A Holistic Logic for Mathematical Reasoning

摘要

We introduce a holistic logic which provides a definition of "intelligent" systems by requiring that they are capable of evaluating functions, that is, determining function values (outputs), beyond the limits of any given Turing machine or any given formal system. Such systems are called creative systems. The core of holistic logic is the language CL which is used to represent mathematical theorems, proofs, and proof methods, that is, heuristics for the discovery of proofs. CL models the ordinary representation in mathematics textbooks. Holistic logic was verified by experiments with the SHUNYATA program. In the first experiments, SHUNYATA, starting from scratch, constructed simple proof methods, that is, heuristics, by analyzing proofs of simple mathematical theorems and used these heuristics to construct proofs of new theorems in the same or in other theories. In further experiments SHUNYATA produced proofs of significant theorems in mathematical logic and analysis on the basis of rather simple heuristics which model reasoning processes of mathematicians and can be constructed by an analysis of simple preceding proofs. The experiments suggest the hypothesis that mathematical knowledge, that is, proofs and heuristics, arise in a self-developing process which starts from any universal programming language and any input and cannot be reduced to a Turing machine but to the language and the input from which it starts. Holistic logic can also be applied for programming inputs and outputs in natural systems, for example, Physarum polycephalum and other swarms such as ants, bees, and some bacteria which can solve complex logistic and transport problems.