Towards computer aided mathematics

In his autobiography1 Bertrand Russell characterizes mathematics as follows: “It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect and yet true”. Actually the perception of mathematical research as an artistic discipline has a long history and a significant number of today’s mathematicians share this view. In contrast, however, Russell himself dedicated large parts of his life to defending logicism, that is, the view that mathematics is reducible to logic, and together with Alfred Whitehead he proposed in his influential Principia Mathematicae an axiomatic system to build all mathematics upon. The two viewpoints—mathematics as an art versus logicism—may appear contradictory at first. They are not though, if we separate the different aspects of mathematical practice. The invention and shaping of new mathematical structures based on mathematical knowledge as well as on aesthetic and social criteria or the discovery of the essential arguments in complex mathematical proof, for instance, are activities that typically require human ingenuity. On the other hand the verification and grounding of already pre-structured and established chunks of mathematics in foundational systems or the search for simple (sub-)proofs are examples of tasks that often require far less ingenuity. Some overoptimistic and improperly reflected predictions in the field of artificial intelligence and automated reasoning on the mechanization and automation of mathematics have unfortunately generally questioned the role of human ingenuity in mathematics without making the above distinction clear. Unlike in chess, however, where human intelligence is no longer dominating over machine intelligence, it seems to me that human ingenuity will remain dominant in many essential aspects in mathematics research and education for a long time to come. Taking our distinction above into account this does not mean, however, that there is no need for assistance systems for mathematics and Russell would presumably