Generalizing the Oracle Method

Although important, the main focus of this paper is not the specific oracle outlined above. The larger point is that if an operation in the mind is non-physical, this does not preclude it from being modeled. Specifically, oracles seem to work for modeling a wide variety of non-physical operations. There are probably other operations which will require other formalisms, but formalisms should not be avoided simply because the formalism is not physically computable. 

So how does one make a general application of oracles to modehng the mind First of all, it is important that the operation under consideration be well-defined in terms of what it is doing. It is not worthwhile to simply state something occurs here —such is not a well-specified description. In the example above, specific preconditions (a decision problem, a program, its input, and a set of existing axioms) and a 

Second, specific reasons must exist in order to believe that the proposed process is incomputable. Since solving the halting problem is known to be incomputable and adding axioms is incomputable by definition (otherwise they would be theorems), then specific evidence indicates that the proposed process is incomputable. 

The hard part then comes in testing the theory. Because the results are incomputable, and not even likely reducible to a probability distribution, testing it is more difficult. In the case of computable causes, a specific end-point prediction can be established by computation, and then the result can be validated against that computation. In this case, the result is not computable, and therefore validation is more difficult. Validation will often be based on the qualitative description of the process rather than a quantitative prediction. Parts of it may still be quantifiable, but only with difficulty. For instance, to test the example presented, a method of identifying and counting the number of axioms within a person s mind is needed in order to come up with a quantifiable prediction. However, since this is not possible, it can only be tested based on secondary quantifications. Thus, testability on proposed oracles becomes much more dialectic. 

For engineering, oracles can be used to better identify and measure complexity. If axioms become quantifiable, and the number of axioms required to solve problems becomes quantifiable, then this can transform the practice of complexity estimation. One such method to use these ideas to calculate software complexity is given in Bartlett (2014). 

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