by Wells, David
These results differ, but not absurdly – and Euler had used ingenious approximations on the way. If his arguments were genuinely nonsensical, would he have got such comparable figures? That's most unlikely! The interpretation of his results was another matter, and it took over a hundred years for consistent and satisfactory interpretations of divergent series to be accepted.
Euler well knew that his ‘sum’ of a divergent series was no ordinary sum, as adapted to fit infinite series. He was deliberately changing the meaning of the word. He did so because using his new conception, his ‘experimental’ evidence was sufficiently convincing. Not surprisingly, Euler's repeated boldness led to occasional mistakes but this was not one of them and his successes far outweighed his failures. As Morris Kline puts it,
Like his predecessors, Euler's work lacks rigour, is often ad hoc, and contains blunders, but despite this, his calculations reveal an uncanny ability to judge when his methods might lead to correct results.
[Kline 1983: 307]
The exploration of mathematical miniature worlds is deeply scientific – and can therefore leave mathematicians suspended in uncertainty for long periods, like any scientist. The idea that 1 − 1 + 1 − 1 + 1 − 1….. = 1/2 is either simple-minded, a silly suggestion only a child or a mystic would make, or it is the brilliant intuition of a great mathematician (Leibniz) or the clear and logically satisfying result of deep insight into what the terms ‘convergence’ and ‘divergence’ could and should mean, once you decide you cannot lose the aesthetic delights and practical applications of series that on the face of it are pathologically absurd and dangerous.
This process of developing your idea of what an entity ought to be, closely parallels the physicists’ development of the concept of an atom or an electron, or the wave–particle model of light. The final results of mathematical exploration are pelucidly clear but on the way they can be confusing and messy. Fortunately, they are exciting too.
A practical use for divergent series
Euler himself illustrated one use of divergent series. First he proved that,
tends to a limit, known as Euler's constant, or γ for short. Euler then found a series for γ:
where the Bs are the Bernoulli numbers. Unfortunately, this series for γ diverges: fortunately, it has the remarkable property that the error from stopping at any particular term is less than the next term in the series. Since the early terms get very small indeed before the later terms get larger, Euler was able to use just the first 10 terms of the series to show that,
Gauss, another walking calculator, went rather further than Euler and added the last four figures in parentheses [Bromwich 1931: 324–5]. What a wonderful demonstration that the road to insight and understanding in mathematics must pass through the gates of imagination!
10 Mathematics becomes game-like
Divergent series are an example of an idea so novel and so difficult to understand that early work on divergent series was a mixture of uncertain manipulation, brilliant intuition, apparent failure, genuine errors, and incomprehensible success. It was only after many brilliant mathematicians had struggled with the idea that the dark and obscure landscape of divergent series was illuminated, and mathematicians finally decided what a divergent series ‘really is’ and how they could safely be handled.
Euler's relation for polyhedra
A similar confusion arose over another of Euler's interests: polyhedra. He noticed what Descartes had also seen earlier that for the simplest polyhedra, the number of vertices plus the number of faces exceeded the number of edges by 2: V + F = E + 2.
Thus, in Figure 10.1 a cube has 8 vertices, 6 faces and 12 edges, and 6 + 8 = 14 + 2. The octahedron on the right has 6 vertices, 8 faces and also 12 edges, while the irregular polyhedron below which is a square pyramid face-to-face with a cube, has 9 vertices, 9 faces and 16 edges, and 9 + 9 = 16 + 2.
Figure 10.1 Diagram of cube, octahedron and pyramid ON cube
Naturally such simple observations cry out for an explanation, but all the early proofs were defective. Every proof that was offered turned out to have a destructive counter-example because no one had a clear idea of how to define a polyhedron.
Could a polyhedron have holes? Several holes? Could it consist of two parts meeting only at a point, like two pyramids joined at a common vertex? Could it be of two parts joined along an edge? What if a polyhedron intersects itself? How would you then count the edges, and faces? All the early proofs were undermined by ‘weird’ counter-examples.
If a polyhedron is defined as a solid form bounded by polygons, then the ‘doughnut’ in Figure 10.2 fits the definition. But it has 16 faces and 16 vertices but only 32 edges, so V + F = E. The hole-in-the-middle seems to have got rid of the original ‘2’.
Figure 10.2 Polyhedral doughnut
Only when the idea of what a polyhedron ‘is’ had been clarified to the satisfaction of mathematicians, was it possible to go ahead and draw conclusions about them with confidence.
This point has another side: the idea of a polyhedron was created in such a way that Euler's very simple and elegant relationship, was true. It turned out that the simplest and most aesthetically pleasing definition of a polyhedron fitted mathematicians’ original intuitions of what a polyhedron ‘ought to be’. Yet the process of throwing up counter-example after weird counter-example was not wasted, far from it, because it was only through being tested in the fire, as it were, that mathematicians were forced to create the ‘best’, the ‘proper’ definition – just as the best and proper definition of angle measure turned out to be the radian (p. 99).
Lakatos on Proofs and Refutations Imre Lakatos (1922–1974) made this example famous in his book Proofs and Refutations [1976]. This is an excellent account of the relevant history, but his conclusion, that mathematics always develops like a science, was false. He took a sequence of episodes in the history of mathematics and falsely generalised to mathematics as a whole. Most theorems go through no such historical process as he recounted.
Lakatos made the mistake of seeing mathematics as far too scientific – so that all theorems were open to potential refutation – and had no conception of mathematics as game-like.
The subtitle of Proofs and Refutations was The Logic of Mathematical Discovery. Note the definite article. He claimed that this was a ‘general method of mathematical discovery’. Criticising the deductivist style he wrote that:
The whole story vanishes, the successive tentative formulations of the theorem in the course of the proof-procedure are doomed to oblivion while the end result is exalted into sacred infallibility.
[Lakatos 1976: 27]
Did he really believe that all theorems undergo ‘successive tentative formulations’? They do not. The problems he chose to analyze all involved questions of definition, that is, processes involved in the very construction of game-like mathematics, of the informal becoming formal. Of mathematics that is already adequately formalised, he said nothing at all.
In the process of formalising definitions of concepts and objects, failures of proof may be simple errors of reasoning but they may also be indications that the formalisation is itself flawed.
Vast tracts of ‘mathematical discovery’, however, do not consist of the development of such ‘informal theories’. How did Lakatos account for discovery outside his ‘informal theories’? He did not, because he had already denied them.
Lakatos seems not to have realised, or perhaps he could not accept, that if you chose to play a game – or if you invent a game yourself – certain implications must follow from the rules [Wells 1993b].
The invention-discovery of groups
In contrast to divergent series and polyhedra, the creators of group theory never encountered such problems. Early group theorists made many errors but – unlike the case of Euler's relation – they were game-like errors, not errors of definition.
One of the streams that led to the creation of group theory was the idea of the permutations of the roots of
an equation. Here is an example with 5 objects which do not have to be the roots of an equation – they could be just about any objects at all. If we start with the ordered set,
S = {a
b
c
d
e}
and they are permuted into this order
{b
c
a
e
d}
then we can see that a → b, b → c and c → a completing a cycle: a → b → c → a…At the same time, d and e just change places, d → e and e → d. We can represent the first cycle by (a b c) and the second by (d e), and the whole permutation from the 1st row to the 2nd is then clearly defined as (a b c)(d e).
Sometimes an element of a permutation does not move at all. In this case, if
{a
b
c
d
e}
permutes into
{d
b
e
c
a}
then we can write the whole permutation as (a d c e)(b) showing that 4 letters form a cycle and b remains unchanged.
We can if we wish apply a permutation repeatedly. If we write (a b c)(d e) = X, then the results of applying X again and again are: S =
{a b c d e}
XS =
{b c a e d}
XXS = X2S =
{c a b d e}
XXXS = X3S =
{a b c e d}
XXXXS = X4S =
{b c a d e}
XXXXXS = X5S =
{c a b e d}
XXXXXXS = X6S =
{a b c d e}.
So X6S = S. This is not a great surprise, since the permutation (b c a) when used 3 times, returns (a b c) to (a b c) and each switch (d e) ‘undoes’ the effect of the previous switch, so to ‘undo’ {b c a e d} by going forwards we must apply it 2 × 3 − 1 = 5 more times.
This is beginning, just, to look like an algebra of permutations and indeed there is a very natural way to ‘multiply’ permutations. If we write (a b c)(d e) = X and (a d c e)(b) = Y. Then XY is defined perfectly clearly to be the effect of Y followed by the effect of X, so we have: S =
{a b c d e}
YS =
{d b e c a}
XYS =
{e c d a b}.
If we calculate YXS by performing permutation X first and then permutation Y, the result is, S =
{a b c d e}
XS =
{b c a e d}
YXS =
{b e d a c}.
In this case, XY does not equal YX so the permutations X and Y are not commutative. However, all permutations of n objects are associative, meaning that given three permutations X, Y and Z, (XY)(Z) = (X)(YZ). Also, each permutation has an inverse, which is the permutation which ‘undoes’ the permutation immediately. So the inverse of,
X = (b c a)(e d)
is
X′ = (c b a)(d e)
because X′XS = S. It is also true that XX′S = S so the order of operations of X and X′ makes no difference. X and X′ do commute.
We are now close to defining a group of operations, (which does not have to be a permutation of objects). The defining rules, or axioms, of a group are that you start with a set of states and every operation leads to another state of the set. There must also be an identity operation, usually labelled I, which changes nothing. For permutations of 5 objects, the identity is the permutation I = (a)(b)(c)(d)(e). Also each operation, P has an inverse, P′ such that PP′ = P′P = I, and finally, the operations are associative.
Arthur Cayley proved that every finite group is isomorphic – meaning essentially identical to – some permutation group, so the apparently simple permutation groups are actually representative of all the groups that exist. So the permutation groups are not as simple as they seem, far from it, and group theory itself is extraordinarily complex – a perfect example of how a very simple idea leads to a vast and very rich miniature world.
The point we wish to emphasise here, however, is that because of the simple and straightforward nature of the idea of a permutation, none of the difficulties and obscurities that attended the birth of a sound theory of divergent series or of polyhedra, ever appeared.
Different parts of mathematics differ greatly in their origins, and in their initial clarity or obscurity and so different mathematicians also have different views of mathematics and how it should be approached.
Atiyah and MacLane disagree
This argument between Saunders Maclane, author with Garrett Birkhoff of the standard textbook, A Survey of Modern Algebra, who recorded the incident, and Michael Atiyah, illustrates one difference very well:
I adopted a standard position – you must specify the subject of interest, set up the needed axioms, and define the terms of reference. Atiyah much preferred the style of the theoretical physicists. For them, when a new idea comes up, one does not pause to define it, because to do so would be a damaging constraint. Instead they talk around the idea, develop its various connections, and finally come up with a much more supple and richer notion…However I persisted in the position that as mathematicians we must know whereof we speak…This instance may serve to illustrate the point that there is now no agreement as to how to do mathematics…
[MacLane 1983: 53]
Well, frankly there never was. From Archimedes to Diophantus and on to Fermat, Euler and Newton…mathematicians have succeeded with different styles and different approaches. The work of Atiyah is much more geometrical than that of MacLane's who is an algebraist. This is Atiyah's view of geometry:
Geometry is that part of mathematics in which visual thought is dominant whereas algebra is that part in which sequential thought is dominant. This dichotomy is perhaps better conveyed by the words “insight” versus “rigour” and both play an essential role in real mathematical problems.
[Atiyah 2003: 29]
It is no accident that Atiyah turned in his later years to problems of theoretical physics. We could say that MacLane thinks of mathematics from the start as game-like – and can afford to do so because of the kind of hard-edged mathematics that he does – whereas Atiyah, as MacLane notes, is more like a physicist, who is exploring a mathematical landscape that is much less clear cut, and so he hesitates and explores further as he attempts to achieve the precision and clarity which MacLane has to begin with.
Andrew Wiles expressed very well his own sense of doing mathematics as exploring in the dark:
Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into furniture, but generally you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of – and couldn't exist without – the many months of stumbling around in the dark, that precede them.
[Griffiths 2000]
Mathematics and geography
Mathematicians are explorers of many miniature mathematical worlds. Explorers often find the objects or phenomena that they discover novel and surprising and they do not always describe them accurately. Indeed, just because they are novel and surprising, early explorers may mis-describe them, misunderstand them, and give most misleading reports.
It is only after much further study that the ‘true nature’ of the kangaroo, or the manatee, or carnivorous plants are determined. The same is true of mathematicians exploring their miniature worlds. A perspective that naturally brings us to the subject of the mathematician as a scientist.
11 Mathematics as science
Introduction
The great Victorian scientist T. H. Huxley once made the mistake of claiming that math
ematics was a deductive science modelled on Euclid in contrast to the natural sciences which were inductive, based on observation, experiment and conjecture. James Sylvester (1814–1897), an equally great mathematician, in a famous address to the British Association for the Advancement of Science in 1869, disagreed:
[Huxley] says ‘mathematical training is almost purely deductive. The mathematician starts with a few simple propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them’…we are told [by Huxley] that Mathematics is that study which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation.
Sylvester was appalled:
I think no statement could have been made more opposite to the undoubted facts of the case, that mathematical analysis is continually invoking the aid of new principles, new ideas and new methods, not capable of being defined by any form of words, but springing directly from the inherent powers and activity of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer physical world…that it is unceasingly calling forth the faculties of observation and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verification, and that it affords boundless scope for the exercise of the highest efforts of imagination and invention.
