Plato and Pythagoras: The Minds Behind The Modern World?

It is the 21st century. It is a world of order, but a world of chaos. A world of power, but a world without security. A world of progress, but a world without direction. The present is too close to us. We cannot see where we are. Surrounded by the ordinary, we are unable to see how it is strange. Our encyclopedias tell us the size of the universe, the origin of life and the structure of the brain. We believe these stories to be true, but others have seen a different world. Not merely of different values, but a different world altogether. Trapped in our way of seeing, it's not easy to uncover these other worlds, but on our journey to glimpse them, we may catch sight of our own. Let me tell you a story. We're all prisoners in a cave. We can't move. Our limbs are strapped. Even our heads are fixed. There's light from a fire casting shadows on the wall in front of us. The sound is bouncing and echoing off the wall. This is the only reality we've ever known. It's as if we're looking at a kind of screen. We think it is the world. Then, unexpectedly, one of us breaks free and stumbles out of the cave into the sun. He's totally disorientated, dazzled by this experience. He doesn't know what he's seeing at all. It takes him a long time to learn to see things as they really are. To understand that he previously lived all his life in a state of illusion. That's a curious story. It was first told about 24 centuries ago in Athens. It's one of the founding texts of Western culture. It's the story of how someone traveled from illusion to reality. He somehow discovered the real truth of things. And it's by Whether we like it or not, this ancient story is part of all of us. It's part of the way we think. The modern world is Plato's legacy in the sense that, without his concentration on mathematics as the path to reality, we wouldn't have had modern science and therefore we wouldn't have had modern technology. If it weren't for centuries of mathematicians playing around with invisible objects in the way in which Plato recommended, we would not have computers, we would not have the kinds of buildings and machines that computers have made possible. The past may be the key to the present, but it is elusive. We see it through our spectacles. We give it our meaning. As we look out from the present, we must move away from this world on a journey into another. Our story does not start with Plato, but 150 years earlier, some two and a half thousand years ago, with a more shadowy, legendary figure. We all learned his name at school, but were given no idea of his importance. His name is Pythagoras. Exiled Romanian Shakti Ungaru, professor of the History of Mathematics at Tel Aviv University, revisits Crotone, a town in southern Italy, once a major city in greater Greece. As I left Romania, I used the first available opportunity to come to Crotone, the actual place in which the master organized his school and developed his ideas. It was here that Pythagoras came up with one of the most incredible and absurd and yet one of the most powerful and fruitful ideas without which western thought, the way we know it, is inconceivable. The crucial Pythagorean idea consisted in the view that the material world is actually made up, constructed out of numbers, and that the proper way to understand it is to unravel the numerical relationships lying at the essence of things. In an age of computers and checkout tills, accountants and commodity brokers, to say that all is number is a very different thing than it was for Pythagoras. For us, numbers are mundane, prosaic, matter of fact. They're tools that help us deal with things, instruments that help us measure and count and manipulate the world around us. For Pythagoras, numbers had almost magic powers, for they held the secret of the universe. They were no less than the very essence of the world. And it is because we are Pythagoras's children, steeped into his ways of thinking, that it is quite impossible to conceive ourselves in a world for which the Pythagorean patterns of explanation are unavailable to us. Pythagoras's claim that the world was based on numbers was unlike anything anyone had said before. There had of course been religious myths before the Greeks, but no one had tried to give an explanation, a theory of the world. Yet here was an account of the world that described it in terms of an abstract theoretical notion. Other Greeks had suggested the world was made of water or fire, but you could not point to numbers, you could not find them in the world. It seemed they lay outside of the physical world. Little wonder then that for Pythagoras they were sacred and mystical. Pythagoras was born on the Greek island of Samos, off the coast of Turkey, at the time of the tyrant Polycrates. It is said to have travelled to Egypt before settling in Italy, where he established his own community in Crotone, where it is claimed for a time he ruled the city. His world was of course very different from ours. It was a primitive society, lacking technical prowess. But it is not the material differences that separate us from him, but the way we think. According to an ancient legend, one day Pythagoras was passing a blacksmith's, and it occurred to him that the notes made by the hammers against the anvil seemed to vary with the size of the hammer. On returning home he is said to have tried this out, comparing the notes from different lengths of string, and concluded that if musical notes were numerical, perhaps the world was also. Jocelyn Godwin teaches music at Colgate University in New York State. He believes that Pythagoras' discovery not only led Pythagoras to the conclusion that the world was number, but has provided us with the basis of all music. I imagine him discovering it like this. He takes a string, let's say a string two feet long, this string, and he asks himself, what is the simplest mathematical operation I can do with this string? Cut it in half. What does a string one foot long sound like? And if you listen to the relation of those two notes, you hear something very familiar. You hear an octave. What you have there is the framework in which music can happen. So how does it happen? We imagine Pythagoras taking the next step. He divides his string in three. And the resulting note produced by this length of string is higher again than the octave, by an interval which we call the perfect fifth. But you can really make music with a perfect fifth, which you can't with an octave. For example, I can take a chain of perfect fifths like this. I take those seven fifths and I pack them together within a single octave and what do I get? The scale that we know, the white notes of the keyboard, are derived from that very simple mathematical operation of dividing a string into three. So Pythagoras found that out of the fifths are made the scale, the basis of all melody. The music we play is Pythagorean because it's based on the scale as far as its melody is concerned. It's based on the chords that come from those simple numbers as far as its harmony is concerned. And its rhythm, in most cases, doesn't go beyond the number four. Pythagoras found this a remarkable discovery because it showed him that what the ear hears as most satisfying and most restful, those simple intervals, are also the simplest numbers of mathematics. To him, this seemed a revelation that there's something underlying the universe which is attainable by us both through music and through mathematics. For Pythagoras, music is a metaphor for the way the world works. Everywhere we look we see the interplay of concord and discord, the interplay of harmony and disharmony. We see people and things coming together like the notes of a fugue. Two and a half thousand years after Pythagoras formed his own academy in Crotone, his name is still taught in the local school as it is in schools throughout the world. But it's not Pythagoras' belief that the world is number or his discovery that music is mathematical that is taught. Rather, his name is associated with geometry and a theorem about triangles. If mathematics is at the root of music, we can imagine Pythagoras wondering whether it was at the root of everything. And so he looked to the other senses, to the world of sight, to the world of vision, to geometry, to find a parallel numerical relationship to the one he had uncovered in music. And that is exactly what the Pythagoreans believed they had found with their famous theorem. For as every schoolgirl and boy knows, the Pythagoras theorem states that the square of the longest side of a right-angled triangle is equal to the sum of the squares of the other two sides. It is nice and dandy to say that the world is made up of numbers. It is an entirely different matter to show what that means. To show what that means is to prove the underlying numerical relationships that are responsible for the phenomena. For us, proofs are something we learn at school. But for the Pythagoreans, they were a radical new discovery. Not only had they demonstrated that the world was based on number, but through proof, they believed they discovered the key to the hidden secret world that governed the universe. Proof was precisely the means that gave one access to the underlying reality, to the substance that made up the cosmos. We live in a world of geometry just as much as we live in a world of numbers. We surround ourselves with right angles and straight lines. Almost every object around us is symmetrical and made from elementary geometrical shapes. For us, shape and number are quite separate. In Pythagoras' world, they were inseparable. And we can demonstrate it here by setting it up... Keith Critchlow argues that we've much to learn from the aspects of Pythagoras we've discarded. A professor of art and architecture, he was involved in the attempt to complete the largest Gothic cathedral in the world, St John the Divine, on the fringe of Harlem in New York City. We tend to think of shape as an aesthetic matter, but for Pythagoras, shape, like music, was number. Pythagoras believed that the world could be explained in terms of mathematics. The world we see and experience had inside it mathematical form. The way we believe he did this was to say that the number one could be treated as a point. A geometric point is the number one. Number two could be represented by two such figures, and that is the definition of a line. Two points making a line. So the first relationship is a line. By bringing in the number three and three points, we get the first shape, a triangle. Now, we can bring in the fourth one to make a square by doing this, or we can make it into a pyramid by putting the fourth one on top, and we get our first pyramid. So Pythagoras associated numbers one, two, three and four with basic shapes. The circle, the line, the triangle and the square. And for him, these numbers, these shapes, made up the world. The history of architecture could be seen to be the history of employing these basic shapes that Pythagoras proposed. If we look at this skyline here, what do we see? We see nothing but geometry. It may be occasional tree, but nearly a complete landscape of geometry. It is both its blessing and its disaster. Pythagoras would have been disappointed, let's put it like that, with New York. He'd have been disappointed because he said the three shapes which need to be used in harmony, the circle, the triangle and the square. Now, if we look at this landscape, we see a complete lack of the circle and very, very little of the triangle. Pythagoras proposed that these had a psychological value. Now, the difference between New York we're looking at and the cathedral we're standing on is the cathedral embraces these shapes. It is fundamental to a cathedral which is a building about the whole. That's what it sets out to be. It is and has to employ these shapes. And that's why a building like the cathedral is of vital necessity to New York. In a sense, the skyline of New York is Pythagorean in the geometric sense. And there are examples within what is basically a right angle and an upright geometry where we see things like the Empire State Building, which is immensely beautiful because there's a change in shape and there's other buildings, one or two in the skyline, and they fulfill another very important role that Pythagoras suggested. It seems strange to us, maybe a little unlikely, but he said that these basic shapes must be kept in balance. The evidence is right the way through architectural history, through the Renaissance, through Gothic England and France. They always employed these basic shapes in combination and they were healthy at that point. So one could suggest there is the implication of being unhealthy if you only use one shape and you do not harmonize with the other shapes. It's another reason why the cathedral here and the dome on the cathedral and what the cathedral will finally be will be an immensely important addition to the whole body, the geometric body of New York. From our modern perspective, Pythagoras' view of numbers, if we can catch sight of it, strikes us as mystical and somewhat unlikely. But it is at the source not only of how our world looks, but how it works, for it is at the basis of our technology and our science. According to Dr Atkins, a physical chemist working at Oxford, there's no doubt of our debt to Pythagoras. Pythagoras opened the eyes of the world. He had a hunch, and it was little more than a hunch, that mathematics would prove to be the key to the description of nature, the description of everything around us. And in the course of 2,000 years, we've come to see that mathematics is the key to understanding just about everything. Pythagoras had the hunch, but it was Newton who turned it into the extraordinary edifice that we have today. Newton was really the first person to show that mathematics could be used to describe the world. He invented the mathematics that he needed as he went along. Well, I would almost take the view that he discovered it. The mathematics was around, and he prized it out of the universe. Take this game of croquet, for example. It might not look like mathematics, but you can describe its motion using mathematics. The speed of the ball, its position, the direction in which it's travelling and its mass, the impact of the mallet on the ball. All these things can be summarised once and for all in a single, very simple mathematical expression. Essentially, that the force is equal to the mass times the acceleration. The history of science has been the development of laws that have uncovered the numerical relationships between things, which Pythagoras had only guessed existed. Mathematics is a kind of very concise picture of the landscape. When you look at the pattern in which a tree grows, it's almost as though you see the mathematics forcing itself on your attention from inside the fabric of the universe. We see the way that a plant grows, that a flower opens. In each case, we find that it represents a particular type of mathematics. You may think that maths can never come up with the unexpected, so how can it possibly account for the rich complexity of the world that surrounds us? How can it account for the complexity of trees and of flowers and of people? But mathematics is capable of generating unexpected things, as we can see in this pattern, in which each point represents a number, and the colour, the speed with which the numbers, multiplied by themselves, approach infinity. Although generated by a simple rule, the pattern contains almost limitless complexity. Keep in mind the general shape of these two big spheres with little lumps growing on them. And as I proceed, you will see this pattern appearing time and time again. What I'm going to do is to magnify the region inside that rectangle. You can see that we've still got the black and the blue regions, but that the structure on the line is becoming clearer. I'll do that again. I'll calculate the area within the rectangle, and you can see the complexity that's embedded in the complete diagram beginning to appear. Let me magnify this region, and we're getting the complexity unfolding in front of us. This is quite unlike an abstract painting. If you look at an abstract painting, what happens is that by the time you get to the magnification that shows the flakes of paint, you really lose all interest. But here, I can go on exposing the structure of this extraordinarily beautiful picture a million times or endlessly. So although the complexity of nature does not look as if it could be generated by mathematical rules, patterns like this show that simple numerical relationships repeated endlessly can produce quite unexpected results. Pythagoras had a vision. He had the vision that mathematics was the right way of describing the world. Indeed, he thought it was probably the only way of describing the world. But he went beyond that, and he had an extraordinary view that maybe the world was only mathematics, that all the leaves and the trees and the animals, the planets, they were just mathematics, nothing more than mathematics. Everyone thinks of matter now as being composed of atoms, but the closer we look, the less there seems to be, that the atoms become almost replaced by mathematics as we peer into them. So if I were to ask anyone what they meant by substance, they might say it's stuff that you could touch. Then I might press them to go a little further, and they might say, well, it's a collection of atoms. But why should I stop there? Why don't I press them even further? And then they will say, well, it's made of electrons. They might say, well, it's made of nuclei, so I can go on pressing. But when I press them to the point of saying, what do you mean by a proton? What do you mean, really, truthfully, deeply, by an electron or a quark? All they can do is not say, well, it's a little piece of rock in the middle of it all. All they can do is to point to a collection of mathematical expressions and say, this is what I mean. And the question that may seem weird, but I think is not absurd, is the question that Pythagoras himself raised 2,000 years ago, that as well as just being a description, is it perhaps the case that when we touch this object, we're really touching the mathematics, that is mathematics the deep structure of the world? For Pythagoras, mathematics uncovered the divine order of the world. And today, in Crotone, a Christian procession celebrates a similar belief in a divine order to the world. But it was Plato who extended Pythagoras' world of numbers to politics and religion. Plato, a Greek aristocrat, came to Crotone 80 years after Pythagoras' death. No written remains exist of Pythagoras' teachings, but Plato's works remain almost intact and have echoed through the centuries of learning that form the culture of the West. Pythagoras' view of mathematics was adopted by Plato. But Plato went further and saw mathematics as part of a perfect, pure world, an ultimate reality which contrasted with the world of appearances, of shadows that surround us. In so doing, he set up the very framework of our thought. He gave us so much by erecting a set of distinctions and a vocabulary which facilitated both Christianity and modern science. He made what we think of as the civilisation of the West possible. Composer Malcolm Williamson was master of the Queen's music and wrote a ballet based on Plato's famous analogy of the cave. Plato is as relevant today, at least as relevant today, as when it was written. And I can render it, I believe, in musico-balletic terms in a way that I couldn't rewrite it, I couldn't put it down as drama with a spoken word. For Plato, we live in a cave of shadows. Since, as prisoners in the cave, we've known nothing else, we believe these shadows to be reality. Look, you're the reality, there's the firelight, and this is just a shadow. You must be doing the same as what she's doing. Do you realise that? That's right, yes, good. But you're reflecting from over there, you see Edwin and his girlfriend there, you're an engaged couple, but you're just the shadows. And prisoners, will you look at these as if they are real things? I know they're only the shadows of the puppets cast by the firelight, but you think that this is all that life is. This is the only world you know. You've never seen these puppets behind you. And now comes the terrible shock when one of you decides to break loose and see if there really is another reality. This perfectly ordinary man, one of many prisoners, first of all thinks that the shadows are reality, an ultimate reality. Then he moves up to seeing that the puppets are real and the shadows are only from the puppets. Then he sees the fire that causes that. And then he goes into the upper world and he sees in the pool this beautiful, beautiful image, which is the moon, and he thinks, at last, I've found the ultimate reality. But then his experience with water shows him that it's a reflection, yet another illusion, and he moves from that to looking at the moon itself, which he thinks is the ultimate. Plato seems to be saying that there is an ideal world, an ideal good, which he symbolises by the sun. We are living in this world of illusions where we think the illusory is real. Behind the unreal, the illusory, the shadow, there is something more real, and behind that there is something more real. So for Plato, behind every appearance, behind every shadow, was an ideal form. These forms were arranged in a hierarchy. The forms of simple objects like tables and chairs were at the bottom, while abstract forms like beauty and justice were close to the top. The highest form was the good. For Plato, we live in a world of illusion and must strive to reach a higher, purer, more perfect world. But how was this world to be reached? How could we break our chains and escape from the cave? Plato's answer was through thought, reasoned argument and reflection. And only a few, the guardians of the people, the philosopher kings, could have access to this higher world. Plato's story about the world has not only greatly influenced Christianity, but it has provided us with the very way we think of religion. Something to aspire to, something beyond, something pure and good and not material. The whole way we see Christianity is coloured by Plato's vision of the world. We take it for granted that religions oriented towards God, life after death, an eternal world above, about which you need to hold the right beliefs. For the monks who lived in this place, the fusion of Christianity and Platonism had become complete. The Bible's earth and heaven had been amalgamated with Plato's world of sense and world of ideal forms. The Bible's flesh and spirit had become for the monks the distinction between the body and the soul. For Plato, the soul was part of the world of forms and the way we were able to have insight into the higher world. Plato held that the forms of thought are themselves immortal and unchanging. And if our minds can handle them and we can familiarise ourselves with them, then we belong to that world. We're immortal too. To a surprising extent, we still think of the soul in that way nowadays, as if it were a spiritual substance attached to our bodies. Like Plato, we still think of the soul as something that is part of us and yet is part of another world beyond. An immortal addition to our mortal selves. But it is not only the church who have adopted Plato's vision of a true, perfect world that we can catch sight of behind or beyond this one. It is a vision that has dominated art, literature and poetry. Just as the early Christians were attracted to Plato's world because they liked his doctrine of the soul, so the Romantic poets were attracted to Plato for rather similar reasons. William Wordsworth came here to Tintern Abbey because it seemed to him that Plato's teaching made art possible. Plato had shown him that nature is not just a dead machine but is everywhere full of reminders of a world of ideal forms, a spiritual world beyond this world of sense. So Wordsworth came to this place where history and eternity and nature all meet and are one in order to get his own soul into harmony with the place. And then once he'd found that harmony, he could write poetry. These beauteous forms, to them I have owed that blessed mood in which the heavy and weary weight of all this unintelligible world is lightened until we are laid asleep, embody and become a living soul while with the power of harmony and the deep power of joy we see into the life of things. In general, any work of art is a reflection of a reality and any work of art is striving to realize something that is going to capture the essence of the moment. So it's a matter of capturing something that's going to last out of something that is impermanent. I think every century, every style, every period, every era has to come to terms in one way or another with the Platonic idea of the essence of things. Plato was critical of art because for him, art meant imitation. Today, art has a very different sense, incorporating the Platonic ideal. For art is now seen as something that captures the deeper meaning that lies beneath the surface. Right here on this wall, we have a Picasso and he's trying in this particular picture, which is just entitled the accordion to get the essence of the geometry of the visual image. So he's capturing the geometrical contours, which are the essence of the accordion as he sees the object. Picasso's painting is influenced by the Platonic ideas because he's looking for the reality that underlies the fleeting essence of what we see and behold in our daily experience. I think that we all have to aspire to try to make some order out of our lives. Some reasonable and find some abiding values amidst the constant change that we have about us. It's just a matter of getting away from the feeling that life is one damn thing after another to the place where we find that there are some things that are important and that we evolve and that we are hopefully moving in a direction where we can find some sanity amidst a chaotic world. But the fine ideal of a perfect world that we can try to glimpse through poetry, literature and art and that we can aspire to in our morality and behaviour harbours within it the seeds of danger and destruction. Plato's crucial idea is that there's a single knowable ultimate truth of things out there quite independent of us but waiting to be discovered by us. The trouble is that if there is just one absolute truth in that way and we can learn it then an elite group of people can be schooled in it till they all know it alike without any disagreement and then they'll have the right to rule. The elite who've got this knowledge are for Plato, the philosopher kings, for the church it's the hierarchy and the monks. Either way you're initiated into a power elite of people who must guard a deposit of sacred divine unalterable knowledge that's come down from the heavenly world and for the sake of that knowledge people are prepared to persecute. Plato himself advocated the persecution of heretics and the church practiced it. For the youthful Plato the sentencing to death of the great thinker Socrates by the democracy of Athens for leading the youth of the city astray was a great shock. For us almost the world over democracy is seen to be a good thing but for Plato it was a dangerous chaotic form of mob rule. Applying his idea of a true perfect world to society Plato argued that society should be run to ensure that the finest and purest goals were striven for rather than to assuage the immediate demands of the mob. In the Republic Plato's description of his perfect society it was philosophers, the guardians of the people who were to rule. They alone could see the truth, could see what was right. Julius Tomin is a Czech philosopher who now lives in England. In politics as with Christianity Plato's vision of a true abstract perfect world is at one and the same time both a fine laudable and desirable ideal and a dangerous threat. Tomin is a man who has lived through the paradox of Plato's ideas who sees their strength and value and their danger. Standing in front of Marx's tomb with Plato's republic in hand it seems to me as if here Plato's ghost has come alive. The inscription on the tomb which says that for Marx the point is in changing the world catches the very essence of Marxism. What does it mean? It means you must know what is the best world and you must get power to change the society and that's in the very essence of Plato's own project in the Republic. Not even the rulers in Plato's state are entrusted with knowledge. It's Plato who prescribes. You have a perfect parallel in Marx. It's not actually the party members or even the politburo of the central committee who are entrusted with knowing where to go and how to. It's all prescribed in Marx. Like the philosopher rulers and the priests the party elite are the only ones with access to the truth. They have the right to impose it on everyone else for their own good. There is a lot in Marx for which I am positively and deeply indebted. It made philosophy something which mattered for me personally and which I knew mattered to my nation and ultimately mattered to the world. And it was in Czechoslovakia in the 1960s that Tomin was imprisoned, barred from university life and sent to work in a power station. It was in this unlikely setting that he first studied Plato. It gets into your bones when you go day by day into a place which is damp, which is full of draft, which is full of noise and you are doing all your studies there. But at the same time there were glorious moments. There were glorious moments of awareness that it was me and there who had freedom, who had freedom to think, who had freedom to speak. Czechoslovakia seen from within the factory was like a grotesque parallel to the republic and vice versa. Many aspects which Plato devised as features of his ideal state like watertight censorship, perfect regimentation of lives of every citizen, was something which Czechoslovakia realized in a rather shabby, everyday, mundane, highly imperfect way, I think, that one can show in the republic that what Plato really was after was power. And the resulting totalitarianism of the republic is the result of that aspiration. When Czechoslovakia was still a communist state, Tomin decided to set up a clandestine seminar to discuss Plato's work. Presenting Plato to students ready to risk imprisonment for the sake of doing philosophy was a great experience which I wanted others to share. I invited philosophers from Oxford to join me and they came. And each time they came, it was as if Oxford came right into the midst of Prague. And joining Prague and Oxford was an overpowering experience. And soon students from the official university started to join us, disregarding all dangers for their position there. And after several months, the authorities decided that they simply couldn't tolerate such a strong experience in the middle of Prague. And my seminars started to be invaded by police week by week and simply could not take place and I had to disconnect them. By that time, Tomin had been invited to Oxford and Cambridge and when he was forced to leave Czechoslovakia, he decided to come to Oxford. But just as the ideal of truth had led to Tomin's imprisonment in Czechoslovakia, so in Oxford, the same paradox caught Tomin in its web. The problem in the West is that everybody is taught and believed and accepts that he lives or she lives in a free society, that he is subjected to no propaganda. And so the constraints which are placed on one's thought are much more subtle. Just as the authorities in Czechoslovakia had opposed Tomin's views because they threatened their truth, so Tomin argues in the academic world of the West, there was opposition for similar reasons. Although philosophers travel to his seminars in Prague, once in Britain, Tomin believes his views were a threat to established positions and the prevailing truth. Within a year, no university department would give him a job. The goal is to acquire a place which you can acquire only if you write papers and if you write books and in all those papers and books, what is really important is referring to other people's papers and books. And in all this scholarly endeavour, the endeavour to understand Plato is completely lost. The academia here has the capacity, much greater capacity to crush you if you stand in its way or if you stand against it. If you stand against the official academia here, it's an absolutely lonely fight. Plato's legacy is everywhere. The way we structure our thought, the words and categories we use are his. But his legacy is a confusing one. Yes, Pythagoras and Plato have given us technology and science. They've given us the idea of a theory which might explain the workings of the world. They have given us the idea of knowledge and truth. They have given us ideals, social, political, religious. But if we owe our success to Plato, we also owe our chains. In art, religion, science and politics, we have fought to impose one truth. We have sought to maintain our truth over the truths of others. But maybe, after two and a half thousand years, the dominance of the Platonic world, the world of a single objective truth and an ultimate reality, is beginning to break down. Plato built a distinction between objective knowledge and subjective opinion into our way of thinking. I think we can probably get along without that distinction. We might get along without the idea that there is a true world out there to be discovered and corresponded to. On this conception of truth, unlike Plato's, there is no sense to the phrase I have arrived at the truth of the matter. There is nothing to contrast with shadows. There is no such thing as the light which someone might have seen. Nevertheless, Plato's influence on the way we think about mathematics and logic and science and so forth has been so great that we still don't know whether we'll be able to live without him. We don't know if a deeper alternative to his way of thinking is even possible. And so it is. We live in a world of order and a world of chaos. A world of progress and a world without direction. It is the 21st century.