Games and Mathematics

by Wells, David

  Pre-scientific men had no knowledge of the applicabilty of maths outside the business for which counting and arithmetic were invented. Of course, arithmetic works for the purposes of trade – that's why it was created – but why should the counting numbers fit very different areas of life?

  Pythagoras is a paradigm case. He discovered that the strings of musical instruments which produced certain harmonious notes when plucked had lengths in simple ratios. This seems both mysterious and apparently pointless: you don't need to understand mathematics to play a musical instrument and most musicians are not mathematicians. Nobel prize-winning physicist Eugene Wigner once gave a much-anthologised address on ‘The unreasonable effectiveness of mathematics in the natural sciences’. He claimed that,

  the enormous usefulness of mathematics in the physical sciences is something bordering on the mysterious, and that there is no rational explanation for it.

  [Wigner 1960]

  and concluded that:

  The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it, and hope that it will remain valid for future research, and that it will extend, for better or for worse, to our pleasure even though perhaps also to our bafflement, to wide branches of learning.

  A statistician might reply that you can easily fit any number of different curves to any set of data. Indeed, you can, but they will generally be complicated and ugly, whereas ellipses are both simple and elegant. That is the puzzle – that the mathematics used by hard scientists is so astonishingly simple.

  How do scientists use mathematics?

  Physicists and other scientists exploring the natural world are in much the same position as the mathematician inventing – or discovering? – new mathematical objects and concepts. This can be a long process in which the naive preconception that mathematicians are always precise in their thinking and always know exactly what they are talking about, is turned on its head.

  Thus mathematicians found the sums of many infinite series long before they had a clear definition of ‘the sum of an infinite series’. Even more strangely, as we have seen, Euler and others explored divergent series which apparently did not have a sum at all.

  Oliver Heavyside, a great pioneer of electrical theory, was as daring and devil-may-care as most mathematicians were careful and cautious. Divergent series were just one of the novel methods that he used with great success in his work. The reaction of pure mathematicians and even other physicists was mixed. H. T. H. Piaggio, the author of a very popular textbook on differential equations, exclaimed that,

  Heavyside's methods seemed a kind of mathematical blasphemy, a wilful sinning against the light. Yet Heavyside's results were always correct! Could a tree be really corrupt if it always brought forth good fruit?

  [Piaggio 1943]

  On the other hand, one of his papers was rejected by the Proceedings of the Royal Society, an almost unheard of insult: members were generally allowed to published whatever they submitted in the Proceedings, but Heavyside had used divergent series rather too freely [Nahin 1988/2002: 222–3].

  Heavyside claimed that ‘rigorous mathematics is narrow, physical mathematics bold and broad’, and that,

  In working out physical problems there should be, in the first place, no pretence of rigorous formalism. The physics will guide the physicist along somehow to useful and important results, by the constant union of physical and geometrical or analytical ideas. The practice of eliminating the physics by reducing a problem to a purely mathematical exercise should be avoided as much as possible. The physics should be carried right through, to give life and reality to the problem, and to obtain the great assistance which physics gives to mathematics.

  [Nahin 1988/2002: 217, 219]

  This forceful statement both explains how Heavyside was so often correct despite his ‘dubious’ methods – he was kept on the straight and narrow by his powerful intuition – and why many pure mathematicians have no talent for applied mathematics, and vice versa. Heavyside shows a mental cast of mind which Atiyah no doubt would appreciate, but perhaps not MacLane.

  Methods and technique in pure and applied mathematics

  The Indian-Arabic system of numerals in which we count in powers of 10, lends itself to simple sums or algorithms for the basic processes of addition, subtraction, multiplication and division and even – if you were a Victorian schoolboy – for the extraction of square and cube roots. These sums are indeed so very simple that certain seventeenth-century mathematicians decided that they could completely automate them, starting with the almost forgotten Wilhelm Schickard who built his Calculating Clock in 1623, then on through Pascal's adding machine and Leibniz's machine which did all the four basic operations. Leibniz was inspired by the thought that,

  It is unworthy of excellent men to lose hours like slaves in the labour of calculation which would safely be relegated to anyone else if machines were used.

  [Leibniz 1685]

  Today we have electronic calculators for a variety of ‘elementary’ calculations, but applied mathematicians still require methods and techniques to solve problems which cannot be solved by algorithms and yet which do not require great imagination or originality either.

  This is where mathematical methods come into their own. When I was at college we had a giant textbook called Methods of Mathematical Physics by Harold Jeffreys and Bertha S. Jeffreys, both distinguished mathematicians. Today there is a plethora of books like Mathematical Methods for Science Students [Stephenson 1961] which consists of 21 chapters on infinite series and calculus, differential and integral, plus 6 chapters on real numbers, inequalities, determinants, matrices, groups and vectors. All the chapters consist of extremely highly developed mathematics in which all doubts and uncertainties and ambiguities have been removed to leave the tools which science students and indeed anyone else, can use with confidence.

  The subtle proofs, the delicate conditions and qualifications, the dozens, even hundreds, of papers published on curious and weird objects such as Fourier series or the elliptic integrals as mathematicians attempted to understand these strange topics better and better and replace their complexity by clarity and simplicity, are nowhere to be seen.

  Each topic has been reduced more-or-less to a game in which the rules of the game are clear, the allowable moves are unambiguous and the tool-user is more-or-less guaranteed to get ‘the right answer’.

  Happily, the qualifications ‘more-or-less’ are still necessary. It is not illegal to attempt to find the Laplace transform of a strange function for which the standard methods don't work – but for the functions generally met with by science students, these methods will indeed function perfectly.

  This process of reducing insightful and imaginative complexity to routine method, took many years, as we can see in the subject of finding the areas under curves such as the cycloid.

  Quadrature: finding the areas under curves

  What is the simplest way to calculate the area under a simple curve such as y = x3, from, let's say, 0 to 10, from first principles?

  The simplest and most obvious first move – which was used by Cavalieri (1598–1647) in his book, Geometria Indivisibilibus and Wallis (1616–1703) – would seem to be to divide the x-axis from 0 to 10 into a number of equal parts, construct the vertical strips, as in Figure 14.2a, and then calculate the total area of all the strips.

  Figure 14.2a y = x3 divided into strips (underestimate)

  With 10 equal parts, the sum of the areas of the strips underneath the curve is,

  Adding these consecutive cubes is not hard, whether by hand, with a calculator or by using the formula that we have met before:

  Whichever way, A = 2025. This, however, is a significant underestimate, so another smart tactic would be to repeat the sum as in Figure 14.2b, to get an overestimate:

  Figure 14.2b y = x3 divided into strips (overestimate)

&
nbsp; A* = 3025; so we expect the average (A + A*) to be a much better approximation, and indeed it equals 2525 whereas the actual area is, by elementary calculus, 10000/4 = 2500. The error is exactly 1% which seems quite good for such a basic and unsophisticated strategy which can be made much more accurate by dividing the area into many more strips.

  On the other hand, adding cubes is not that simple, and if we want to use the same strategy for finding the area under y = x4 or y = x5 then each time we raise the index we need a new formula for the sums of nth powers of consecutive integers. This is certainly a general method but it is not a simple general method. Fermat had a better idea, a sharper strategy.

  Using his method, you divide the x-axis from 0 to 10 not into equal parts, but into unequal parts by first marking the points 10, 10e, 10e2, 10e3, 10e4, 10e5, and so on, where e is chosen to be less than 1 (Figure 14.3). At first glance this is more complicated than the first method, but that is an illusion. Fermat, like a good chess player, has looked ahead and seen that this leads to a very simple calculation. The sum of the strips is now equal to, counting backwards from the right:

  Figure 14.3 Fermat's unequal strips

  This is an infinite series which may already be a bonus. Infinite series are often easier to add up than their partial sums. In this case the sum is equal to,

  and the sum of this simple geometric series – so much easier to sum than the consecutive cubes – is,

  In order to make the approximate sum more accurate we let e tend to 1, and the area then tends to 104/4, as before [Boyer 1945]. Unlike the original method, however, Fermat's cunning strategy can be easily used for any integral power of x.

  The cycloid

  The cycloid is generated by a point moving on the circumference of a circle rolling along a straight line. The circle on the left of Figure 14.4 starts with a vertical diameter PT. After rotating through 180° the same diameter becomes TP, and after completing a circle of 360° it returns to its initial orientation PT. Meanwhile the point P has traced one complete arc of the cycloid, PPP.

  Figure 14.4 Basic cycloid figure

  The cycloid has so many beautiful features and was the cause of so much acrimonious controversy in the seventeenth century that is was called the Helen of Geometry. Among other properties, it is the curve of quickest descent, or brachistochrone: the fastest path for a ball to move from A to a lower point B under gravity is part of a cycloid; and the tautochrone: a ball in a cycloidal basin has the same period of oscillation wherever it starts.

  Galileo named it the cycloid in 1599 and tried to find its area by cutting a cycloid out of metal sheet and weighing it. This suggested that the area was 3πr2 where r is the radius of the generating circle, but Galileo was not convinced by his own experiment.

  Later Toricelli (1608–1647), the inventor of the barometer, and Fermat and Descartes and others all found the area to be indeed 3πr2. The following simple proof is a cross between those of Roberval (1602–1675) and Pascal [Boyer 1956: 135–6; Coolidge 1949: 100].

  Figure 14.5a Cycloid with three circles

  Figure 14.5a shows that when the circle has rolled to touch the line PU at Q, (which happens to be one quarter of the way from P to U in our figure), the point P has risen to Pʹ. If the original circle had simply rotated about its centre without rolling anywhere, then P would have risen to R. Since, however, it has also moved a distance PQ to the right, we can actually construct the point Pʹ by using the fact that RPʹ = PQ. We also need to notice that the arc RP = arc PʹQ = arc VU.

  Next, looking at the distance the circle still has to roll, we see that PʹV = QU and that both of these lengths is equal to arc VW.

  We now divide the area of the half-cycloid into equal strips of width, horizontally (Figure 14.5b).

  Figure 14.5b Cycloid divided

  The area of the half-cycloid is the sum of all the strips PʹVVʹ which we can think of as,

  The strips VVʹ, in the limit, will be the area of the half-circle, or πr2, but what is the limit of the sum of the strips PʹV? To find it we draw the graph of the arc lengths VW against y, starting with y = 2r when s = 0 and moving to y = 0 when s = πr (Figure 14.6).

  Figure 14.6 Cycloid with graph of y against s

  We don't know, and we don't need to know, what s is as a function of y. Roberval in his proof just called the matching curve, the ‘companion of the cycloid’. It is in fact just what it looks like, a sine curve. What we do need to notice is that since the half-circle is symmetrical about the line y = r, the rate at which s increases will be the same at points symmetrical about the centre line, y = r, and so the curve will have point symmetry about its centre, C, where y and s momentarily increase together, making the gradient of the tangent 45°.

  We conclude that the area under the curve which is the limit of the second sum, is half the area of the rectangle, or × 2r × πr = πr2. Adding the area of the half-circle we get the area of 3πr2/2 for the half cycloid, and 3πr2 for the whole cycloid.

  This dynamic way of thinking about the ball actually rolling along the line and then ‘unwrapping’ the perimeter of the half-circle to create the side of a rectangle, is typical of the way of thinking of many seventeenth-century mathematicians. It had to be highly creative and imaginative – like a brilliancy at chess – because no general methods of finding areas were known. The division of the area into parallel strips was itself a very general idea, but its application was delightfully different in every particular case.

  Although Descartes (and Fermat) had created coordinate geometry, and although Apollonius (292–190 BC) had long ago proved a property of the parabola which was equivalent to the equation ay = x2, mathematicians still tended to think of curves as created dynamically by movement, rather than being defined statically by equations [Boyer 1956: 134–5].

  As it happens, the cycloid cannot be simply defined by an equation of the form y = f(x), but only by parametric equations: x = r(t − sin(t)) and y = r(1 – cos(t)). Today, its area can be found by basic calculus. It is not especially simple but it is not complicated either, requires no imagination or insight and is indeed completely routine, a matter of applying well-known and well-understood techniques.

  The tangent at any point on the cycloid can also be found today by pure technique, but Descartes had no difficulty in finding it by a dynamic argument which we can still appreciate, not least because it is so simple and elegant.

  In Figure 14.7 is half a cycloid as before, and a point Pʹ on the cycloid. The first puzzle is to find the position of the rolling circle PT when P has moved to Pʹ. To solve this, Descartes drew PʹV parallel to PU and then found Q on PU so that QU = PʹV, exploiting the property that we noted earlier. Q is where the rolling circle touches PU and is also what we today would call the instantaneous centre of rotation of the line QPʹ at that moment. We at once conclude that the tangent is perpendicular to QPʹ, as drawn.

  Figure 14.7 Descartes: tangent to cycloid

  In Descartes's time, that idea had not been invented (or discovered), but he drew the same conclusion by a more complicated argument about a polygon, ending up by exclaiming that, ‘the same would happen with a polygon with one hundred thousand million sides, and consequently also with the circle.’ He was right. It would. He continued,

  I could demonstrate this tangent in another way, more beautiful in my view, and more geometric; but I leave it out to spare the trouble of writing it…

  [Descartes 1996, 2: 309, in Jesseph 2007: 423]

  By ‘more geometric’ Descartes meant that his other proof, which he never revealed, did not involve ideas of motion or mechanism. Descartes believed that ‘mechanical’ curves such as the cycloid should be banished from ‘proper’ geometry, unlike Pascal who was happy to welcome them. Fortunately, Descartes did not let his prejudice stand in the way of his curiosity [Jesseph 2007: 423].

  Science inspires mathematics

  Whatever the explanation – if there is one – the success of mathematics in (hard) science has been a boon to
both: scientists have been able to create very powerful theories, while mathematicians have been forced – or inspired – by scientific questions to develop their own theories.

  The calculus could have been used merely to find areas under curves and gradients of tangents more efficiently, but no, it was used at once to solve difficult new problems such as the problem of the shape of a solid which offered least resistance to a flowing liquid, and the problem of the path of light through different media [Chapter 15], and of course in Newton's wonderful theory of universal gravitation. The mutual fertilisation continues to the present day.

  In contrast, the fate of mathematics without science is illustrated by traditional Japanese mathematics, wasan. Japanese mathematics in the pre-feudal era was utilitarian but during the Edo period (1603–1867) it lost its practical character and became the hobby of leisured samurai, merchants and rich peasants who enjoyed wasan much as they enjoyed haiku or the tea ceremony as an art form unconnected to everyday life.