1.1.
We are concerned to understand the motivation for the development of pure mathematics, and it will not do simply to point to aesthetic qualities in the subject and leave it at that. It must be remembered that there is far more excitement to be had from creating something than from appreciating it after it has been created. Let there be no mistake about it, the fact that the mathematician is bound down by the rules of logic can no more prevent him from being creative than the properties of paint can prevent the artist. … We must remember that the mathematician not only finds the solutions to his problems, he creates the problems themselves.

1.2.
One of the most common sources of difficulty to the student of mathematics is that a piece of work is very rarely presented to him in the form in which it was worked out by its originator. It has been condensed, polished up, and rearranged in a logical order, the ideas which gave it birth concealed like the works of a clock behind the clock-face. So does the mathematician appear as a super-mind because we cannot follow the trend of his thought.

How does the mathematician think? It would be difficult indeed to give a coherent answer to this question, but something may be done to dispel popular illusions. One might begin by saying that he must be possessed by an overriding curiosity. In mathematics it is almost more important to be able to ask questions than to be able to answer them. "Who dragged whom, how many times, in what manner, round the wall of what?" quoted Jeremy Stickles in Lorna Doone: surely the propounder of this question was a bit of a mathematician.

1.3.
One of the distinguishing characteristics of the true mathematician is that, out of the multitude of questions that occur to him, he is able to select the ones that are worth answering. We have seen some of the preferences which guide him in his choice, the desire for generality, the pursuit of rigour and precision. The real criterion is whether the study of the problem is likely to be fertile in fresh ideas and to give birth to elegant mathematics: but this is a little like the advice to the Snark-hunters that they should recognise the creature by its taste. The great mathematician, like the great performer in most human activities, has a kind of intuition in such matters.

It is nevertheless possible to single out one or two typical questions that he may ask. In modern mathematics, for instance, we have learnt that when a problem has defied repeated attempts at solution we ought to ask, can it be proved that the problem is insoluble.

1.4.
It is rarely the case that the proof of the impossibility of solving a problem finally ends the matter. Out of the ashes of the old problem a new one arises. We know from the Fundamental Theorem of Algebra that the quintic equation should have five solutions, and if the solutions cannot be expressed by means of the ordinary operations of algebra, how can they be expressed? If Euclid's parallel postulate cannot be proved, what happens in a geometry in which it is denied?

This last question is typical of the modern axiomatic mathematics. If the axioms cannot be deduced from each other, what happens if we change one of them? For instance, what happens if, instead of assuming the axiom $ab = ba$, we substitute for it $ab = -ba$? This is a question which the pure mathematician may legitimately put to himself, but it so happens that the new sort of algebra which results, has important applications in applied mathematics. ... We here see how the exacting and, as it might seem, tiresome process of delving down to the roots of the subject can bear fruit in the form of new and imaginative branches of mathematics.

1.5.
There is perhaps not such a gap between mathematical and other forms of thought as exists in popular imagination. Of course, the great in all walks of life have a way of their own, and it would be presumptuous to attempt to penetrate further and say what it is that distinguishes the truly great mathematical mind. Even our own mental processes are largely a mystery to us. We cannot say what happens to us in the moment of enlightenment, or the moment when against probability we notice the clue which turns out to be the essential link in the chain. All that we can say is that, if we have thought about a problem, particularly if we have asked ourselves sensible questions about it, the solution will often come to us easily when we return to it after a period of leisure, or even will flash upon us at a moment when we are occupied with other things. And this is a phenomenon which appears to be as common in the great discoveries of mathematics and science as in our attempts to remember our next-door-neighbour's sister's married name.

1.6.
Mathematical notation is a language, and something is to be learnt from a comparison with the language of everyday speech. In the development of language, when an idea becomes important, a new word is coined to stand for it. The new word is a shorthand term for the group of words which would previously have been used, and the language gains in conciseness. The same sort of thing happens in mathematics. ... But mathematical notation is more subtle than the gradual accretion of more and more symbols.

1.7.
The human mind is incapable of assimilating vast numbers of unrelated facts. We become interested in things when we can weave a web of connections between them and find the general properties that underlie them. The scientist seeks the same law to explain the fall of an apple and the motion of a planet. The historian studies the letters of a fifteenth-century family because he thinks them typical of their period. Even great literature achieves its fascination because it has application far beyond its allotted span of space and time; we are not interested in Hamlet as a prince of Denmark but as a human being. It is in mathematics that we see at its purest this urge to see the general beyond the particular.

2.1.
Manages to convey to the outsider a more lively picture of the approach and the methods of the author's subject than could reasonably be expected.

2.2.
Mathematics in some form or another is used by almost everyone; but, nevertheless, the mathematician is still regarded by a large number of people with great respect and very often with awe, particularly so when the mathematician claims to be doing research in mathematics. It is to this awestricken section of the community that Mr A Read, lecturer in mathematics in the University of St Andrews, addresses his remarks in A Signpost to Mathematics. Whether or not he succeeds in making clear what prompts the mathematician to think of, and to juggle with, more and more abstruse problems is somewhat doubtful; but he has certainly written an interesting, stimulating and very readable book.