For Those Who Think Mathematics Dreary

Problems and Sums

Skipping the details, we’ll just note that this is what is called a numeric example. A problem is genuine when it takes creative effort to solve it. In fact there are two categories of problems in mathematics: those that are solved by technique and those that require insight. The category into which a problem falls depends on what one knows. In this sense what is a problem for some is for others a routine exposition of a logically connected sequence of reasoning and calculation.

The transformation of a problem that used to take insight into one that only requires technique can be a considerable step forward.

Some nineteenth-century textbooks talk of division as an arithmetic operation that takes a head, a fact that evokes a smile of condescension on the lips of many. If it is no longer an intellectual art, but an activity accessible to children, it is because an algorithm has been found—a kind of recipe, a breakdown of the problem into a sequence of simple actions. Likewise, if certain aspects of musical composition are algorithmised, some musical problems that used to require insight shall become a matter of technique. It’s a different story that people who have invested serious effort into the development of this kind of insight may not be enthusiastic about the possible algorithmisation of some of its elements. They may feel that no algorithm could embrace or replace their insight. According to some scholars, all problems (in mathematics or the other arts) can be solved by appropriate technique and if there are problems for which none is known as yet, that is due to gaps in our system of knowledge—a mere question of time, though it may be a long time. Others maintain that some problems will simply never be solved, however much time and perseverance we put into attacking them.

Problem solving in the technical sense isn’t really the solving of problems. It is what experts do when they don’t know what to do, that is, it is knowing how to handle situations for which there is no recipe. This is where imagination comes in. Not for nothing did the great mathematician David Hilbert say of one of his students: ‘He did not have enough imagination and therefore became a poet.…’ Another famous scholar, Godfried Harold Hardy, describes mathematicians as people who like painters or poets are makers of patterns, except that theirs are more permanent, being made with ideas. Mathematics used to occupy a central place among the arts, and today most people think that it is the art of doing sums. That’s hardly their fault though. With so many reforms in mathematical education worldwide, experts state that if the fine arts were taught in the same way, they would be reduced to studying techniques for clipping stone and mixing paints.

When in the seventeenth century the famous number theorist Pierre de Fermat wrote in the margin of a page in Diophantus’ Arithmetica that he had a proof of the insolubility of the equation xⁿ + yⁿ= zⁿ for n > 2 in integer numbers, but the margin was too small to contain it, few could have realised what impact his remark would have on the development of mathematics for the next 350 years. Today we can’t be sure whether Fermat did in fact have a solution of the problem, which was to become known as Fermat’s Great Theorem. What matters is that Fermat’s Great Theorem stimulated mathematical thinking and research for several centuries. Because of the accessible statement and the huge prize promised to the first solver, many amateurs have also tried their hand at this problem. In 1993 Andrew Wiles, professor of mathematics at Princeton University, announced a proof of Fermat’s challenge during a lecture at Cambridge. Now this is a genuine problem. But it’s not the volume of the solution, taking up about 200 pages as it does, that makes Fermat’s Great Theorem into a problem. It isn’t its difficulty either. Raising the number 2001 to the 2001^th power, especially by hand, isn’t easy at all, but that difficulty is of a technical nature. The difficulty of genuine problems lies in finding an approach, organising the premises, using the statements and facts already known and proven, as well as ordering them into a logical sequence.

The example in the picture Counting in Our Heads and Fermat’s Great Theorem are the two extremes of what the wide public is accustomed to calling a problem. This doesn’t mean that mathematics is in the middle, though. Nor does it mean that one has to solve problems or become a mathematician in order to discover the magic of mathematics. We’ll tell you about that magic. Just read on.

Ptolemy and Cleopatra

It seems that each Swahili word is composed of two parts—one to show the size (e.g., ki- ‘little’) and another to express the basic concept (e.g., -buzi ‘goat’). The same is true of the English expression little goat, but in goat nothing expresses the default size, as the m- in Swahili mbuzi appears to do.

Could we go on filling in the cells in the tables? The guess that mto is ‘river’ is correct. We’d be wrong about kito and kitu though—these words mean respectively ‘gem, jewel’ and ‘thing, something’; ‘little river’ is kijito and ‘little person, dwarf’ is kijitu. This is in accordance with a rule that is not reflected in the problem.

Was there no less laborious way to find the answer? There are dictionaries of Swahili, as well as people who know the language. It isn’t so with all problems, however. When for example Jean-François Champollion applied the same method to decipher the Egyptian hieroglyphic script, he accomplished something that no one had known how to do for two millennia. For a start he compared two names, one () from the Rosetta Stone, the other () from the Philae Obelisk, which appeared to belong respectively to Ptolemy and Cleopatra, judging by the Greek texts that accompanied the two hieroglyphic inscriptions. Champollion conjectured that the recurring characters corresponded to identical sounds, hence also to identical letters in the Greek form of the names—for example the reclining lion, which is in fourth position in Ptolemy and in second position in Cleopatra, was to be read as l, the two eagles in Cleopatra as a and so on.

What kind of problems are these actually, linguistic or mathematical ones? The former, to be sure, because they’re about linguistic objects—the structure, meaning and written representation of words of human languages. But the latter too—what is mathematics all about, if not the discovery of structures, regularities and correspondences?

It may not be a straightforward matter to find a correspondence in a given context. Here are two problems offered by Douglas Hofstadter, a leading researcher in artificial intelligence:

What in 12344321 corresponds to 4 in 1234554321?

Figure 1

Figure 2

These lines are called Koch curves of the first, second, third and fourth generation, respectively. They belong to a more general class of curves known as fractals.

And here are a square and an octagon whose edges are replaced by fifth-generation Koch curves:

Figure 3

Popular Bulgarian embroidery! This was the last place you ever expected to encounter mathematics, was it not?…

Amorous Turtles

The problem can be solved in a variety of ways (including some that apply knowledge of what is called higher mathematics), but one can get a good idea of the movement by the use of a relatively simple computational model which employs turtle robots. By commanding each turtle to turn towards the object of its passion and move one step forwards and repeating this sufficiently many times, we get the following picture:

Figure 4

The path each turtle walks is a part of a famous curve known as a logarithmic spiral. If we connect the consecutive positions of the turtles by segments, we shall notice that they are always in the corners of a square whose side shrinks with the speed of the movement of the turtles:

The logarithmic spiral occurs in nature—it is the shape of the shell of the oceanic snail Nautilus (Figure 6). We also use it in a more prosaic (but still important) context—when plotting the growth of a deposit with fixed compound interest (Figure 7).

Figure 6	Figure 7

When constructing a computational model of a problem, we usually mean to solve an entire class of related problems, not just the one at hand. It is easy to vary the starting conditions. For example, the turtles may love one another crosswise (the first loving the third, the second the fourth). Or we can get one of them to hate its neighbour and flee from it (Figure 8). As one sees, that makes it a leader—all the others follow it.… We can also vary their number (Figure 9).

Figure 8	Figure 9

What is more, the interesting graphics obtained by varying the starting conditions suggest the idea that this model can serve to generate computational variations to the theme of Los Angeles (Figure 10)—a picture by the painter Victor Vasarely, whose pictures give an exceptional illusion of three-dimensionality. We place six turtles in the corners of a regular hexagon and connect the positions of the turtles after every beat by segments, alternating the colour of the trace (Figure 11). From there on we can create variants by modifying one parameter or the other in the initial setting of the problem (for example the strength of the attraction—Figure 12). Finally, we can locate the turtles in randomly chosen points of the screen and let them go (Figure 13). A sufficient number of experiments can even produce realistic images.

Figure 10	Figure 11
Figure 12	Figure 13

We hope that with the right attitude to the teaching of mathematics, when digital technologies are used to stimulate the spirit of discovery, students will come to see mathematics in a new way—as an area in which interesting experiments can be made and hypotheses formulated. Even if they happen to reinvent the bicycle, the students may feel the joy of the process of invention itself and acquire habits of creative thinking.

They may never become so skilled in mathematics as Monet, Mondrian and Vasarely were in painting, but at least they will have a chance of appreciating mathematics as the art it is. No lesser thing, that.