TABLE OF CONTENTS In this chapter we explain the scope and guiding philosophy of the present book, we try to make clear its logical structure and the role which Maple plays in the book. We also wish to orientate the readers on the logical structure which forms the basis of this book.
Mathematics can be compared to a cathedral. We wish to visit a small part of this cathedral of human ideas of quantities and space. We wish to learn how mathematics can be built. Mathematics spans a very wide spectrum, from the simple arithmetic operations a pupil learns in primary school to the sophisticated and difficult research which only a specialist can understand after years of long and hard postgraduate study. We place ourselves somewhere higher up in the lower half of this spectrum. This can also be roughly described as where University mathematics starts. In natural sciences the criterion of validity of a theory is experiment and practice. Mathematics is very different. Experiment and practice are insufficient for establishing mathematical truth. Mathematics is deductive, the only means of ascertaining the validity of a statement is logic. However, the chain of logical arguments cannot be extended indefinitely: inevitably there comes a point where we have to accept some basic propositions without proofs. The ancient Greeks called these foundation stones axioms, accepted their validity without questioning and developed all their mathematics therefrom. In modern mathematics we also use axioms but we have a different viewpoint. The axiomatic...
