A view on the future of symbolic computation

Since approximately 1960, symbolic computation added algebraic algorithms (polynomial algorithms, simplification algorithms for expressions, algorithms for integration, algorithms for the analysis of algebraic structures like groups etc.) to numerics and provided both numerical and algebraic algorithms in the frame of powerful integrated mathematical software systems like Macsyma, Reduce,..., Mathematica, Maple,... Various wonderful tools like graphics, notebook facilities, extensible two-dimensional syntax etc. greatly enhanced the attractivity of these systems for mathematicians, scientists, and engineers. Over the recent decades, sometimes based on very early work in the 19th century, new and deep research results in various branches of mathematics have been developed by the symbolic computation research community which led to an impressive variety of new algebraic algorithms. In parallel, in a different community, based on new and deep results in mathematical logic, algorithms and systems for automated theorem proving were developed. In the editorial for the Journal of Symbolic Computation (1985), I tried to offer this journal as a common forum for both the computer algebra and the computational logic community and for the interaction and merge of the two fields. In fact, in some specific theorem proving methods (as, for example, decision methods for the first order theory of real closed fields and decision methods for geometry), algebraic techniques play an important role. However, we are not yet at a stage where both worlds, the world of computational algebra (the algorithmization of the object level of mathematics) and the world of computational logic (the algorithmization of the meta-level of mathematics) would find there common frame in terms of integrated mathematical software systems. In the talk, I will sketch a view on future symbolic computation that hopefully will integrate numerics, computer algebra, and computational logic in a unified frame and will offer software systems for supporting the entire process of what could be called "mathematical theory exploration" or "mathematical knowledge management". In this view, symbolic computation is not only a specific part of mathematics but, rather, will be specific way of doing mathematics.This will have drastic effects on the way how research, education, and application in mathematics will be possible and the publication, accumulation, and use of mathematical knowledge will be organized. We envisage a kind of "Bourbakism of the 21st century", which will be very different --- and partly in opposition to --- the Bourbakism of the 20th century.