On computable abstractions (a conceptual introduction)

Machines operationally equivalent to a Turing machine have fundamental limitations that prevent them from yielding results that are comparable to the results produced by humans when performing intellectual tasks (e.g., processing of natural language, visual object recognition, derivation of the whole of mathematics from first principles). For instance, while these machines can cover only a subset of the whole extent of real numbers, humans are able to work with the entire set of real numbers (e.g., building other sets on top, selecting any element from the set). Abstractions are essential when a person performs intellectual tasks (both when the person does it alone or using a machine as a support tool). In this case, however, abstractions are either ideas or depend on ideas in the mind of the person, making them unsuitable for mechanical manipulation. In spite of this, it is possible to define abstractions that can be processed entirely through a mechanical process. In this paper, I introduce these " computable abstractions " and the mechanical abstractional process that deals with them, opening the possibility for machines to perform intellectual tasks with results that are intuitively comparable to humans' results.