Concepts of Modern Mathematics

Ian Stewart

• Editorial: Dover Publications
• Año: 2012
• ISBN:
  9780486134956
• Categoría: Mathematics
• Categoría: Algebra

Some years ago, `new math` took the country's classrooms by storm. Based on the abstract, general style of mathematical exposition favored by research mathematicians, its goal was to teach students not just to manipulate numbers and formulas, but to grasp the underlying mathematical concepts. The result, at least at first, was a great deal of confusion among teachers, students, and parents. Since then, the negative aspects of `new math` have been eliminated and its positive elements assimilated into classroom instruction.
In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts underlying `new math`: groups, sets, subsets, topology, Boolean algebra, and more. According to Professor Stewart, an understanding of these concepts offers the best route to grasping the true nature of mathematics, in particular the power, beauty, and utility of pure mathematics. No advanced mathematical background is needed (a smattering of algebra, geometry, and trigonometry is helpful) to follow the author's lucid and thought-provoking discussions of such topics as functions, symmetry, axiomatics, counting, topology, hyperspace, linear algebra, real analysis, probability, computers, applications of modern mathematics, and much more.
By the time readers have finished this book, they'll have a much clearer grasp of how modern mathematicians look at figures, functions, and formulas and how a firm grasp of the ideas underlying `new math` leads toward a genuine comprehension of the nature of mathematics itself.

Chapter 1

Mathematics in General

‘It is difficult to give an idea of the vast extent of modern mathematics’

– A. Cayley, in an address of 1883.

From the sudden conversion of our schools to ‘modern mathematics’ one might gain the impression that mathematics has lost control of its senses, thrown out all of its traditional ideas, and replaced them by weird and whimsical creations of no possible use to anyone.

This is not entirely an accurate picture. At a conservative estimate, most of the ‘modern mathematics’ now taught in schools has been in existence for over a century. In mathematics new ideas have developed naturally out of older ones, and have been incorporated steadily with the passing of time. But in our schools we have introduced a number of new concepts all at once, mostly without any discussion of how they relate to traditional mathematics.

Abstractness and Generality

One of the more noticeable aspects of modern mathematics is a tendency to become increasingly abstract. Each major concept embraces not one but many diverse objects, all having some common property. An abstract theory develops the consequences of this property, which may then be applied to any of the diverse objects.

Thus the concept ‘group’ has applications to rigid motions in space, symmetries of geometrical figures, the additive structure of whole numbers, or the deformation of curves in a topological space. The common property is the possibility of combining two objects of a certain kind to yield another. Two rigid motions, performed in succession, yield a rigid motion; the sum of two numbers is a number; two curves stuck end to end form another curve.

Abstraction and generality go hand in hand. And the main advantage of generality is that it saves work. It is pointless to prove the same theorem four times in different disguises, when it could have been proved once in a general setting.

A second feature of modern mathematics is its reliance on the language of set theory. This is usually no more than common sense in symbolic dress. Mathematics, particularly when it becomes more general, is less interested in specific objects than in whole collections of objects. That 5 = 1 + 4 is not terribly significant. That every prime number of the form ‘4n + 1’ is a sum of two squares is significant. The latter is an observation about the collection of all prime numbers, rather than about any particular prime number.

A set is just a collection: we use a different word to avoid certain psychological overtones associated with the word ‘collection’.¹ Sets can be combined in various ways to give other sets, in the same way that numbers can be combined (by addition, subtraction, multiplication,…) to give other numbers. The general theory of arithmetical operations is algebra: so we also can develop an algebra of set theory.

Sets have certain advantages over numbers, particularly from the point of view of teaching. They are more concrete than numbers. You cannot show a child a number (‘I am holding in my hand the number 3’), you can show him a number of things: 3 lollipops, 3 ping-pong balls. You will be showing him a set of lollipops, or ping-pong balls. Although the sets of interest in mathematics are not concrete – they tend to be sets of numbers, or functions – the basic operations of set theory can be demonstrated by means of concrete material.

Set theory is more fundamental to mathematics than arithmetic–although the fundamentals are not always the best starting point – and the ideas of set theory are indispensable for an understanding of modern mathematics. For this reason I have discussed sets in Chapters 4 and 5. The language of set theory is used freely thereafter, though I have tried not to use anything beyond very elementary parts of the theory. It would be wrong to overemphasize set theory per se: it is a language, not an end in itself. If you knew set theory up to the hilt, and no other mathematics, you would be of no use to anybody. If you knew a lot of mathematics, but no set theory, you might achieve a great deal. But if you knew just some set theory, you would have a far better understanding of the language of mathematics.

Intuition and Formalism

The trend to greater generality has been accompanied by an increased standard of logical rigour. Euclid is now criticized because he didn’t have an axiom to say that a line passing through a point inside a triangle must cut the triangle somewhere. Euler’s definition of a function as ‘a curve drawn by freely leading the hand’ will not allow the games that mathematicians wish to play with functions, and anyway it’s far too vague. (What is a ‘curve’?) One can go overboard for this sort of thing, verbal arguments can be replaced by a profusion of symbolic logic and checked for validity by a blind application of a standard technique. If carried too far (and in this case, enough is too much) this destroys understanding, instead of aiding it.

The demands for greater rigour are not just a whim. The more complicated and extensive a subject becomes, the more important it is to adopt a critical attitude. A sociologist, trying to make sense of masses of experimental data, will have to discard those experiments which have been badly performed, or whose conclusions are dubious. In mathematics it is the same. All too often the ‘obvious’ has turned out to be false. There exist geometrical figures which do not have an area. According to Banach and Tarski² it is possible to cut a spherical ball into six pieces and reassemble the pieces to form two balls, each the same size as the original. On grounds of volume, this is impossible. But the pieces do not have volumes.

Logical rigour provides a restraining influence which is of great value in dangerous circumstances, or when dealing with subtleties. There are theorems which most practising mathematicians are convinced must be true; but until someone proves them they are unjustified assumptions, and cannot be used except as assumptions.

Another place where one must be careful about one’s logic is when proving something impossible. What is impossible by one method may easily be performed by another, so very careful specifications are required. There exist proofs that quintic equations cannot be solved by radicals,³ or that angles cannot be trisected with ruler and compasses. These are important theorems, because they close off unproductive paths. But if we are to be certain that the paths really are unproductive, we must be very cautious with our logic.

Impossibility proofs are very characteristic of mathematics. It is virtually the only subject that can be sure of its own limitations. It has at times become so obsessed with them that people have been more interested in proving that something cannot be done than in finding out how to do it! If self-knowledge be a virtue, then mathematicians as a breed are saints.

However, logic is not all. No formula ever suggested anything on its own. Logic can be used to solve problems, but it cannot suggest which problems to try. No one has ever formalized significance. To recognize what is significant you need a certain amount of experience, plus that elusive quality: intuition.

I cannot define what I mean by ‘intuition’. It is simply what makes mathematicians (or physicists, or engineers, or poets) tick. It gives them a ‘feel’ for the subject; with it they can see that a theorem is true, without giving a formal proof, and on the basis of their vision produce a proof that works.

Practically everybody possesses some degree of mathematical intuition. A child solving a jig-saw puzzle has it. Anyone who has succeeded in packing the family’s holiday luggage in the boot of the car has it. The main object in training mathematicians should be to develop their intuition into a controllable tool.

Many pages have been expended on polemics in favour of rigour over intuition, or of intuition over rigour. Both extremes miss the point: the power of mathematics lies precisely in the combination of intuition and rigour. Controlled genius, inspired logic. We all know the brilliant person whose ideas never quite work, and the tidy, organized person who never achieves anything worthwhile because he is too busy getting tidy and organized. These are the extremes to avoid.

Pictures

In learning mathematics, the psychological is more important than the logical. I have seen superbly logical lectures which none of the audience understood. Intuition should take precedence; it can be backed up by formal proof later. An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true.

In subsequent chapters, I have tried to stress the intuitive side of mathematics. Instead of giving formal proofs I have tried to sketch the underlying ideas. In a proper textbook one should, ideally, do both; few texts achieve this ideal.

Some mathematicians, perhaps 10 per cent, think in formulae. Their intuition deals in formulae. But the rest think in pictures; their intuition is geometrical. Pictures carry so much more information than words. For many years schoolchildren were discouraged from drawing pictures because ‘they aren’t rigorous’. This was a bad mistake. Pictures are not rigorous, it is true, but they are an essential aid to thought and no one should reject anything that can help him to think better.

Why?

There are plenty of reasons for doing mathematics, and anyone reading this is unlikely to demand that the existence of mathematics be justified before he proceeds one page further. Mathematics is beautiful, intellectually stimulating – even useful.

Most of the topics I intend to discuss come from pure mathematics. The aim in pure mathematics is not practical applications, but intellectual satisfaction. In this pure mathematics resembles the fine arts – few people ask that a painting should be useful. (Unlike the fine arts it has generally accepted critical standards.) But the remarkable thing is that – almost despite itself – pure mathematics is useful. Let me give an example.

In the 1800s mathematicians expended a lot of energy on the wave equation; a partial differential equation arising from the physical properties of waves in a string or in fluid. Despite the physical origin, the problem was one of pure mathematics: nobody could think of a practical use for waves. In 1864 Maxwell laid down a number of equations to describe electrical phenomena. A simple manipulation of these equations produces the wave equation. This led Maxwell to predict the existence of electrical waves. In 1888 Hertz confirmed Maxwell’s predictions experimentally, detecting radio waves in the laboratory. In 1896 Marconi made the first radio transmission.

This sequence of events is typical of the way in which pure mathematics becomes useful. First, the pure mathematician playing about with the problem for the fun of it. Second, the theoretician, applying the mathematics but making no attempt to test his theory. Third, the experimental scientist, confirming the theory but not developing any use for it. Finally the practical man, who delivers the goods to the waiting world.

The same sequence of events occurs in the development of atomic power; or in matrix theory (used in engineering and economics); or in integral equations.

Observe the time-scale. From the wave equation to Marconi: 150 years. From differential geometry to the atomic bomb: 100 years. From Cayley’s first use of matrices to their use by economists: 100 years. Integral equations took thirty years to get from the point where Courant and Hilbert developed them into a useful mathematical tool to the point where they became usef...