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Witold Marciszewski Hypercomputational vs . Computational Complexity A Challenge for Methodology of the Social Sciences

1.1.The first term in the title of this essay, hypercomputational , requires elucidation as being quite a novelty (a bit shocking, perhaps) in the language of science. Fortunately, the term computational has a well-established meaning since Turing’s seminal study of 1936. Fortunately, as well, it was the same Turing, in his work of 1938, who offered us a first hint toward the idea of the hypercomputational, the hint being involved in the concept of an oracle— a device (hypothetically postulated) to render values of uncomputable functions; such a rendering is now called hypercomputing. Thus hypercomputational complexity is one which cannot be handled by algorithms, that is, computational devices. Nevertheless, it may be handled with other means. What means? Turing put forward his idea of oracle in order to make more precise the concept of intuition, specially as appearing in the context of G ödel’s discovery of undecidable statements in mathematics. If such a statement is acknowledged as true without any proof (and even without a chance of being proved), the human faculty acting there is what one calls intuition or insight. The same faculty is busy in judging mathematical axioms as true. Turing [1938] attempted at formalizing this informal concept of intuition. More on this subject is to be said later. Here it is enough to express the conjecture that mathematical intuition may deal with the uncomputable. However, as being mathematical, it remains in the realm of numbers, while numbers are capable of being computed; if not computed in the strict Turingian [1936] sense, then in a way called hypercomputing– as suggested recently by a circle of researchers led by Jack Copeland. How should the uncomputable and the hypercomputable be related with each other? It is enough and is safe to tell that the former is included in the latter. That is to say, at least some uncomputable functions can be hypercomputed. To make this a plausible conjecture, let us suppose there are magnitudes in the world which are both continuous and uncomputable. Next, suppose