Concepts of Demonstration

Abstract

The view that mathematics is theorem proving, hence the method of mathematics is the axiomatic method, and the view that mathematics is problem solving, hence the method of mathematics is the analytic method, lead to two different concepts of demonstration: axiomatic demonstration and analytic demonstration. This chapter highlights the limitations of axiomatic demonstration. It maintains that analytic demonstration is the basic concept of demonstration and that, in terms of this concept, it is possible to deal with the question of the depth of demonstrations and theorems. On this basis, the chapter characterizes a view of mathematics, called the heuristic view, in terms of the following features: mathematics is problem solving by the analytic method; mathematical objects are hypotheses human beings make to solve mathematical problems by the analytic method; and analytic demonstration is the basic concept of demonstration.