|
Mathematics |
| The magic number This week saw the discovery of the biggest prime number yet. Big deal? Yes, says Simon Singh, because of primes the internet works, our emails are secure, and the world is a safer place Simon Singh Friday March 04 2005 The Guardian All day yesterday people were asking me just what was so interesting about breaking the world record for giant prime numbers. It was certainly surprising to see this week's breakthrough in primes sandwiched in between articles about Iraq and the Michael Jackson trial, but Dr Nowak and his 7,816,230-digit prime number did indeed deserve their place on the front page because this discovery of a new biggest prime symbolises mankind's progress in confronting a challenge of epic proportions. It is an intellectual struggle that dates back to the Ancient Greeks and which holds some of the deepest, most beautiful mysteries imaginable. First, a quick reminder. A prime number is simply one that cannot be divided by any other number except 1 and itself. So 21 is not a prime number, because it can be divided by 3 and 7, but 3 and 7 are both primes because nothing will divide into either of them. Hence, the primes are the building blocks of mathematics, the numerical equivalent of atoms. Just as a molecule of water can be broken down into two atoms of hydrogen and one atom of oxygen, so can a big chunky number such as 90 be broken down into its prime atoms 2, 3, 3 and 5, because 2 x 3 x 3 x 5 =90. Consequently, a complete understanding of prime numbers would lead to a more profound understanding of all numbers. One of the first people to explore primes was Euclid in around 300BC in Alexandria. He noticed that primes become increasingly rare as numbers increase. For example, between 10 and 20 there are four primes (11, 13, 17, 19), but between 110 and 120 there is only one (113). He wondered whether the primes eventually became extinct or whether they go on forever? Is there a biggest prime or is there an infinite number of primes? In one of the most staggeringly bril liant and gorgeous breakthroughs in the history of human thought, Euclid proved that there is an infinite number of primes. He started by assuming the opposite, namely that there is a finite list of primes. Let's assume that 2 and 3 are the only primes in the world. However, if we multiply them together (2x3) and add 1 we get 7. Clearly 2 and 3 will not go into 7, so we have a new prime. But still our list of primes is not complete, because we can multiply all our known primes (2x3x7) and add 1 and we get 43, and once again we have discovered yet another prime. The argument needs a little refinement, but Euclid was basically saying that with any finite list of primes it is always possible to multiply them together and add 1 and demonstrate that the list is incomplete. If there is an infinite number of primes, then why is it so hard to find newer, bigger primes? The primes become increasingly rare, until eventually there are vast deserts of numbers where none exist. In between the deserts there will be an oasis where one prime sits quietly, but finding the location of these oases is a hit and miss affair. The location of primes is apparently unpredictable. And this leads to the greatest prime mystery in the world, namely the Riemann hypothesis. In 1859 the German mathematician Bernhard Riemann made a conjecture about the approximate distribution of primes, but after almost 150 years nobody has yet been able to prove its veracity. It is undoubtedly the single greatest outstanding conundrum in mathematics. For pure mathematicians, proving the Riemann hypothesis would provide a firm foundation for their subject and allow them to explore the rest of mathematics with renewed vigour, but this probably seems too abstract for most people. Researching into primes might still seem like rather an arcane pursuit. What's the point? The mathematical motivation for proving the Riemann hypothesis and understanding primes is merely the search for truth. Pure mathematicians are simply curious and are intrigued by such challenges. Proving theorems is akin to climbing mountains - you prove them because they are there. Or proving theorems is like writing a symphony - the result is something that lifts the human spirit. But if you insist on something that benefits society on a more material level, then research into prime numbers can still justify itself. Modern encryption relies on the strange property that multiplying prime numbers is relatively easy (7 x 13 = ?), but working out what two prime numbers multiply together to give a certain result is much harder (? x ? = 323). Indeed, with very large numbers it becomes virtually impossible to solve such problems, and this leads to effectively unbreakable codes. Thanks to the mathematics of primes and these codes it is possible to send credit card details over the internet, which gives rise to e-commerce, more efficient businesses, lower inflation, stronger economies and a wealthier society. And thanks to primes our emails can be encrypted and made safe from prying eyes. Prime numbers mean that our privacy can be protected. And on a global scale, these prime number codes allow every government and army in the world to defend themselves against eavesdroppers and phone-tap pers. America's National Security Agency (NSA) is the biggest employer of mathematicians in the world. And if that still isn't enough, and you want to have a direct personal financial benefit, then primes can deliver again. RSA, an encryption corporation in the United States, offers $20,000 to anybody who can work out which two primes multiply together to give 3107418240490043721350750035888567930037346022842727545720161948823206440518081504556346829671723286782437916272838 033415471073108501919548529007337724822783525742386454014691736602477652346609. Solving this problem would help gauge the strength of today's codes. Or, you can try to break Nowak's record for the biggest prime. Download some free software and join the Great Internet Mersenne Prime Search (Gimps). You will become one of 40,000 Gimps around the world and if you happen to be the Gimp that discovers a prime with more than 10m digits then you can claim a reward of $100,000 from the Electronic Frontier Foundation. For the really big bucks then you just have to prove the Riemann hypothesis. The Clay mathematics institute in Massachusetts is offering $1m for what will be the most important proof in modern mathematics. And not only will you become rich, you will become famous and achieve the closest thing to immortality. Scientific theories are often proved incorrect or at least refined over the course of time, but mathematical theorems last for ever. We laugh at Pythagoras's ideas about medicine, but we still learn his mathematical theorem at school. Or, as the British mathematician GH Hardy put it, "Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. 'Immortality' may be a silly word, but probably a mathematician has the best chance of whatever it may mean." Simon Singh is the author of Fermat's Last Theorem. His latest book is Big Bang, a history of cosmology. |
| On the Mathematics of
Multiple Universes www.scienceweek.com For much of history, people thought that the Sun and planets were all there was to the Universe -- apart, that is, from a few point-like stars nailed to a crystal sphere. Much later, at the dawn of the twentieth century, astronomers, aided by a new generation of giant telescopes, discovered that the Sun was merely one among a hundred billion or so other suns swirling in a great whirlpool of stars called the Milky Way. Later still, and here we are talking about a mere seventy-odd years ago, it was discovered that the Milky Way was but one "galaxy" among ten billion others fleeing from one another like so many pieces of cosmic shrapnel in the aftermath of a titanic explosion: the Big Bang. "The history of astronomy," as the novelist Martin Amis has aptly re-marked, "is the history of increasing humiliation." For a century now, the human race has been shrinking, its importance in the cosmic scheme of things dwindling as the Universe grows, not in small, manageable steps but in sickening lurches of the human mind. And if you think that today, finally, we have reached the end of the road, prepare yourself for another body blow. More and more physicists are convinced that our Universe is not alone but merely one among an infinity of others, drifting like bubbles on the river of time. Among those who think this is physicist Max Tegmark of the University of Pennsylvania. Tegmark imagines a "multiverse" in which the individual universes dance to the tune of different laws of physics. It's a remarkable idea, and Tegmark has arrived at it by a remarkable route: by pondering an abstract and esoteric question. Why is mathematics so damned good at describing our universe? Three and a half centuries ago, Isaac Newton (1642-1727) discovered that the manner in which massive bodies move when they are subjected to forces can be perfectly described by simple mathematical equations. Following in his footsteps, generations of physicists have appealed to mathematics for a compact description of the world. Their spectacular success in penetrating nature's inner workings leaves little doubt that, as Galileo (1564-1642) observed, "Nature's great book is written in mathematical symbols". The strong implication is that God is a mathematician (!) and if physicists ever succeed in finding a complete description of the Universe -- one that neatly summarizes all fundamental features of reality -- such a "theory of everything" will be a mathematical theory. We can phrase this a little more precisely. The edifice of mathematics is built from what mathematicians call "formal systems". Familiar examples of formal systems are arithmetic and flat-paper geometry, sometimes known as Euclidean geometry. But mathematicians know of many others, such as Boolean algebra and group theory. A formal system consists of a set of givens, or axioms, and the consequences, or theorems, that can be deduced from them by applying the rules of logic. For instance, the axioms of Euclidean geometry include the statement that "parallel lines never meet", while the theorems that can be deduced from the axioms include such statements as "the internal angles of a triangle always add up to 180 degrees". Tegmark, who is a Swede and therefore happiest when surrounded by pine trees, likes to think of mathematics as a towering Christmas tree with shiny boxes hanging from all of its branches. In each box is a different formal system. Some boxes have already been opened and their contents examined, but many have not, providing continuing employment for the world's mathematicians. Nobody knows what wonders remain to be unboxed. "However, one thing can be said for sure," says Tegmark. "One of the unopened boxes must contain the theory of everything." This prompts a serious question. Why should this box correspond to our Universe? After all, every box on the tree of mathematics contains a formal system and, apart from obvious differences in their complexity, there is nothing to distinguish one box from another. "Why should one box be privileged above all others?" says Tegmark. "Why should one mathematical structure, and only one, out of all the countless mathematical structures, be endowed with physical existence?" Try as he might, Tegmark has been unable to answer this question. Rather than seeing this as a failure, however, his stroke of genius is to see it as a virtue. Perhaps there is a good reason why there appears to be nothing special about the box containing the theory of everything. Perhaps that reason is that there is nothing special about it. That there is nothing special about a circumstance, most notably the position of Earth in the Universe, has proved to be a powerful guiding principle in science. In the sixteenth century, the Polish astronomer Nicholas Copernicus (1473-1543) maintained that movement of the Sun and planets across the sky could best be explained if Earth were not, as most people supposed, the central pivot about which the cosmos turned, but if instead it were just another planet circling the Sun. In the twentieth century, astronomers extended the "Copernican principle" to the wider cosmos revealed by their giant telescopes. Just as Copernicus had maintained that there was nothing special about the place of Earth in what we now know as the Solar System, so modern astronomers maintained that there was nothing special about the place in the Universe of the Milky Way, the galaxy that contains Earth and the Sun. On this apparently flimsy bedrock -- the dull, unexceptional nature of the Milky Way -- is founded the entire edifice of "cosmology". Cosmology is the science that has taught us that we live in a universe that is expanding and cooling in the aftermath of a violent explosion that took place twelve to fourteen billion years ago. Tegmark's brainstorm is to take the tried-and-tested Copernican principle and extend it to the tree of mathematics. It is his contention that there is absolutely nothing special about the box that contains the theory of everything -- nor, for that matter, about any of the other boxes on the tree of mathematics. Every single one has equivalent status. All, in other words, correspond to universes. Pause for a moment to take in what this means. Tegmark is saying that, in addition to a universe that dances to the tune of the theory of everything -- the one we find ourselves living in --there is a universe governed by the laws of arithmetic, a universe ruled by the laws of two-dimensional geometry, and so on, ad infinitum. Out there in the big wide multiverse, perhaps beyond the farthest limits yet probed by our telescopes, there are universes that dance to the tune of all conceivable mathematical equations. Consider the implications of this audacious idea. In the 1930s, the Austrian physicist Eugene Wigner (1902-1995) famously remarked on "the unreasonable effectiveness of mathematics in the physical sciences". Nobody has come up with a satisfactory explanation of why it is so effective at encapsulating the essence of the Universe. No doubt even Newton in his time puzzled over the matter. But if Tegmark is right, suddenly it is abundantly clear why mathematics is so good at describing physics. The answer is almost laughably trivial. Mathematics is so good at describing physics because mathematics is physics. According to Tegmark, every last formal system on the tree of mathematics is endowed with physical existence. Every last formal system corresponds to an actual universe. Tegmark has extended the Copernican principle beyond our Universe to an infinite set of universes. Not only is there nothing special about our status within the Universe, he claims; there is nothing special about the status of our Universe within the infinity of universes that constitutes the multiverse. Adapted from: Marcus Chown: The Universe Next Door: The Making of Tomorrow's Science. Oxford University Press 2003, p.97. More information at: http://www.amazon.com/exec/obidos/ASIN/0195168844/scienceweek |