by Wells, David
The spirit of system versus problem solving
On the other hand, some differences in style seem to be embedded in the individual personality not the culture. Euclid set a two-thousand-year-old standard for systematic exposition which was envied by early modern scientists and philosophers who thought that by copying the style of Euclid they could also produce Truth with a capital T. Eighteenth-century critics referred derogatively to the Spirit of System which tried to deduce a vast number of conclusions from a minimal set of premises, a project which outside of geometry they judged to be absurd and impossible.
We have already met the Bourbaki group of French mathematicians. They were system builders par excellence. Their guiding ambition was to provide ‘a complete treatment of modern mathematics’ [Felix 1960: 65]. ‘The organizing principle will be the concept of a hierarchy of structures, going from the simple to the complex, from the general to the particular [Mathias 1992].
Benoit Mandelbrot, famous for the Mandelbrot set, refers to Bourbaki as an extremist movement which was destroyed by the coming of the ‘rebirth of experimental mathematics, prompted by the arrival of the electronic computer’ [Frame & Mandelbrot 2002: Chapter 4]. Alan Bishop also criticised the formality of so much modern mathematics in the Bourbaki mould and gave a list of features he considered bad:
The experts now routinely equate the panorama of mathematics with the production of this or that formal system. Proofs are thought of as manipulation of strings of symbols. Mathematical philosophy consists of the creation, comparison, and investigation of formal systems. Consistency is the goal. In consequence meaning is debased, and even ceases to exist at a primary level.
[Bishop 1985: 2]
In contrast, the phenomenon that was Paul Erdös showed no interest in general systems or in deep abstraction. He was a problem solver and problem poser of genius. However, some of his problems did lead to new fields of mathematics, such as probabilistic number theory (with Mark Kac) and random graphs (with Alfréd Rényi), illustrating how a focus on the specific and contingent leads naturally to the more general and abstract.
The truth – it can hardly be a surprise – is that mathematics depends on both generalists and problem solvers. Bourbaki upset their opponents not because generality itself is bad – it can be dazzlingly powerful – but because they were extremists. One of the greatest mathematicians of all time illustrates the Middle Way. We have already quoted Hilbert's view that, ‘He who seeks for methods without having a definite problem in mind seeks in the most part in vain.’ Hilbert was a brilliant problem solver with great ambitions. Addressing the International Congress of Mathematicians in 1900, he presented 23 problems that in his opinion were the most important unsolved problems at the start of the twentieth century, [Hilbert 1900] and claimed that:
As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development…It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains wider and freer horizons.
Yet Hilbert also proposed to create a proof theory by which all of mathematics might be formalised and all proofs reduced to calculation in a formal system. It is significant that this project failed – mathematics is not so easily tamed – but by seeing why it failed, we now understand mathematics better.
Visual versus verbal: geometry versus algebra
Another striking stylistic difference is between a preference for the visual or verbal. Bourbaki were very verbal. Felix Klein was very visual, and insisted that his students make models:
[Klein] had a strong power of geometric visualization, and all his investigations were essentially governed by appropriate geometric pictures…if you look at the drawings in his papers on automorphic functions, you will be astonished by the beauty of these figures, most of which are made up of very simple basic figures like triangles with curved sides. This beauty rests precisely on the fact that these figures illustrate the underlying mathematical relationships in an extremely simple and transparent way. Since Klein builds on these figures, all the results he derives possess that self-evidence which, as we said before, is the goal of mathematical research.
[Krull 1987: 50] [See also Mumford 2002]
Twenty years ago, Ian Stewart wrote in New Scientist that,
Attitudes to mathematics are changing; austere and formal pedantry is once more giving way to ideas, and it is becoming increasingly permissible to draw pictures that help to explain those ideas…Geometric thinking is back in vogue these days, though not in our schools. A month ago I was talking to an American mathematician who predicted a revival of geometry, resulting from the massive effort in computer graphics.
[Stewart 1985]
Mandelbrot would no doubt agree. He records how he one day sat in school as his teacher explained a problem, algebraically.
Mandelbrot's hand shot up, ‘Sir, you don't need to make any calculations. The answer is obvious.’ He described a geometrical approach that yielded a fast, simple solution. Where others would have used a formula, he saw a picture. The teacher, skeptical at first, checked. Correct. And Mandelbrot kept doing the same thing, in problem after problem, in class after class.
[Mandelbrot 2006]
Mandelbrot has a very visual way of thinking: ‘I would say to myself: This construction is ugly, let's make it nicer. Let's make it symmetric. Let's project it. Let's embed it. And all that, I could see in perfect 3-D vision. Lines, planes, complicated shapes’ [Mandelbrot 2006]. His work has subsequently been as far from pure abstraction as you can get:
Indeed, my work is unabashedly dominated by awareness of the importance of the messages of our senses. Fractal geometry is best identified in the study of the notion of roughness. More specifically, it allows a place of honour to full-fledged pictures that are as detailed as possible and go well beyond mere sketches and diagrams…But those pictures then went on to help me and many others generate new ideas and theories. Many of these pictures strike everyone as being of exceptional and totally unexpected beauty…In front of our eyes, the visual geometric intuition built on the practice of Euclid and of calculus is being retrained with the help of new technology.
[Frame & Mandelbrot 2002: Chaper 2]
So, contrary to the impression that pupils might get from their school textbooks, mathematicians do not all think in the same way and we might even say that mathematics is more than just ‘one’ subject, mathematicians’ styles of thinking being so extremely varied.
Women, games and mathematics
No one can fail to notice that most mathematicians and chess players – including almost all the greatest – have been men. Not only do few women reach the heights of either profession, but fewer women choose to play on the lower slopes, as it were. Why? Part of the explanation must be an historical lack of opportunity for women but there may be other factors – such as style. We might hypothesise that many women have a distinct style of thinking compared to many men. Sherry Turkle and Seymour Papert have written that,
When we looked closely at programmers in action we saw formal and abstract approaches; but we also saw highly successful programmers in relationships with their material that are more reminiscent of a painter than a logician. They use concrete and personal approaches to knowledge that are far from the cultural stereotypes of formal mathematics.
They then suggest that many women, more than men, are likely to think in such ways.
Several intellectual perspectives suggest that women would feel more comfortable with a relational, interactive, and connected approach to objects [in contrast to] men with a more distanced stance, planning, commanding, and imposing principles on them.
[Turkle & Pappert 1990]
Turkle and Pappert imply both social and cognitive differences. Since women are generally more field-dependent than men it may be important that, ‘field-dependent secondary students experienced more mathematics an
xiety than did field-independent learners’ [Hadfield & Maddux 1988].
Academic studies of women, girls and maths typically focus on their confidence, or fear of success, on learned helplessness, or whether mathematics is seen as useful, plus, invariably, differences in spatial ability, which do show up though the situation is far more complex than the naive claim that, ‘Men visualise better than women!’ [Fennema & Leder 1990].
Let's approach the question from the other side: what is special about male mathematicians? They typically appear competitive, very ambitious and driven. Reuben Fine, a one-time challenger for the world chess championship who retired from chess to become a psychoanalyst, and Benjamin Fine, claim that the mathematician is a healthy narcissist for whom mathematical creativity is an effective defence against anxiety [Fine & Fine 1977].
Thanks for the compliment! It may well be that many (especially pure) mathematicians do find that mathematics is a successful psychological defence against anxiety or anomie. What could be more meaningful, beautiful, and yet self-contained (and isolating if that's what you want) than mathematics – apart from chess! That line of thought suggests that mathematics will be especially attractive to individuals with a certain style of psychological defence, which could explain why some people are so attracted to chess, or mathematics, while others are repelled.
Mathematics and abstract games: an intimate connection
Abstract games are an extraordinary cultural phenomenon, and so is the mathematics, as was said until quite recently, meaning both pure mathematics and all its many applications. Yet they are both embedded, together with traditional puzzles and mathematical recreations, in an aspect of society that has been there from its very origins. Let's return to an old theme that also could not be more modern.
Knots have been around from the dawn of civilisation, and earlier, since they can form naturally in the twisting of vines and creepers. The 5400-year-old Ice Man discovered in the Alps in 1991 was remarkably preserved with all his equipment, which included a leather quiver with fourteen arrow shafts, a belt-purse, leather clothing, and other items, all of them sewn or knotted [Turner & van de Griend 1996: 34].
Figure 20.2 Three knots with 3, 4 and 5 crossings
These drawings of the only knots with fewer than six crossings (Figure 20.2) already look abstract. Given a knot in a piece of string, you can copy it to another piece of string, or a rope, or even a ribbon or length of hair, because it is the abstract features that you copy, not the irrelevant details. Follow the rules – twist, put this end through the loop, push that loop through here, pull the other end through…and pull it tight – and the knot appears.
The rules for creating each knot are never arbitrary, nor are the rules for knitting or crocheting. As we have noted, there are only a limited number of basic types of knitting – or of tying a tie [Fink & Mao 2001]. This fact is a pointer to the presence of mathematics behind these everyday activities.
Figure 20.3 Figure of ocean plait
The ocean plait is a glorified endless knot (Figure 20.3).
The three stitches (Figure 20.4), taken from a Victorian book on sewing and sewing machines, are endless knots in a different sense. It is to be expected that knots can readily be created by machinery, repeating the same sequence of mechanical game-like moves as often as necessary.
Figure 20.4 Three machine stitches from Victorian book
Knots and stitches, knitting and crochet, braids and weaving – and string figures – all display a hidden side of traditional culture. They can all be analysed mathematically – and they have been recently – yet for thousands of years they were not seen as mathematics but simply passed on (and changed and developed – new knots are still occasionally discovered) from one generation to the next.
Knots have long been puzzling. Alexander was challenged to undo the Gordian knot – he famously ‘solved’ the puzzle by cutting it. He was obviously a poor mathematician. The great physicist Paul Dirac was smarter. One day, observing a colleague's wife knitting, he realised that there was a second way to form the stitches, which he then proceeded to explain to the astonished woman. He had re-invented purl and plain.
Gauss sketched several knots in his notebooks for 1794, and wrote a paper on electrodynamics which involved linked wires in space. His pupil Johann Listing wrote in 1847 the first book on Topology, in which discussed the theory of knots. [Turner and van de Griend 1996: x, 262] They are now exploited by chemists, molecular biologists and physicists as well as mathematicians.
The latest apparition in the rarefied regions of modern cosmology is a competitor to string theory called loop quantum gravity in which space time is a network of abstract links and elementary particles are braids in this fabric. Who would have thought that a schoolgirl's plaits could be linked to the microstructure of the universe [Castelvecchi 2006]?
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