What Is Mathematics? is one of the great classics, a sparkling collection of mathematical gems, one of whose aims was to counter the idea that "mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician." In short, it wanted to put the meaning back into mathematics. But it was meaning of a very different kind from physical reality, for the meaning of mathematical objects states "only the relationships between mathematically 'undefined objects' and the rules governing operations with them." It doesn't matter what mathematical things are: it's what they do that counts. Thus mathematics hovers uneasily between the real and the not-real; its meaning does not reside in formal abstractions, but neither is it tangible. This may cause problems for philosophers who like tidy categories, but it is the great strength of mathematics - what I have elsewhere called its "unreal reality." Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely in either.
I first encountered What Is Mathematics? in 1963. I was about to take up a place at Cambridge University, and the book was recommended reading for prospective mathematics students. Even today, anyone who wants an advance look at university mathematics could profitably skim through its pages. However, you do not have to be a budding mathematician to get a great deal of pleasure and insight out of Courant and Robin's masterpiece. You do need a modest attention span, an interest in mathematics for its own sake, and enough background not to feel out of your depth. High-school algebra, basic calculus, and trigonometric functions are enough, although a bit of Euclidean geometry helps.
One might expect a book whose most recent edition was prepared nearly fifty years ago to seem old-fashioned, its terminology dated, its viewpoint out of line with current fashions. In fact, What Is Mathematics? has worn amazingly well. Its emphasis on problem-solving is up to date, and its choice of material has lasted so well that not a single word or symbol had to be deleted from this new edition.
In case you imagine this is because nothing ever changes in mathematics, I direct your attention to the new chapter, "Recent Developments, " which will show you just how rapid the changes have been. No, the book has worn well because although mathematics is still growing, it is the sort of subject in which old discoveries seldom become obsolete. You cannot "unprove" a theorem. True, you might occasionally find that a long-accepted proof is wrong-it has happened. But then it was never proved in the first place. However, new viewpoints can often render old proofs obsolete, or old facts no longer interesting. What Is Mathematics? has worn well because Richard Courant and Herbert Robbins displayed impeccable taste in their choice of material.
Formal mathematics is like spelling and grammars matter of the correct application of local rules. Meaningful mathematics is like journalism-it tells an interesting story. Unlike some journalism, the story has to be true. The best mathematics is like literature-it brings a story to life before your eyes and involves you in it, intellectually and emotionally. Mathematically speaking, What Is Mathematics? is a very literate work. The main purpose of the new chapter is to bring Courant and Robins?stories up to date-for example, to describe proofs of the Four Color Theorem and Ferrnat's Last Theorem. These were major open problems when Courant and Robbins wrote their masterpiece, but they have since been solved. I do have one genuine mathematical quibble (see ? of "Recent Developments"). I think that the particular issue involved is very much a case where the viewpoint has changed. Courant and Robins?argument is correct, within their stated assumptions, but those assumptions no longer seem as reasonable as they did.
I have made no attempt to introduce new topics that have recently come to prominence, such as chaos, broken symmetry, or the many other intriguing mathematical inventions and discoveries of the late twentieth century. You can find those in many sources, in particular my book From Here to Infinity, which can be seen as a kind of companion piece to this new edition of What Is Mathematics?. My rule has been to add only material that brings the original up to date-although I have bent it on a few occasions and have been tempted to break it on others.
What Is Mathematics?
Unique.
Ian Stewart
Coventry, June 1995
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