Launch not beyond your depth, but be discreet,
And mark that point where sense and dullness meet.A. Pope (1966: An Essay on Criticism)
The writer of any book dealing with mathematics who wishes to reach a broad audience invariably faces a dilemma: How to describe the mathematics involved. No matter how well motivated the intervening notions, nor how lengthily described, the question that eventually will pose itself is what to do regarding proofs.
Working mathematicians are generally reluctant to dispense with them, and I am no exception. For, on the one hand, a proof of a statement shows its necessity, its truth with respect to an underlying collection of assumptions. And, on the other hand, in doing so, it usually conveys an intuition on the nature of the objects occurring in the statement. This intuition is of the essence. It decreases the confusion that the alternation of definitions and statements in the mathematical discourse naturally creates.
Occasionally, however, the understanding afforded by a proof does not compensate for the effort of its reading. This may be so because one already has a form of the intuition mentioned above (and would, therefore, feel annoyed by having to “prove the evident”) or because the proof is too involved and fails to convey any intuition. In these cases the task of following the proof's details becomes boring.
In this trade-off between boredom and confusion different readers find different solutions by choosing subsets of proofs to be read that best suit their circumstances. To make these choices possible, this book gives proofs for many of its (mathematical) statements. To further make it easier, some observations are now given.