Computational Mathematics, Computational Logic, and Symbolic Computation

摘要

"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.