The progress of science from a computational point of view: the drive towards ever higher solvability

Abstract

This essay's content is rendered by the titles of the successive sections. 1. Effective solvability versus intuitive solvability. -2. Decidability, i.e. effective solvability, in predicate logic. The speedup phenomenon -3. Contributions of the secondorder logic to the problems of solvability -4. The infinite progress of science in the light of Turing's idea of the oracle. The term "oracle" is a technical counterpart of the notion of mathematical intuition.A more detailed summary can be obtained through juxtaposing the textboxes labelled with letters A...F. Conclusion: in the progress of science an essential role is played by the feedback between intellectual intuitions (intuitive solvability) and algorithmic procedures (effective solvability).Keywords: effective solvability, intuitive solvability, decidability, speedup, secondorder logic, solvability, oracle, mathematical intuition, progress of science, algorithmic procedures. Wir müssen wissen, wir werden wissen.David Hilbert, the punch line of his lecture [1930] The advance of science is not comparable to the changes of a city, where old edifices are torn down to give place to new, but to the continuous evolution of zoologic types which develop ceaselessly, where an expert eye finds always traces of the prior work of the centuries past. Henri Poincaré, "The Value of Science", Introduction.1.1. The progress of science, characteristic of the Western civilization, embraces two processes: (i) the increase of the amount of methodically checked information, based on a reliable evidence, (ii) the development of more and more efficient methods and means of problem-solving via information processing. The notion of science, as conceived in this essay, is pertinent to all academic disciplines in which a methodological research is carried on. Not only to empirical sciences, as it is frequently understood in the English literature. Instead, I use the term "science" rather in the sense of German "Wissenschaft".The computational approach, as here understood, extends across at several areas. Not only computer science but also mathematical logic, some issues concerning the measure of information, considerations on the interaction between mechanical and non-mechanical (intuitive, creative) ways of problem-solving.The choice of the term "solvability" in the title is premeditated to connect its meaning with the meanings of "decidability" and "intuition".A. Effective solvability. A decision problem P is said to be decidable or effectively solvable when P can be solved by the universal Turing machine. 14 W. Marciszewski properties (in the above formula the property φ defines a classaccording to the axiom AA"). Contributions of the second-order logic to the problems of solvability and decidability3.1. This section continues the subject of "Gödelian speedup" discussed in 2.4 and 2.5. Here the emphasis is to be put on the mechanical aspect of decidability, the closest to the computational perspective by the fact that "decidable" in the technical sense means decidable by Turing machi...