Dover Publications, 1971

**Introduction**

A conversation I had a long time ago comes into my mind. One of our writers, a dear friend of mine, was complaining to me that he felt his education had been neglected in one important aspect, namely he did not know any mathematics. He felt this lack while working on his own ground, while writing. He still remembered the co-ordinate system from his school mathematics, and he had already used this in similes and imagery. He felt that there must be a great deal more such usable material in mathematics, and that his ability to draw from this rich source. But it was all, so he thought, quite hopeless, as he was convinced of one thing: he could never penetrate right into the heart of mathematics.

I have often remembered this conversation; it has always suggested avenues of thought to me and plans. I saw immediately that there was something to do here, since in mathematics for me the element of atmosphere had always been the main factor, and this was surely a common source from which the writer and the artist could both draw. I remember an example from my schooldays: some fellow students and I were reading one of Shaw's plays. We reached the point where the hero asked the heroine what was her secret by means of which she was able to win over and lead the most unmanageable people. The heroine thought for a moment and then suggested that perhaps it could be explained by the fact that she really kept her distance from everyone. At this point the student who was reading the part suddenly exclaimed: "That is just the same as the mathematical theorem we learnt today!" The mathematical question had been: Is it possible to approach a set of points from an external point in such a way that every point of the set is approached simultaneously? The answer is yes, provided that the external point is far enough away from the whole set:

I did not wish to believe the writer's other statement, namely that he could never penetrate right into the heart of mathematics, that for instance he would never be able to understand the notion of the differential coefficient. I tried to analyse the introduction of this notion into the simplest possible, obvious steps. The result was very surprising; the mathematician cannot even imagine what difficulties the simplest formula can present to the laymen. Just as the teacher cannot understand how it is possible that a child can spell c-a-t twenty times, and still not see that it is really a *cat*; and there is more to this than to a cat!

This again was an experience that caused me to do a great deal of thinking. I had always believed that the reason why the public was so ill-informed about mathematics was simply that nobody had written a good popular book for the general public about, say, the differential calculus. The interest patently exists, as the public snaps up everything of this kind that is available to it; but no professional mathematician has so far written such a book. I am thinking of the real professional who knows exactly to what extent things can be simplified without falsifying them, who knows that it is not a question of serving up the usual bitter pill in a pleasanter dish (since mathematics for most is a bitter memory); one who can clarify the essential points so that they hit the eye, and who himself knows the joy of mathematical creation and writes with such a swing that he carries the reader along with him. I am now beginning to believe that for a lot of people even the really popular book is going to remain inaccessible.

Perhaps it is the decisive characteristic of the mathematician that he accepts the bitterness inherent in the path he is traveling. "There is no royal road to mathematics", Euclid said to an interested potentate; it cannot be made comfortable even for kings. You cannot read mathematics superficially; the inescapable abstraction always has an element of self-torture in it, and the one to whom this self-torture is joy is the mathematician. Even the simplest popular book can be followed only by those who undertake this task to a certain extent, by those who undertake to examine painstakingly the details inherent in a formula until it becomes clear to them.

I am not going to write for these people. I am going to write mathematics without formulae. I want to pass on something of the feel of mathematics. I do not know if such an undertaking can succeed. By giving up the formula, I give up an essential mathematical tool. The writer and the mathematician alike realize that form is essential. Try to imagine how you could express the feel of a sonnet without the form of the sonnet. But I still intend to try. It is possible that, even so, some of the spirit of real mathematics can be saved.

One way of making things easy I cannot allow; the reader must not omit, leave for later reading, or superficially skim through, any of the chapters. Mathematics can be built up only brick by brick; here not one step is unnecessary, for each successive part is built on the previous one, even if this is not quite as obviously so as in a boringly systematic book. The few instructions must be carried out, the figures must really be studied, simple drawings or calculations must really be attempted when the reader is asked to do so. On the other hand I can promise the reader that he will not be bored.

I shall not make use of any of the usual school mathematics. I shall begin with counting and I shall reach the most recent branch of mathematics, mathematical logic.

**Contents**

- Playing with Fingers
- The "Temperature Charts" of the Operations
- The Parcelling Out of the Infinite Number Series
- The Sorcerer's Apprentice
- Variations on a Fundamental Theme

Postscript on Geometry with Measurement - We Go Through All Possibilities
- Colouring the Grey Number Series
- "I Have Thought of a Number"

- Diverging numbers
- Limitless Density
- We Catch Infinity Again
- The Line is Filled Up
- The Charts Get Smoothed Out
- Mathematics is One

Postscript about Waves and Shadows - "Write it Down" Elements
- Some Workshop Secrets
- "Many Small Make a Great"

- And Still There are Different Kinds of Mathematics

Postscript about the Fourth Dimension - The Building Rocks
- Form Becomes Independent
- Awaiting Judgement by Metamathematics

Postscript on Perception Projected to Infinity - What is Mathematics Not Capable of?