On the White Box Integration of Computer Algebra Algorithms into a Deduction System

Introduction Mathematics is one of the oldest sciences, it is as old as human culture and can be found in almost every aspect of everyday life. Although its pure foundations are abstract, mathematics finds its application in the description of real world problems. In fact real world problems today are not what they used to be 3000 years ago, and while the world has changed, the appearance of mathematics has as well. From the very beginnings, which might have been e.g. calculating with stones, each stone representing a cow [30], the appearance of mathematics grew more and more complex and abstract. During the centuries sciences as geometry, astronomy and physics got into the focus of mathematical interest and drove mathematics itself to the development of a wide variety of theories, concepts, representations and techniques. While mathematics turned out to be an appropriate tool to handle various scientific problems, every new challenge fertilised the development of mathematics itself, and this is what characterised the role of mathematics for a long time: a science that founded its existence upon its application within other sciences. However, the sheer plentitude of emerging mathematical ideas as well as these ideas becoming more and more abstract created a strong need to find general ways to properly describe mathematical problems. This has led not only to an upcoming formalism in mathematics, but also initiated a development which had much deeper consequences, namely the negotiation of mathematics with itself. In the 17th century Locke's 'Nihil est in intellectu, quod non prius fuerit in sensu' (nothing is in the mind, that has not been in the senses before) was countered by Leibniz' 'Nisi intellectus ipse' (except for the mind itself) [48]. Although Leibniz was referring to the human mind in a philosophical sense, these words also illustrates the change of attitude towards doing mathematics starting in these times. Now mathematics was no longer only a tool to challenge problems from other scientific domains, but there was also a consciousness for the pure innate nature of mathematics decoupled from its application. Leibniz started to work not only on the application of mathematics, but was also researching the principles of mathematical reasoning and tried to model methods of reasoning by application of a calculus, and his call 'Calculemus!' (Let us compute!) [47] was the first step towards the idea of reasoning by means of computation. Two centuries later the work of …