Games and Mathematics (25 page)
Read Games and Mathematics Online
Authors: David Wells
It looks very much here as if the lines JH and KG are parallel, and when Daniel repeated his experiment by drawing several differently shaped triangles, these lines did always seem to be parallel – but appearance is no proof.
By using weighted averages, which Daniel was taught especially so that he could use them for this problem, he could calculate the locations of every point in
Figure 12.3
in terms of the vertices, as J has been calculated in the figure. In particular, he was able to treat JH and KG as
vectors
and check if they really are parallel.
Figure 12.3
in terms of the vertices, as J has been calculated in the figure. In particular, he was able to treat JH and KG as
vectors
and check if they really are parallel.
Figure 12.3
Schultz figure showing weighted averages
Schultz figure showing weighted averages
It turns out JH and KG are not parallel, but that 4JH/5 − KG = AB/20. Since JH, KG and AB all ‘point in much the same direction’ we can see why JH and KG seem to the naked eye to be genuinely parallel, though they are typically a fraction of a degree apart. Only as the triangle becomes more and
more obtuse does it become obvious that they are not parallel [Wells & Schultz
2008
].
more obtuse does it become obvious that they are not parallel [Wells & Schultz
2008
].
This negative conclusion can be easily checked with any dynamic geometry program, but that raises a further question: if Daniel had started by using a computer, he might have spotted the apparent parallelism, paused, wondered for a moment, moved one of the vertices to check his suspicion, realised that the appearance was false and moved on, all in a few seconds. Would he have thus gained or lost? I think he would have lost. The appearance is very surprising and deserves to be investigated further, by calculation, because
only
exact calculation – which amounts to a
proof
– will reveal what is actually going on ‘behind the appearance’.
only
exact calculation – which amounts to a
proof
– will reveal what is actually going on ‘behind the appearance’.
Logic merely sanctions the conquests of intuition.[Jacques Hadamard]
Hadamard (1865–1963) was a truly great mathematician but not all mathematicians would agree with his judgement. Gauss published six proofs of his
quadratic reciprocity
theorem during his lifetime and a seventh was found in his papers. Yet he was no doubt convinced it was true by experiment, and his first proofs were not erroneous, so why the multiplication?
quadratic reciprocity
theorem during his lifetime and a seventh was found in his papers. Yet he was no doubt convinced it was true by experiment, and his first proofs were not erroneous, so why the multiplication?
Proofs do far more than logically certify that what you suspect, or conjecture, is actually the case. Proofs need ideas, ideas depend on imagination and imagination needs intuition, so proofs beyond the trivial and routine force you to explore the mathematical world more deeply – and it is what you discover on your exploration that gives the proof a far greater value than merely confirming a ‘fact’.
Of course, the proof must be solid. William Thurston is a brilliant geometer with a wonderful intuition and imagination. Here he is on what he means by a good proof:
I’d like to spell out more what I mean when I say that I proved this theorem. It meant that I had a clear and complete flow of ideas, including details, that withstood a great deal of scrutiny by myself and by others. Mathematicians have many different styles of thought. My style is not one of making broad sweeping but careless generalities, which are merely hints of inspiration: I make clear mental models, and I think things through…My proofs have turned out to be quite reliable. I have not had trouble backing up claims or producing details for things I have proven. I am good at detecting flaws in my own reasoning as well as in the reasoning of others.
Not all mathematicians have been as rigorous as Thurston. Saunders MacLane reported that in Italy during the years 1880–1920 a number of results in algebraic geometry were published without careful proving. The situation became so bad that it was jokingly said that a triumph for an Italian algebraic geometer consisted in proving a new theorem and simultaneously proposing a counter-example to the theorem. As a result Italian results in algebraic geometry were discredited until several mathematicians, including Emmy Noether whom we met earlier, cleared up the disputed points by applying much more rigorous standards of proof [Hanna
1996
: 2].
1996
: 2].
A too-scientific approach to mathematics also has its risks.
Induction
can refer both to a sound mathematical method of proof, and to the use of scientific induction within mathematics. As we have seen, induction within mathematics is common enough and rightly so but if you don't prove what you've induced, you could be in trouble. Ramanujan, genius though he was, occasionally fell into this trap, and was criticised by Hardy who believed that nothing but absolute rigour would suffice in number theory, if mistakes were to be avoided. Littlewood wrote of Ramanujan:
Induction
can refer both to a sound mathematical method of proof, and to the use of scientific induction within mathematics. As we have seen, induction within mathematics is common enough and rightly so but if you don't prove what you've induced, you could be in trouble. Ramanujan, genius though he was, occasionally fell into this trap, and was criticised by Hardy who believed that nothing but absolute rigour would suffice in number theory, if mistakes were to be avoided. Littlewood wrote of Ramanujan:
His intuition worked in analogies…and…by empirical induction from particular numerical cases…The clear-cut idea of what is meant by a proof…he perhaps did not possess at all. If a significant piece of reasoning occurred somewhere, and the total mixture of evidence and intuition gave him certainty, he looked no further.
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.
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.
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
Computers and mathematical proof
Computers and ‘proof’
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.
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.
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].
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.
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 problemsExperimental 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.
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.
n
), which he introduced in his book,
Godel, Escher, Bach: an Eternal Golden Braid
[Chapter
5
] which starts,
n
> 2,
, 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].
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 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.
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
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.
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
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.
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:
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,
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.
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.
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