In the Abstract of Prof. Primiero’s lecture „Computational Hypotheses and Computational experiments” (at Warsaw University of Technology, November 19, 2019) I found the following, much thought-provoking claim; let it be called PC (Primiero’s Claim).<\/p>\n
PC: The analogy between the scientific method and the problem-solving process underlying computing still is a tempting proposition.<\/em><\/p>\n This is a strong temptation, indeed. I am one who did succumb to it. This is due to my vivid interest in Turing’s [1939] idea of oracle<\/em> as discussed in my paper „The progress of science from a computational point of view: the drive towards ever higher solvability”<\/a> — in the journal Foundations of Computing and Decision Sciences<\/em>, 2019 (Vol.44, No.1) — Section 4 dealing with Turing’s [1939] idea of oracle.<\/p>\n What is there said seems to provide some premises to discuss the analogy claimed by PC. To wit, that similar rules control the problem solving-processes (hence a progress of science) in empirical sciences and those called by Prof.Primiero computational sciences<\/em>. It seems to me (a point for discussion) that the latter term can be conceived as equivalent with what is called „decision sciences” (see the quoted journal’s title), i.e., theories of decidability (to be called computability as well).<\/p>\n A crucial Turing’s (1939) claim concerning decidability in mathematics (see quotations by Marciszewski 2019), which continues his revolutionary result of 1936\/37 (on the existence of uncountable numbers), is the following. When there is a problem undecidable at a given evolutionary stage of axiomatized and formalized theory, it can prove solvable with inventing Such an adding of axioms and the concepts involved — to be briefly called conceptualization<\/em> — somehow motivated by mathematical intuition, do not enjoy a merit of infallibility. This have to be checked as guesses which may fail, as is happens with some presages; hence their metaphorical name „oracles” as suggested by Turing. In this respect they resemble\u00a0 empirical hypotheses being in need of testing. While in empirical teories hypotheses are tested with experiments, in computational science they are tested with their efficiency to produce right algorithms, and those, in turn — with their ability to be transformed into effective programs.<\/p>\n Thus the axioms of Boolean algebra produce, e.g., various algorithms to solve the problems of validity of propositional formulas, while those in turn, can be used to construct programs for automated theorem proving (the so-called provers).<\/p>\n Very interesting examples of testing such guesess can be found in the evolution of arithmetic. This, however, is a new subject, to be discussed in a next post. And still in another post one should consider thought-provoking analogies in problem-solving between mathematics and empirical sciences, e.g., astronomy.<\/p>\n","protected":false},"excerpt":{"rendered":" In the Abstract of Prof. Primiero’s lecture „Computational Hypotheses and Computational experiments” (at Warsaw University of Technology, November 19, 2019) I found the following, much thought-provoking claim; let it be called PC (Primiero’s Claim). PC: The analogy between the scientific … Czytaj dalej
\nappropriate new axioms. Those, in turn are due to a creative concept-forming\u00a0 to be involved into the axioms, and so expressed by primitive terms of the theory in question.<\/p>\n