Cover image for What counts : how every brain is hardwired for math
What counts : how every brain is hardwired for math
Butterworth, Brian.
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New York : Free Press, [1999]

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xv, 416 pages : illustrations ; 25 cm
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QA141.5 .B786 1999 Adult Non-Fiction Central Closed Stacks

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Though he admits to not being particularly good at math, Butterworth (cognitive neuropsychology, U. College, London), the founder of the Mathematical Cognition journal, contends that we all possess an inherent "numerosity" sense--developed to different degrees of course. The author bases his case on empirical research and historical speculation. Annotation copyrighted by Book News, Inc., Portland, OR

Reviews 2

Publisher's Weekly Review

Are our brains "hardwired" to count and conceptualize numbers, or are counting, and other mathematical activities something that we learn, like playing the piano? Butterworth, editor of the journal Mathematical Cognition, is convinced that evidence points to the existence of circuits in the brain devoted to identifying what he calls "numerosities," or, more simply, the number of objects in a collection of things. To this network of specialized circuits, or "Number Module," Butterworth explains, each person adds the mathematical knowledge of his or her culture. Thus, people who "aren't good in math" have trouble not because they're dumb or not applying themselves, but because their Number Module is different from the prevailing one. Not surprisingly, Butterworth has strong views on how to teach mathematics, and these form a prominent part of his book. He also shows how a person's brain can change to devote more resources to respond to mathematical stimuli. For example, a study of Braille proofreaders based on brain-scan maps has demonstrated that the part of the brain devoted to this activity grows in size after six hours work. But give the proofreaders a few days off, and their brains shrink back to normal. Butterworth's prose is marred by repetition, and his digressions to explain various well-known math puzzles and peculiarities, such as Pascal's triangle, often aren't germane to his argument (do we really need a proof of G”del's theorem here?). But these are minor caveats about a provocative book that makes an important addition to the recent flurry of titles regarding how our minds work. Teachers as well as readers curious about the brain and psychology will be challenged by the ideas expounded here. (Aug.) (c) Copyright PWxyz, LLC. All rights reserved

Choice Review

Is mathematics purely a social construct? Butterworth resoundingly says "no." He starts his argument from a historical and anthropological perspective to show that all peoples in all times have made use of some form of cardinal and ordinal arithmetic. He then proceeds through psychological studies of infants and apes to show that at least some sense of number is innate. Psychological and neurological studies of people with various disabilities allow him to tease out the basic number process and to locate it in the left parietal lobe of the brain. This basic process is insufficient for more complicated mathematics; higher mathematics has to be constructed on top of this simple base. The ways in which various people create or construct mathematical methods leads to a discussion of teaching mathematics; this discussion is the weakest part of the book because it ignores most of the work that has been done in the instructional area. Butterworth finishes by discussing infinity and G"odel's theorem. The results in this important book will interest all those who teach or try to understand mathematics. General readers; undergraduates through professionals. P. Cull; Oregon State University



Chapter One Thinking by Numbers This grand book, the universe...cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics. Galileo The World by Numbers Of all the many abilities that have raised us from cave-dwellers using stone tools to creators of great cities and modern science, one of the most important is the ability to use numbers. It is also among the least understood. Traditional studies of the role of numbers have been concerned largely with their contribution to the development of science. However, numbers have affected almost all the aspects of our life which are most characteristically human. We have used numbers from the beginning of the historical record, and perhaps from way back in the prehistory of our species. This book is an attempt to explain why we all think about the world in terms of numbers. Nowadays, we use numbers routinely. We use them to count things, to tell the time, as statistical data, to gamble, to buy and to sell. Even barter needs numbers: I'll give you six knives if you give me two pigs. We use them to rank competitors, in addresses (house numbers and postal codes), to grade examination candidates. Blood pressures, temperatures, and IQs are given numerical values. Cars and their engines, TV stations and telephone lines, all have their number labels. Goods in shops have bar codes. People have social security numbers, bank account numbers, and passport numbers. Your height, weight, and age are all denoted by some numerical multiple of a standard unit. The importance of numbers lies not just in their obvious utility, but also in the way they have shaped how we think about the world. It is the language in which we formulate scientific theories. Numbers, as Einstein said, are the 'symbolic counterpart of the universe'; they are crucial to the measurements we take as the fundamental evidence for our theories. 'When you cannot measure...your knowledge is of a meagre and unsatisfactory kind,' to quote the great nineteenth-century scientist Lord Kelvin, who, not coincidentally, invented the scale for measuring absolute temperature. To get some idea of just how fundamental numbers are, try to imagine our world without them. There would be no money, no counting, no income tax. Without numbers, trade would be restricted to face-to-face barter. I could see the knives you wished to trade, and you could see my pigs, but trade at a distance would be extraordinarily difficult: how could I convey the number of pigs I would be willing to trade for your knives, and the number of knives I would be willing to accept for my pigs? Without numbers, there would be none of our familiar sports, such as football, baseball, and tennis, since they use numbers to define how many players may be in a team, and to keep the score. Many sports obsessively keep numerical records; thus athletics is a good example. Qualification for the Olympic Games depends on exceeding a numerical standard. This tradition goes back to the original Olympian Games, whose very period -- the Olympiad -- is a definite number of years, four. Numerical records were kept of some events. Phayllus of Croton was credited with a leap of 55 feet (16.8 metres), which you might think raises the question of just how good the Greeks were at measurement. Some modern competitions, such as the heptathlon, are defined by the number of events in which the athletes take part, seven in this case. Without numbers, we could not frame basic theories of physical nature, such as Kepler's laws of planetary motion, Newton's laws of motion, or Einstein's E = mc2. Chemists would be at a distinct disadvantage without the numerically ordered periodic table of the elements. The study of human nature has also depended on numbers to quantify mental attributes such as intelligence, reading age, or degree of introversion. The basic laws of perception are another good example of the use of numbers in the study of human nature. To see a bright light get brighter you need a bigger absolute increase in energy than to see a dim light get brighter; to hear a loud noise get louder you need a bigger absolute increase in energy than for a quiet noise. The scientist, with or without numbers, would ask whether there is some general law that will predict how big an increase is needed to make a noticeable difference to brightness or to loudness. It turns out that there is, and it was discovered in 1834 by Ernst Weber. The law states that a barely noticeable change in brightness depends on a proportional increase in energy, not an absolute increase. What is the value of the increase that yields a difference which is just detectable? Experimentation tells us that an increase of roughly 1% in energy gives us a noticeable difference in brightness and about 10% is needed for a noticeable difference in loudness. Weber's law could not be formulated, perhaps not even thought about, without numbers. Clearly, numbers are extraordinarily useful, but there is still the question of how we came to describe and represent our world in terms of numbers. Is there something about the world that would oblige us to invent numbers if we didn't have them already? Many useful but difficult inventions, such as the alphabet, double entry bookkeeping, or the printed circuit,6 were invented just once and were diffused around the world. One possibility, then, is that there was one ancient Einstein who invented numbers. Once he had made the breakthrough, it became clear how valuable the idea was and so it was eagerly adopted by neighbours, and then neighbours of neighbours, and so on. This scenario implies that cultures distant from the inventor would get numbers later than those nearby, or perhaps not at all. This is what has happened with the alphabet, for example. Indeed, some Amazonian tribes still haven't learned to read or write. On the other hand, many inventions seem to have been less difficult, like plant domestication, pottery, and fire, and arose independently in many areas. Is the idea of numbers a relatively easy invention that many societies could have developed by themselves? But even easy inventions depend on local circumstances. The invention of settled agriculture with crops and domesticated animals has depended on having plants and animals that are suitable and easy to domesticate. Although trading creates a need for numbering (as I discuss in Chapter 2), it is not clear what would make the invention of numbers harder or easier. A third possibility is that the idea of numbers was not invented at all. Rather, they are something about us, an intrinsic part of human nature, like the ability to see colours, or schadenfreude. Today using numbers for trading, ordering, and labelling seems very easy, convenient, and natural. In a way this is very surprising, since numbers are, in the words of Adam Smith, 'among the most abstract ideas which the human mind is capable of forming'. Numbers are not properties of objects. You cannot touch, see, or feel them. They are not like the properties of an orange. If an object is an orange it will have a characteristic colour, texture, size, shape, smell, and taste. You can check each of these properties to see whether a candidate object is an orange or, for example, a ball or a lemon. But a collection of five things doesn't possess characteristic colour, shape, or taste. What all such collections have in common is their fiveness, and this is abstract. To understand numbers -- to understand, for example, the difference between five and four -- is to understand something very abstract indeed. To use numbers, therefore, we should require extensive training, as for other abstract concepts. Think how long it takes to learn and apply the principles of chemical reactions or the Linnaean taxonomic categories. This is not just a matter of acquiring the Latin names, since in traditional societies, which of course use the vernacular, learning to classify plants and animals is a very long job which starts in earliest childhood. And yet, as I show in Chapter 2, everyone can count or tally up small collections of objects, and can carry out simple arithmetical operations, whether they are Cambridge graduates or tribesmen in the remote fastnesses of the New Guinea highlands. Even cultures which have no words for numbers still count, calculate, and trade. Despite its abstractness, the idea of number seems to be universal. What is it about our brains that makes us all familiar with this idea? The Universality of Numbers and the Mathematical Brain The philosopher and psychologist Jerry Fodor divides cognitive abilities into two sorts: the highly specialized, which he calls 'cognitive modules', and the general purpose, which he calls 'central processes'. Cognitive modules extract just one type of information from the senses. For example, one module extracts information about the colour of objects, while another module works out the shape of an object. Fodor calls this property of modules 'domain specificity'. Because they are specialized, they can operate fast, but to do so their operation must be automatic. For example, you cannot see a red flower without processing the information about its colour -- you cannot choose not to see its redness. These fast, automatic, domain-specific modules confer an adaptive advantage on the organism: seeing colours in this way speeds identification of prey and predators. Through our genes we inherit instructions for building these modules in the brain (and the nerve cells in the sense organs). The modules consist of distinct neural circuits, which in many cases will be ready to go as soon as the organism is born, or very soon after -- as with colour vision in human infants. Since we are born with functioning colour vision circuits, we do not need to learn to see in colour. Fodor believes that key aspects of our linguistic abilities are modular in this sense. Of course it is true that we have to learn English (or French or Chinese), but the basic mechanisms for doing so are built into our brains at birth, and many of the operations are fast and automatic. For example, you cannot see the letter string red without identifying the word and evoking its meaning. Central processes are the converse of cognitive modules. They operate slowly; we can choose whether to operate them or not; we are not born with brain circuits specialized to do them; and they need learning. The ability I am exercising now -- typing English words into my computer -- falls into this category. Typing English depends on learning. It's not automatic: I don't do it if I don't want to. And the ability to type English is so recent that there has been no time for evolution to select brains that are particularly good at it. This means that we are not born with specialized typing circuits: we have to rely on training and practice to become proficient. Memory is another central process for Fodor. It is not domain-specific -- we can remember anything, if we work at it. And reasoning, too, is central, since we can reason about anything. So what is numerical ability, a cognitive module or a central process? Fodor is quite clear that it is a central process. Calculation is slow and effortful; reasoning seems to play an important role, unlike seeing the world in colour; it seems to depend on formal instruction; it does not seem to be innate because there are very wide variations of numerical ability -- as compared with the ability to see colours -- and there may even be people who have been denied instruction who cannot use numbers at all. Finally, we do not appear to be born with specialized brain structures for numeracy. The Number Module I shall try to show that Fodor was wrong in every particular. I shall argue that the human genome -- the full set of genes that make us what we are -- contains instructions for building specialized circuits of the brain, which I call the Number Module (Figure 1.1). The job of the Number Module is to categorize the world in terms of numerosities -- the number of things in a collection. It makes us, and other creatures who possess it, sensitive to the number of things in a collection. Compare seeing the world in terms of colour with seeing it in terms of number. Both processes operate automatically: we cannot help but see the cows in the field as brown and white, nor can we help seeing that there are three of them. Both processes fail to develop normally in some individuals. Just as there are people born colour-blind, there are also, as I argue in Chapter 7, people born with a kind of number-blindness. What makes human numerical ability unique is the development and transmission of cultural tools for extending the capability of the Number Module. These tools include aids to counting, such as number words, finger-counting, and tallying, and also the accumulated inventions of mathematicians down the centuries -- from numerals to calculating procedures, from counting-boards to theorems and their proofs. Our Mathematical Brain, then, contains these two elements: a Number Module and our ability to use the mathematical tools supplied by our culture. This may seem obvious, but it is contentious. Another idea is that we possess an innate sensitivity to quantities, but not to numerosities. For example, Stanislas Dehaene, in his book The Number Sense,10 following a twenty-year tradition of work with animals and infants, postulates a brain mechanism, called 'The Accumulator', which interprets numbers as approximate quantities, rather like the level of liquid in a bottle. Different numbers are represented by different levels. It works by recording events -- a drop of liquid for each event. So the three cows in the field would first be coded as three events, and represented by the level of the total amount of the three drops of the imaginary liquid. Another possibility is that numbers are just like anything else we learn, such as the periodic table, the capitals of African states, or how to use WordPerfect in DOS. Good numerical ability depends on just those capacities that make us good at school subjects: good teaching, high intelligence, hard work, and an excellent memory. There is nothing specialized about this ability. Testing the Number Module. All scientific hypotheses are empirically testable. That, according to philosopher of science Karl Popper, is what distinguishes scientific hypotheses from the propositions of mysticism and religion. The sceptical reader has no doubt already formulated some tests that my hypothesis of a Number Module in everybody's brain will need to pass before it is accepted as credible. First, if the Number Module is built into all our brains when we are born, then everybody should show evidence of being able to carry out tasks that depend on categorizing the world in terms of numbers of things -- what I call numerosities -- whether or not they have had the opportunity for instruction, formal or informal, in numbers and arithmetic. Everybody should be able to match the numerosities of collections of things on the basis of one-to-one correspondence between the members of the collections, and they should be able to tell which of two collections is the larger. Not much, you may think, but these two operations form the basis of everything else we know about numerosities. And we must be sure that the universality of the ability is not explainable in terms of the spread of an invention combined with the fact that humans are just about smart enough to learn how to use it. Compare numbers and the alphabet. The alphabet was invented just once, presumably after a lot of hard work by intelligent and dedicated scribes, in the Middle East around 1700 bc. Few probably have the capacity to invent an alphabet, but most of us are smart enough to learn how to use one. Second, if specialized brain structures are in place when we are born, though of course they will mature and develop over time, then we should see evidence of capacities that depend on numerosity even in the infant. Third, we should be able to locate these specialized structures in the brain. Certain localized brain damage could affect number skills only. Imaging of brain activity should show the same localized hot-spots whenever the brain is calculating. Fourth, if we have a Number Module, we should be able to find evidence that will satisfy Fodor's criteria for modules: fast and automatic numerical operations, at least for basic operations such as identifying or comparing numerosities. Fifth, if this numerical capacity is part of all our brains, then we need to explain why some people are good or very good at arithmetic, while others are bad or hopeless. Sixth, if we inherit a Number Module, then it is encoded in our genome, which in turn we inherited, with mutations, from our ancestors. Does this mean that there is a set of genes that code for the building of the specialized brain circuits? Does it mean that our near ancestors, in the last ice age, counted as well? What about our more distant ancestors, Homo erectus, or great apes, other mammals, birds, reptiles, worms? Seventh, how does the way we talk about numbers and the way we write them affect the development of number skills that go beyond basic numerosity? Does the Mathematical Brain hypothesis have anything to say about how we can help our children learn more advanced arithmetic, or how we should design our education system? Finally, there is something strange and emotive about numbers and counting. Many cultures from Africa to orthodox Jews forbid counting people. Some numbers are regarded as lucky or unlucky. Some people suffer anxiety at the thought, not to mention the practice, of arithmetic. Does this have anything to do with our Mathematical Brain? The chapters that follow fully investigate all these issues. Types of Number and Their Uses The first step is to clarify what numbers can mean. Perhaps because we use number words and Arabic numerals all the time, we may unreflectingly think that each time we use, say, the number word 'five', we mean the same thing. After all, five is five: it is always half of ten and a tenth of fifty; it always comes after four and before six; it's the number of fingers on the average hand; it's the number of tanners in half a crown. In fact, this is far from the case. The number word 'five' has many distinct and different meanings, just as the word 'orange' has distinct meanings -- the fruit and the colour. Of course, the two are related, the colour meaning is derived from the (typical) colour of the thing, but the colour isn't the fruit: you cannot eat the colour. Languages sometimes distinguish explicitly at least some of the different number meanings. English and other European languages distinguish ordinal meanings by using the special terms first, second, third, fourth, and so on, rather than one, two, three, four. Let me start by explaining how number terms -- words and numerals -- have different meanings. But fundamental to human thinking about numbers are the meanings that answer the question, 'How many?' These are numerosities. My hypothesized Number Module is a brain mechanism that identifies numerosities. Numerosities Think of a collection of things, any collection of any things. This collection will have a number, or what I prefer to call a 'numerosity' to distinguish it from other meanings of number terms. Many of our familiar ideas of number are properties of numerosities, and the relations that hold among numerosities. For example, numerosities are completely ordered by the size of the collections that represent them. Let us call a collection of cups with the numerosity four, a four-collection, and a collection of saucers with the numerosity three, a three-collection. Four is larger than three, because you can take each cup, pair it with a saucer, and still have one cup left over. Another example is adding: the sum of two numbers means finding the numerosity of their collections combined. Since collections are simply combined (technically, we are forming the union of the two collections), it does not matter in which order the collections are taken: this is why 4 + 3 = 3 + 4. Other numerical operations and properties can be built from these foundations. One fundamental property is so obvious that it is frequently overlooked. Every numerosity, every number, is the sum of other numerosities: 2 is the sum of 1 and 1; 3 is the sum of 2 and 1; 4 is the sum of 2 and 2, and of 3 and 1; and so on. (In the simplest number system, 1 is not a sum.) This property -- sometimes called additive composition because numbers can be added together to form other numbers -- turns out to be critical in understanding how the child comes to understand numbers, as I show in Chapter 3. Although numerosities depend on collections, they are properties of collections and cannot be defined in terms of them, rather as 'yellow' is a property of a banana but is not definable in terms of bananas. Of course, you can use a banana as a representative of yellow things. In the same way I can point to a particular collection, for example, the collection of fingers on my left hand, to illustrate what a collection with a numerosity of five is like. A numerosity is the number you get when you count a collection. In this sense, counting means putting each thing in the collection in one-to-one correspondence with a number. In practice, this normally means putting each thing in one-to-one correspondence with a number word, where the last number word counted denotes numerosity of the collection of things counted: 'one, two, three, four, five: five pigs'. Counting is the key to numerosity. Children use counting practice to build a sense of numerosity into a full numerosity system. Counting also helps children discriminate among numerosities since collections that have different counts have different numerosities. To understand numerosities, you need to have a grasp of two other ideas. First is the idea of an object -- something that can be individuated. In most cultures, anything that can be individuated can form a collection that has a numerosity. They can be things that are clearly visible as individuals, such as pigs, wives, or handkerchiefs; they can be things which have a visible manifestation, but are rather more complicated, such as teams in the Premier League, or hops on one leg; they may be invisible yet distinct to the senses, such as notes in a bar, bars in a Blues; they may be abstract, such as days in my holiday or resolutions for the New Year. However, there are exceptions. For example, the language of the Hopi of New Mexico allows only collections of things you can see to have a numerosity, so you can say 'five dogs', 'five stones', or 'five brides', but you cannot say 'five days'. So instead of saying, 'I will be there in five days', you have to use a locution equivalent to the English sentence, 'I will be there on the fifth day'. The Hopi language distinguishes grammatically, as do European languages, between numbers for numerosities and numbers purely to denote order, ordinal numbers, as we shall shortly see. I have allowed myself a simplification when I talk of 'the idea of numerosity' as if possession of it is all-or-nothing. So, for example, if you possess the concept, then not only can you tell that two collections A and B have the same numerosity, but you can also tell whether A has a larger numerosity than C. What is more, the concept should apply to collections of any number of members. Yet, when we come to examine the abilities of animals and infants, we shall see that they might be able to tell whether two sets are the same, but not which of two sets is bigger, and animals and children can deal only with small numerosities. This raises the question of whether animals and children really do possess the concept, but are limited in applying it, or whether they possess some other concept. Ordinal Numbers Ordinal numbers are used for ordering things, like houses, placings in a competition, and so on. Unlike numerosities, the essential property of ordinal numbers is in the ordering, not in the size of the collection, though of course each ordinal corresponds to a numerosity. What is important is that there is a definite relation among the numbers: each number is either before or after any other number, so that all the numbers are completely ordered. Knowing the numbers of two houses in the same street means, usually, that we know their relative position in the street. A higher number is farther up (or farther down) the street than a lower number, though this will depend on the ordering principle: most streets in Britain have even numbers along one side and odd numbers along the other. Some house numbering, incidentally, is completely useless as an address. In Venice, for example, buildings were renumbered in 1841, by district, called a sestier ('sixth'), not by street. The sestier of St Mark's begins with number 1 in St Mark's Square at the Doge's Palace and is numbered consecutively up to 5562 on the wall of the Fondaco dei Tedeschi (an important place in the history of arithmetic, as we shall discover in the next chapter) near the Rialto Bridge. There is no way of working this out -- you just have to know it. To help, the Venetians put up a notice saying Ultimo numero del Sestier de San Marco, but you do not find this out until you get there. If you want to find an address, say San Marco 3832, you will have no idea where it is unless you buy Jonathan del Mar's Indicatore Anagrafico di Venezia, which tells you in which street every number is. Should you not have this invaluable guidebook, and end up in Campo San Angelo, where you notice the number 3555 and decide to follow the numbers in ascending order to get to 3832, you will walk through the Calle de la Madona, via Rio Terà dei Assassini and several other side alleys, to Calle de la Verona, then on through the Calle de la Mandola, several other side streets, to the Rio terà de la Mandola, and several more side streets. Then, after a quarter of an hour of steady walking, you will find 3832 back in Campo San Angelo. For ordering things, number words or numerals can be replaced by any set of symbols that are completely ordered, for example letters of the alphabet, provided the ordering principle is well defined and understood by the users. There is a curious ordering principle to be found in a street in the London suburb of Willesden. The original houses were given the names Belmont, Rayleigh, Overton, Newlyn, Delmore, Everon, Shirley, Beechcroft, Uplands, Rutland, Yelverton. The initial letters spelled out the word BRONDESBURY, the name of the street in which they were situated. The postman, the inhabitants, and their visitors understood the ordering principle, so there was no difficulty in finding the right house. A striking feature of the terms for the names of the ordinals corresponding to one and two is that they have a quite different origin. The words for 'first' in English, French, Gothic, and Greek are derivations of the Indo-European preposition pro-, meaning 'before', which through historical sound change became fr- and then fir-. 'Second' comes from the Latin secundus, meaning 'following', from the verb sequi ('to follow'), while in the Gothic, as in Latin alter, the word for second means 'the other'. Similarly, Finnish and Basque, which are unrelated to other European languages, do not derive their words for first and second from the corresponding cardinals, the terms for the numerosities. All this suggests that the conceptual origin of ordering is separate from that of counting and numerosity. Numerical Labels A very striking feature of the modern world is the need to label things, places, and people uniquely. Numerals are very convenient for this. First, there are enough of them for each thing labelled to have its own number. Second, there are only ten symbols, all of which are part of the standard repertory of symbols on keyboards all over the world. For historical reasons some labelling conventions still incorporate letters as well as numbers. Most national systems of car registration numbers, British National Insurance identifications, and British and Canadian postcodes all mix letters and numbers. Telephone systems in the USA and Britain used to have three-letter district codes followed by a four-digit number. My old telephone number was WIL 4025, which located me in Willesden. Numbers starting MAY indicated fashionable Mayfair, though others were more obscure, such as AMB for Ambassador, which was never the name of a district of London. The letters have long since been replaced by numbers. Numerical labels are not ordered by size. My telephone number is 4259 7461, but it is never read out as genuine numbers are read. When asked for my telephone number I never say, 'Forty-two million, five hundred and ninety-seven thousand, four hundred and sixty-one', and I do not claim that it larger than my friend David's, which is 'only' 2673 6579. In fact, labels are not ordered at all. For this reason, labelling can be done in other ways, provided each label is unique. Car registrations in California (and now in Britain) can be chosen by the user, as long as no one else is using the same registration. One linguist I know has the registration LINGUA, and a neurologist has BRAINS. TV channels used to be given names -- BBC, ITV, NBC, CBS -- and could still. It is just more convenient to give them numbers, because the ordering may help viewers find the desired channel, provided they can remember that BBC 1 is on Channel 31 on the remote control (unless you re-program it). Although numerical labels are designed to be unique -- no two people have the same telephone number nor the same National Insurance number -- they are felt to be depersonalizing, in a way that names are not, even though names, such as John Smith, or even Brian Butterworth, are not unique. Number Six in the TV series 'The Prisoner' continually protested, 'I am not a number!' Fractions Fractions are ratios of two numerosities, two whole numbers. The most familiar even have special names, such as 'half' and 'quarter'. For some reason, other simple ratios use the words from the ordinal series -- 'third', 'fifth', 'sixth', and so on. In fact, even 'quarter' comes from the Latin word quartus, meaning 'fourth'. Fractions are often used for measures, but not all measures can be expressed as fractions, as we shall see. Measure Numbers Measure numbers answer questions like 'How much?', 'How long?', 'How heavy?', where there is an agreed scale or dimension. Historically, the dimension was divided into easily grasped chunks of everyday experience. For length, parts of the body formed the chunks. In Biblical times people used the width of the finger, of the palm, the span of the hand, the length of the forearm from the elbow to the tips of the fingers (the cubit), and so on. Of course, people differ in their dimensions, which would create unacceptable confusion today, but even in Renaissance Italy the braccia ('arm length') was standardized only city by city, so a Florentine braccia may have been shorter than a Milanese. Even British standardized Imperial measure was based on bodily chunks. The inch was the length from the tip to the knuckle of the thumb. The foot is self-explanatory, though a thousand years ago it was standardized on barleycorns as these were less variable than human feet: a foot was 36 barleycorns 'taken from the middle of the ear'. A yard was the length from King Edgar's nose to the tip of the middle finger of his outstretched arm. These convenient and readily visualized chunks meant that lengths for most practical purposes could be expressed as whole-number multiples of these units. For example, the Turkish kilim in my study is 2 yards 1 foot 7 inches long. Today we would express this length as a multiple of a single standardized unit type that has no basis in the dimensions of the human body -- in this case 224 centimetres. But even with chunks, some answers needed more precision than whole numbers could provide. Suppose a yard of timber cost four pennies; how much timber could you get for one penny? The solution was to use fractions of the smallest unit -- one-half, one-quarter, two-sevenths, and so on. These fractions are the ratios of two whole numbers, and are readily understood because they map neatly onto the numerosity concept. One can easily envisage a length divided into, say, four equal segments; that is, as a four-collection. It is then quite straightforward to think of a sub-collection of one or two or three segments. You may recall learning these as 'proper' fractions. Pythagoras' theorem, for which Pythagoras is said to have sacrificed a hundred oxen to the gods, states that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. This led directly to the problem of 'incommensurables' -- unmeasurables -- whose discoverer was allegedly drowned by Pythagoreans for upsetting their philosophy that 'all is numbers'. Here is the problem. If the sides of a right-angled triangle are equal to 1, then the square on the hypotenuse is equal to 2, and the length of the diagonal is thus the square root of 2: a number not expressible as a ratio of two whole numbers. This means that measure numbers can be very different from the other numbers we have considered. These measure numbers are familiar enough as 'decimals', that is, as whole numbers followed by more numbers after the decimal point, for example 2.50, 2.666, 3.141 592 62. They are ordered according to a different principle from the counting numbers 0, 1, 2, 3,...used for numerosities. For any two (different) whole numbers, one will be larger; and for any whole number there will be one and only one successor. For any two of the measure numbers, one will be the larger; however, there is no unique next number. After 2.50, there is 2.51, but between these two there is 2.501; and between 2.50 and 2.501 there is 2.5001, and so on. In fact, there are infinitely many 'next' numbers between, say, 2.50 and 2.51, or indeed between any two decimals. The German mathematician Leopold Kronecker once said, 'God made the integers. All else is the work of man.' Of course, he was wrong on both counts, but he did articulate the idea that whole numbers are somehow different from other types of number. Cyclical Numbers There are also numbers that repeat in a regular cycle. The most familiar is clock time, where 4 is both before and after 5, and addition doesn't work in the usual way (11 o'clock and 5 o'clock add up to 4 o'clock -- the sum is 'smaller' than the numbers you add together). It is possible to express the arithmetical rules of cyclical numbers in terms of ordinary arithmetic, by using the idea of the modulus. Clock time has a modulus of 12 (or 24 on a 24-hour clock), which means that you carry out your arithmetic operation -- addition, subtraction, and so on -- on the remainders of the numbers after you have divided them by 12 (or 24). So when 11 is divided by the modulus, 12, it leaves remainder 11, 5 leaves remainder 5, and their sum is 16, whose remainder after dividing by 12 is 4. So the arithmetic of cyclical numbers can be translated into the arithmetic of ordinary numerosities. However, they have the unusual property that there is a largest number, the modulus. Days and dates are also cyclic. For us, days repeat in a seven-item cycle, and dates in a cycle of 365 or 366 days. It wasn't always quite so neat. The Chaucer scholar Paul Strohm17 describes the situation at the end of 14th century: Rather than considering the present as a year in a sequence -- 1399, for example -- the medieval person would have dated the year in terms of the present king's reign; in this case, the twenty-third year of King Richard, second of that name since the conquest...Weeks and days were normally measured by the liturgical calendar. Henry IV's deposition of Richard II occurred on Monday of the feast of St Michael the Archangel (rather than on 29 September). Even the time of day followed a different rhythm: Clocks were a rarity, and most of those in existence were astrological, measuring the movements of the heavens rather than the hours of the day. Thus people were as likely to identify the time of day by an adjacent liturgical service, such as prime or lauds, as by a particular hour. Infinite Numerosities Over the centuries, mathematicians have found that other types of number can be constructed from the natural numbers. Some of these we already grasp intuitively, without needing mathematicians to construct them for us. One example is infinity (or, at least, the smallest infinity). We all know that there is no largest number. As children, we played a game in which one of us would name the biggest number they could think of, and another would try to top it. We quickly got bored with this game since it soon became clear that any number I came up with, for example 'quadzillion zillion zillion', you could always top it with 'and one'. Now, any number like this is finite. One way of expressing this finiteness is to say that you could count up to the number, starting from 1, in a finite time, albeit perhaps a very long time indeed. However, the collection of all numbers is not countable precisely because however many numbers you have counted, there is always that number 'and one'. Evidence from children as young as five suggests that they understand that there is no largest number,18 and so they have at least the beginnings of a grasp of infinity. However, very few of us understand the properties of even this smallest infinity without specialist training. There are many other types of number that have been described by mathematicians, but they take us well beyond the main argument of the book. I have focused on seven types of number familiar in everyday life. In Chapter 9, 'Hard numbers and easy numbers', I explain in more detail why some kinds of number problems are hard for us, and others easy. Essentially, those that can be expressed straightforwardly in terms of collections and their numerosities are easy, while those where the connection is more obscure we find difficult. So, for example, proper fractions such as 1/2 or 3/4 are easy because 1/2 can be expressed as sub-collection of a collection with 2 members; while 3/4 can be expressed as sub-collection of a collection with 4 members. Improper fractions, such as 3/2 or 5/4, don't translate so readily into these concepts, and so even children doing maths at secondary school have trouble with them. Decimals are also relatively difficult, both conceptually and practically, and these, as we have seen, have a quite different structure from the numerosities. The argument I present in this book is a very simple one. We are born with brain circuits specialized for identifying small numerosities. I call these circuits the Number Module, which is the inner core of all our numerical abilities. Onto this inner core we build more advanced abilities, largely by learning from the culture around us what is already known about number and mathematics. This means that my numerical abilities, and yours, depend on three things: the innate inner core, the mathematical knowledge of the culture in which we live, and the extent to which we have acquired this knowledge, as was shown in Figure 1.1. Had we been born into a culture with very little mathematical knowledge, our abilities would typically be much less than those we have the potential to acquire from our mathematically more advanced culture. Of course, with little opportunity or desire to acquire mathematics, our mathematical abilities would be less than if we had devoted much time to study under the guidance of skilled and dedicated teachers. But, I argue, even the most idle and uninterested person born into the least mathematical culture will still categorize the world in terms of numerosities. At the same time, my hypothesis implies that there will be people who are born without the Number Module -- that is, without an innate ability to recognize small numerosities. This, I predict, will be a serious handicap to acquiring the cultural resources needed for good mathematical abilities. Even if you are not convinced by anything else in this book, I hope that you will be convinced that our ability to use numbers is fundamental to the way we think about the world, that it is the basis of much of what we call civilization, and that to understand our common humanity we need to understand how we understand numbers. Copyright © 1999 Brian Butterworth. All rights reserved.