Computational Mathematics, Computational Logic, and Symbolic Computation (Invited Lecture)

“Computational mathematics” (algorithmic mathematics) is the part of mathematics that strives at the solution of mathematical problems by algorithms. In a superficial view, some people might believe that computational mathematics is the easy part of mathematics in which trivial mathematics is made useful by repeating trivial steps sufficiently many times on current powerful machines in order to achieve marginally interesting results. The opposite is true: Many times, computational mathematics needs and stimulates deeper mathematics, i.e. deeper mathematical theorems with more difficult proofs than “pure” mathematics. This is so because, in order to establish an algorithmic method for a given mathematical problem, i.e. in order to reduce the solution of a given problem to the few operations that can be executed on machines, deeper insight on the given problem domain is necessary than the insight necessary for establishing the reduction of the given problem to powerful nonalgorithmic abstract mathematical operations as, for example, choice functions and the basic quantifiers of set theory. Computational mathematics comes in two flavors: “numerical mathematics”, in which the original problems are replaced by approximate versions and one is satisfied with approximate solutions to the approximate problems, and “exact algorithmic mathematics” in which the original problems are solved by algorithms in the original domains or isomorphic representations of these domains. Exact algorithmic mathematics can be divided into “discrete mathematics”, in which the objects in the underlying mathematical domains are finitary, and “computer algebra”, in which the objects in the underlying mathematical domains according to their original defintion are infinite and the possibility of an isomorphic finitary representation in itself is a non-trivial mathematical question. For many mathematical problems it can be mathematically proved that exact algorithmic solutions are not possible or are possible only by algorithms with a certain complexity. Even in these cases, algorithmic mathematics can and should go on by considering either approximate versions or special cases of the problem. Mathematical logic is the mathematical meta-theory of mathematics. The characteristic feature of mathematics is its method of gaining knowledge from given knowledge by reasoning. Hence, the meta-theory of mathematics is essentially the theory of reasoning. As any other mathematical theory, one can and should ask the question of how much of reasoning can be made algorithmic. The