德谟克利特Democritus

德谟克利特以其原子论最为著名，但他也是一位优秀的几何学家。关于他的生平所知甚少，但我们知道留基伯是他的老师。

德谟克利特年轻时肯定访问过雅典，主要是为了拜访阿那克萨戈拉，但德谟克利特抱怨他在那里鲜为人知。据第欧根尼·拉尔修在公元二世纪的著作[5]所述，他说：-

我来到雅典，却无人识我。

德谟克利特对雅典之行感到失望，因为当时已年迈的阿那克萨戈拉拒绝见他。

正如⟦N1⟧在[3]中指出的：-

今天他若重游此地，会发现多么不同：从东北方向进城的主要道路经过令人印象深刻的德谟克利特核研究实验室。

除了前往雅典的那次旅行外，德谟克利特还进行过许多次旅行。伯特兰·罗素在[9]中写道：-

他广泛游历南方和东方各地以寻求知识，他也许在埃及度过了相当长的时间，并且他肯定访问过波斯。然后他回到德谟克利特，并留在那里。

德谟克利特本人写道（但一些历史学家质疑这段引文的真实性）（见[5]）：-

在我的所有同代人中，我在旅行中走过的地方最多，同时进行了最详尽的探究；我见过最多的气候和国家，聆听过数量最多的博学之士。

他的旅行确实带他到了埃及和波斯，正如伯特兰·罗素所暗示的，但他几乎肯定也旅行到了巴比伦，并且有些人声称他旅行到了印度和埃塞俄比亚。他肯定是一个学识渊博的人。正如托马斯·利特尔·希思在[7]中所写：-

……没有哪个主题是他未曾作出显著贡献的，从数学和物理学，到伦理学和诗学；他甚至被称为“智慧”。

尽管人们对他的生平所知甚少，但对他的物理学和哲学却了解颇多。关于他的物理学和哲学理论，我们主要有两个知识来源。首先，亚里士多德详细讨论了德谟克利特的思想，因为他强烈反对德谟克利特的原子论思想。第二个来源见于伊壁鸠鲁的著作中，但与亚里士多德相反，Epicurus是德谟克利特原子论的坚定信奉者。Epicurus的这部著作由Diogenes Laertius保存在他公元二世纪的著作[5]中。

德谟克利特并非第一个提出原子论的人。他的老师留基伯曾提出一个原子体系，Clazomenae的阿那克萨戈拉也是如此。事实上，原子论的痕迹可以追溯得更早，也许可以追溯到毕达哥拉斯学派关于正多面体在宇宙构成中起根本作用的观念。然而，德谟克利特对物理世界提出了比他的任何前辈都远为精细而系统的看法。他的观点在[2]中被概括如下：-

德谟克利特断言，空间，或者说虚空，与实在，或者说存在，有同等的权利被视为存在。他把虚空设想为真空，一个无限的空间，其中运动着构成存在的无数原子（即物理世界）。这些原子永恒而不可见；绝对微小，小到其大小无法再减小（因此得名atomon，即“不可分”）；绝对充实且不可压缩，因为它们没有孔隙，完全填满它们所占据的空间；并且是同质的，仅在形状、排列、位置和大小上有所不同。

以此作为物理世界的基础，德谟克利特可以把世界上的一切变化解释为原子运动的变化，或原子堆积方式的变化。这是一个非凡的理论，它试图基于少量观念解释整个物理学，并且还使数学在物理学中发挥根本作用，因为德谟克利特所提出的整个结构是定量的，并服从数学定律。德谟克利特理论中的另一个基本观念是，自然的行为像一台机器，它不过是一个高度复杂的机械装置。

于是，德谟克利特有一些问题需要回答。温暖、颜色和味道等性质如何纳入原子论？对德谟克利特来说，原子仅在数量上有所不同，而所有质的差异都只是表面现象，并且源于观察者因原子构型不同而产生的印象。温暖、颜色、味道这些属性只是约定俗成的——实际存在的只有原子和虚空。

德谟克利特的哲学包含一种早期形式的能量守恒。在他的理论中，原子是永恒的，运动也是如此。德谟克利特通过原子随机运动并碰撞形成更大的物体和世界，解释了宇宙的起源。在他的理论中没有神明干预的位置。相反，他假定了一个一直存在、并将永远存在、且充满随机运动原子的世界。涡旋运动由于原子的碰撞而发生，并在由此产生的涡旋运动中，仅仅由于原子质量不同而造成了原子分化为不同层次。这不是一个通过某种超自然存在的设计或目的而产生的世界，而是一个通过必然性，也就是从原子自身的本性而产生的世界。

德谟克利特在其原子论哲学之上建立了一种伦理理论。他的体系是纯粹决定论的，因此他不能承认个人的选择自由。对德谟克利特来说，选择自由是一种幻觉，因为我们并未意识到一项决定的所有原因。德谟克利特相信[3]：-

……灵魂要么受到扰动，以致其运动以剧烈的方式影响身体，要么处于静止状态，此时它和谐地调节思想和行动。免于扰动是带来人类幸福的条件，而这正是伦理的目标。

德谟克利特描述至善，他将其等同于愉悦，如下：-

……一种灵魂平静安宁地生活，不受恐惧、迷信或任何其他情感扰动的状态。

他想消除对神的信仰，他认为这些神只是被引入来解释当时尚无科学解释的现象。

关于德谟克利特对数学的贡献，确知者甚少。正如⟦E1⟧所述：-

关于德谟克利特的数学，所知甚少（尽管著述颇多）。

我们确实知道德谟克利特写了许多数学著作。第欧根尼·拉尔修（见[5]）列出了他的著作，并给出特拉叙卢斯作为这些信息的来源。他写了On numbers, On geometry, On tangencies, On mappings, On 无理数，但这些著作都没有留存下来。不过，我们从其他参考文献中知道了一点。托马斯·利特尔·希思[7]写道：-

在1906年幸运发现的阿基米德的《方法》中，我们被告知德谟克利特是第一个陈述以下重要命题的人：圆锥的体积是同底等高的圆柱体积的三分之一，棱锥的体积是同底等高的棱柱体积的三分之一；也就是说，德谟克利特在欧多克索斯首次科学证明这些命题之前约五十年或更早就已经阐述了它们。

关于德谟克利特还有另一条引人入胜的信息，由普鲁塔克在他的Common notions against the 斯多亚派中给出，其中他报告了德谟克利特提出的一个两难问题，如斯多葛派克律西波斯所报道的（见[7]、[10]或[11]）。

如果一个圆锥被一个平行于底面的平面切割[他指的是一个无限接近底面的平面]，我们该如何看待形成截面的那些面？它们是相等的还是不相等的？因为，如果它们不相等，它们将使圆锥不规则，有许多凹陷，像台阶一样，还有不平整；但是，如果它们相等，截面将相等，圆锥将显得具有圆柱的性质，由相等的而非不相等的圆组成，这非常荒谬。

这个两难问题中有重要的思想。首先注意，正如托马斯·利特尔·希思在[7]中指出的，德谟克利特有立体是无限多个平行平面之和的思想，他可能用这个思想来求圆锥和棱锥的体积，如阿基米德所报道的。德谟克利特的这个思想可能后来引导阿基米德将同样的思想应用于产生巨大效果。这个思想最终将导致积分理论。

在[7]、[8]、[10]和[11]中有许多讨论，涉及德谟克利特是否区分了几何连续统与其原子体系的物理离散。托马斯·利特尔·希思指出，如果德谟克利特把他的原子理论搬到几何线上，那么对他来说就不存在两难，因为他的锥体确实带有原子大小的台阶。托马斯·利特尔·希思肯定相信，对德谟克利特来说线是无限可分的。其他人，例如见[10]，得出了相反的结论，认为德谟克利特对应用数学问题作出了贡献，但由于他的原子理论，他无法处理由此产生的无穷小问题。

Democritus of Abdera is best known for his atomic theory but he was also an excellent geometer. Very little is known of his life but we know that Leucippus was his teacher.

Democritus certainly visited Athens when he was a young man, principally to visit Anaxagoras, but Democritus complained how little he was known there. He said, according to Diogenes Laertius writing in the second century AD [5]:-

I came to Athens and no one knew me.

Democritus was disappointed by his trip to Athens because Anaxagoras, then an old man, had refused to see him.

As Brumbaugh points out in [3]:-

How different he would find the trip today, where the main approach to the city from the northeast runs past the impressive "Democritus Nuclear Research Laboratory".

Certainly Democritus made many journeys other than the one to Athens. Russell in [9] writes:-

He travelled widely in southern and eastern lands in search of knowledge, he perhaps spent a considerable time in Egypt, and he certainly visited Persia. He then returned to Abdera, where he remained.

Democritus himself wrote (but some historians dispute that the quote is authentic) (see [5]):-

Of all my contemporaries I have covered the most ground in my travels, making the most exhaustive inquiries the while; I have seen the most climates and countries and listened to the greatest number of learned men.

His travels certainly took him to Egypt and Persia, as Russell suggests, but he almost certainly also travelled to Babylon, and some claim he travelled to India and Ethiopia. Certainly he was a man of great learning. As Heath writes in [7]:-

... there was no subject to which he did not notably contribute, from mathematics and physics on the one hand to ethics and poetics on the other; he even went by the name of 'wisdom'.

Although little is known of his life, quite a lot is known of his physics and philosophy. There are two main sources for our knowledge of his of physical and philosophical theories. Firstly Aristotle discusses Democritus's ideas thoroughly because he strongly disagreed with his ideas of atomism. The second source is in the work of Epicurus but, in contrast to Aristotle, Epicurus is a strong believer in Democritus's atomic theory. This work of Epicurus is preserved by Diogenes Laertius in his second century AD book [5].

Certainly Democritus was not the first to propose an atomic theory. His teacher Leucippus had proposed an atomic system, as had Anaxagoras of Clazomenae. In fact traces of an atomic theory go back further than this, perhaps to the Pythagorean notion of the regular solids playing a fundamental role in the makeup of the universe. However Democritus produced a much more elaborate and systematic view of the physical world than had any of his predecessors. His view is summarised in [2]:-

Democritus asserted that space, or the Void, had an equal right with reality, or Being, to be considered existent. He conceived of the Void as a vacuum, an infinite space in which moved an infinite number of atoms that made up Being (i.e. the physical world). These atoms are eternal and invisible; absolutely small, so small that their size cannot be diminished (hence the name atomon, or "indivisible"); absolutely full and incompressible, as they are without pores and entirely fill the space they occupy; and homogeneous, differing only in shape, arrangement, position, and magnitude.

With this as a basis to the physical world, Democritus could explain all changes in the world as changes in motion of the atoms, or changes in the way that they were packed together. This was a remarkable theory which attempted to explain the whole of physics based on a small number of ideas and also brought mathematics into a fundamental physical role since the whole of the structure proposed by Democritus was quantitative and subject to mathematical laws. Another fundamental idea in Democritus's theory is that nature behaves like a machine, it is nothing more than a highly complex mechanism.

There are then questions for Democritus to answer. Where do qualities such as warmth, colour, and taste fit into the atomic theory? To Democritus atoms differ only in quantity, and all qualitative differences are only apparent and result from impressions of an observer caused by differing configurations of atoms. The properties of warmth, colour, taste are only by convention - the only things that actually exist are atoms and the Void.

Democritus's philosophy contains an early form of the conservation of energy. In his theory atoms are eternal and so is motion. Democritus explained the origin of the universe through atoms moving randomly and colliding to form larger bodies and worlds. There was no place in his theory for divine intervention. Instead he postulated a world which had always existed, and would always exist, and was filled with atoms moving randomly. Vortex motions occurred due to collisions of the atoms and in resulting vortex motion created differentiation of the atoms into different levels due only to their differing mass. This was not a world which came about through the design or purpose of some supernatural being, but rather it was a world which came about through necessity, that is from the nature of the atoms themselves.

Democritus built an ethical theory on top of his atomist philosophy. His system was purely deterministic so he could not admit freedom of choice to individuals. To Democritus freedom of choice was an illusion since we are unaware of all the causes for a decision. Democritus believed that [3]:-

... the soul will either be disturbed, so that its motion affects the body in a violent way, or it will be at rest in which case it regulates thoughts and actions harmoniously. Freedom from disturbance is the condition that causes human happiness, and this is the ethical goal.

Democritus describes the ultimate good, which he identifies with cheerfulness, as:-

... a state in which the soul lives peacefully and tranquilly, undisturbed by fear or superstition or any other feeling.

He wanted to remove the belief in gods which were, he believed, only introduced to explain phenomena for which no scientific explanation was then available.

Very little is known for certainty about Democritus's contributions to mathematics. As stated in the Oxford Classical Dictionary :-

Little is known (although much is written) about the mathematics of Democritus.

We do know that Democritus wrote many mathematical works. Diogenes Laertius (see [5]) lists his works and gives Thrasyllus as the source of this information. He wrote On numbers, On geometry, On tangencies, On mappings, On irrationals but none of these works survive. However we do know a little from other references. Heath [7] writes:-

In the Method of Archimedes, happily discovered in 1906, we are told that Democritus was the first to state the important propositions that the volume of a cone is one third of that of a cylinder having the same base and equal height, and that the volume of a pyramid is one third of that of a prism having the same base and equal height; that is to say, Democritus enunciated these propositions some fifty years or more before they were first scientifically proved by Eudoxus.

There is another intriguing piece of information about Democritus which is given by Plutarch in his Common notions against the Stoics where he reports on a dilemma proposed by Democritus as reported by the Stoic Chrysippus (see [7], [10] or [11]).

If a cone were cut by a plane parallel to the base [by which he means a plane indefinitely close to the base], what must we think of the surfaces forming the sections? Are they equal or unequal? For, if they are unequal, they will make the cone irregular as having many indentations, like steps, and unevennesses; but, if they are equal, the sections will be equal, and the cone will appear to have the property of the cylinder and to be made up of equal, not unequal, circles, which is very absurd.

There are important ideas in this dilemma. Firstly notice, as Heath points out in [7], that Democritus has the idea of a solid being the sum of infinitely many parallel planes and he may have used this idea to find the volumes of the cone and pyramid as reported by Archimedes. This idea of Democritus may have led Archimedes later to apply the same idea to great effect. This idea would eventually lead to theories of integration.

There is much discussion in [7], [8], [10] and [11] as to whether Democritus distinguished between the geometrical continuum and the physical discrete of his atomic system. Heath points out that if Democritus carried over his atomic theory to geometrical lines then there is no dilemma for him since his cone is indeed stepped with atom sized steps. Heath certainly believed that to Democritus lines were infinitely divisible. Others, see for example [10], have come to the opposite conclusion, believing that Democritus made contributions to problems of applied mathematics but, because of his atomic theory, he could not deal with the infinitesimal questions arising.