his father had fostered in him in his early years, that helped him to develop a remarkable talent for observation. Aristotle and the members of his group began to collect observations while in Assos, in particular in zoology and biology. Barnes writes in [6] that Aristotle's:-

... studies on animals laid the foundations of the biological sciences; and they were not superseded until more than two thousand years after his death. The enquiries upon which those great works were based were probably carried out largely in Assos and Lesbos.

Aristotle probably begun his work Politics on Assos as well as On Kingship which is now lost. He began to develop a philosophy distinct from that of Plato who had said the kings should be philosophers and philosophers kings. In On Kingship Aristotle wrote that it is:-

... not merely unnecessary for a king to be a philosopher, but even a disadvantage. Rather a king should take the advice of true philosophers. Then he would fill his reign with good deeds, not with good words.

However, Aristotle's time in Assos was ended by political events. The Persians attacked the town and Hermias was captured and executed. Aristotle escaped and stopped on the island of Lesbos on his way to Macedonia. It was more than a passing visit for he remained there for about a year and must have had the group of scientists from Assos with him for they continued their biological researches there.

Macedonia was now at peace with Athens, for Philip had made a treaty in 346 BC. In 343 BC Aristotle reached the Court of Macedonia and he was to remain there for seven years. The often quoted story that he became tutor to the young Alexander the Great, the son of Philip, is almost certainly a later invention as was pointed out by Jaeger, see [16]. Grayeff in [12] suggests that Philip saw in Aristotle a future head of the Academy in Athens. Certainly this would have suited Philip well for Speusippus, the then head of the Academy, was strongly opposed to Philip and strongly encouraging Athens to oppose the rise of Macedonia.

The treaty between Athens and Macedonia began to fall apart in 340 BC and preparations for war began. The following year Speusippus died but Aristotle, although proposed as head of the Academy, was not elected. The position went to Xenocrates and Philip lost interest in his support for Aristotle. He moved back to his home in Stagirus and took with him to Stagirus his circle of philosophers and scientists.

Aristotle did not marry again after the death of his wife but he did form a relationship with Herpyllis, who came from his home town of Stagirus. It is not clear when they first met but together they had a son, Nicomachus, named after Aristotle's father.

Philip was now at the height of his power but, as so often happens, that proved the time for internal disputes. Aristotle supported Alexander, Philip's son who soon became king. Alexander decided on a policy similar to his father in regard to Athens and sought to assert his power by peaceful means. Alexander protected the Academy and encouraged it to continue with its work. At the same time, however, he sent Aristotle to Athens to found a rival establishment.

In 335 BC Aristotle founded his own school the Lyceum in Athens. He arrived in the city with assistants to staff the school and a large range of teaching materials he had gathered while in Macedonia; books, maps, and other teaching material which may well have been intended at one stage to support Aristotle in his bid to become head of the Academy. The Academy had always been narrow in its interests but the Lyceum under Aristotle pursued a broader range of subjects. Prominence was given by Aristotle to the detailed study of nature and in this and all the other subjects he studied [6]:-

His own researches were carried out in company, and he communicated his thoughts to his friends and pupils, never thinking to retain them as a private treasure-store. He thought, indeed, that a man could not claim to know a subject unless he was capable of transmitting his knowledge to others, and he regarded teaching as the proper manifestation of knowledge.

Whether the works that come down to us are due to Aristotle or to later members of his school was questioned by a number of scholars towards the end of the 19th century. The reasons are discussed by Jaeger [16], but in this work Jaeger argues that the apparent differences in the approach by Aristotle in different works can be explained by his ideas developing over a number of years. Grayeff [6] examines certain texts in detail and again claims that they represent developments in the ideas of Aristotle's school long after his death. He writes:-

According to a tradition which arose about two hundred and fifty years after his death, which then became dominant and even today is hardly disputed, Aristotle in these same years lectured - not once, but two or three times, in almost every subject - on logic, physics, astronomy, meteorology, zoology, metaphysics, theology, psychology, politics, economics, ethics, rhetoric, poetics; and that he wrote down these lectures, expanding them and amending them several times, until they reached the stage in which we read them. However, still more astounding is the fact that the majority of these subjects did not exist as such before him, so that he would have been the first to conceive of and establish them, as systematic disciplines.

After the death of Alexander the Great in 323 BC, anti-Macedonian feeling in Athens made Aristotle retire to Chalcis where he lived in the house which had once belonged to his mother and was still retained by the family. He died the following year from a stomach complaint at the age of 62.

It is virtually impossible to give an impression of Aristotle's personality with any certainty but the authors of [2] write:-

The anecdotes related of him reveal him as a kindly, affectionate character, and they show barely any trace of the self-importance that some scholars think they can detect in his works. His will, which has been preserved, exhibits the same kindly traits; he makes references to his happy family life and takes solicitous care of his children, as well as his servants.

Barnes [6] writes:-

He was a bit of a dandy, wearing rings on his fingers and cutting his hair fashionably short. He suffered from poor digestion, and is said to have been spindle-shanked. He was a good speaker, lucid in his lectures, persuasive in conversation; and he had a mordant wit. His enemies, who were numerous, made him out to be arrogant and overbearing. ... As a man he was, I suspect, admirable rather than amiable.

We have commented above on the disputes among modern scholars as to whether Aristotle wrote the treatises now assigned to him. We do know that his work falls into two distinct parts, namely works which he published during his lifetime and are now lost (although some fragments survive in quotations in works by others), and the collection of writings which have come down to us and were not published by Aristotle in his lifetime. We can say with certainty that Aristotle never intended these 30 works which fill over 2000 printed pages to be published. They are certainly lecture notes from the courses given at the Lyceum either being, as most scholars believe, the work of Aristotle, or of later lecturers. Of course it is distinctly possible that they are notes of courses originally given by Aristotle but later added to by other lecturers after Aristotle's death.

The works were first published in about 60 BC by Andronicus of Rhodes, the last head of the Lyceum. Certainly [2]:-

The form, titles, and order of Aristotle's texts that are studied today were given to them by Andronicus almost three centuries after the philosopher's death, and the long history of commentary upon them began at this stage.

What do these works contain? There are important works on logic. Aristotle believed that logic was not a science but rather had to be treated before the study of every branch of knowledge. Aristotle's name for logic was "analytics", the term logic being introduced by Xenocrates working at the Academy. Aristotle believed that logic must be applied to the sciences [6]:-

The sciences - at any rate the theoretical sciences - are to be axiomatised. What, then, are their axioms to be? What conditions must a proposition satisfy to count as an axiom? again, what form will the derivations within each science take? By what rules will theorems be deduced from axioms? Those are among the questions which Aristotle poses in his logical writings, and in particular in the works known as Prior and Posterior Analytics.

In fact in Prior Analytics Aristotle proposed the now famous Aristotelian syllogistic, a form of argument consisting of two premises and a conclusion. His example is:-

(i) Every Greek is a person.
(ii) Every person is mortal.
(iii) Every Greek is mortal.

Aristotle was not the first to suggest axiom systems. Plato had made the bold suggestion that there might be a single axiom system to embrace all knowledge. Aristotle went for the somewhat more possible suggestion of an axiom system for each science. Notice that Euclid and his axiom system for geometry came after Aristotle.

Another topic to which Aristotle made major contributions was natural philosophy or rather physics by today's terminology. (I [EFR] show my age and the traditional nature of St Andrews University if I remark that in the 1960s a pass in 'General Natural Philosophy' formed part of my degree.) Aristotle looks at matter, change, movement, space, position, and time. He also made contributions to the study of astronomy where in particular he studied comets, geography with an examination of features such as rivers), chemistry where he was interested in processes such as burning, as well as meteorology and the study of rainbows.

As well as important works on zoology and psychology, Aristotle wrote his famous work on metaphysics. This, according to Aristotle, studies [2]:-

... the most general or abstract features of reality and the principles that have universal validity. ... metaphysics studies whatever must be true of all existent things just insofar as they exist, [and] it studies the general conditions which any existing thing must satisfy.

Although Aristotle does not appear to have made any new discoveries in mathematics, he is important in the development of mathematics. As Heath explains in [14]:-

The importance of a proper understanding of the mathematics in Aristotle lies principally in the fact that most of his illustrations of scientific method are taken from mathematics.

Clearly Aristotle had a thorough grasp of elementary mathematics and believed mathematics to have great importance as one of three theoretical sciences. However, it is fair to say that he did not agree with Plato, who elevated mathematics to such a prominent place of study that there was little room for the range of sciences studied by Aristotle. The other two theoretical sciences, Aristotle claimed, were (using modern terminology) philosophy and theoretical physics.

Heath notes in the introduction to [14] some of the mathematics referred to by Aristotle in his works:-

... Aristotle was aware of the important discoveries of Eudoxus which affected profoundly the exposition of the Elements by Euclid. One allusion clearly shows that Aristotle knew of Eudoxus's great Theory of Proportion which was expounded by Euclid in his Book V, and recognised the importance of it. Another passage recalls the fundamental assumption on which Eudoxus based his 'method of exhaustion' for measuring areas and volumes; and, of course, Aristotle was familiar with the system of concentric spheres by which Eudoxus and Callippus accounted theoretically for the independent motions of the sun, moon, and planets. ...

The incommensurable is mentioned over and over again, but the case mentioned is that of the diagonal of a square in relation to its side; there is no allusion to the extension of the theory to other cases by Theodorus and Theaetetus...

Heath [14] also mentions the mathematics which Aristotle, perhaps surprising, does not refer to. There is:-

... no allusion to conic sections, to the doubling of the cube, or to the trisection of an angle. The problem of squaring the circle is mentioned in connection with the attempts of Antiphon, Bryson, and Hippocrates to solve it; but there is nothing about the curve of Hippias ...

While Heath [14] discusses the many mathematical references in Aristotle, the book [5] attempts to construct (or reconstruct) a work on Aristotle's view of the philosophy of mathematics. As Apostle writes in [5]:-

... numerous passages on mathematics are distributed throughout the works we possess and indicate a definite philosophy of mathematics, so that an attempt to construct or reconstruct that philosophy with a fairly high degree of accuracy is possible.

We end our discussion with an illustration of Aristotle's ideas of 'continuous' and 'infinite' in mathematics. Heath [15] explains Aristotle's idea that 'continuous':-

... could not be made up of indivisible parts; the continuous is that in which the boundary or limit between two consecutive parts, where they touch, is one and the same...

As to the infinite Aristotle believed that it did not actually exist but only potentially exists. Aristotle writes in Physics (see for example [15]):-

But my argument does not anyhow rob mathematicians of their study, although it denies the existence of the infinite in the sense of actual existence as something increased to such an extent that it cannot be gone through; for, as it is, they do not need the infinite or use it, but only require that the finite straight line shall be as long as they please. ... Hence it will make no difference to them for the purpose of proofs.