1amjng Mil INI INI] gravity. I Eowever, the story of Newton's law rests heavily on Greek work on the ellipse and Kepler's analysis of observational data. A sub-theme of the book is the practical uses of mathematics. I li'ic I have provided a very eclectic sample of applications, both past .hkI present. Again, the omission of any topic does not indicate that it lacks importance. Mathematics has a long, glorious, but somewhat neglected history, and the subject's influence on the development of human ru I lure has been immense. If this book conveys a tiny part of that story, it will have achieved what I set out to do. Coventry, May 2007 in Tokens, Tallies and Tablets Mathematics began with numbers, and numbers are still fundamental, even though the subject is no longer limited to numerical calculations. By building more sophisticated concepts on the basis of numbers, mathematics has developed into a broad and varied area of human thought, going far beyond mything that we encounter in a typical school syllabus. Today's mathematics is more about structure, pattern and form than it is about numbers as such. Its methods are very general, and often abstract. Its applications encompass science, industry, commerce - even the arts. Mathematics is universal and ubiquitous. It started with numbers Over many thousands of years, mathematicians from many different cultures have created a vast superstructure on the foundations of number: geometry, calculus, dynamics, probability, topology, chaos, complexity and so on. The journal Mathematical Reviews, which keeps track of every new mathematical publication, classifies the 11 43 taming [HI. INI INI II TOKENS, TALLIES AND TABLETS subject niio nearly a hundred major areas, subdivided into several thousand specialities. There are more than 50,000 research mathematicians in the world, and they publish more than a million pages < il new mathematics every year. Genuinely new mathematics, Mi.ii is, not just small variations on existing results. Mathematicians have also burrowed into the logical foundations i >l thei r subject, discovering concepts even more fundamental than numbers - mathematical logic, set theory. But again the main motivation, the starting point from which all else flows, is the concept of number. Numbers seem very simple and straightforward, but appearances are deceptive. Calculations with numbers can be hard; getting the right number can be difficult. And even then, it is much easier to use numbers than to specify what they really are. Numbers count things, but they are not things, because you can pick up two cups, but you can't pick up the number 'two'. Numbers are denoted by symbols, but different cultures use different symbols for the same number. Numbers are abstract, yet we base our society on them and it would not function without them. Numbers are some kind of mental construct, yet we feel that they would continue to have meaning even if humanity were wiped out by a global catastrophe and there were no minds left to contemplate them. Writing numbers The history of mathematics begins with the invention of written symbols to denote numbers. Our familiar system of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to represent all conceivable numbers, however large, is a relatively new invention; it came into being about 1500 years ago, and its extension to decimals, which lets us represent numbers to high precision, is no more than 450 years old. Computers, which have embedded mathematical calculations so deeply into our culture that we no longer notice their presence, have been with us for a mere 50 years; computers powerful enough and 11 i enough to be useful in our homes and offices first became 1.1' spread about 20 years ago. Without numbers, civilization as we now know it could not exist. i.....bers are everywhere, hidden servants that scurry around behind the scenes - carrying messages, correcting our spelling In n we type, scheduling our holiday flights to the Caribbean, keeping track of our goods, ensuring that our medicines are safe and l lh (five. And, for balance, making nuclear weapons possible, and iiiding bombs and missiles to their targets. Not every application I 'I mathematics has improved the human condition. I low did this truly enormous numerical industry arise? It all began with little clay tokens, 10,000 years ago in the Near East. liven in those days, accountants were keeping track of who i wiied what, and how much - even though writing had not then I ii en invented, and there were no symbols for numbers. In place of number symbols, those ancient accountants used small clay tokens. Some were cones, some were spheres and some were shaped like eggs. There were cylinders, discs and pyramids. The archaeologist I lenise Schmandt-Besserat deduced that these tokens represented basic staples of the time. Clay spheres represented bushels of grain, cylinders stood for animals, eggs for jars of oil. The earliest tokens date back to 8000 bc, and they were in common use for 5000 years. As time passed, the tokens became more elaborate and more specialized. There were decorated cones to represent loaves of bread, and diamond-shaped slabs to represent beer. Schmandt-Besserat realized that these tokens were much more than an accountancy device. They were a vital first step along the path to number symbols, arithmetic and mathematics. But that initial step was rather strange, and it seems to have happened by accident. It came about because the tokens were used to keep records, perhaps for tax purposes or financial ones, or as legal proof of ownership. The advantage of tokens was that die accountants could quickly arrange them in patterns, to work out how many animals 13 TAMING THE INFINITE or how much grain someone owned or owed. The disadvantage was that tokens could be counterfeited. So to make sure that no one interfered with the accounts, the accountants wrapped the tokens in clay envelopes - in effect, a kind of seal.They could quickly find out how many tokens were inside any given envelope, and of what kind, by breaking the envelope open. They could always make a new envelope for later storage. However, repeatedly breaking open an envelope and renewing it was a rather inefficient way to find out what was inside, and the bureaucrats of ancient Mesopotamia thought of something better. They inscribed symbols on the envelope, listing the tokens that it contained. If there were seven spheres inside, the accountants would draw seven pictures of spheres in the wet clay of the envelope. At some point the Mesopotamian bureaucrats realized that once they had drawn the symbols on the outside of the envelope, they didn't actually need the contents, so they didn't need to break open the envelope to find out which tokens were inside it. This obvious but crucial step effectively created a set of written number symbols, with different shapes for different classes of goods. All other number symbols, including those we use today, are the intellectual descendants of this ancient bureaucratic device. In fact, the replacement of tokens by symbols may also have constituted the birth of writing itself. Tally marks These clay marks were by no means the earliest examples of number-writing, but all earlier instances are little more than scratches, tally marks, recording numbers as a series of strokes — such as | | | | | | | | | | | | | to represent the number 13. The oldest known marks of tins kind — 29 notches carved on a baboon's leg bone - are about 37,000 years old.The bone was found in a cave in the Lebombo mountains, on the border between Swaziland and TOKENS, TALLIES AND TABLETS Tally marks have the advantage that they can be built up one at a time, over long periods, without altering or erasing previous marks. They are still in use today, often in groups of five, with the fifth stroke cutting diagonally across the previous four. z 11 ii I ircsence of tally marks can still be seen in modern numerals. Our symbols 1, 2, 3 .....lerived from a single stroke, two horizontal strokes linked by a sloping line, and n in :i! horizontal strokes linked by a sloping line. ■ ii itli Africa, so the cave is known as the Border Cave, and the bone 1.1 lie Lebombo bone. In the absence of a time machine, there is no way to be certain what the marks represented, but we can make inlormed guesses. A lunar month contains 28 days, so the notches may have related to the phases of the Moon. There are similar relics from ancient Europe. A wolf bone found in former Czechoslovakia has 57 marks arranged in eleven groups of live with two left over, and is about 30,000 years old.Twice 28 is 56, so this might perhaps be a two-month long lunar record. Again, there seems to be no way to test this suggestion. But the marks look ileliberate, and they must have been made for some reason. Another ancient mathematical inscription, the Ishango bone from Zaire, is 25,000 years old (previous estimates of 6000-9000 years were revised in 1995). At first sight the marks along the edge < )f the bone seem almost random, but there may be hidden patterns. ,Ua numbers between 10 and 20, namely is 60. Another row contains 9,11, 19 and 13, 17 and 19, whose sum is 60. Another row contains 9, 11, ? and 21, which also sum to 60. The third row resembles a , aethod sometimes used to multiply two numbers together by repeated 14 15 taming THE INFINITE TOKENS, TALLIES AND TABLETS n 11 li ii f 21 Z T 13 17 19 * lllillfll!1 19 9 =■ —^ The Ishango bone showing the patterns of marks and the numbers they may represent. doubling and halving. However, the apparent patterns may just be coincidental, and it has also been suggested that the Ishango bone is a lunar calendar. 10 The first numerals The historical path from accountants' tokens to modern numerals is long and indirect. As the millennia passed, the people of Mesopotamia developed agriculture, and their nomadic way of life gave way to permanent settlements, in a series of city states - Babylon, Eridu, Lagash, Sumer, Ur. The early symbols inscribed on tablets of wet clay mutated into pictographs - symbols that represent words by simplified pictures of what the words mean - and the pictographs were further simplified by assembling them from a small number of wedge-shaped marks, impressed into the clay using a dried reed with a flat, sharpened end. Different types of wedge could be made by holding the reed in different ways. By 3000 bc, the Sumerians had developed an elaborate form of writing, now called cuneiform or wedge-shaped. The history of this period is complicated, with different cities becoming dominant at different times. In particular, the city of liabylon came to prominence, and about a million Babylonian clay tablets have been dug from the Mesopotamian sands. A few hundred of these deal with mathematics and astronomy, and they ■how that the Babylonian knowledge of both subjects was extensive. In particular, the Babylonians were accomplished astronomers, and they developed a systematic and sophisticated '.yinbolism for numbers, capable of representing astronomical data to high precision. The Babylonian number symbols go well beyond a simple tally system, and are the earliest known symbols to do so. Two different types of wedge are used: a thin vertical wedge to represent the number 1, and a fat horizontal one for the number 10. These wedges are arranged in groups to indicate the numbers 2—9 and Z0-50. However, this pattern stops at 59, and the thin wedge then takes on a second meaning, the number 60. The Babylonian number system is therefore said to be 'base 60', or sexagesimal.That is, the value of a symbol may be some number, or 60 times such a number, or 60 times 60 times such a 7 «7 4t Tf -or Iff < )ls for 1000, seven symbols for 100, two symbols for 10 I I..... symbols for 1. nous caused the Egyptians severe headaches. At various i ii m Is, ihey used several different notations for fractions. In the Old little>m (2700-2200 bc), a special notation for our fractions, . '/,,„ '/32 and l/6+, was obtained by repeated halving. These i ii is used parts of the 'eye of Horus' or 'wadjet-eye' hieroglyph. -i. , lMl „ ions formed from parts of the wadjet eye 21 taming THE INFINITE TOKENS, TALLIES AND TABLETS The best known Egyptian system for fractions was devised during the Middle Kingdom (2000-1700 bc). It starts with a notation for any fraction of the form 1/n, where n is a positive 11 iw vtr.The symbol ^> (the hieroglyph for the letter R) is written over the top of the standard Egyptian symbols for n. So, for £--* example, '/u is written f\\. Other fractions are then expressed by .i< UI i ng together several of these 'unit fractions'. For instance, % - % + % Special symbols for special fractions Interestingly, the Egyptians did not write 2/s as '/s + l/s.Their rule seems to have been: use different unit fractions. There were also different notations for some of the simpler fractions, such as Vi, 2/3 and %. The Egyptian notation for fractions was cumbersome and poorly adapted to calculation. It served them well enough in official records, but it was pretty much ignored by subsequent cultures. Numbers and people Whether you like arithmetic or not, it is hard to deny the profound effects that numbers have had on the development of human civilization.The evolution of culture, and that of mathematics, has v< ine hand in hand for the last four millennia. It would be difficult m i disentangle cause and effect - I would hesitate to argue that mathematical innovation drives cultural change, or that cultural .....Is determine the direction of mathematical progress. But both ■ 'i i In isi statements contain a grain of truth, because mathematics .mil i 11111111- co evolve. I In ie is a significant difference, though. Many cultural < hanges in • I' uly apparent. New kinds of housing, new forms of transport, ......iw ways to organize government bureaucracies, are relatively ■ 'i" i* his to every citizen. Mathematics, however, mostly takes place I" I.....I the scenes. When the Babylonians used their astronomical Observations to predict solar eclipses, for instance, the average 1 in was impressed by how accurately the priests forecast this .....ishing event, but even the majority of the priesthood had little no idea of the methods employed. They knew how to read 111ili is listing eclipse data, but what mattered was how to use 11...... How they had been constructed was an arcane art, best left '1' specialists. Some priests may have had good mathematical educations — all 11 i i ited scribes did, and trainee priests took much the same lessons i. .cribes, in their early years - but an appreciation of mathematics \\ .isn't reaUy necessary to enjoy the benefits that flowed from new discoveries in that subject. So it has ever been, and no doubt .ilvvays will be. Mathematicians seldom get credit for changing our world. How many times do you see all kinds of modern miracles credited to computers, without the slightest appreciation that computers only work effectively if they are programmed to use sophisticated algorithms, that is procedures to solve problems, and that the basis of almost all algorithms is mathematics? The main mathematics that does lie on the surface is arithmetic. And the invention of pocket calculators, tills that tot up how much you have to pay and tax accountants who do the sums for you, for a fee, are pushing even arithmetic further behind the scenes. But at least most of us are aware that the arithmetic is there. We are wholly dependent on numbers, be it for keeping track of legal obligations, levying taxes, communicating instantly with the far side of the planet, exploring the surface of Mars or assessing the latest wonder drug. All of these things trace back to ancient Babylon, and to the scribes and teachers who discovered effective ways to record taming the infinite tokens;, iallii s and tahlf.ts What numbers do for us Most upmarket modern cars now come equipped with satnav -satellite navigation. Stand-alone satnav systems can be purchased relatively cheaply. A small device, affixed to your car, then tells you exactly where you are at any moment and displays a map - often in fancy colour graphics and perspective - showing the neighbouring roads. A voice system can even tell you where to go to reach a specified destination. If this sounds like something out of science fiction, it is. An essential component, not part of the small box attached to the car, is the Global Positioning System (GPS), which comprises 24 satellites orbiting the Earth, sometimes more as replacements are launched. These satellites send out signals, and these signals can be used to deduce the location of the car to within a few metres. Mathematics comes into play in many aspects of the GPS network, but here we mention just one: how the signals are used to work out the location of the car. Radio signals travel at the speed of light, which is roughly 300,000 kilometres per second. A computer on board the car - a chip in the box you buy - can work out the distance from your car to any given satellite if it knows how long the signal has taken to travel from the satellite to your car. This is typically about one tenth of a second, but precise time measurement is now easy. The trick is to structure the signal so that it contains information about timing. In effect, the satellite and the receiver in the car both play the same tune, and compare its timing. The 'notes' coming from the satellite will lag slightly behind those produced in the car. In this analogy, the tunes niiilhl go like this: CAR ... feet, in ancient times, walk upon England's... SATELLITE ... And did those feet, in ancient times, walk... I lure the satellite's song is lagging some three words behind the same siuni in the car. Both satellite and receiver must generate the same 'song', and successive 'notes' must be distinctive, so that the timing difference is easy in observe. Of course, the satnav system doesn't actually use a song. The signal is .1 series of brief pulses whose duration is determined by a 'pseudo-random code'. This is a series of numbers, which looks random but is actually based on some mathematical rule. Both the satellite and the receiver know the rule, so they can generate the same sequence of pulses. numbers and calculate with them. They used their arithmetical skills for two main purposes: down-to-earth everyday affairs of ordinary human beings, such as land-measurement and .iccountancy, and highbrow activities like predicting eclipses or recording the movements of planets across the night-time sky. We do the same today. We use simple mathematics, little more than arithmetic, for hundreds of tiny tasks — how much anti-parasite treatment to put into a garden fishpond, how many rolls of wallpaper to buy to paper the bedroom, whether it will save money to travel further for cheaper petrol. And our culture uses sophisticated mathematics for science, technology and increasingly lor commerce too. The inventions of number notation and arithmetic rank alongside those of language and writing as some of the innovations that differentiate us from trainable apes. .'■I 25