The Methodology of Mathematics | by Ronald Brown and Timothy Porter

• Introduction

We start with some general questions to which we believe it is helpful for students to be able to formulate some kind of answers. The question for teachers of mathematics at all levels is to what extent, if at all, the training of mathematicians should involve professional discussion of, and assessment in, possible answers to these questions, such as those given or suggested here.

• Some basic issues for mathematicians

1. Is mathematics important? If so, for what, in what contexts,and why?
2. What is the nature of mathematics, in comparison with other subjects?
3. What are the objects of study of mathematics?

• What is the importance of mathematics?

It is not generally recognised how much of a part mathematics plays in our daily lives. Some of the mathematics is of course quite old: every day we use numbers, graphs, addition and multiplication. It is easy to forget that the invention of these was at one time a great discovery. The replacement of Roman numerals by Arabic numerals, and so the possibility of a good bookkeeping system, is said to have led to the prosperity of Venice in the 14th century. It is also of interest here to note the importance of pedantry in mathematics. A key aspect of the Arabic system is its use of the number zero. At first it seems absurd to count the number of objects in an empty box. The surprise is how essential this is for an adequate numeration system, in which the number 0 is used as a place marker. The lack of this concept of zero held up the progress of mathematics for centuries.

On a higher level, without the mathematics of error correcting codes we would not have had the beautiful pictures of Jupiter from the Voyager II. This mathematics is also essential in many aspects of telecommunications and of computers, and in particular for CD players. There is an amusing story about this last application [7]. Negotiations between Sony and the Dutch company Philips about the standards for CD were held by top management. The Japanese considered Philip's proposal for error correction inferior to theirs, and in the end the Japanese proposal was accepted. Back in Eindhoven, the embarrassed managers called in their science directors to declare that the company did not have sufficient expertise in this area called "coding theory" and to find out where in Europe the real experts could be found. To their dismay, the answer was: "in Eindhoven!", in the person of the Dutch number theorist Van Lint!

Without the mathematics of cryptography, there would not be possible the current level of electronic financial transactions crossing the world, and involving billions of dollars. Currently, the mathematics of category theory, a theory of mathematical structures, is being used to give new insights into future logics and algebras for the design of the next generation of programs and software.

The enormous applications of mathematics in engineering, in statistics, in physics, are common knowledge. It is also imagined that the role of mathematics is being taken over by the use of supercomputers. It is not so generally realised that these supercomputers are the servants of mathematical and conceptual formulations: the electronics is marvellous in that it does the calculations so quickly and accurately. For example, body scanners are an application, a realisation, of a piece of 19th century mathematics expressing how to reconstruct a solid object of varying density from views through it of an X-ray, where the only measurement is the change of intensity as the ray passes through the body, for a large number of varying positions of the ray. The theories of the big bang, of fundamental particles, would not be possible without mathematics.

• What is the nature of mathematics?

There is here a mystery. The Nobel prize-winner E. Wigner has written a famous essay "The unreasonable effectiveness of mathematics in the natural sciences" [8]. For us, the key word is "unreasonable". He is talking about the surprise that the use of mathematics is able to give predictions which are in accord with experiment to the extent of nine significant figures. How is such astonishing accuracy possible?

It seems likely that a full "explanation" of the success of mathematics would need more understanding of language, of psychology, of the structure of the brain and its action, than is at present conceivable. Even worse, the development of such understanding might need, indeed must need, a new kind and type of mathematics. It is still important to analyse the scope and limitations of mathematics. It is also reasonable that such an analysis should be a necessary part of the education and assessment of a student of mathematics.

• What are the objects of study of mathematics?

This has already been answered to some extent. Mathematics does not study things, but the relations between things. A description of such a relation is what we mean by a "concept". Thus we talk about the distance between towns, and might feel this is less "real" than the towns themselves. Nonetheless, relations between things, and our understanding of these relations, is crucial for our operation in and interaction with the world. In this sense, mathematics has the form of a language. It must be supposed that our ability to operate with concepts, with relationships, had and maybe continues to have an evolutionary value.

It is also curious in this respect that the achievements of mathematics are generally held by mathematicians to be the solution of some famous problem. Certainly such a solution will bring to the solver fame and fortune, or at any rate a certain fame within the world of mathematicians. Yet the history of mathematics and its applications shows that it is the language, methods and concepts of mathematics which bring its lasting value and everyday use. We have earlier mentioned some examples of this. At a more advanced level, we can say that without this language, for example that of groups and of Hilbert spaces, fundamental particle physics would be inconceivable.

Some of the great concepts which have been given rigorous treatments through this mathematicisation are:
number, length, area, volume, rate of change, randomness, computation and computability, symmetry, motion, force, energy, curvature, space, continuity, infinity, deduction.

Very often the problem to make some mathematics is, in the words of a master of new concepts, Alexander Grothendieck, "to bring new concepts out of the dark" [6]. It is these new concepts that make the difficult easy, which show us what has to be done, which lead the way.

More important is the way mathematics deals with and defines concepts, by combining them into mathematical structures. These structures, these patterns, show the relations between concepts and their structural behaviour. As said before, the objects of study of mathematics are patterns and structures.