A Formal Background to Mathematics 2a: A Critical Approach to Elementary Analysis

This volume is devoted to a treatment of some of the fundamentals of elementary analysis (including what is usually described as "calculus") and a critique of the expositions to be found in typical high school and other conventionally informal texts. Attention is concentrated upon elementary analysis, since it is the major portion of mathematics at the high school stage and, more importantly, is the portion fraught with more of the fundamentally difficult concepts (limits and convergence in various guises). Other topics, such as fragments of linear algebra, probability and statistics, sometimes treated in high school texts, have been ignored on tile grounds that they are not so heavily laden with basic mathematical difficulties (though some exhibit problems which are not basically mathematical). Also omitted from the main text is all consideration of the use and abuse of calculators as an alleged aid in "getting across" the concepts of limit and calculus (but see the Epilogue (iv)). Another omission is the absence of any reference to t;le poss i bil it) (taken up by some writers) of adopting the concepts and techniques of non-standard analysis with the same end in view. (This is a bold and interesting idea; but its execution seems to me to demand extreme caution, if it is not to back-fire and foster more misunderstanding than it eliminates.)