Mathematical Theory Exploration

Abstract

Mathematics is characterized by its method of gaining knowledge, namely reasoning. The automation of reasoning has seen significant advances over the past decades and, thus, the expectation was that these advances would also have significant impact on the practice of doing mathematics. However, so far, this impact is small. We think that the reason for this is the fact that automated reasoning so far concentrated on the automated proof of individual theorems whereas, in the practice of mathematics, one proceeds by building up entire theories in a step-by-step process. This process of exploring mathematical theories consists of the invention of notions, the invention and proof of propositions (lemmas, theorems), the invention of problems, and the invention and verification of methods (algorithms) that solve problems. Also, in this process of mathematical theory exploration, storage and retrieval of knowledge plays an important role. The way how one proceeds in building up a mathematical theory in successive, well structured, layers has significant influence on the ease of proving individual propositions that occur in the build-up of the theory and also on the readability and explanatory power of the proofs generated. Furthermore, in the practice of mathematical theory exploration, different reasoning methods are used for different theories and, in fact, reasoning methods are expanded and changed in the process of exploring theories, whereas traditional automated reasoning systems try to get along with one reasoning method for large parts or all of mathematics.