by Wells, David
The result was that Ramanujan's work contained false as well as true conclusions [Littlewood 1963: 87–8]. Traditional Japanese wasan mathematics also illustrated this weakness.
Imperfect induction [was] one of the characteristic features of the Japanese mathematics…mathematics as a branch of natural science…never thought of demonstration…In cases [like this] it was only customary to take the relation for granted from its existence in a few first instances…it resulted therefore that correct propositions intermingled with false results, which even the best mathematicians were not able to detect.
[Mikami 1961: 166, 168]
Scientific experiment is a wonderful exploratory tool for the mathematician but it is no more than a tool. The soul of mathematics is proof, through game-like concepts, tactics and strategies, which the sciences lack. The modern electronic computer is another wonderful tool for experiment and as an aid to making game-like moves, but which must also be treated with care.
13 Computers and mathematics
The computer has given us the ability to look at new and unimaginably vast worlds. It has created mathematical worlds that would have remained inaccessible to the unaided human mind, but this access has come at a price. Many of these worlds, at present, can only be known experimentally.
[Borwein et al. 2009]
We have already seen that the game of Nine Men's Morris and a crucial question about Hex have been solved using computers. Computers have an obvious and powerful role in exploring mathematical recreations such as pentomino puzzles which are hard for human brains to grasp but which can be attacked head on by brute force computing power. Today their incredible power is also used for summing sequences, finding relationships between sets of numbers which we suspect are connected in some way, and creating amazing visual images – the Mandelbrot set is the best-known example – so that mathematicians can, literally, see what they are doing. Not only can computers display geometrical figures and show them moving as the parameters of the problem are changed but they can display the behaviour of series and sequences in graphical form – and observation within this new and dynamic world can not only suggest conjectures and calculates data, but also ideas for proofs.
It is plausibly no coincidence that two of the greatest mathematicians of all time, Leonard Euler and Karl Friedrich Gauss, were both calculating prodigies who kept their arithmetical abilities into adulthood. Gauss, especially, relied upon generating data and pattern spotting for many of his results in number theory. As he explained, he obtained many of his results, ‘through systematic experimentation’ [Mackay 1994] adding on another occasion that, “I have had my results for a long time, but I do not know yet how I am to arrive at them” [Asimov & Shulman 1988: 115].
Two of the best-known experimentalists, David Bailey and Jonathan Borwein, work at the Centre for Experimental and Constructive Mathematics at Simon Fraser University in Canada. They use the term “experimental mathematics” to mean methods that includes the use of computation for:
(1) Gaining insight and intuition.
(2) Discovering new patterns and relationships.
(3) Using graphical displays to suggest underlying mathematical principles.
(4) Testing and especially falsifying conjectures.
(5) Exploring a possible result to see if it is worth formal proof.
(6) Suggesting approaches for a formal proof.
(7) Replacing lengthy hand derivations with computer-based derivations.
(8) Confirming analytically derived results.
[Bailey & Borwein 2000: 2–3]
The authors rightly place at the top of their list the achievement of ‘insight and intuition’ without which you can get results but, like Gauss, you don't know how to ‘arrive at them’. Experiment is vital to mathematics, but it has its limitations.
‘Using graphical displays’ is another crucial point: because computers can show geometrical figures moving, the computer as a visual aid is helping to take some mathematicians – and pupils – back to the days when seventeenth-century mathematics, including Descartes and Newton, naturally thought of curves as created dynamically, by combinations of movements.
The editors of the new journal Experimental Mathematics, Epstein and Levy [1995: 674] explain that they do value proofs, that an experimental result that has been proved is more desirable than a mere conjecture and that the use of computers ought to enhance mathematics, not undermine proof. Just so. They also point out that even today not all experiments are done on a computer. Some still use pencil and paper, or involve building physical models.
Hofstadter on good problems
Experimental mathematics has finally come into its own – hurrah!
[Hofstadter, personal communication 1989]
In 1989 I did a very small survey on ‘What is a good problem?’ Douglas Hofstadter is a computer scientist and mathematician with a strong feeling for the beauty of mathematics and the role of experiment. This was his response:
A good problem is one that mixes order and chaos in a deep and subtle way, and that fires the imagination for that reason. Perhaps another way to say this is that in solving a good problem, one discovers some wonderful and totally unexpected regularity when one expected nothing and on first sight saw only a jumble. The example of Morley's theorem in geometry is an excellent one.
[Hofstadter, personal communication 1989]
Indeed it is, but Hofstadter's answer is perfectly illustrated also by the Hofstadter sequence, Q(n), which he introduced in his book, Godel, Escher, Bach: an Eternal Golden Braid [Chapter 5] which starts,
Like the Fibonacci sequence, the terms are all defined by the previous terms: Q(1) = Q(2) = 1 and for n > 2,
This looks complicated and it is: in the Fibonacci sequence each term is the sum of two previous terms, but in Hofstadter's sequence, the two immediately previous terms only tell you how far back you have to go to find the terms you are going to add.
This is a perfect example of a rich problem which invites scientific exploration using a computer. The definition is unusually complex and not surprisingly the sequence seems very irregular, which suggests using experiment to calculate a large number of values and then to look for patterns in the hope that these will suggest conjectures, generalisations and analogies.
It turns out that amidst its ‘chaotic’ behaviour there are indeed intriguing hints of regularity. The first 2000 Q numbers are scattered above and below the line n/2 in ‘bursts of increasing amplitude and length’ which Klaus Pinn calls generations. Generation k is descended mostly from generation k−1 but with just a few members from generation k−2. These ‘bursts’ initially appear at n = 3, 6, 12, 24, 48, 96…. doubling each time, which is also reminscent of chaotic behaviour [Pinn 1998] [MathWorld: Hofstadter's Q-sequence].
Hofstadter naturally used a computer to explore a situation, much like physicists or chemists often use experiments to ‘see what happens’.
Computers and mathematical proof
Computers have been used to aid the proof of complex theorems, but the first examples were not encouraging. The four-colour theorem, first proposed in 1852, says that any plane map can be coloured with at most four colours so no two regions with the same colour share a border.
Figure 13.1 Four-colour theorem for divided circle
The map in Figure 13.1 proves that four colours are necessary for four regions if they all share boundaries with each other and strongly suggests that far more complex maps – such as the map in Figure 13.2 – would need many more colours. Not so! four colours are always sufficient. Simply adding more regions does not, it seems, make it harder to colour a map.
Figure 13.2 More complex map
The four-colour theorem was finally proved, or ‘proved’, in 1976 by Appel and Haken but only by reducing the problem to about 10000 special cases which they checked on a computer. Many mathematicians were unimpressed: even if there were no bugs in the computer program, the proof was not illuminating. They were agreeing, in
effect, with Hermann Weyl [page 74]. The computer was blind and its calculations added nothing to a deeper understanding of the problem, while most mathematicians were already convinced that four was the correct answer. More recently, a second proof has been published, but it has the same defect – it depends on computer calculations.
Computers and ‘proof’
Inevitably, some mathematicians have drawn extreme conclusions from the power of computers. This is what Doron Zeilberger, a prominent experimentalist, has to say about a famous conjecture:
We show in a certain precise sense that the Goldbach conjecture is true with probability larger than 0.99999 and that its complete truth could be determined with a budget of 10 billion.
[Borwein et al. 2009: 11] [Zeilberger 1993: 980]
The title of Zeilberger's work is ‘Theorems for a Price: Tomorrow's Semi-rigorous Mathematical Culture’ [1993]. The introduction of ‘price’ and ‘cost’ leads immediately to consideration of a trend that has overtaken the business world and is now intruding rapidly on academia: a focus on productivity and efficiency. Hence Zeilberger's even more outrageous claim that,
It is a waste of money to get absolute certainty, unless the conjectured identity in question is known to imply the Riemann Hypothesis.
[Zeilberger 1993]
If the cost is $10000000 then perhaps it is a ‘waste of money’ but to most mathematicians – still – mathematics has nothing to do with money, but is a search for insight and understanding. What we can safely conclude, however, is that the incredible feats which can be achieved with the aid of computers are already shifting, albeit slightly and controversially, our understanding of what it is to do mathematics.
At the same time, the very computers that were created by formal styles of thinking and promoted by attempts to reduce informality to logic and structure, are now so powerful that they allow styles of doing mathematics that are highly visual and experimental so that the very informality that has seemed inimical to mathematical progress and a threat to its rigour and long-term success, is now blossoming. Sylvester would have been delighted!
The claims made by some enthusiasts – I would call them extremists – that mathematics in the future will become an experimental science like physics, strike me as off the mark: but I am happy to contemplate a world in which all three aspects of mathematics, the game-like, the scientific and the perceptual, grow in symbiosis, with every conceivable style of mathematician finding his or her own landscape to explore and mine, and in which the games of mathematics flourish as never before.
Finally: formulae and yet more formulae
J. E. Littlewood, reviewing the collected papers of the Indian genius Ramanujan, quoted a couple of his extraordinary formulae (one is reproduced on page 213) and then remarked,
But the great day of formulae seems to be over. No one, if we are again to take the highest standpoint, seems able to discover a radically new type…
[Littlewood 1986: 95]
I suppose it depends on what you mean by ‘the highest standpoint’ and ‘radically new’. Here is one of a multitude of possible counter-examples to Littlewood's pessimistic claim, created by a combination of experimental mathematics and computer power:
This result, now known as the BBP or Bailey–Borwein–Plouffe formula, has the remarkable property that by using very little computer memory and rather little calculation you can calculate the next few digits in the binary expansion of π without calculating all the previous digits first. So we now know that the 400 billionth binary digit of π is 0.
Just as remarkably, the formula was found not by logical reasoning but by exploiting first some clever observations and a strong sense of analogy, and then programming a computer to search for a formula for π of a certain type. The full story is told in Experimental Mathematics: Plausible Reasoning in the 21st Century by Jonathan Borwein and David Bailey.
Littlewood also wrote that, referring to Ramanujan's papers,
The moral seems to be that we never expect enough; the reader…experiences perpetual shocks of delighted surprise.
[Littlewood 1986: 96]
This could surely be said also of this formula for π and many other results from the new field of experimental mathematics!
14 Mathematics and the sciences
As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.
[Einstein 1921]
Scientists abstract
When Galileo did his famous experiment of rolling a marble down a straight chute and timing its descent, he abstracted away the friction of the marble against the chute and the inevitable errors in his measurement of the time taken. When he argued that the flight of a cannonball could be analysed into a steady motion horizontally and an entirely separate accelerating motion vertically, he ignored the resistance of the air which would have had negligible effect on his marble in a chute but greatly distorts the flight of a cannonball from a perfect parabola.
Galileo's experiments were concrete and physical but his conclusions, which he believed to be Laws of Nature ordained by God, were precise and mathematical. His results fit Einstein's dictum. The abstract model can be analysed with great confidence to make predictions – but the predictions will not fit reality perfectly. Plato compared the use of mathematics for the study of reality to, ‘fitting a sandal to the foot’. It's a good analogy: the sandal should fit as closely as possible – if it doesn't it will be uncomfortable! – but no one mistakes one for the other [Young 1928: 204].
Mathematics anticipates science and technology
The ancient Greeks’ study of the conic sections as slices of circular cones is surely strange. Making a model of a cone is not easy with limited materials and anyway why should anyone think of slicing one? We do not know – perhaps someone spotted that an asymmetrical slice was itself symmetrical and couldn't resist investigating further – but anyway, the results were beautiful and astonishing and surely very difficult, not least because you have to think in three dimensions. Figure 14.1 (somewhat simplified) is from Book 1, Proposition 11 of the Conics of Apollonius of Perga (262–190 BCE) ‘the Great Geometer’.
Figure 14.1 Apollonius figure
The figure shows a circular cone and a slice through E, F, K and D which is parallel to the edge through N, creating a parabola. FG is the axis of this parabola and L, a point on the axis is the mid-point of the chord through L and K. MN is the perpendicular to both LK and FG, cutting the cone at M and N.
This figure is very far from being the most complex in the work of Apollonius who often argues about plane conics by going back to their definitions as slices of cones and arguing in three dimensions: result, brilliant pure mathematics.
In this figure, Apollonius proves that, in our notation, KL2 = ML·LN. Very elegant, but what have such properties to do with science? Many centuries later, Kepler (1571–1630) was trying to use the voluminous data collected by Tycho Brahe to work out the orbits of the planets as they revolved (according to Copernicus) round the sun. He knew about the conics because Apollonius had been translated into Latin in 1536. Kepler concluded from his analysis of Brahe's data that the orbits were ellipses, and proposed his three laws of motion. Newton, whose friend Edmund Halley had published an edition of Apollonius in 1710, also studied the Greek geometer and used the geometry of the conics in his Principia Mathematica.
If the Greeks had ignored sliced cones, Kepler might have merely thought that his orbits were squashed circles – and modern astronomy would have been stuck on its launching pad.
The success of mathematics in science
As for us who are wiser or more timid, let us be content to view most of these calculations and vague suppositions as intellectual games to which Nature is not obliged to conform.
Jean d’Alembert [1963: 24–5] [Wells 2003: 102]
The modern hard sciences, starting with physics and astronomy, have long been highly mathematical which suggests th
at the world itself is essentially mathematical also. ‘God ever geometrises’ as Plato put it. But why? Why is maths successful in the hard sciences, but not in the relatively feeble soft sciences? Here is Einstein in 1921:
At this point an enigma presents itself, which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?
[Einstein 1921]
The success of mathematics in science reflects analogies in the physical world that are totally unexpected: for example, between the ellipses which are slices of cones or cylinders but which are also the paths of the planets, to a good approximation. This analogy was present within the physical world before it was analysed by mathematics. If the paths of the planets were easier to observe, more primitive men might have spotted that the curved slice of a cylindrical pole and the path of a planet appeared to be the same shape, without recognising that shape as mathematical. Why do such physical analogies exist – and why are they so precise?
