Requirements for Building upLarge Mathematical Knowledge

Diierent requirements for the representation of the factual knowledge for the use in mechanised reasoning systems are represented. In particular, we distinguish diier-ent types of knowledge: axioms, deenitions (for introducing concepts like \set" or \group") and theorems (for relating the concepts) for formulating concepts and their relationships, as well as proofs, examples and counterexamples. The consistency of mathematical knowledge bases cannot be proved in general, but it is possible to restrict the points where inconsistencies may be imported to very few cases, namely to that of axioms. Deenitions and theorems should not lead to any inconsistencies because deenitions form conservative extensions and theorems are proved to be consequences. In order to build up large knowledge bases, diierent formalisms and the transitions between theses should be supported. 1 Requirements Let us rst motivate why standard logic, rst-order as well as higher-order logic although being a main means for the representation of mathematical knowledge, is in itself insuucient for such a representation. One main point is that the basic notion of logic is that of a formula, but that in mathematics diierent kinds of formulae are distinguished, in particular one distinguishes between axioms, deenitions, and theorems and these classes are further subdivided (for instance, the theorems in main theorems, theorems, lemmata, corollaries). In addition, in mere logic a knowledge base consists of an unstructured set of formulae, whereas text books are well-structured and mathematicians spend a lot of time in the nal presentation of the mathematical content. Building up a mathematical knowledge base is very diierent from just adding formulae to an existing set. For instance, a very important constraint for deenitions is, that all concepts which are used in the deenition|with the exception of the deenien-dum of course|must already be known. Analogously all concepts in theorems must be known. But even if all concepts are known, a deenition has to fulll further extra-logical requirements , for example, it is normally given in a form as abstract as possible. Of course, not only formulae play a role in mathematics, but mathematical knowledge incorporates proofs, examples, and counterexamples as well. A knowledge base has to support these kinds of knowledge as well. An important diier-ence between a logical and a mathematical representation of knowledge is given by the fact that mathematical knowledge bases are always assumed to be consistent, even if that cannot be proved. Which requirements should a knowledge representation formalism for …