On advancing frontiers of science
A pragmatist approach

Part A is an abstract of the paper bearing the same title. Its full text – in “Our Pub” Library.  Part B is added as a supplement.

A. The pragmatist approach, as stated in this essay, takes into account two features of knowledge, both having an enormous potential of growth: the scope of science, whose frontiers can be infinitely advanced, while firmness of its propositions grows with consolidating once attained frontiers. An opposite view may be called limitativist as it conservatively sticks to some a priori limiting principles which do not allow progressing in certain directions. Some of them influence science from outside, like ideological constraints, other ones are found inside science itself.

The latter can be exemplified by the principles like those: (1) there can be no action at a distance; (2) there are no necessary truths; (3) there are no abstract objects. The first might have happened to limit physics with rejecting the theory of gravitation. The second entails that arithmetical propositions are either devoid of (clasical) truth or are not necessary; this would limit arithmetic to the role of a mere calculating machine, without giving any insights into reality. The third principle, for instance, limits logic to the first-order level (since in the second one variables range over abstract sets). The history of ideas shows that such limiting principles, had they been obeyed, would have hindered some great achievements of science. This is why we should not acknowledge any such principle as necessarily true, that is, winning in confrontation with any view contrary to it. Such principles on equal terms should compete with other propositions in obtaining as high degree of epistemic necessity as they may prove worth of.

To the core of pragmatist approach there belongs treating epistemic necessity as a gradable attribute of propositions. In accordance with ordinary usage, “necessary” is a gradable adjective, having a comparative form. The degree of epistemic necessity of a scientific statement depends on how much it is needed for the rest of the field of knowledge (Quine’s metaphor). The greater damage for knowledge would be caused by getting rid of the point in question, the greater is its epistemic necessity. At the top of such a hierarchy are laws of logic and arithmetic. Among physical laws at a very high level there is the law of gravitation, owing both to its universality, that is, a colossal scope of possible applications (advancement of frontiers), and its having been empirically confirmed with innumerable cases (consolidation of frontiers). Such a success has proved ible owing to the bold transgression of the limiting principle 1 (see above), and this has resulted in so high a degree of unavoidability.

A motto for  pragmatism can be found with the Chinese saying: “Black cat or white cat: if it can catch mice, it’s a good cat“. Thus, for a pragmatist, either the axiom of choice, or a higher order logic, is a good cat, as it does enable results vital for science, not attainable  otherwise.

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B.  There  is a computational motivation for pragmatism that can be rendered as follows. Let us not be discouraged by philosophical principialists’ aversions to certain theories, if only these theories either will make possible, or will dramatically speed up, some computational procedures crucial for science. In the paper, two such procedures are considered, one of them presupposing the axiom of choice, the other one — higher-order logics.

The former is hidden at the bottom of skolemization or (in Hilbert’s approach) of the use of epsilon operator. Such devices are meant to algorithmise proofs with predicate logic (e.g., in form of Beth semantic tableaux) up to the highest possible degree (with regard to undecidability of logic). Just owing to such algoritmisation, we get able to harness computers that they work for us provig theorems or assisting us in proving.

Kurt Gödel (1936) pioneered the idea that (1) some proofs, which in the first-order logic cannot be carried out, get feasible in the second-order logic, and (2) other ones which at the first-order level would require time available neither to humans nor to computers, become tractable in an accessible time when performed at higher levels. In my paper I am to report on a very instructive exemplification of point 2 as given by George Boolos with a human-made formalization (“A Curious Inference”, 1987), and to hint at its continuation through a current research in mechanized theorem proving.

To sum up, this case illustrates that the difference between limits and frontiers in science gets in a way reflected in a crucial difference between robots and human beings. A robot equipped with algorithms based, say, on the first-order logic is strictly limited, being unable to perform proofs requiring higher-orders. It is up to a human researcher, who feels himself like a computational pragmatist, to create the second-order logic, and some rooted in it problem-solving programs; and still, face the risk of being accused of platonism by learned collegaues. If the second-order logic fails to algorithmically solve problems arising in it, the pragmatic human mind is free to give life to an algorithmised third-order logic, and so on — potentially — up to infinity to which there tends such a sequence of succsessively advancing frontiers.

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