We give below his last chapter Aftermath:-
Bertrand Russell said that mathematics is the science in which we do not know what we are talking about, and do not care whether what we say about it is true. This paradox means in the first place that mathematicians are people who put together patterns of certain kinds. The patterns must be made up of something, but the ordinary mathematician does not usually concern himself about what the something really is. It may be different things in different cases or to different people, but whether it is so is a question for philosophers. As mathematicians we do not know what it is that we are talking about. As to not caring whether what we say is true, perhaps this means that many different kinds of primary axioms could form the starting points of mathematical systems. The mathematicians would only be concerned to follow out their consequences, not to enquire about the comparative validity of different sets of axioms.
In this book we have given elementary introductions to various branches of mathematics, but no attempt at a complete survey has been made. Algebra and geometry are to form the subjects of further volumes of this series, so that very little has been said about them here. Dynamics and subjects of that kind, usually known as applied mathematics, have only been mentioned casually. The main subject which we have dealt with is what mathematicians call analysis. This is a rather vague expression for those parts of mathematics in which the ideas of limit, variation, function, and so on, are uppermost. The experts in these subjects sometimes describe themselves as analysts. An analyst should be able to handle such things as integrals and infinite series just as well as if they were the simple expressions of elementary algebra. The expression " + ..." is not uncommonly used in mathematical writings to mean something which the writer proposes to ignore, in the hope that it does not really matter very much. Analysts also use this expression, but they should know, each time they use it, exactly what they mean by it.
Mathematics is a highly technical subject. If you take down a book from a mathematician's shelves and open it at random, it is very likely that on the first page which you read, you will not be able to understand anything at all. Still I hope that anyone who has read this book would feel that, even if he was reading a foreign language, it was not written in an unknown script. He might be able to form some idea of the sort of thing that was going on, even if he could not actually follow the details of the working. At present even a mathematician cannot usually follow the writings of other mathematicians without a special study of their particular subjects. The time when any one person could know the whole of mathematics is long past. The accumulated stock of mathematical knowledge is very large, and is still growing rapidly. All that any mathematician can do now is to concentrate on those topics which he finds specially interesting. In this way it is possible to reach the limits of knowledge on fairly narrow fronts, and to make progress there, while remaining comparatively ignorant of other parts of the subject.
It is impossible in a book of this kind to teach mathematical technique. There is no short cut to this. Anyone who wants to be able to solve mathematical problems must go through the ordinary routine. This is what mathematical text books are for. There are many good ones at the present time. Much of the fascination of mathematics lies in the scope it gives for the use of complicated techniques. One has to take trouble to learn to use them, but most people who have done so seem to have found it well worth while.
I find that many people, even those working in other branches of learning, do not know whether mathematics is, like science, active at the present day, or whether it is merely a routine which has come down to us from time immemorial. Actually, I suppose that more people are engaged in mathematical research now than at any previous time. Some topics become exhausted, of course; so far as I know, no one now discovers anything new about trigonometry, for example. At the same time new subjects open out before us, so that there always seems to be plenty to do.
Progress has been continuous for several hundred years now, and shows no signs of slackening. Probably, as long as there are mathematicians, some of them will be finding out something new about their subjects.
The question is sometimes asked, what have mathematicians discovered in modern times which would have been completely new and strange to the Greeks? One of the best answers to this is, the theory of functions of a complex variable. This is a subject which we have not been able to touch on here, but which has occupied a very large part of the time of mathematicians during the last century and a half. Briefly, this is what it is about. We have introduced here the idea of a function, and the idea of a complex number. Now put the two ideas together. We can define functions in which the independent variable is not a "real number" but a "complex number" (x, y), or x + iy, in the sense of Chapter X. Such a number (x, y) or x + iy is usually denoted simply by z. Then in formulas such as x2 or x3 we can just replace the x by z and think instead about z2 or z3. In this way we are led to consider functions of a complex variable. The theory of such functions contains many very remarkable theorems, particularly those due to the great French mathematician Cauchy (1789-1857). Cauchy's theory of functions of a complex variable would have surprised the Greeks very much, and surely it would have delighted them too.
Perhaps the most surprising thing about mathematics is that it is so surprising. The rules which we make up at the beginning seem ordinary and inevitable, but it is impossible to foresee their consequences. These have only been found out by long study, extending over many centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, Cauchy or Riemann; few careers can have been more satisfying than theirs. They have contributed something to human thought even more lasting than great literature, since it is independent of language.
It is sometimes supposed that mathematicians have extraordinarily remote and mysterious minds, or that they are people who can think quite easily about the inconceivable. This is not so. Some of the patterns which they make up are extremely complicated, so that it almost passes the power of the human mind to see whether they fit together correctly or not. But essentially their patterns are of the same sort as the simple ones of which we have given some examples here.
Mathematicians are often asked why they spend their lives trying to solve such curious problems. What good is it to know that every number is the sum of four squares? Why do you want to know about prime-pairs? What does it matter whether is rational or irrational?
A mathematician faced with these questions is in much the same position as a composer of music being questioned by someone with no ear for music. Why do you select some sets of notes and have them repeated by musicians, and reject others as worthless? It is difficult to answer except to say that there are harmonies in these things which we find that we can enjoy. It is true of course that some mathematics is useful. The invention of logarithms was welcomed by astronomers because it reduced the labour of their calculations. The theory of differential equations enables engineers to think about such things as the flow of water in pipes. The theory of linear operators enables the physicist to think about the atom. But the so-called pure mathematicians do not do mathematics for such reasons. It can be of no practical use to know that is irrational, but if we can know, it would surely be intolerable not to know. Pure mathematicians do Mathematics because it gives them an aesthetic satisfaction which they can share with other mathematicians. They do it because for them it is fun, in the same way perhaps that people climb mountains for fun. It may be an extremely arduous and even fatal pursuit, but it is fun nevertheless. Mathematicians enjoy themselves because they do sometimes get to the top of their mountains, and anyhow trying to get up does seem to be worth while.
I once heard a lecture by a physicist in which he derided what he thought were the futilities of pure mathematics; but then he referred to some theorem of pure mathematics which, fifty years after its discovery, had found an application in relativity, and this seemed to him little short of miraculous. But such cases are not uncommon. The ellipse was studied for centuries before it was found to be the orbit of a planet. To express astonishment at this is to mistake the nature of mathematics. Mathematicians are engaged in discovering and mapping out a real world. It is a world of thought, but it is of a kind on the basis of which the physical world is, to a certain extent, also constructed.