Amazing Quantum Theories That Make Us Rethink How the Universe Works

The 1905 special theory of relativity was limited to measurements made by equipment moving without acceleration. The general theory of relativity eliminates this special restriction. To achieve this, however, Einstein had to revolutionize our concepts of space and time even more deeply. Building on the insights of the earlier theory, he now argued that space can be bent by matter. Thus, we have reached a third stage in the philosophy of space and time. The atomists first proclaimed that space, or nothing, existed in order to solve the problem of change. Newton revived this doctrine when, despite the criticisms of Aristotle and Descartes, he needed absolute space for his broader theory. Einstein now completes his revolution with the momentous claim that space changes. That is, space has evolved from a static structure proposed to In Newton's classical physics, space tells bodies how to move. A body given an initial shove will follow a straight line in space unless deflected by some other force. Einstein thought that this one-way influence was peculiar. If space acted on bodies, should not bodies act on space? The germ of Einstein's later theories of space and time, his general theory of relativity, is that this circle is completed. Central idea of the theory of general relativity, space-time tells bodies how to move, and bodies tell space-time how to curve. But what is a curved space or curved space-time? If space is emptiness, or even a kind of nothingness, how could that curve? What would space bend into? Physicists begin to answer these questions by insisting that only observable entities may play a role in physics. Since space is invisible, and we see no lines of latitude and longitude etched across the night sky, the straight lines through space can only be discovered by following bodies around and observing their paths. Thus, physicists define a straight line through space as the path traversed by a body moving without interference of any kind, that is, moving inertially. Suppose, however, that two bodies are shoved along two adjacent parallel paths in the same direction, but later bump into each other. Newton would say that some force must have pushed them together. If they were moving inertially without disturbance, they would have followed two parallel paths, and since parallel lines never intersect, the bodies would never collide. Einstein disagreed and suggested another possibility. Suppose that the bodies bump into each other because space itself is curved. Suppose that space is crisscrossed in all direction by lines that inertial bodies follow, but that the fabric of space can be bent, twisted, and curved. Then bodies coasting along parallel paths may indeed bump into each other if the paths converge. It does sound strange to say that straight lines curve, or that parallel lines intersect, but this is just a superficial oddity of physicist language. They adhere to the definition above even when parallel straight lines intersect, and prefer simply to say that they have enlarged our notion of what straight and parallel mean. Einstein's concept of curved space is often illustrated by the rubber sheet analogy. Strictly speaking, it is space-time that is curved, but this analogy helps us picture a curved space. Imagine a large sheet of supple rubber stretched out and secured along its edges, perhaps like the surface of a large drum in an orchestra. A heavy lead weight placed in the center of the sheet will sink down into a smooth circular well. This gives us a picture of what it means to say that mass causes space to curve. More interestingly, suppose that children begin shooting marbles back and forth along the rubber sheet. Suppose a marble is shot straight, but not directly at the lead weight. Instead of following a straight line, its path will bend towards the lead weight. Either it continues on away from the weight after its deflection, or it will spiral down into the well. There is no significant attraction between the lead weight and the marble. The lead weight affects the marble by distorting the spatial surface that it travels along. This nicely shows how Newton's gravitational forces are replaced in Einstein's theories of curved spaces. For Newton, bodies move towards each other when they exert attractive forces. For Einstein, they influence each other indirectly by affecting the space between them. Each body curves the space in its environment. When other bodies coast along straight lines, they veer toward the center of curvature. Some say that in general relativity, gravitational forces are geometrized away. Suppose further that the lead weight on the rubber sheet was moved rapidly up and down. If the sheet were large enough and flexible enough, ripples of upward and downward motion would spread out across the sheet. Perhaps they would look like the circular waves in the surface of a pond caused by a falling stone. Similarly, Einstein's theory predicts that moving masses will cause traveling distortions in the fabric of space, gravity waves. Thus, the rubber sheet analogy illustrates the way mass curves space, the curvature of straight inertial paths and gravity waves. The world was convinced of the truth of Einstein's outlandish conception of curved space in 1919. In the aftermath of Germany's defeat at the end of the First World War, the English astronomer Arthur Eddington organized an expedition to observe a total eclipse in South Africa. British confirmation of the theories of Einstein, a Swiss national working in Berlin, would at once advance physics at the cause of world peace. How might observations made when the sun was eclipsed by the moon confirm the curvature of space? Light follows straight lines, but according to Einstein, these lines should bend around and towards heavy masses. Thus, starlight traveling closely by and almost grazing the sun should be slightly deflected towards it. Ordinarily, the sun's brilliance would drown out the starlight and make it impossible to observe this deflection. But during a total eclipse, the moon would just block out the sun's own light. The images of stars around the edge of the darkened moon should be slightly out of place because its light was bent on its way to Earth-bound telescopes. During the summer and early autumn of 1919, Eddington measured his blurry photographs and calculated his results. Word began to leak out that his team had confirmed Einstein's predictions about the degree of deflection within experimental error. Back in his office, according to a well-known story, when Einstein was first handed the historic telegram with a preliminary announcement of the results, he glanced at it, set it aside, and continued talking to a doctoral student. When she asked about its contents, Einstein's calm report caused great jubilation. Einstein, however, remained calm. Somewhat amused, he said simply, but I knew all the time that the theory was correct. When pressed to admit that the experiment might have given negative evidence, he went on, in that case, I'd have to feel sorry for God, because the theory is correct. Eddington announced his results publicly in London at a meeting of the Royal Society, whose president some 200 years earlier had been Isaac Newton. The next day, the 7th of November 1919, the London Times carried the now-famous headlines on its front page. Revolution in Science, New Theory of the Universe, Newton's Ideas Overthrown. From that day, Einstein was a celebrity, and space was curved. Although historians have since cast some doubt on Eddington's first crude results, the bending of starlight is now routinely observed by astronomers. The Hubble telescope orbiting Earth has produced some beautiful images of what is now called gravitational lensing. This occurs, for example, when one distant galaxy sits far behind another nearer galaxy. As light streams around the nearer galaxy, the rays may be bent towards us from several different angles. The telescope will see multiple copies of the distant galaxy arrayed around the nearer one. Physicists now accept that a substantial number of experiments confirm Einstein's ideas of a curved spacetime. Gravity waves, however, have been a significant exception. Even movements of very large bodies produce extremely weak waves, and they have so far proved impossible to detect here on Earth. In 1974, however, Taylor, Hulse, and their colleagues produced the first serious evidence for the existence of waves rippling through the fabric of space. They studied the motion of pairs of stars orbiting rapidly around each other, and showed that they lose energy at just the rate that would be expected if gravity waves were emanating outwards. In 1993, they won the Nobel Prize for this discovery. Several large ground-based projects are now underway to detect gravity waves originating in outer space. The idea that space is curved seems bizarre. Einstein emphasised that the train of thought that led to this conception depended heavily on certain thought experiments. In these imaginary situations, Einstein used his reason and intuition to investigate physics without mathematics or experiment, and at times this method made Einstein appear more like a modern philosopher than a hard-nosed physicist. Two of these now-famous thought experiments illustrate one route Einstein took to the discovery of general relativity, and also are good examples of the breathtaking beauty of Einstein's arguments. Suppose that some surveyors live on the surface of a large spinning wheel, and make a series of length measurements. Since acceleration means a change in speed, or a change in direction away from straight line motion, every part of the wheel except its very centre is accelerating as it rotates. Such surveyors might, for example, measure the radius of the wheel from its centre to its rim, and also its circumference. For an ordinary stationary wheel, the circumference is about three times longer than the diameter. That is, the length of the circumference divided by the length of the diameter is the number pi, which is about 3.141. But Einstein claimed this would not be true for the spinning wheel. Remember, he said, the lesson of special relativity. Faster speeds mean shorter lengths. That is, there is length contraction in the direction of travel. Special relativity applies only to measurements made with rulers and clocks moving at steady speeds in the same direction. But, Einstein argued, it also applies approximately to accelerating bodies. During noticeably short moments of time, even an accelerating body moves at about the same speed. A sports car may accelerate to high speeds within 20 seconds, but during each of those 20 seconds, its speed is more or less the same. Thus, during short moments, we can apply special relativity to measurements made by accelerating rulers and clocks, and predict that they too will find length contraction and time dilation. Consider how this affects the surveyors on the spinning wheel, which is, by assumption, a well-formed circle. They measure the radius by laying their rulers along one of the spokes. Since these rulers will lie at right angles to the direction of motion, they are not contracted at all. However, rulers laid along the circumference will experience length contraction as the wheel spins. More copies of the ruler would fit into the circumference. This means that the measured length of the circumference divided by the measured length of the radius will give an answer different to that for the stationary wheel. Instead of a number about 3, the result will be much larger. The rulers along the circumference shrink, while those along the radius stay the same, and so the ratio increases. Nonetheless, the wheel is a circle. It is the same distance from its hub to its circumference in every direction. This is remarkable. According to Einstein, laws of geometry known since ancient times do not apply to rapidly accelerating bodies. In sum, Einstein's thought experiment with the spinning wheel taught him that acceleration distorts geometry. Building on this insight, a second thought experiment led Einstein deeper, toward the core ideas of general relativity. Imagine that an experimenter is in a inside of a lift, and that the room is located in outer space far from any planet or any other source of gravity. Suppose further that a hook and long rope is attached to the outside of the room at its top, and that God or some other being steadily pulls the room upwards. This will cause the room to move up faster and faster with constant acceleration. Anything floating or unsupported in the room will settle to the floor, and remain there as long as the acceleration continues. Einstein realized that for the experimenter enclosed in the room, the upward acceleration would feel the same as a gravitational force pulling down. In fact, no experiment performed inside the room could distinguish between an upward acceleration and downward gravitational pull. Perhaps objects fell to the floor because God was pulling the room up, or perhaps some unknown planet had swum into the neighborhood and its gravity was pulling everything downwards. No experiment could decide. In sum, Einstein's thought experiment taught him that acceleration is equivalent to gravitation. If no experiment can detect a difference, he concluded, they are the same as far as physics is concerned. The reason for this equivalence is that gravity affects all bodies in the same way. It is a universal force. A gravitational force will pull all the bodies in the room down at the same rate, and thus is indistinguishable from an upward acceleration, which tries to pull all the bodies upwards. In contrast, electric forces affect only bodies that carry electric charges. Thus, an electric attraction pulling down on the room would affect only some of the objects in the room. The experimenter inside would quickly see that only charged bodies were affected and conclude that an electric force was present. It is the universality of gravity that makes it equivalent to an upward acceleration. Einstein's line of reasoning reached a stunning climax when he combined the lessons of these two thought experiments. Put somewhat crudely, he triumphantly argued, this is the line of reasoning that Einstein used to argue that matter tells spacetime how to bend. Matter is the source of gravitation, and gravitation bends spacetime. Thus, spacetime is curved and behaves like a flexible rubber sheet. This is a bewildering fantastic series of ideas. No one would take them seriously if Einstein had not used them to make some very surprising predictions that experiments later confirmed. We have met one example above. If we assume that light follows straight lines, then the paths taken by light should be distorted by any nearby matter that distorts the geometry in its neighborhood. This is the effect confirmed by the eclipse expedition and by gravitational lensing. Light does not follow straight lines, but the lines are bent by large masses. Many other predictions followed. Returning to the spinning wheel, Einstein argued that a clock on the rim would run more slowly than a similar clock left at the center of the wheel. This is just another application of the idea that faster speeds mean longer hours. Since the rim is accelerating, Einstein argued that acceleration also causes time dilation. But since acceleration is equivalent to gravitation, he concluded that gravitation causes time dilation. Clocks that feel stronger gravity pulling them down run more slowly. In 1960, Pound and Replica confirmed this claim directly by placing two similar clocks at the top and bottom of a tower on the campus of Harvard University. Since the clock at the bottom was nearer to the earth and therefore felt its gravitational pull more strongly, Einstein's theory predicted that this lower clock would run more slowly. Although the difference was very, very slight because the clocks were so near to each other, delicate measurements found just the effect Einstein predicted. In fact, Einstein took his ideas one step further. Rather than saying that mass causes gravity, which distorts the geometry of space, he asserted more simply that mass is directly associated with distorted geometry. In other words, rather than picturing gravity as something contained within and distorting space, he said they were one and the same thing. He identified gravity with the curvature of space. The rubber sheet analogy helps make this clear. The marbles rolling past the lead weight are deflected by the curvature of the rubber sheet. There is no need to assume some gravitational force or field that exists over and above the sheet and pulls on the passing marbles. Thus, for Einstein, gravity just is the curvature of space. As before, the forces have been geometrized away. This insight that gravitation was equivalent to an acceleration was key in Einstein's struggle to generalize his earlier theory of special relativity. Remember that this was limited to special cases, namely to measurements made with unaccelerated or inertial rulers and clocks. In the early theory, Einstein simply did not know how to handle acceleration. Newton thought that acceleration was so strange that it provided the best evidence for absolute space. Einstein rejected absolute space but simply ignored the problem of acceleration in the early theory. Now his thought experiment suggested a new way to think about acceleration. Instead of pointing to the existence of absolute space, acceleration was the equivalent of a gravitational field. Thus, instead of accepting Newton's talk of a mysterious invisible space containing all bodies, Einstein could say that all effects of acceleration in the room were due to gravitation. Thus, Einstein's thought experiment with the small room was also his answer to Newton's bucket argument. Einstein agreed that the surface of water was concave but denied that this was due to accelerated motion relative to absolute space. The concavity was evidence for the presence of some gravitational field pulling the water out to the sides of the bucket. This claim is very much alive and at the center of contemporary debate. The universe is exceptionally large. Although light is fast, it takes it about three years to travel to the nearest star to Earth. A typical galaxy like our Milky Way galaxy contains about 100 billion stars. Light crosses from one end to the other in about 50,000 years. In human terms, galaxies are incredibly huge, but they are only tiny specks compared to the universe. There may be 50 million galaxies in the universe. Some cosmologists speculate that a beam of starlight would take 100 billion years to travel through the entire universe. If that is true, then even the Hubble telescope orbiting around Earth can see only a small portion of the whole universe. In ancient times, the Roman poet Lucretius argued vividly that the universe must be infinite. Suppose, he said, that there was an end to space, then we could stand at the brink and hurl our spear across the edge. If the spear rebounded, we would know that there was something beyond the end. If the spear did not rebound, then it carried on into more space. In either case, there must be something beyond the brink, something beyond the supposed end of space. Thus, he concluded, it is not really an end, and space must continue on infinitely. Newton, too, thought that his absolute space was infinite. But once Einstein had persuaded us that space can curve and bend, questions about the shape and size of the universe were opened to experimental tests. Interestingly, Einstein raised the possibility that the universe might have a limited finite size, but not have any boundary. That is, the universe may be finite and unbounded. This does not make sense at first, but can be explained by a simple analogy. Consider the surface of a balloon. The surface itself is a two-dimensional curved space. Although the surface has a limited size, say about 50 square centimetres, the surface has no boundary. An ant crawling over the balloon could move forwards endlessly without meeting a boundary. Thus, a curved space that turns back onto itself can be finite even though it has no boundaries. We live in a four-dimensional space-time, and Einstein considered the possibility that our universe is curved in such a way that it is finite and unbounded, like the balloon. This would have a surprising consequence. Just as an ant on the balloon crawling in a straight line would return to its starting point, astronauts travelling in a straight line for an exceptionally long time would find themselves back on Earth. This seems crazy, but there are experiments that might provide evidence that this is the case. Some astronomers are pointing their powerful telescopes in diametrically opposite directions and hope that they might see the same distant galaxy in both directions. Most think there is little chance of success, but we might be surprised. We reverence ancient Greece as the cradle of western science. Here, for the first time, the world witnessed the miracle of a logical system which proceeded from step to step with such precision that every single one of its propositions was absolutely indubitable. I refer to Euclid's geometry. This admirable triumph of reasoning gave the human intellect the necessary confidence in itself for its subsequent achievements. Some seek truth and some doubt it, some are dedicated to seeking progress in our knowledge of reality, and some find it all too absurd. In the European tradition, the battles between these two warring tribes took place in the shadow of a great fortress. Defenders of truth could raise their fingers over the heads of the sceptics and point upwards to mathematics. A shining crystal palace of certainty, surrounded by thick walls of deductions and demonstrations. But the revolution in theories of space and time during the 20th century finally levelled this fortress. Sceptics have overrun even mathematics. Much of the fashionable relativism that has flourished during the past 30 years took heart from this downfall of mathematics, and with good reason. Geometry was the glory of ancient Greece. Building on its beginnings in Egypt and Babylon, the ancient Greeks pursued geometry with extraordinary passion and precision for a thousand years, and first built up the fortress of mathematics on firm foundations. For them, geometry was the study of the real properties of real things. It was a branch of metaphysics. Like their other theories, it was object-oriented. They did not, in the first place, think of triangles and spheres as inhabiting a geometric space. Instead, they typically drew concrete figures in the sand or on their slates, and investigated their properties and inner relations. The Greeks did more than reveal the beauty of geometry in a thousand different theorems. They organised this knowledge, and invented a new method that grounded it securely in the simplest truths. This axiomatic method is one of our most precious inheritances. Even in ancient times, it was applied broadly to physical science, optics, and philosophy by Proclus. This method is essentially the same as that used in the step-by-step arguments here. It begins by identifying the basic premises, and then carefully illuminates the reasoning that leads from them to some new truth in the conclusion. One historian has asserted that all the prominent mathematicians in later antiquity were students or scholars in Plato's academy, or students of these and so on. The most prominent of these lineal descendants was Euclid, who lived in Alexandria, Egypt, around 300 BCE, about 50 years after Plato's death. Euclid's famous book on geometry, The Elements, is one of the great books of European culture. Its beauty is twofold. By brilliantly creating and arranging his proofs, he was able to show that the mathematical discoveries of the preceding centuries could be traced back to five premises. Just five ideas formed the foundations of the entire crystal palace. Secondly, Euclid concluded his treatise with the touch of a real virtuoso. Plato had written of the beauty and symmetry of regular solids, which became known later as the platonic solids. These are three-dimensional geometric figures like the cube, whose size and angles are all the same. Oddly, even though there are an infinity of differently shaped sides that might be fitted together in infinitely different ways, there are only five different regular solids. There are only five ways of building perfectly symmetrical solids with flat sides. At the end of his treatise, Euclid leapt from two-dimensional plain figures into the realm of solids. In the great climax of Greek geometric thought, he proved that no future mathematician would ever discover another perfect solid. That the five platonic solids were perfect and complete. Historians do find it difficult to judge how much of The Elements is original, and how much Euclid merely collected and systematized the efforts of early geometers. After Euclid, scribes neglected copying the scrolls of early geometers, and in time their works were lost altogether. Euclid's basic premises contained a time bomb, which later was to topple the edifice he so carefully erected. His five premises were supposed to be the most basic and simple truths, grounded in the deepest simplicities of being. But consider each in its turn. The first four are plausible basic truths, but the fifth suddenly extends common sense into the great unknown. How would we prove what happens at infinity? What experience or intuition could help us here? Already in ancient times, there was discomfort with Euclid's fifth postulate. Beginning with the Renaissance in the 15th century, and continuing into the scientific revolution in the 17th century, the prestige of mathematics rose again, and new suspicions were levelled at his mysterious fifth postulate. Many ambitious mathematicians sought to outdo Euclid by cleaning up his axioms by finding fewer or simpler ones that would serve as a more secure foundation for the queen of sciences. Some 300 years of searching produced only mounting frustrations. Mathematicians could neither do without Euclid's fifth postulate or its equivalents, nor show that it was wrong or faulty. It was there, ugly and stubborn, like a wart on the face of mathematics. In desperation, several innovators finally adopted an indirect strategy. Instead of seeking for a deeper truth behind or underneath the fifth postulate, they would assume that Euclid's postulate was false, and, using this reversed axiom together with the other four, would derive contradictions. That is, they would perversely assume that all parallel lines intersect. Surely they thought such an abomination would lead to contradictions. By leading this false assumption into a contradiction, that is, by reducing it to an absurdity, they would reveal its falsity. Surely this would prove that Euclid's postulate could not be false, and was a necessary foundation stone for a true and consistent mathematics. This result was astonishing and led to some very deep thinking on the part of Europe's leading mathematicians. They had combined the false premise with Euclid's four other simple premises and begun proving things, hoping to squeeze out a contradiction. But one theorem led to another, and soon they had a large number of new theorems but no contradiction. They continued on, hoping the bad apple they had started with would spoil the whole barrel of new theorems, but could find no contradictions at all. To their surprise, they had soon built up an entire alternate mathematics, as if God had created a psychotic alternative universe and they had stumbled on its strange system of geometry, a world where parallel lines met each other. Mathematicians soon realized that they could build pictures or models of these strange worlds. The simplest was the two-dimensional surface of a sphere. Consider a globe plastered over with a map of Earth. The lines of longitude cut the equator at right angles, and all intersect at the north and south poles. But one definition of parallel lines is that they cut another line at the same angle. Thus, at the equator, the lines of longitude are parallel, but they nonetheless intersect at the poles. In fact, all the largest circles that can be drawn on the globe are parallel, according to this definition, and yet intersect. Unwittingly, the mathematicians had created a new axiom system to describe the geometry of circles drawn on the surface of a globe. The strange universe they had created was not Euclid's, but it was free of contradictions. It was the geometry of a curved space. Suppose that we experience only a small bit of the universe, and thus believe that parallel lines do not intersect. Suppose that parallel lines extended across the universe far enough do intersect, like lines of longitude at the poles. Suppose we live in a universe that is not described by Euclid's geometry. Suppose the universe is curved. These thoughts must have made Gauss dance. Although Gauss's thoughts seemed a mere fantasy about lines extending across the cosmos, he brilliantly saw that his ideas could be tested here on Earth. The key was that triangles on a curved space differ from those on a flat space. Both are constructed of three line segments meeting at three angles. In the flat space described by Euclid, the sum of those angles was always two right angles, 180 degrees. But this is not true in curved space. The sum of the three angles in a triangle in curved space can be more than two right angles. To see this, consider the large triangle on the globe constructed from two lines of longitude that intersect at right angles at the north pole. If the part of the equator that lies between them is taken as the third side, then a triangle is formed between the north pole and the equator. Every one of its angles is a right angle, thus the sum of its three angles is three right angles. The curved space permits lines meeting at large angles to fit together to form a triangle. Gauss was so seized by this fact that he soon persuaded his friends to test his ideas. They knew that small triangles would appear to be flat and have angles that added up to two right angles. But triangles that were large enough might begin to reflect any curvature in space and have angles that added up to more than that. Gauss and his friends hauled telescopes and other apparatus to the tops of three mountains in Germany. With lights flashing on two of them and the telescope on the third, they were able to measure the angles between the mountaintops. Repeating this, they found all three angles. Alas, after careful collaborations and calculations, they discovered that, within experimental error, the sum of the three angles was two right angles. Euclid was right. His geometry described the true structure of the world. Defeated, Gauss trudged back down the muddy slopes, not realising that he had simply been two centuries too early. Although his challenge to Euclid was stillborn, the new geometries blossomed into an exciting subspeciality of mathematics. Soon, new paradises of exotic geometries were being invented and explored in the minds of mathematicians all over Europe. Euclid still reigned, but subversive energies lurked in the shadows. As we have seen, Einstein worked to extend his special theory into the more general theory, which included gravity for eight long years, from 1907 to 1915. This was a titanic struggle of late nights, frustrating mistakes, and time-wasting failed ideas. At one point, Einstein began grappling with the idea that gravity might distort geometry. This was a breathtaking intuition, but Einstein was a poor mathematician and knew too little of advanced mathematics to express his ideas. Desperate, he finally appealed for help to a friend from his days at university. Now a professional mathematician, the friend surprised Einstein by saying that an entire sub-branch of mathematics had been developed to describe the geometry of curved spaces. The mathematical language Einstein had been groping for in the dark had already been worked out in detail, but Einstein had not known. This new high-powered language for describing curved spaces had been worked out by mathematicians merely for its sheer beauty. A few of them, like Clifford and Riemann, speculated that Gauss was perhaps correct, and that further experiments should be done to investigate the geometry of the physical universe. But in the main, curved spaces were studied as pure abstractions. Once in Einstein's hands, however, the new language helped him make the final breakthroughs to his general theory. The experiments celebrated as confirmations of Einstein's theory also confirmed that Gauss was indeed correct. The geometry of curved spaces, with triangles whose angles did not add to two right angles, was the geometry of our world. The lesson is a profound one. The axiomatic method was born out of a desire to isolate the essence of ideas and their relations. Euclid's masterpiece applied the method and surprisingly revealed that assumptions about parallels were a cornerstone of geometry. Two thousand years later, this paid off in a wholly unexpected way. It inspired the development of non-Euclidean geometries and brought Einstein's general relativity to fruition. The step-by-step analysis of arguments is a precious inheritance. Anyone first encountering the idea of curved space often asks what it is curved into. Generally, things bend by moving into some unoccupied space. But if the whole universe consists of curved space, what could it bend into? Physicists usually reject this question, and this section explores one of the motivations for this neglect. We usually think of ourselves as three-dimensional creatures living in a three-dimensional world. But suppose there was a two-dimensional universe called Flatland, populated by dismally thin two-dimensional creatures, the Flatlanders. They might slide around their world very contentedly and never feel that they have been deprived of a dimension, just as we feel cozy in our three dimensions. Flatlanders might develop a sophisticated geometry and indeed might decide that their space was curved, as if they were, unbeknown to themselves, living on the surface of a balloon. They might notice, for example, that the angles in large triangles did not add up to two right angles. Their measurements would, of course, all take place within their world, and would have to describe the curvature of their space in a language that referred only to their two-dimensional world. It would have to be an intrinsic description. Gauss invented a way of describing a curved space that depended on features and measurements confined within that space. This language is complete and does not require any reference to what may be outside the space. The Flatlanders, after all, may be correct that their universe is only two-dimensional, that there is not even nothing outside their horizontal world. Geometers say that Gauss's intrinsic descriptions need not refer to any outer, embedding space. Physicists have adopted Gauss's viewpoint. Curved spaces are entirely autonomous. Since a curved space can be described internally, curvature does not presuppose any outside space for it to curve into. In fact, this is motivated by their empiricism. If physical theory does not need to mention a structure, if measurements and predictions do not depend on it, then the structure has no place in physics. Embedding spaces are superfluous. So far we have two intersecting stories. Einstein's monumental struggles to extend special relativity led him to believe that space was somehow curved. He did not know that investigations of Euclid's fifth postulate had led mathematicians to the development of an elaborate new language for curved spaces. When Einstein adopted this language in his new general theory, he was able to make precise and surprising predictions. Their confirmation persuaded us that the world was correctly described by these exotic geometries. That Euclid was overthrown. These facts tumbled philosophy into a new world and must affect the way every philosopher thinks. They mark a new age in the history of reason. Amid the busy, hurly-burly of our ordinary lives, the philosopher's search for permanent truth, for deeper, lasting understanding, seems not only impractical, but also hopelessly naive. From Plato onwards, however, philosophers could always point at mathematics as an ideal of certainty and knowledge. They could say that truth was attainable, that they had proud, noble, valuable examples here ready at hand. Under the flag of Euclid's geometry, philosophers could rally the troops and fend off the skeptics. For some 2,000 years, Euclid's axioms were thought true. They were precise insights into the nature of being, into the structure of our world. The edifice of inferences and theorems he built on his axioms was a model for deriving truth from truth, an ideal for reason everywhere to struggle towards. General relativity led to a re-evaluation of Euclid. Before Einstein, the exotic geometries of curved spaces were often thought to be mere fantasies, games that mathematicians might play during their idle hours. But Einstein showed that experiment favoured these non-Euclidean geometries. It was the world that was fantastic. Thus, Euclid's axioms were not insights into being. They were reinterpreted as mere historical accidents. They were arbitrary starting points chosen by a mathematician with a limited imagination. Just as people who do not travel much might think that the Earth is flat, mathematicians studying only circles and lines on their little slates might think that Euclid's axioms were obvious. If they had first studied larger systems like triangles formed by mountains or stars, they might have observed that three angles in a triangle do not always add to two right angles. They might have chosen different axioms. To add insult to injury, even Euclid's reasoning was found to be flawed. With the rise of more powerful logics in the 20th century, the elements was carefully scrutinised and found to contain a catalogue of minor errors and omissions. Although all of Euclid's theorems did follow from his axioms, his proofs were sometimes sloppy or incomplete by modern standards. Euclid was cut down to human size. Mourning the loss of their hero, philosophers retreated. They had thought of axioms as truths, as the deepest, most secure foundations for their systems. Now axioms became mere assumptions, mere premises. They were provisional starting points which another philosopher might or might not accept. Instead of pillars driven into bedrock, axioms were merely floating ideas selected for convenience on given occasions and disposed of tomorrow. Non-Euclidean geometries of course did not replace Euclid with a new crystal certainty. Once Einstein had shown that experiment would decide which geometry described the physical world, he made all geometry uncertain. Future experiments and deeper theories may show that even these curved spaces are superficial, special cases. Deep down, our world may be neither Euclidean nor non-Euclidean. There is no longer any certainty. Mathematicians pioneered this new view of axioms, as they were the first to be acutely aware that Euclid had competitors. Among philosophers, Russell loudly propagandised for the new viewpoint in his Principa Mathematica, where he emphasised that his assumptions were doubtful and provisional. He claimed only that they were sufficient to derive the conclusions he needed, and not that they were necessarily true in any deeper sense. After Einstein and early work by the philosopher Hans Reichenbach, Euclid's downfall was widely accepted. Axioms became assumptions. For many philosophers, these developments represent real progress, a welcome sign that philosophy has thrown off its dreams of achieving truths that escape from human fallibility. Complete certainty was always beyond our grasp, they say, and the myth that mathematicians had founded simply confused philosophers. We are not diminished by Euclid's downfall, we are liberated. Living with a sense that all knowledge is as fragile or as strong as we make it is a mark of maturity.