The SHUNYATA program constructs proof methods by analyzing proofs of simple theorems in mathematical theories such as group theory and uses these methods to form prooh of new theorems in the same or in other theories. Such methods are capable of generating proo& of theorems whose complexity represents the state of the art in automated theorem proving. They… (More)
A fundamental question is whether Turing machines can model all reasoning processes. We introduce an existence principle stating that the perception of the physical existence of any Turing program can serve as a physical causation for the application of any Turing-computable function to this Turing program. The existence principle overcomes the limitation… (More)
We give an effective procedure that produces a natural number in its output from any natural number in its input, that is, it computes a total function. The elementary operations of the procedure are Turing-computable. The procedure has a second input which can contain the Gödel number of any Turing-computable total function whose range is a subset of the… (More)
This paper constructively proves the existence of an effective procedure generating a computable (total) function that is not contained in any given effectively enumerable set of such functions. The proof implies the existence of machines that process informal concepts such as computable (total) functions beyond the limits of any given Turing machine or… (More)
This paper discusses " computational " systems capable of " computing " functions not computable by predefined Turing machines if the systems are not isolated from their environment. Roughly speaking, these systems can change their finite descriptions by interacting with their environment.