The language of mathematics

Abstract

Pure mathematics is concerned with the exploration of mathematical concepts arising initially from the study of space and number. In order to capture and communicate mathematical ideas we must make statements about mathematical objects and much mathematical activity can be described as the formulation of mathematical statements and then the determination of whether or not such statements are true or false. It is important to be clear about what constitutes a mathematical statement and this is considered in this first chapter. We begin with simple statements and then examine ways of building up more complicated statements. Mathematical statements It is quite difficult to give a precise formulation of what a mathematical statement is and this will not be attempted in this book. The aim here is to enable the reader to recognize simple mathematical statements. First of all let us consider the idea of a proposition . A good working criterion is that a proposition is a sentence which is either true or false (but not both) . For the moment we are not so concerned about whether or not propositions are in fact true. Consider the following list. (i) 1 + 1 = 2. (ii) π = 3. (iii) 12 may be written as the sum of two prime numbers. (iv) Every even integer greater than 2 may be written as the sum of two prime numbers. […]