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Scientific Theory and Computational Irreducibility

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Posted by: Gunnar Tomasson

The concepts of “Scientific Theory” and “The Realm of Science” outlined in my note of 11/22/03 on “Einstein’s Unfinished Revolution” and identification of the latter as the domain of Hermann Weyl’s “Realistic Mathematics” imply that, just as Gödel’s Incompleteness Theorem does not apply within that domain, so Stephen Wolfram’s concept of “Computational Irreducibility” is inapplicable to “The Realm of Science”.

For Wolfram’s concept is predicated on – and relates to - what Ludwig Wittgenstein termed “the illusion that the so-called laws of nature are the explanations of natural phenomena,” translated into the modern notion that a “scientific theory” is to be judged by its PREDICTIVE power alone rather than by its LOGICAL coherence AND adequacy of its representation of what Einstein termed its “empirical contents and their mutual relations” as indicated by its PREDICTIVE power with respect thereto.

The difference between the two sets of criteria is reflected in Einstein’s comments on the General Theory of Relativity cited in the earlier note: “The chief attraction of the theory lies in its LOGICAL completeness. If a single one of the conclusions drawn from it proves wrong, it must be given up; to modify it without destroying the whole structure seems to be impossible.”

That is to say, the theory’s representation of empirical phenomena must be judged inadequate on discovery of a single phenomenon which is not consistent therewith.

Also, as detailed in my note of 11/8/03 on “Einstein’s ‘Scientific Testament’”, Einstein recognized that the theory’s axiomatic premises themselves might prove untenable – this would affect neither its predictive power nor logical coherence but the adequacy of its axiomatic representation of “The Realm of Science”.

The following extract from a draft version of my note of 11/22/03 addresses related issues in less formal fashion.

THEORY AND PREDICTION

The detailed observations of Tycho Brahe and Kepler of solar system orbital patterns represented “the evidence of experiments” which “[found] their representation in the conclusions of [Newton’s] theory.” This fact was reaffirmed through the discovery in 1846 of Neptune, whose existence had been signaled by observed perturbations of the orbit of Uranus (discovered in 1781) which could not be explained on Newtonian principles. Yet, these principles were later ‘falsified’ when non-Newtonian aspects of Mercury’s orbit, which had been construed to signal the existence of an inner planet (Vulcan), were instead judged to reflect ‘relativistic’ effects.

By the same token, the fact that “the evidence of experiments” on Mercury’s orbital pattern accords with the General Theory of Relativity is no warrant in logic for using “the conclusions of [Einstein’s] theory” as presumptive evidence of the existence of never-observed aspects of the Cosmos such as Black Holes and Big Bang Creation.

It is “vain fiction” to suggest otherwise; for, as Einstein once observed, “as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” (‘Geometry and Experience’, lecture 1921, reprinted in ‘Ideas and Opinions’, Dell/Laurel Paperback, 1976, p. 228)

In the field of particle physics, Pauli’s ‘prediction’ in 1931 of the yet-to-be-discovered neutrino within the conceptual framework of the Standard Model mirrored that of the planet Neptune within the Newtonian Model of solar system orbital mechanics – yet, the Standard Model breaks down with respect to whatever ‘Vulcan’ phenomena may be encountered at scales smaller than that of the Planck Length.

Gunnar





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