I should note that I cannot say Jaki himself, RIP, would agree with my claims about the "imperfect production" of known natural laws in physical reality, at least the Jaki of The Relevance. I have not transcribed a few quotations which suggest Jaki does believe formal laws do appear in nature, but I am admitting that, so I'm not trying to "cook the book" in my favor. Having said that, I must emphasize how Jaki frequently reminds the reader that, despite how well theoretical laws appear in nature, "what's actually happening" in nature––as exposed by subsequent progress in the vastness, precision, and sheer mathematical abstraction of the physicist's toolkit––is always, elusively just beyond the edge of physical theory.
That qualification is important since it might be the way in which Jaki's and my views on this matter (heheh) align to some extent: physics must always admit that even its best formulas are only approximations of some deeper, actual reality in Creation. As such, physical theory––precisely as theory––will always be idealized and thus always retain a disparity between the empirical and theoretical. I would even go so far as to suggest that, if mankind someday does reach a "final theory" (albeit, as Jaki has argued on countless occasions, but especially in God and the Cosmologists, not a theory absolutely true and conceptually necessary [i.e., in violation Gödelian limitedness and future empirical falsification]), this will only show us that a complete description of nature is not ultimately a mathematical reality. A bizarre claim, perhaps, but all it means is this: insofar, on the one hand, as Gödelian limitedness preempts a complete and apodeictic formalization of nature and, on the other hand, insofar as the "final" theory actually refers to nature, then nature will actually exceed the formal "grasp" of any final mathematical theory (cf. the quotation on p. 136).
But we don't even need to imagine a final theory to see how reality transcends, or at least evades, complete mathematization. At every moment we are faced with an essential dimension of created reality which wholly escapes physico-mathematical analysis, namely, "now." (This is a point made by H. Bergson to A. Einstein which Fr. Jaki refers to in more than one place.) Fr. Jaki is also keen on reminding us that another equally fundamental dimension of reality which physics has no way of "claiming" is the sheer actuality of existence––"is" (cf. the quotation from p. 136). How does one "quantify" the sheer fact of existence?
In any event, apart from some emphases in bold (and perhaps some transcription errors), I will let Jaki and his sources speak for themselves. Then I will offer some concluding thoughts to dovetail with my most recent previous post on this topic.
pp. 66–67 …any past or future configuration of a mechanical system can in principle always be exactly calculated. In this sense there could be no dark corners in a mechanical system: it was by definition an open book, theoretically at least, with no mysteries, paradoxes, or uncertainties.
70 [Maxwell's] famous equations, once called by Einstein the most important event in physics since Newton, were in fact stripped in their final form of all the scaffolding of mechanical analogies. Indeed, the gap between physical representation and mathematical formulas was so enormous in these equations that all efforts aimed at their interpretation in terms of mechanical concepts ended in failure.
93 The wave and particle dualism shed further light on the fact that the absolute determinism and precision of which classical physics professed to know so much was not only unattainable but could not even be demonstrated to exist in nature.
101 …Galileo… voiced his astonishment time and again on seeing how closely natural processes follow the patterns of geometry. That he attributed more geometrical patterns to nature than he could demonstrate worried him little. He blamed the discrepancies between mathematically expressed laws of physics and actual observations upon the shortcomings of the calculator, who was unable to eliminate all the “material hindrances” present in physical phenomena. For him there existed a perfect one-to-one correspondence between the abstract world of geometry and the real world of things. … His famous law defining the distance traveled by falling bodies as a function of the square of the time of fall rested more on geometry than on actual experiments. … [His] admiration for the Pythagoreans … [gave him] a robust confidence that all truths incorporated in the universe… were written in the language of mathematics in characters that were “triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it….”
102–103 Berkeley raised the question: “Do not mathematicians submit to authority, take things upon trust, and believe points inconceivable? Have they not their mysteries, and what is more, their repugnances and contradictions?”
105 …an attempt [to determine the role of integral numbers in nature] was the Bode-Titius law giving the relative distances of the planets. It can be written as 4+3X2^(n-2), where n takes on the values 1, 2, 3… starting with n=1 for Mercury, the innermost planet. Although the formula breaks down for Mercury, it gave with surprising accuracy the distances of all other planets known in 1772 and even predicted the mean distance of the asteroids and distances of Uranus. On the other hand the formula failed utterly for Neptune and Pluto.
118–119 …Gibbs felt prompted to offer this little aside: “A mathematician may say anything he pleases, but a physicist must be partially sane.” At the same time, however, … [the] partial differential coefficients Gibbs introduced had no physically realizable notion. In the situation that he investigated, the entropy and volume of a system were supposed to remain constant while the mass of the system was changing. Such a procedure, however, is purely mathematical, for there is no way of adding or subtracting mass from a system without changing its entropy.
119 …an extensive study of stable and unstable phenomena … showed him [Maxwell] that unstable configurations in the physical world, such as a rock on a mountain top, or a match starting a forest fire, were actually flaws in the deterministic picture of physics. … “If … cultivators of physical science… are led in the pursuit of the arcana of science to study the singularities and instabilities, rather than the continuities and stabilities of things, the promotion of natural knowledge may tend to remove that prejudice in favor of determinism which seems to arise from assuming that the physical science of the future is a mere magnified image of the past.”
120 …as W. Heitler notes, “no further conclusions should be derived from this picture and questions of what the 'radius' of such a ball [i.e., an orbital electron] would be, etc., are void of any physical meaning.”
121–122 …Hertz aptly said [about the supposed incorrigibility of mechanics], “that which is derived from experience can be annulled by experience.” … As J. von Neumann noted about quantum mechanics: “One can never say that it has been proved by experience but only that it is the best known summarization of experience.”
124 Einstein: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” …indicates that out of the large number of mathematical systems, mathematics itself does not have the criterion to choose the simplest one that at the same time would translate perfectly the assumed basic simplicity of the laws of nature. In other words, the confidence that mathematics might find such a criterion can be supported only by a sort of faith in mathematics not by strict arguments.
125–126 As is well known, quantum electrodynamics has to fall back on the technique of renormalization, which, to use the succinct characterization of Dirac, “has defied all the attempts of the mathematicians to make it sound.” Renormalization, to be sure, is a highly successful technique. … Still, it is highly unsatisfactory. It almost amounts to cheating, as it replaces infinite quantities… arrived at by the theory, with the very small quantity established by observation. … Renormalization in quantum electrodynamics is therefore basically an ad hoc procedure, and as such it can offer little in the way of understanding the physical reality.
130–131 The setback suffered by the thoroughgoing formalists [such as Hilbert, Whitehead, and Russell] in the hands of Gödel's theorem should help prevent our forgetting that the mind thrives on sensory experience. … This is why he [Gödel] insisted so emphatically that a decision about the Euclidean or non-Euclidean geometry of the universe could be made ultimately on an experimental basis alone. Again, it was this “sensory” substratum of the geometry physics has to use, that kept suggesting to him that the scientific explanation of physical reality can never be final. … The concreteness of nature… is rich beyond comprehension in aspects in features. This is why even the most bizarre sets of mathematical postulates and geometrical axioms can prove themselves isomorphic with some portion of the observational evidence and useful in systematizing it. … This is why the physicist might even be overcome by a mood of skepticism concerning the uniqueness of coordination between his mathematical tools and the actual features of the universe. … Consequently, the formulation of new mathematical theories useful for physics will very likely go on indefinitely.
134 Dirac: “it is quite unnecessary that any satisfying description of the whole course of the phenomena should be given.” … What is … implied there [by Dirac] is that as mathematics grows more effective in coping with the problems of physics, it also becomes more evident how limited is that aspect of the phenomena that can be grasped, ordered, and correlated by mathematics.
135 Only a limited range of the full reality of things can ever be accommodated in the molds of mathematics…. …even Russell recognized this when he stated that “physics is mathematical not because we know so much about the physical world, but because we know so little: it is only its mathematical properties that we can discover.”
136 Whitehead: “There is no valid inference from mere possibility to matter of fact, or in other words, from mere mathematics to concrete nature.” For contrary to the dreams and hopes of latter-day Pythagoreans, numbers depend on the concreteness of things instead of generating those things.
233 …the saying, “The greater the sphere of our knowledge, the larger is the surface of its contact with the infinity of our ignorance.”
250–252 [Coulomb's] best torsion balance was sensitive enough to measure 1/100,000 of a grain, providing thereby a firm support for the view that the same inverse square law governs all known forces in nature. … Actually this law stood as a symbol for the unification of all branches of physics that nineteenth-century physicists hoped to establish.
259–260 Instances that show how persistent determination to reduce experimental error demanded major revisions of modern physical theories could be listed to no end. …quantum theory runs into infinities that cannot be removed only by ad hoc renormalization techniques, a procedure wholly unsatisfactory, although it yields values in almost perfect agreement with experimental data. Thus the renormalized Dirac theory clearly points beyond itself, for its morass of arbitrarily “tamed” infinities cannot be a satisfactory answer.
266 …Schrödinger had to ignore deliberately certain details of spectroscopic evidence and submit his wave equation as a first approximation.
268 …better vacuums mean the elimination of “obstacles,” of disturbing factors, and a gradual approximation of the “ideal conditions” which a physical experiment should always emulate.
272 The continuous progress in achieving precision then came to be interpreted as an ever more perfect realization of the ideal case of strict determinism. Furthermore, as the gradual approach toward the ideal was a reality, classical physicists, with apparently unassailable logic, assumed the factual existence of the ideal case.
275 What the uncertainty principle means essentially is that the determination of the “state of the world at an instant” is not possible in terms of mechanistic physics. And this limitation holds also for that proverbial “superior spirit” to which Laplace and others liked to refer. …made it abundantly clear that the [sic] mechanical intelligibility does not exhaust the whole range of intelligibility. Therefore one should not conclude on the basis of the indeterminacy principle that “the world is not a world of reason, understandable by the human intellect.” [citing P. W. Bridgman, “The New Visions of Science,” Harper's Magazine 158 (1929): 450.]
280 Pascal: “Truth is so subtle a point that our instruments are too blunt to touch it exactly. When they do reach it, they crush the point and bear down around it, more on the false than on the true.”
Just because all or any natural phenomena can be described mathematically does not mean they are in fact obeying a mathematical formula. To believe otherwise would be to commit a fallacy similar to post hoc ergo propter hoc (i.e., correlation≠causation). And just because a law holds for a range of phenomena does not mean all the phenomena in that range will always manifest the law. The intersection of two laws of nature, as Aristotle argued, result in chance occurrences. Ric Machuga refers to this is a case of per accidens causation resulting from per se causation (cf. "Machuga" parts of these posts for more). As such, the constant interplay of immutable natural laws will not fail to result in chance occurrences, such that, imagining ourselves far, far in the future with a massive database of nearly all past chance intersections recorded for millennia, we could even find a mathematical description of those chances occurrences with universal validity (universal, because the observed 'constituents' of the mathematical description occurred throughout the cosmos). Despite the universality of the law––call it the "normative law of chance" (NLC)––it is absurd to say nature was obeying NLC “all along,” since NLC is patently a sheer construct from a horde of prior data. Yet, to some extent, it seems that all physical law is of this character. To what extent on a case by case basis––aye, there's the rub.
As to the objection that this “normative chance law” is not a real law because it cannot predict later chance occurrences, we can reply two ways: first, by the nature of the case, NLC is already a stochastic law, and so perhaps it could be reformulated into a statistical law, like the laws of quantum mechanics. As a statistical law, NLC would escape the charge of failing to make absolute predictions, for it would be bound to make some correct predictions within its modified statistical range of accuracy. Moreover, its merit as a perfect account of past events would render it as empirically robust "in one direction" as Newtonian mechanics was prior to the advent of Einsteinian relativity and quantum mechanics. After that advent, of course, "in a new direction," Newtonian mechanics lost its validity in any more than a statistical, large-body sense. Such could well be the fate of NLC. Recall the Bode-Titius law (p. 135): while it suffered admitted empirical discrepancies, it perfectly accounted for an impressive range of phenomena and made successful predicitons, to boot. Is the Bode-Titius a law of nature? Is NLC? Why or why not?
Second, the predictive capacity of “normal laws” (as opposed to my beloved but perhaps repugnant NLC) abides only in conjunction with ceteris paribus conditions, that is, only on the condition that the rest of the natural system remains as it was prior to making the prediction. The current fundamental laws of physics, for example, simply break down when applied prior to the Planck threshold. As Templeton-Prize winning physicist-theologian Michael Heller explains in Creative Tension, once we cross the Planck threshold (i.e., theorize above 1 x 10^-33 cm, 1 x10^93 g/cm^3, 1 x 10^-44 sec), we are able to work with points in Poincaré fields, but beneath that threshold, the proto-singularity is atemporal and aspatial, two adjectives which simply have no place in classical, Einsteinian, and quantum physics. Similarly, the laws of Newtonian mechanics are only valid in a certain range of phenomena (namely, those 'smaller' than that which general relativity describes and 'larger' than what quantum mechanics describes). Despite the inability of Newton's laws to make accurate predictions at, say, the quantum level, we don't ipso facto deny the validity of those laws. Likewise, the law of gravitation could very well be decreasing based on the larger expansion or contraction of the cosmos over eons. Therefore, the predictive power of Newton's inverse square law is only good as long as we limit ourselves, anthropocentrically, to the ceteris-peribus status quo as we know it now. Along these lines, my (hypothetical) NLC need not make any novel predictions, because, being a universal law about universal intersections of sub-laws, its surrounding ceteris paribus conditions might change as some new intersection of sub-laws relevantly altered the (past) universal conditions from which the law had been derived.