VIII.3 The Ubiquity of Mathematics

T. W. Körner

1 Introduction

We live surrounded by mathematics: when we open a door or use a nutcracker, we exploit ARCHIMEDES [VI.3] law of the lever; when a bus goes around the corner, we experience at first hand NEWTON’S [VI.14] law that a body continues to travel in uniform motion in a straight line unless acted on by an external force; when we use a rapidly accelerating elevator, we can feel for ourselves the equivalence of gravitational and accelerational inertia that lies at the heart of GENERAL RELATIVITY [IV.13]; and when we run a tap fast into a kitchen sink we see a thin and flat circle of water with a clear boundary, which is the chaotic “hydraulic jump” between two well-behaved solutions of a certain PARTIAL DIFFERENTIAL EQUATION [I.3 §5.4].

Because mathematics and physics are so interlinked, almost everything we see involves mathematics. With the help of elementary calculus, we know that a baseball, after it leaves the bat, will have a trajectory in the shape of a parabola. This calculation assumes that there is no air resistance, but a more complicated calculation can take air resistance into account too. If a chain hangs between two points, then the curve it forms can again be explained mathematically. This time, the technique used is the CALCULUS OF VARIATIONS [III.94]: the curve is the one that minimizes the potential energy of the chain, and the calculus of variations allows you to work it out. (It is called a catenary. The rough idea of the calculation is to consider small perturbations of the chain. Since the potential energy is minimized, we know that however we perturb it, we cannot decrease the potential energy. This information can be used to derive a differential equation that determines the curve. In general, the differential equations that arise from this technique are called the Euler–Lagrange equations.) Even the way that wet sand behaves when you walk across it involves interesting mathematics, as Reynolds realized in 1885. Typically, the sand just around where you tread dries out—if you have not noticed this strange phenomenon, then have a look next time you are on a beach. The reason this occurs is that when the tide goes out the sea tends to leave the grains of sand extremely well-packed. If you then tread on the sand, you disturb this packing, creating a less well-packed part of the sand near where you tread. This has more room for water, so it draws water in and down, temporarily drying out the sand around your foot.

It would be easy to give hundreds more examples of physical phenomena that can be analyzed mathematically. However, if one accepts that physics governs the universe and that mathematics is the language of physics, then it is not surprising that these applications exist. Therefore, this article will focus on the appearance of mathematics in other areas, and in particular geography, design, biology, communication, and sociology.

2 Uses of Geometry

If you travel about on Earth’s surface, then you need to make small adjustments to your watch as you move from one time zone to another. There is one exception to this, however: if you cross the international date line, then you have to make a big adjustment (assuming, that is, that your watch shows not just the time but the date as well). Why is it necessary to have a discontinuity of this kind? Well, suppose that it is midnight on a Tuesday in Lisbon, for example, and imagine a path that goes westward right around the globe. If the time changes along this path are all small ones that reflect where one is in relation to the sun, then the time of day goes back by one hour for every 15 degrees of longitude that we move. Therefore, when one gets back to Lisbon it is midnight on Monday. (Remember that we are talking about a mental path here, and not an actual journey.) Something is clearly not right. The practical consequences of this theoretical problem were first felt by the tattered remnants of Magellan’s first circumnavigation of the globe who had to do penance for performing religious ceremonies on the wrong day!

Here is another argument for the necessity of the date line. Let us ask exactly when the year 2000 began. The answer depends, of course, on what part of the world you are talking about, and more particularly on its longitude, but for any part the answer is midnight at the beginning of January 1. In other words, in any particular place the year began when the Sun was (approximately) over the opposite side of the world. It follows that at any given time at most a small fraction of the world was celebrating the very beginning of the year 2000. Therefore, at least somewhere had to go first, which means that parts of the world just to the east of it had missed their chance and had to wait almost 24 hours. Thus, again we see that there has to be a discontinuity.

These phenomena reflect the fact that a certain continuous map has no continuous inverse. The map in question takes a real number w to the point w (cos w, sin w), which lives in the unit circle. Notice that if we add 2π to w then we do not affect the values of cos w and sin w. Now let us try to invert the map. This means that for each point (x,y) in the unit circle we must pick some w such that cos w = x and sin w = y. This w is the angle that the line from 0 to (x,y) makes with the horizontal, with the all-important proviso that you can add any multiple of 2π to it. So the question becomes, can we choose the appropriate multiple in a continuous way? Again, the answer is no, since if you go around the circle once and let the angle vary continuously, you find that you have added 2π to it.

The above fact is one of the simplest theorems of TOPOLOGY [IV.6], the branch of mathematics that you turn to if you want to know about the existence or nonexistence of continuous functions with given properties. Another situation where continuous functions are useful is when one is creating a map (in the geographer’s sense) of the world. Such maps are more convenient if they are drawn on a flat piece of paper, so a preliminary question we might ask is whether there is a continuous function from the surface of the sphere to the plane such that any two different points in the sphere go to different points in the plane. Not only is the answer no, but Borsuk’s antipodal theorem tells us that there must be some pair of antipodal points (that is, points of the sphere that are exactly opposite each other, such as the North and South Poles) that go to the same point in the plane.

However, perhaps we do not mind too much about continuity. If we take our sphere and make a cut from the North Pole to the South Pole, then we can open it up at the cut and flatten it out onto a plane. (To see this, imagine that it is made of particularly stretchy rubber.) Alternatively, we could cut the sphere into two hemispheres and draw maps of each hemisphere separately.

Now another problem arises: it does not seem to be possible to draw a map of even half the world without distortions. This is not a topological problem, but a geometrical one, in the sense that we are interested in finer properties of Earth’s surface—shape, angle, area, and so on—than those that are preserved by continuity. Because the sphere has positive CURVATURE [III.78], no part of it can be mapped to the plane in a length-preserving manner, so some distortion is necessary. However, we have a certain amount of freedom to decide what kind of distortion we are prepared to accept and what kind we would like to avoid. There is, it turns out, a conformal map from the sphere (minus the poles) to a cylinder (which one can cut and roll out so that it fits into a plane)—it is the famous “Mercator projection.” A conformal map is one that preserves angles, so the Mercator projection is particularly useful for navigation purposes: if it looks as though you need to head north-northwest, then you really do. A disadvantage of the Mercator projection is that as you move away from the equator, the countries look bigger and bigger (though the angle-preserving property means that in close-up they are always the right shape). There is another projection that distorts shapes but preserves area. To work out the details of these projections, one must use mathematics—and in particular solve differential equations.

Here are a few simple applications of geometry to everyday life. If you have ever wondered what the best shape is for a manhole cover, then mathematics can come to your aid. Of course, it depends what one means by “best,” but if you often need to lift manhole covers, then you may be annoyed if they keep falling down the manholes. Can this be avoided? If the cover is rectangular, then the length of any side is less than the length of the diagonal, so it can drop down the hole, but if it is circular, then its width is the same in all directions and this is not possible.

Does this mean that only circular manhole covers are safe from dropping down their manholes? Actually, no. If you draw the three vertices of an equilateral triangle and join each pair of them by a circular arc centered on the third, then you obtain a sort of “curved triangle,” known as the Reuleaux triangle. (This is commonly misspelt “Rouleaux” in the mistaken belief that it has something to do with rolling. Actually, it is named after a nineteenth-century German engineer called Franz Reuleaux.)

Have you ever wondered why coins are the shapes they are? Most of them are circular, but the British fifty pence piece, for example, is a slightly curved polygon with seven sides. A moment’s thought makes it clear that for any odd number n ≥ 3 you can have a Reuleaux polygon with n sides, and the fifty pence piece is indeed a Reuleaux heptagon. This is convenient for slot machines: it means that you can have a slot into which the coin only just fits, however you push it in.

What about the best shape for a conveyor belt? If we construct it in the obvious manner, then one of its two sides will be exposed and the other not. Eventually, the exposed side will wear out, but the other side will be in pristine condition, since it will not have been used at all. However, as any mathematician will tell you, not all surfaces have two sides. The most famous example of a one-sided surface is the MÖBIUS STRIP [IV.7 §2.3], which is obtained from a flat strip of paper by twisting one end through 180 degrees and joining it to the other end. If you have a long enough conveyor belt for it to be practical to give it a twist somewhere, then you can wear out both sides equally (this makes sense locally even if globally the belt now has just one side), thereby doubling the use you get out of the belt. (You might think it simpler just to turn the belt over after a while, but the Möbius-strip design has been taken seriously enough to be patented, and similar designs have been used as typewriter ribbons and in tape recorders.)

3 Scaling and Chirality

Why are Arctic mammals large? Is it just a fluke that they have evolved that way? This does not sound like a mathematical question, but some simple mathematics can easily convince us that it is not a fluke at all. Since the Arctic is cold and animals need heat, animals that are better at preserving heat are more likely to thrive. The rate at which an object loses heat is proportional to its surface area, but the rate at which it generates heat is proportional to its volume. So if you double the size of an animal in every direction, then the rate at which heat is generated goes up by a factor of eight, while the rate at which it is lost goes up by a factor of only four. That is, larger animals find it easier to preserve heat.

But why, in that case, are Arctic animals not much bigger still? This can be explained by a similar scaling argument. If you scale an animal up by a factor of t, then its volume, and hence its weight (animals, being made predominantly of water, tend to have roughly the same density), will multiply by t³. The animal has to support this weight with its bones. The amount of force you need to snap a bone is roughly proportional to the area of a cross-section of that bone, and areas go up by a factor of t². So if t is too large, the animal will not be able to support its own weight. It does have the option of increasing the relative thickness of its bones, but if t is very large then its legs will be too thick for this to be a practical solution.

A similar sort of scaling argument explains why, if you drop a mouse down a 1000 foot mine shaft, then, to quote Haldane, “on arriving at the bottom, [it] gets a slight shock and walks away.” In this case, air resistance is roughly proportional to surface area, while the gravitational pull is proportional to mass, and therefore to volume. It follows that, the smaller you are, the smaller your terminal velocity, and the less you are bothered by a fall.

A simple fact with many scientific ramifications is that two shapes can be reflections of each other without being rotations or translations. For example, if you see a hand without seeing the body to which it is attached, then you can tell whether it is a right hand or a left hand. (If you can shake it naturally with your right hand, then it is a right hand.) This phenomenon is known as chirality. a shape is chiral if it cannot be obtained from its mirror image by rotation or translation.

The notion of chirality appears in many parts of science. For example, many elementary particles have a fundamental property known as “spin,” which means that they often have right-handed versions and left-handed versions. In pharmacology, it is now understood that many molecules are chiral, and that the two different versions can have radically different properties. An example that had tragic consequences is the drug thalidomide: one form of it is effective against morning sickness while the other causes birth defects. Unfortunately, in the late 1950s several thousand pregnant women were given a 50:50 mixture of the two forms. Less harmful examples of the importance of chirality abound. For instance, there are many chemicals that smell or taste different when you look at their reflected versions. (This may seem paradoxical, but the explanation is simple: the sensors in our noses and mouths also contain molecules with chirality.)

So far we have been considering rigid motions, but some shapes are chiral in the stronger sense that not even a continuous motion in space is enough to turn them into their mirror images. Two interesting examples are the TREFOIL KNOT [III.44], which comes in a “right-handed” and a “left-handed” version (the proof that these two versions are genuinely distinct is not at all easy), and the Möbius strip, which was mentioned earlier. The rough reason that the Möbius strip is chiral is that when you do the twist, you do it either according to the “corkscrew rule”—that is, twisting it as if you were pushing a corkscrew into the cork—or the opposite way. If you try to visualize it, you may be able to convince yourself that the direction of twist is not altered by continuous deformations, and also that the mirror image of a Möbius strip that obeys the corkscrew rule is a Möbius strip that does not obey the corkscrew rule.

4 Hearing Numerical Coincidences

Legend has it that Pythagoras, passing a blacksmith hammering a set of iron bars in a particularly pleasing way, was led to discover the laws of harmony. In modern terms, these laws say that two sounds go together particularly well (at least in the European tradition) if their frequencies are in the ratio r to s for some pair of small integers r and s: the smaller the better. As a result, people have tried to devise musical scales that have as many of these pleasing intervals as possible.

Unfortunately, there are limits to how well you can do. If you take a very simple ratio such as 3/2, which corresponds to what musicians call a perfect fifth, then its powers—9/4, 27/8, 81/16, and so on—get successively more complicated. However, by great good fortune it happens that 2¹⁹ is rather close to 3¹². To be precise, 2¹⁹ = 524 288 and 3¹² = 531 441, which is a difference of about 1.4%. It follows that (3/2)¹² is close to 2⁷. Since doubling a frequency raises the note by an octave, this says that twelve perfect fifths make an interval close to seven octaves. This allows one to build up a scale in which the fifths are approximately perfect.

There are many ways of doing the approximation. Early choices of musical scale would make some of the fifths perfect, at the expense of others. The modern compromise adopted by Western music for the last 250 years is to distribute the inaccuracies equally. If successive notes in a musical scale have frequencies in the ratio 1 to α, then starting from a frequency u the notes will have frequencies u, αu, α²u, and so on. If you want k notes in the scale, then αk should equal 2 (so that after k steps you have gone up by an octave). This means that all smaller powers of α must be irrational, so that all the other intervals in the scale are inharmonious! However, when k = 12, the fact that 3¹² and 2¹⁹ are close has the consequence that α⁷, which equals 2^7/12, is close to 3/2 (more precisely, it is just over 1.4983), which means that all the fifths are close to perfect.

Tuning systems are discussed in more detail in MATHEMATICS AND MUSIC [VII.13 §2].

5 Information

Few things illustrate better how the abstract mathematical theory of one generation can become the common sense of the next than the following two closely related ideas: that all information can be expressed as a series of 0s and 1s, and that the “quantity of information” carried by a book, a picture, or a sound is proportional to the number of 0s and 1s required to express it.

A famous theorem of Shannon (described in RELIABLE TRANSMISSION OF INFORMATION [VII.6 §3]) tells us that the rate at which information can be transferred by signals depends on the range of frequencies available. For example, it is the change from signaling electrically along copper wires (with a narrow range of frequencies) to signaling by light (with a very wide range) that has allowed the massive data transfers required by the Internet. The sound waves we hear belong to a very narrow range of frequencies, while the light waves that we see belong to a wide range, and this is why we need much more memory on our computers to store an hour of film than an hour of music. Similarly, it may feel as though visual perception is a passive process—we point our eyes in a certain direction, they behave a bit like video cameras, and we just watch the video—but because light carries so much information, our brains actually have to resort to a wide variety of tricks to deal with it. What we think we see is actually a theatrical representation of reality that our brains have cunningly manipulated. This is why there are optical illusions, and why they continue to work even when you know how they work. By contrast, since sound carries so little information, our brains can process it in a much more direct way (though still not completely direct—there are aural illusions too, and the brain has tricks that help us to pick out the information we are actually interested in from all the sound waves that enter our ears).

When information is transmitted, there are almost always faults in the transmission system, so that our messages are not transmitted perfectly. How do we then recover the messages? Here is a Victorian parlor trick that shows how in a very simple case. One begins by writing down all sequences (x^l, x², … , x⁷) such that every xi is either 0 or 1 and such that the numbers x¹ + x³ + x⁵ + x⁷, x² + x³ + x⁶ + x⁷, and x⁴ + x⁵ + x⁶ + x⁷ are all even. An example of such a sequence is (0, 0, 1, 1, 0, 0, 1).

If you think of these sequences as vectors in the vector space (that is, the seven-dimensional space where the scalars belong to the field of integers mod 2), then you will readily convince yourself that these three properties of a sequence are independent linear conditions, so the set of sequences in question is a four-dimensional subspace of . Therefore, there are sixteen such sequences. A member of the audience is asked to take one of them and change it in one place. The magician can at once identify which digit has been changed. Let us see how this works if we change the third digit of the sequence above, so we now have the sequence (y₁, . . . ,y₇) (0, 0, 0, 1, 0, 0, 1).

The first step is to note that y_l + y₃ + y₅ + y₇ and y₂ + y₃ + y₆ + y₇ have become odd, while y4 + y₅ + y₆ + y₇ is still even (since it is y₃ that has changed). Now the only number that belongs to the first two of the sets {1, 3, 5, 7}, {2, 3, 6, 7}, and {4, 5, 6, 7} but not the third is 3. This tells us that x3 is the variable that has been changed. How are the sets chosen so that this sort of argument will always work? The answer becomes clearer if we use the binary representations of the integers instead and put in a couple of leading zeros. Then the sets are {001, 011, 101, 111}, {010, 011, 110, 111}, and {100, 101, 110, 111} and we see that the ith set is the set of integers with a 1 in the ith digit from the end. So if we know which of the three parities have been changed, then we know the binary representation of the place where the sequence was altered. Therefore, we can reconstruct the original sequence.

This trick, rediscovered by Hamming, is the ancestor of all the error-correcting methods (also discussed in RELIABLE TRANSMISSION OF INFORMATION [VII.6]) that allow our CDs and DVDs to perform flawlessly even if they are slightly scuffed.

The fact that there is a precise mathematical way of measuring information content is of considerable importance in genetics. It has been suggested that the amount of information carried by our DNA, though very large, is much smaller than the information required to describe our bodies completely. This would explain what experimental evidence also corroborates: that the DNA carries a set of general instructions, but the fine detail of our anatomy, such as our fingerprints and the precise arrangements of our capillaries, is partly a matter of chance. So, for example, if it were possible to rerun the growth of the fertilized egg that ended up as you, the result would be broadly similar to you, but small environmental differences would result in a different set of fingerprints and a different arrangement of capillaries.

Under certain circumstances, it is not enough just to transmit information: it must also be protected. If we send our credit card number over the Internet, we want to do so in such a way that it would be very hard for an eavesdropper to find that number. A mathematical way of doing this is described in CRYPTOGRAPHY [VII.7§5].

Here is a slightly different but closely related problem. Suppose that Albert has a secret that he would like to share with Bertha (and only Bertha) in a conversation that everyone can hear. What is he to do? A first step is to think of any piece of information that they can share secretly—it turns out to be a short step from this to sharing a particular piece of information. The following procedure achieves this. First, Albert shouts out a large integer n and an integer u. Next, he chooses a large integer a, which he keeps secret (including from Bertha—obviously, since he does not yet know how to share secrets with her), and shouts out the value of u^a modulo n. Bertha then chooses an integer b, which she keeps secret, and shouts out the value of u^b modulo n. Now Albert is in a position to work out u^ab = (u^b)^a modulo n, since Bertha has told him u^b and he knows a. Similarly, Bertha can use her secret number to work out u^ab = (u^b)^a modulo n. Albert and Bertha now both know the number u^ab modulo n. This is a good example of a shared secret, because all that the eavesdroppers know is u^a, u^b and n, and when n is large there is no known way of calculating u^ab modulo n from u^a and u^b modulo n, apart from methods that take far too long to be practical.

Now suppose that Albert wants to send a credit card number N to Bertha. Assuming that 1 ≤ N ≤ n, then all he needs to do is shout out the number u^ab + N modulo n. Bertha then subtracts the secret number u^ab and obtains N. (Albert should convey only one secret this way, or he will reveal information. For instance, if he sent another credit card number M using the same u^ab, then the eavesdroppers would know the value of M – N. But if he and Bertha choose new numbers n, u, a, and b and use those to share the value of M, the eavesdroppers will effectively know nothing about the pair (M, N).)

Why do we believe that it is “hard” to calculate u^ab from u^a and u^b? What if tomorrow somebody discovers a simple trick for doing it? Surprisingly, even though we cannot be absolutely sure that the problem is hard, there are very precise ways of discussing the question. In particular, there are extremely plausible conjectures, the truth of which would imply that it really is impossible to calculate u^ab in a short time. These issues are discussed in considerable detail in COMPUTATIONAL COMPLEXITY [IV.20].

6 Mathematics in Society

A street in which all houses have front gardens is much prettier than a street in which all those front gardens have been converted into parking places. For some people, aesthetics are more important than convenience, so the effect of converting all the front gardens in a street may well be to reduce the values of all the houses. However, if you convert just one front garden, then it will increase the convenience for that household without making too much of a difference to the look of the street, so the value of that house will increase and the values of all the other houses will decrease slightly. Thus, for each individual house owner there is a financial incentive to convert the front garden, even though if everybody does so then everybody will lose financially.

Clearly, to avoid this unfortunate result the households must cooperate. Nash has shown how, starting from simple assumptions about fairness, there must be a system of mutual payments—for example, a household that wishes to convert its front garden might have to pay a charge that was shared between the other households—which will change their incentives in such a way that they will no longer want to ruin the street.

If the households do not wish to cooperate, Nash has shown that they come to a (usually less favorable) agreement which it is not in the interest of any single individual to break. A simple example of a situation in which no single individual may wish to change but a group acting in concert may wish to change is given by the following game. Suppose that three people hand to an umpire an envelope containing either the word “yes” or the word “no.” If two players have written the same thing and the third has not, then those two players get $400 each and the third player gets nothing. However, if all three have written the same, then all three players get $300. Suppose that the players meet before the game and agree that they will all write “yes” (in order to maximize their average gain). Then no single player will gain by writing no instead, but if two players decide to change then they will both gain.

Nash’s ingenious argument starts with an agreement that is not necessarily in equilibrium, and allows the parties to the agreement to modify their actions very slightly in a way that would improve their own situation if nobody else changed their actions. (However, since the other parties are changing their actions, the total change maybe preferable to nobody.) This results in a function that takes agreements to agreements. This function turns out to obey the conditions of THE KAKUTANI FIXED POINT THEOREM [V.11 §2], from which it follows that there is an agreement that no single individual wishes to change. (See MATHEMATICS AND ECONOMIC REASONING [VII.8], particularly section 4, for a further discussion of Nash’s theorem. Another situation where individual and collective self-interest do not necessarily coincide is the flow of traffic (see THE MATHEMATICS OF TRAFFIC IN NETWORKS [VII.4 §4]).)

Not all applications of mathematical thought to social problems have such satisfactory outcomes. Suppose that there is to be an election (or, more generally, that society has to make a choice between various possibilities) with n candidates and m voters. Let us use the term “voting system” to mean any method of putting the n candidates in order given the preferences of the individual voters. Kenneth Arrow has shown that, under normal circumstances, there is no good voting system. More precisely, he has identified a small set of very reasonable sounding properties that one would wish a voting system to have, and shown that no voting system has all these properties. To give two examples of these properties, it is surely desirable that the final ranking of the candidates should depend on more than just the ranking of one individual voter, and one would also expect that if every voter prefers one candidate x to another candidate y, then x should be ranked higher than y. Instead of listing the other properties, we present a simpler result, known as Condorcet’s paradox, that gives some of the flavor of Arrow’s theorem. (Indeed, Arrow’s theorem can be regarded as a descendant of Condorcet’s paradox.) Consider three voters A, B, and C with the following preferences.

Observe that the majority of the voters prefer x to y, a majority prefer y to z, and a majority prefer z to x. Therefore, majority preference is not a TRANSITIVE RELATION [I.2§2.3]. One consequence of this is that if voters are first asked to vote between two of x, y, and z and there is then a run-off between the winner of the first vote and whichever of x, y, and z is left, then the remaining candidate will always win.

Probability is another branch of mathematics that plays a central role in modern society. In earlier societies people worked until they died. Today people can stop working and live off their savings. You can, of course, just live off the interest of your savings but this means that you will die with a large sum unspent. Alternatively, you can assume that you will live a certain number of years and run down your savings, reaching zero at precisely the moment you expect to expire. This will not be satisfactory if you live longer than you expect. The solution is to make a bet with a wealthy corporation. You pay them your capital and in return they pay you a certain sum every year until you die. If you die early then they have won their bet, and if you die late then they have lost. By taking a large number of such bets and relying on results like the STRONG LAW OF LARGE NUMBERS [III.71§4], the corporation can be almost certain of making a profit in the long run. In effect you have paid a certain amount to transfer the risk (from the financial point of view) that you might live a long time from yourself to the corporation.

One of the earliest ways for mathematicians to make money was to become actuaries—that is, advisers on the appropriate price for transfer of risk in the situation described above. Nowadays, all sorts of risk (Will next year’s coffee crop fail? Will the euro fall against the dollar?) are bought and sold and have to be priced. A discussion of risk pricing in general can be found in THE MATHEMATICS OF MONEY [VII.9].

7 Conclusion

In the past, mathematics has had a dramatic impact on physics and engineering. At one time this led to hopes that biological and sociological phenomena would eventually come to be explained mathematically as well. Later, such hopes came to seem unrealistic: it was understood that these areas contain “emergent phenomena” that are not easily amenable to a reductionist approach and are therefore genuinely harder to describe mathematically than the phenomena studied in the “harder sciences.” However, mathematicians are now beginning to grapple with such phenomena: as even the simple examples in this article have shown, one can apply mathematics to many areas outside its traditional domain, and doing so can be extremely illuminating.