Informal Physical Reasoning Processes

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 of the outputs of Turing machines to lists, that is, recursively enumerable sets. The principle is illustrated by productive partial functions for productive sets such as the set of the Goedel numbers of the Turing-computable total functions. The existence principle and productive functions imply the existence of physical systems whose reasoning processes cannot be modeled by Turing machines. These systems are called creative. Creative systems can prove the undecidable formula in Goedel's theorem in another formal system which is constructed at a later point in time. A hypothesis about creative systems, which is based on computer experiments, is introduced.

An Effective Procedure for Computing "Uncomputable" Functions

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 Goedel number of any Turing-computable total function whose range is a subset of the set of the Goedel numbers of all Turing-computable total functions. We prove that the second input cannot be set to the Goedel number of any Turing-computable function that computes the output from any natural number in its first input. In this sense, there is no Turing program that computes the output from its first input. The procedure is used to define creative procedures which compute functions that are not Turing-computable. We argue that creative procedures model an aspect of reasoning that cannot be modeled by Turing machines.

Informal Concepts in Machines

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 formal system, that is, these machines can, in a certain sense, "compute" function values beyond these limits. We call these machines creative. We argue that any "intelligent" machine should be capable of processing informal concepts such as computable (total) functions, that is, it should be creative. Finally, we introduce hypotheses on creative machines which were developed on the basis of theoretical investigations and experiments with computer programs. The hypotheses say that machine intelligence is the execution of a self-developing procedure starting from any universal programming language and any input.

Beyond Turing Machines

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.