Three windows on mathematics: the axiomatic method, Lenat\u27s Program AM, and Lonergan\u27s cognitional analysis

Mathematics is seen differently by mathematicians, computer scientist, and philosophers. The perspectives and exigencies of each field determine the different views. Mathematicians are describing their own work. They generally accept axiomatic formalization as the best method to describe a mathematical system and to insure its accuracy. Chapter one describes what the formalization entails and also what is meant by a pro- position. Computer scientists bring a different perspective. While computers have been used for many years to perform mathematical computations, it is only recently that computer programs have begun to do mathematical exploration. In order to model this exploration, the programmer must have a working model of what mathematics is and how it is done. Chapter two examines one such program, Douglas Lenat\u27s AM, describing both the design and operation of the program and the conception of mathematics embodied in it. Philosophers begin with the experience of doing mathematics, either from introspection or from the description of mathematicians. They add to that descriptions given by physicists, historians, psychologists, etc., and the operations of what is called common sense. In all this they may try to find what is common to all the experience, and what is unique in it. Chapter three describes one such study, taken from the work of Bernard Lonergan. Lonergan develops a structure of the cognitional processes that is applicable to all disciplines. This structure, with its levels of experience, understanding, reflection, and decision, is found in mathematics, the scientific method, and in everyday experience. The chapter concludes with Lonergan\u27s reflections on symbols, proposition, and the source of mathematical objects. The fourth chapter compares the three perspectives to see what they have in common and how their different interests lead to different emphases. Lonergan\u27s cognitional structure provides the general framework for uniting the other two viewpoints. Lenat\u27s work corresponds most closely to the process of inquiry and understanding; axiomatic formalization to the process of reflection and verification. Lonergan\u27s description enabled this author to understand more clearly what the AM program was and was not doing, and to become aware of a possible limitation in particular implementations of artificial intelligence programs. While artificial intelligence programs and languages assign a single meaning to each term, human learning and experience modifies the meanings of words and expressions. This inflexibility may limit the operation of some programs, especially discovery programs like AM. Finally in chapter five we conclude by showing how the three perspectives complement and correct one another