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  The Greeks began the development of formal logic. We can find the first intimations of this in Plato, and the first systematisation in Aristotle. Logic is frequently misunderstood. It is about the form, rather than the content, of arguments. It tries to decide which forms of argument are valid (and therefore good arguments) and which are invalid (bad arguments). Here is a simple example. Take the argument that ‘if all humans are primates, and all primates are mammals, then all humans are mammals’. Does this structure of argument work generally? In other words, if all A’s are B, and all B’s are C, are all A’s C? If the answer to that is ‘yes’ (and it is), then this is a valid argument, and it works whatever A, B and C are, as long as all A’s are B and all B’s are C. Can we also therefore conclude that all C’s are A (in our example, that all animals are humans)? If the answer is no (and it is) then this second argument is invalid and ought not to be used. Logic studies the validity of arguments at this general level. Although this is a simple case, logic can be very helpful in working out whether or not more complex arguments are good ones, and so deciding between theories.

  Euclid

  We can also find the Greeks doing something very important with geometry, and here the key name is Euclid (fl. 300 BC). Euclid attempted to give geometry stable foundations and a thorough, rigorous structure. He began with definitions (‘a point is position without magnitude, a line length without breadth’, etc.), postulates (assumptions about points and lines) and axioms (assumptions needed to generate proofs, such as ‘things which are equal to the same thing are equal to each other’). He then proceeded to prove, in an absolutely rigorous manner, a great number of geometrical theorems. Some of these were relatively straightforward, but he also managed to prove many complex theorems and discovered a great deal about the structure of geometry. The great beauty of Euclid’s geometry was that if you agreed with the definitions, postulates and axioms, then the proofs compelled you to believe the more complex theorems. For one science at least, here was a definitive procedure for resolving disputes and making progress.

  Attempts were made to ensure that Euclid’s system became even more rigorous, by reducing the number of assumptions (definitions, postulates, axioms) which were not proved but which had to be accepted. The fewer assumptions, the more definitive the system. Attention centred on Euclid’s fifth postulate.

  Figure 3: Euclid’s fifth postulate. Given a straight line and a point on a plane, how many lines can be drawn through that point parallel to the line? Euclid’s answer was one, and only one.

  Here we have a point and a straight line on a plane. The question is: how many straight lines can be drawn through the point which are parallel to the line that we already have? Possible answers are 0, 1 or more than 1. The intuitive answer, and the one assumed by Euclid, is that there is one, and only one, parallel line. It was felt by many geometers that this postulate ought to be provable from the other postulates, definitions and axioms. Attempts to do so failed, however. So, too, did attempts at indirect proof. Indirect proof takes the alternative answers to the one that you want to prove, and attempts to show that they lead to contradictions. If all of the alternatives lead to contradictions, then you have indirectly proved your answer. So people attempted to show that the condition of no parallel lines, or of more than one parallel line, led to a contradiction, but they failed to do so. There is an interesting Greek assumption here. The world is comprehensible, so there must be a non-contradictory way of describing it. If you rule out all of the contradictory answers, what you are left with must be right!

  It was eventually realised, in the nineteenth century, that there was a subtle assumption about the nature of space involved here. Euclid’s geometry, with one parallel line, described flat, homogeneous space (the sort of space we are familiar with from a flat piece of graph paper). A geometry with no parallel lines was self-consistent, and described curved surfaces like that of a sphere. Think of a globe. All lines of longitude are at 180° to each other at the equator but all meet at the poles. This is a positive curvature. With a negative curvature (such as a saddle), in which lines run away from each other, there may be many lines drawn through a point which never meet. These alternative geometries were of only theoretical interest until the advent of Einstein’s theory of general relativity, when it was realised that space might have a positive or a negative curvature. Euclid’s geometry is a major achievement, and remains valid for Euclidean (non-curved) space. It is also an important expression of the way in which the Greeks sought both secure starting points for their science and ways of resolving disputes over their theories.

  Science and Technology

  A society in possession of science should be conscious of a distinction between science and technology. We find no such distinction in any of the pre-Greek societies. The Greeks, though, distinguished between empeiria, meaning a knack or skill acquired through practice, and episteme, meaning knowledge, which required the ability to give reasons why something was the case. The person having empeiria might be able to manipulate the world, but he would not be able to explain why what he was doing should work. A typical example for the Greeks was the difference between someone who knew a few folk remedies for disease, and a doctor who knew the nature of the body and could explain why, how, and in what circumstances those folk remedies would be effective. Plato in particular was keen on this distinction. He made a contrast between the capabilities of someone with a basic empirical or practical acquaintance with a subject, and the theoretical and synoptic knowledge that an expert might be expected to have. The Greeks were conscious of doing something different from – and more sophisticated than – technology, and were also aware that their attitudes to myth and religion were different from those of other societies.

  Great Achievement Assured

  Let us sum up the achievements of the early Greeks. Their society was rather different from other ancient societies, not being a hierarchy or having a centralised state religion. There was a greater tolerance of debate, and also an affluence which allowed well-to-do people the time to contemplate philosophical and scientific questions. The Milesians began to think about the world in a radically different way. They believed their world to be a cosmos, a well-ordered entity, which could be understood and explained by people using words and numbers. They distinguished between the natural and the supernatural, believing the cosmos to be entirely natural and subject to regular behaviour. Gone was any notion of mythical explanation. The Milesians dealt in theories, and tried to improve on each other’s theories. They produced the first theories of matter and the first cosmologies and cosmogonies that are couched in natural terms. Their theories may appear naïve to modern eyes, but that is only to be expected of thinkers who were only just beginning to produce scientific theories. The key point is that they were theories rather than myths.

  The Hippocratics rejected the idea that diseases are punishments sent by the gods, asserting that all diseases have a physical cause. They also produced the first criticism of magical practices on a general level. We also see the Greeks begin to pursue more theoretical questions in science. The Eleatics posed problems about change and motion, and whether space, time and matter are continuous or atomistic. The atomists answered that matter, at least, comes in small, unsplittable packages called atoms, and so developed the first two-level theory of appearance and reality. We also see the development of mathematics and geometry beyond its practical phase. With the Pythagoreans, we find the first investigation of the way in which mathematics may be linked to physics, and the first attempts to prove geometrical theorems. The Greeks became interested in how to resolve debates about their theories, introducing notions of demonstration and proof. They were also conscious of a distinction between science and technology. However crude the initial theories of the Greeks, they were scientific theories as opposed to myths, and from there the Greeks made rapid progress to more sophisticated theories, something unseen in other ancient cultures. The Greeks carried through their vision of a
new way to explain and investigate the world with exemplary thoroughness and enthusiasm.

  3 Men of the World

  Nature does nothing without purpose or uselessly.

  Aristotle, Politics, book 1, 1256b, 20–1

  The great majority of the ancient Greeks whom we have met so far are known as ‘pre-Socratics’. The history of Greek philosophy divides into three periods: the pre-Socratics, who worked before Socrates (469–399 BC); the period of the three great Athenian philosophers, Socrates, Plato (427–348 BC) and Aristotle (384–322 BC); and the period of the Hellenistic philosophers, who all worked after Aristotle. Socrates, though he was immensely important in the history of philosophy, was more interested in ethics than science. Plato and Aristotle are of great significance, though. Aristotle’s views came to dominate both the ancient world and Western thought in general until the scientific revolution of the seventeenth century. Even thinkers who were not specifically Aristotelians borrowed many of his ideas. His views were later synthesised with Christian theology to form a philosophy known as ‘scholasticism’, and it was this that the scientific revolution sought to transform. Plato’s views, while sharing a good deal of ground with Aristotle, were a consistent alternative to Aristotelian ideas. Plato’s thinking underwent a revival in the form of ‘neo-Platonism’, both in the later ancient world and during the Renaissance.

  Athens in this period formed the intellectual hub of Greece, not only in philosophy and science but in the arts and politics as well. Politically, it was a melting pot, veering rapidly from the first experiments in democracy, to various forms of oligarchy (rule by the few), to dictatorship. Athens was also frequently at war with other city states, most notably Sparta. Freedom of expression, and a spirit of open debate and criticism, allowed philosophy, science and the arts to flourish. Theatre and poetry prospered, great rhetorical speeches were made on the issues of the day, fine buildings were put up, and philosophers debated openly, often to audiences. Philosophical symposia were held, in which the great philosophical issues were discussed. These could be polite dinner parties, or rather wild drinking sessions. Plato’s book The Symposium records one of these events, and discusses the nature of love. There were representatives of many philosophical viewpoints on all the major issues of Greek thought. In relation to science, the main views were those of the atomists, Leucippus and Democritus, and of Empedocles and Anaxagoras, and of the Milesians, Thales, Anaximander and Anaximenes. Plato tended to group these people together, calling them the ‘physiologoi’, those who explained in terms of nature. There were also the views of the Eleatics, Parmenides and Zeno. Plato and Aristotle built on the work of these philosophers in interesting new ways, and their thinking was seminal in forming the philosophical and scientific legacy of the Greeks.

  Plato

  Plato was born in 427 BC, probably in Athens, and in his youth was a devoted pupil of Socrates. Socrates professed to know nothing himself, but was remarkably good at conducting philosophical investigations. He was brilliant at showing pompous people who thought that they had definitive answers to philosophical questions that in fact they did not – that they needed to think more deeply. A typical Socratic question was: ‘What is courage?’ Socrates would then refute other people’s ideas about courage without giving a definition himself, prompting people to rethink their views. No respecter of dignity or position, Socrates got himself into trouble. The death of Socrates – he was executed by the Athenians on trumped-up charges of impiety while defending his philosophical views – affected Plato deeply. Plato devoted himself to philosophy, and remains one of the truly great philosophers.

  Plato had an eventful life. He served as a soldier in some of Athens’ wars and was often embroiled in political matters. He became disenchanted with Athenian politics and politicians, and argued that until philosophers became rulers, or rulers philosophers, things would never go well for the state. Later in his life, he attempted to put some of these ideas into practice. He travelled to Syracuse in order to educate the young Dionysus II, in the hope of making him a philosopher-king. However, after intrigue and betrayal the project failed and Plato barely escaped with his life. In Athens, he founded the Academy, a school and research group for philosophers, which was the first of its type.

  Plato’s philosophical writings are remarkable for their artistic as well as their philosophical merit. He wrote his works as dialogues, discussions of philosophical questions between the characters. Often the main protagonist is Socrates, and there is much academic debate about how far Plato’s works record actual conversations and how far they are fictional. Whatever the answer to that question, Plato gives us a unique insight into the sorts of debate that were going on in Athens. He has a remarkable ability to make these debates come alive, and a notable talent for the characterisation of the protagonists. These characters are often related to the philosophical position that they are given to argue, and the dialogues are laced with humour and dramatic turns to the debate. Plato’s works are literature and philosophy at the same time, something very rare in philosophy.

  Plato was more important for his attitude to the investigation and conception of the world than for any specific theories. His steadfast belief was that the order of the cosmos could not have come about by chance. Plato was keen to contrast the potential chaos of the world with the apparent good order of the cosmos. The order of the heavens, the beauty of the world around us, the nature of living beings, all stood in stark contrast to a possible chaos in which the elements were randomly distributed. He also imagined a chaos in which matter itself was not organised, so that there were no recognisable elements like earth or water, just a mess. There was still no conception of gravity, but Plato and others believed in a like-to-like principle whereby like things were brought together. This was not an attractive force as such, more a principle by which things were sorted into groups from an initial chaos. Plato saw that such a principle would only sort similar things, and would not produce the order of the cosmos. Characteristic of the cosmos, according to Plato, was the proper proportion of unlike things together. Talking about his opponents’ views, he said:

  Let me put it more clearly. Fire, water, earth and air all exist due to nature and chance they say, and none to skill, and the bodies which come after these, earth, sun, moon and stars, came into being because of these entirely soulless entities. Each being moved by chance, according to the power each has, they somehow fell together in a fitting and harmonious manner, hot with cold or dry with moist or hard with soft, all of the forced blendings happening by the mixing of opposites according to chance. In this way and by these means the heavens and all that pertains to them have come into being and all of the animals and plants, all of the seasons having been created from these things, not by intelligence, they say, nor by some god nor some skill, but as we say, through nature and chance.

  Opposites such as hot and cold could not have been brought together by a like-to-like principle, nor was it likely that they would come together in a harmonious manner by chance. Some god or skill must have achieved this. This is an early example of an ‘argument from design’. If we look at the workings of a mechanical watch, they appear too complex and well organised, too clearly arranged for a purpose, to have come about by chance. Therefore we believe there to have been a watch-maker. Design arguments state that the world is similarly complex, organised and arranged, so that we must suspect the presence of a designer. Plato believed that a god organised the cosmos out of primordial chaos. The nature of this god is of considerable importance. Plato conceived of him as a skilled craftsman, a demiourgos. It was because the god was skilled that he could form an organised cosmos out of chaos. When he did so, he did the best job possible, taking into account the limitations of his materials. Matter acted by necessity, similar things grouping together, and did not produce what reason and intelligence would produce. So Plato speaks of reason being imposed upon necessity, as far as possible, to produce the order of our world.

  Plato�
��s demiourgos was without jealousy and malice, and acted to create the best cosmos that he possibly could. Plato’s conception of his god was radically different from previous Greek notions of the gods. The gods of Greek mythology were, in short, a pretty rough lot. They schemed, they plotted, they committed adultery and murder and generally acted in an amoral and unpredictable manner. As Plato complained, they seemed to have all the moral shortcomings of human beings. Plato’s demiourgos acted only in the best manner possible. He did not interfere with the world once it was set up.

  The demiourgos was also a geometrical god. That is, he employed the principles of geometry and mathematics to give the best possible proportion and harmony to the world and its constituents. Specifically, he imposed shape and number on the primordial chaos in order to create the cosmos. The world, then, had an underlying geometrical and mathematical structure. There was a purity and timelessness about mathematics and geometry which appealed to Plato. Truths about the physical world are subject to change (today it is 20°C, yesterday it was not); the truths of mathematics and geometry are not. Since Plato believed that knowledge should be indisputable and unchanging, mathematics and geometry were for him perfect examples of what knowledge should be like. Because his god constructed the world using geometry and mathematics, we should also be able to use them in order to understand the structure of that world. Likewise, because the demiourgos has created the best world possible – simple, elegant and aesthetically pleasing – our explanations should be simple, elegant and aesthetically pleasing.