The Elements of Mathematics from a Modern Viewpoint I

This book covers all the mathematics taught in the US, from grade 4 to the end of high school, starting with Elementary Number Theory to Probability Theory and Statistics. The topics are listed on the front cover. Set Theory is taught early: in chapter 5 of the 14 chapters. It is then used systematically as the basic mathematical language. At the end of the nineteenth century, mathematics underwent a profound change, due to the advent of set theory, which was promoted by the German mathematician Georg Cantor. This change was essential. But Cantor's ideas were not readily accepted. In fact, they were opposed by prominent mathematicians of the time, such as Henri Poincare and Leopold Kronecker. Gradually, however, set theory became the fundamental language of practically all of mathematics. Set theory is useful even at the most elementary level, such as what you find in the pre-algebra books used in the U.S. But in many current elementary algebra books written for the schools, set theory is either ignored, or treated as a supplementary subject, without logical connection with other subjects. These books often seem to be written under the conception that they have to mention certain topics and briefly talk about them, so that the student would have encountered them at least once. The likely outcome of such a policy is that at least some of the students may quickly decide that mathematics is not something that can be understood. In addition, the succession of topics is typically haphazard, without any consideration of their difficulty or of the fact that a deep understanding of some topics is essential to understand other topics. You just have to remember those strange names and you are fine if you can solve the problems. The trouble with that policy is that because there is no discernible structure on which to hang the concepts, they are easily forgotten. Obviously, set theory should not be taught as a beginning subject. The right time to introduce the concepts of set theory is when they may help understanding fundamental mathematical concepts, such as: equivalence relations, order relations, functions, the coordinate plane, or the graphs of linear and other functions. Volume II of this book contains the solutions for most of the about 1000 problems proposed here.