Representation of Knowledge in a Geometry Machine E. W. Elcock

Department of Computer Science University of Western Ontario PART 1 In their book Mathematics and Logic Kac and Ulam (1971) comment: "The point of view as it has evolved through centuries is that one need not know what things are as long as one knows what statements about them one is allowed to make. Hilbert's famous Grundlagen der Geometrie begins with the sentence: 'Let there be three kinds of objects; the objects of the first kind shall be called "points", those of the second kind "lines", and those of third "planes". That is all, except that there follows a list of initial statements (axioms) that involve the words "point', "line" and "plane", and from which other statements involving those undefined words can now be deduced by logic alone. This permits geometry to be taught to a blind man and even to a computer!" Leaving aside the attitude implicit in Kac & Ulam's use of the word'even' in the phrase even to a computer', it has become clear that programs to prove theorems in ...