From Schrödinger’s Cat to Thomistic Ontology

February 1, 2018

Wolfgang Smith

Editor’s Note:  Originally presented at Gonzaga University in 1998 as the Templeton Lecture on Christianity and the Natural Sciences, and subsequently published in Ancient Wisdom and Modern Misconceptions, the following sets forth in concise terms the basics of Dr. Smith’s ontological resolution to the quantum reality problem.

Wolfgang Smith Ancient WisdomLet me call your attention, first of all, to an as yet largely unobserved fact: while the scientific worldview continues to consolidate its grip upon society, something quite unexpected has come to pass. The decisive event occurred almost a century ago in fact, back in the early decades of the twentieth century. Since then that so-called scientific worldview — which to this day reigns as the official dogma of science — no longer squares with the known scientific facts. What has happened is that discoveries at the frontiers of science do not accord with the prevailing Weltanschauung, with the result that these findings present the appearance of paradox. It seems that on its most fundamental level, physics itself has disavowed the very worldview proclaimed in its name. This science, therefore, can no longer be interpreted in the customary ontological terms; and thus, as one quantum theorist has put it, physicists have “lost their grip on reality.”1Nick Herbert, Quantum Reality  (Garden City: Doubleday, 1985), p. 15. But obviously this fact has not been publicized, and as the aforesaid physicist observes, constitutes indeed “one of the best-kept secrets of science.” It needs however to be pointed out that, strictly speaking, physics did not “lose” its “grip on reality”: in light of the new findings the fact is rather that modern physics never had  such a “grip” in the first place. This Baconian science, rigorously conceived — that is to say, interpreted without recourse to the customary penumbra of scientistic beliefs — reduces quite simply to a positivistic discipline. And this explains Whitehead’s famous description of that science as “a kind of mystic chant over an unintelligible universe,”2Nature and Life  (New York: Greenwood Press, 1968), p. 15. as well as the admission by one of the leading quantum theorists that “no one understands quantum mechanics.” To be sure, the incomprehension to which Feynman alludes refers to a philosophic  plane: one understands the mathematics of quantum mechanics and its connection with empirical procedures, but not the ontology.

Broadly speaking, physicists have reacted to this impasse in three principal ways. The majority, perhaps, have found comfort in a basically pragmatic or “operational” outlook — the fact that “it works” — while some persist, to this day, in the patently futile attempt to fit the positive findings of quantum mechanics into the pre-quantum scientistic ontology. The third category, lastly, which includes some of the most eminent names in physics, convinced that the pre-quantum ontology is now defunct, have cast about for new philosophic postulates, in the hope of arriving at an acceptable conception of physical reality. There appear to be a dozen or so worldviews presently competing for acceptance in the upper reaches of the scientific community, which to the uninitiated seem to range quite literally from the bizarre to the outright ridiculous.

It is not my objective, in this lecture, to regale you with yet another ad hoc  philosophy designed to resolve or explain away quantum paradox. I intend rather to do the very opposite: to show, namely, that there is absolutely no need for a new philosophic Ansatz, that the problem at hand can in fact be resolved quite naturally on strictly traditional  philosophic ground. What I propose to show is that the quantum facts, divested of scientistic encrustations, fit perfectly into a very ancient and venerable ontology: the Thomistic, namely, which as you know, traces back to Aristotle. Rejected by Galileo and Descartes, and subsequently marginalized, this reputedly outmoded medieval speculation, it turns out, resolves the issue instantly. No need for ad hoc  postulates that stagger our understanding: the keys for which physicists have been groping since the advent of quantum theory, it turns out, have been readily at hand for well over two thousand years.


*   *   *

First formulated in 1925, quantum mechanics has shaken the foundations of science. It appears as though physics has, at long last, broken through to its own fundamental level; it has discovered what I shall henceforth term the physical universe — a realm which seems to defy some of our most basic conceptions of objective reality. It is a world (if we may call it such) that can be neither perceived nor imagined, but only described in abstract mathematical terms. The most useful and widely accepted of these representations is the one formalized in 1932 by the Hungarian mathematician John von Neumann. In this model the state of a physical system is represented by a vector in a so-called complex Hilbert space. This means, basically, that a state can be multiplied by a complex number, and that two states can be added, and that non-zero linear combinations of states, thus formed, will again be states of the physical system. Now, it is this fundamental fact, known as the superposition principle, that gives rise to quantum strangeness. Consider, for instance, a physical system consisting of a single particle, and then consider two states, in which the particle is situated, respectively, in two disjoint regions A and B, which can be as widely separated as we like. A linear combination of these two states with non-zero coefficients will then determine a third state, in which apparently the particle is situated, neither in A nor in B, but somehow in both regions at once. Now, one may say: “State vectors actually describe, not the physical system as such, but our knowledge concerning the physical system. The third state vector, thus, simply signifies that, so far as we know, the particle can be in A or in B, with a certain probability attached to each of the two possible events.” A grave difficulty, however, remains; for the state of the physical system corresponding to the third state vector can in fact be produced experimentally, and when one does produce that state one obtains interference effects which could not be there if the particle were situated in A or in B. In some unimaginable way the particle seems thus to be actually in A and B at once.

What happens, then, if one measures or observes the position of the particle in the third state? It turns out that the act of measurement instantly throws the system into a new state. The detected  particle, of course, is situated either in A or in B, which is to say that only unobserved particles can bilocate. All this, to be sure, is very strange; but let me emphasize that from a mathematical point of view all is well, and that in fact the theory functions magnificently. As I have said before, what puzzles physicists is not the mathematics, but the ontology.

Thus far I may have conveyed the impression that superposition states are rare and somehow exceptional. What is indeed exceptional, however, are states in which a given observable does  have a precise value (the so-called eigenstates); yet even in that case it happens that the system remains necessarily in a superposition state with respect to other observables. The quantum system, thus, is always in a state of superposition; or more precisely, it is at one and the same time in many different states of superposition, depending upon the observable one has in view. On the quantum level superposition is not the exception, but indeed the fundamental fact.

At this point one might say: “There is no reason to be unduly perplexed; superposition applies, after all, to microsystems too minute to be perceptible. Why worry then if ‘weird things’ happen on the level of fundamental particles and atoms? Why expect that one can picture things or happenings which are by nature imperceptible?” Most physicists, I believe, would be happy to adopt this position, if it were not for the fact that superposition tends to bleed into the macroscopic domain. It is this quantum-mechanical fact that has been dramatized by Erwin Schrödinger in the celebrated thought-experiment, in which the disintegration of a radioactive nucleus triggers the execution of the now famous “Schrödinger cat.” According to quantum theory, the unobserved nucleus is in a superposition state, which is to say that its state vector is a linear combination of state vectors corresponding to the disintegrated and undisintegrated states. This superposition, moreover, is transmitted, by virtue of the experimental setup, to the cat, which is consequently in a corresponding superposition state. In plain terms, the cat is both dead and alive! The hapless creature remains, moreover, in this curious condition until an act of observation “collapses its state vector” as the expression goes, and thereby reduces that state vector to one or the other classical eigenstate. The cat is then either dead or alive, period.

The mystery here, of course, has nothing to do with the nature of cats, but pertains rather to the role of measurement in the economy of quantum mechanics. Now, measurement is a procedure in which a given physical system is made to interact with an instrument so that the resultant state of that instrument indicates the value of some specific observable associated with that system. For example, a particle is made to collide with a detector (a photographic plate, perhaps) which registers its position at the moment of impact. Prior to this interaction, the particle will in general be in a superposition state involving multiple positions; we must think of it as spread out over some region of space. Its evolution or movement prior to impact, moreover, is governed by the so-called Schrödinger equation, which is linear, and hence preserves superposition, and is moreover strictly deterministic, which means that given an initial state, the future states are then uniquely determined. At the moment of impact, however, this deterministic Schrödinger evolution is superseded by another quantum-mechanical law, a so-called projection, which singles out one of the positions represented in the given superposition state — apparently for no good reason! — and instantly assigns the particle to the chosen location. Now, this simple scenario exemplifies what happens generally in the act of measurement: a physical system interacts with an instrument or measuring apparatus, and this interaction causes the Schrödinger evolution of the system to be superseded by an apparently random projection. It is as though the trajectory of a particle, let us say, were suddenly altered without any assignable cause. Why does this happen? Inasmuch as the instrument is itself a physical system, one would expect that the combined system, obtained by including the instrument, should itself evolve in accordance with the corresponding Schrödinger equation; but in fact it does not! What is it, then, that distinguishes the kind of interaction we term measurement from other interactions between physical systems, in which Schrödinger evolution is not  superseded?

Quantum theory holds many puzzles of this kind; the “scandal” of superposition assumes many forms. I would like to mention one more of these enigmas, which strikes me as particularly significant. One might think of it as a simplified version of the Schrödinger cat paradox. In the words of Roger Penrose, the problem is this: “The rules are that any  two states whatever, irrespective of how different from one another they may be, can coexist in any complex linear superposition. Indeed, any physical object, itself made out of individual particles, ought to be able to exist in such superpositions of spatially widely separated states, and so be ‘in two places at once’! . . . Why, then, do we not experience macroscopic bodies, say cricket balls, or even people, having two completely different locations at once? This is a profound question, and present-day quantum theory does not really provide us with a satisfying answer.”3The Emperor’s New Mind  (Oxford University Press, 1989), p. 256. As you may know, these matters have been debated for a very long time, and various interpretations of the mathematical formalism have been proposed in an effort to make philosophic sense out of the theory. However, as Penrose observes: “These puzzles, in one guise or another, persist in any interpretation of quantum mechanics as the theory exists today.”4Ibid., p. 296. After more than half a century of debate it appears that no clear resolution of the problem is yet in sight. One thing, however, one crucial point, has been consistently overlooked; and that is what I must now explain.

*   *   *

As one knows very well, it was the seventeenth-century philosopher René Descartes who laid the philosophic foundations of modern physics. Descartes conceived of the external or objective world as made up of so-called res extensae, extended things bereft of sensible qualities, which can be fully described in purely quantitative or mathematical terms. Besides res extensae  he posits also res cogitantes  or thinking entities, and it is to these that he consigned the sensible qualities, along with whatever else in the universe might be recalcitrant to mathematical definition. One generally regards this Cartesian partition of reality into res extensae  and res cogitantes  as simply an affirmation of the mind-body dichotomy, forgetting that it is much more than that; for not only has Descartes distinguished sharply between mind and body, but he has at the same time imposed an exceedingly strange and indeed problematic conception of corporeal nature, a conception, in fact, that renders the external world unperceived and unperceivable. According to René Descartes, the red apple we perceive exists — not in the external world, as mankind had believed all along — but in the mind, the res cogitans;  in short, it is a mental phantasm which we have naively mistaken for an external entity. Descartes admits, of course, that in normal sense perception the phantasm is causally related to an external object, a res extensa;  but the fact remains that it is not the res extensa, but the phantasm that is actually perceived. What was previously conceived as a single object — and what in daily life is invariably regarded as such — has now been split in two; as Whitehead has put it: “Thus there would be two natures, one is the conjecture and the other is the dream.”5The Concept of Nature  (Cambridge University Press, 1964), p. 30. Now, this splitting of the object into a “conjecture” and a “dream” is what Whitehead terms “bifurcation”; and this, it turns out, constitutes the decisive philosophic postulate which underlies and determines our customary interpretation of physics. Beginning with his Tarner Lectures (delivered at Cambridge University in 1919), Whitehead has insistently pointed out and commented upon this fact. “The result,” he declared, “is a complete muddle in scientific thought, in philosophic cosmology, and in epistemology. But any doctrine which does not implicitly presuppose this point of view is assailed as unintelligible.”6Nature and Life, op. cit., p. 6. I am here to tell you that today, after seventy years of quantum debate, the situation remains fundamentally unchanged. Just about every other article of philosophic belief, it would seem, has been put on the table and subjected to scrutiny, while bifurcation continues to be implicitly presupposed as if it were a sacrosanct dogma revealed from on high. And so “the muddle in scientific thought” continues, and has only been exacerbated by the demands of quantum theory.

That’s the bad news; the good news is that the situation can be remedied. In a recent monograph I have shown that physics can indeed be interpreted on a non-bifurcationist basis, with the result that quantum paradox disappears of its own accord.7The Quantum Enigma  (Tacoma, WA: Angelico Press/Sophia Perennis, 2005).  A useful summary with helpful commentary has been given by William A. Wallace in “Thomism and the Quantum Enigma,” The Thomist, vol. 61 (1997), pp. 455-467. No need for such far-flung notions as the “many worlds” hypothesis or any other ad hoc  stipulation; to resolve the semblance of paradox one needs but to relinquish a certain philosophic postulate foisted upon us by Galileo and Descartes. Quantum paradox, it appears, is but Nature’s way of repudiating a spurious philosophy.

*   *   *

We need thus to take a second look at quantum mechanics, but this time from a non-bifurcationist point of view. Now, to deny bifurcation is to affirm the objective reality of the perceived entity: the red apple, thus, is once again recognized as an actual external object. That perceptible entity, moreover, is to be distinguished from what may be called the “molecular apple,” a thing, clearly, which can not be perceived, but is to be known only through the methods of physical science. One is led thus to distinguish between two kinds of external objects: corporeal  objects, which can be perceived, and physical  objects, which can only be observed indirectly through the modus operandi  of the experimental physicist. The two ontological domains are of course closely related, failing which there could be no science of the physical at all. The basic fact is this: Every corporeal object X is associated with a physical object SX from which it derives all of its quantitative  attributes. The red apple, for example, derives its weight from the molecular. The crucial point, however, is that the two are not the same thing; X and SX belong in fact to different ontological planes: to different worlds, one could almost say.

The bifurcationist, obviously, does not recognize this distinction, inasmuch as he denies the existence of the corporeal object X; but in so doing, he implicitly identifies X with SX. The bifurcation interpretation of physics entails thus a reduction of the corporeal to the physical: and therein — in that reductionism, I say — lies the fundamental fallacy of the prevailing Weltanschauung.

The amazing fact, however, is this: Whereas classical physics tolerates that ontological error, quantum mechanics does not. It happens that the new physics itself distinguishes between X and SX; it insists in fact upon that distinction — which is precisely what perplexes the physicist. By its very structure — that is to say, in its categorical distinction between the physical system and its observables — quantum mechanics affirms in its own way the ontological distinction between the physical and the corporeal planes. To be precise, while the system itself belongs to the physical domain, the act of measurement terminates evidently on the corporeal: in the perceptible state, namely, of a corporeal instrument. It is true that the corporeal instrument I is associated with a physical system SI: the point however, once again, is that the two are by no means the same. What distinguishes measurement, thus, from a physical process is the fact that it realizes an ontological transition from the physical to the corporeal domain. No wonder quantum theory entails two very different “laws”: for it has become apparent that Schrödinger evolution operates within  the physical domain, whereas projection (operative in the act of measurement) refers to a transit from the physical into the corporeal. In the language of metaphysics one can say that the former applies to a “horizontal” whereas the latter refers to a “vertical” transition. So too one sees that the discontinuity of state vector collapse mirrors an ontological  discontinuity, which is of course the reason why that phenomenon is inherently incomprehensible from a reductionist point of view. Metaphysically speaking, state vector collapse is inexplicable on a physical basis because it results from the act of a corporeal  entity.

These considerations strongly suggest that the superposition principle needs to be amended for subcorporeal  systems, i.e., for the SX of a corporeal object X, since it is altogether reasonable to suppose that the state vector of SX can admit only superpositions consistent with the perceivable properties of X. And that is doubtless the reason why cricket balls don’t bilocate, and why cats cannot be both dead and alive. Penrose is absolutely right: if cricket balls and cats were “made of individual particles,” they would indeed be able to exist in unrestricted states of superposition; the point, however, is that they are not in fact thus made. From a non-bifurcationist point of view, corporeal objects, as we have seen, are not simply aggregates of particles, but something more. We need therefore to inquire what it is that differentiates X from SX; and for this we shall turn to the Thomistic ontology.

*   *   *

We must begin where St. Thomas himself began: namely, with the fundamental conceptions of Aristotle. The first step, if you will, in the analysis of being, is to distinguish between substances  and attributes: between things that exist in themselves and things that exist in another. Having thus distinguished between what is primary and what is secondary, one proceeds to the analysis of the primary thing. The problem is to break substance into its components: to split the atom of substance, as one might say; and for this one evidently requires the conception of things more primitive than substances, things “out of which” substances are made. Aristotle solved this problem with one of the great master-strokes in the history of philosophy: the distinction between potency  and act. The customary definition of these terms is simple and quite unimpressive at first glance: that which is capable of being a certain thing, but is not that thing, is that thing in potency, whereas that which already is, is so in act. A seed is a tree in potency, and a tree is a tree in act. Aristotle goes on to define matter, or prime matter, to be exact, as that which is in potency to substance, to substantial being. Prime matter as such has consequently no being; but it has nonetheless a capacity or an aptitude for being, one can say. Now, what actualizes this capacity is indeed an act, and that act is called a form, or more precisely, a substantial form. Substance has thus been split into two components: into matter  and form. It is the form, moreover, which contributes to the substance its essential content, its quiddity or “whatness,” what the Germans so expressively call its Sosein. And yet that form is not itself the substance, is not itself the existent thing; for the form without matter does not exist.

It is at this point of the analysis that the genius of St. Thomas Aquinas becomes manifest. And here we come to a second master-stroke in the history of philosophy: he recognized that substantial form is itself in potency to something else: to an act, namely, which is not a form; and that is the act-of-being  itself. To put it in his own words: “The act-of-being is the most intimate element in anything, and the most profound element in all things, because it is like a form in regard to all that is in the thing.”8Summa Theologiae, I, Quest. 8, Art. 1. Now, that innermost element constitutes the “point of contact,” as it were, between created being and its uncreated Source, which is God. The act-of-being belongs thus in the first place to God, who creates and sustains the universe; but yet it also belongs to created substance as its innermost reality. We may think of it as radiating outwards, through the substantial form, to the very accidents by which the being communicates itself to us.

Every being, moreover, is endowed with a certain efficacy, a certain power to act outside itself, which likewise derives from its act-of-being, and thus from God. And yet that efficacy, that power, is distinctly its own. As Etienne Gilson has beautifully explained: “The universe, as represented by St. Thomas, is not a mass of inert bodies passively moved by a force which passes through them, but a collection of active beings each enjoying an efficacy delegated to it by God along with actual being. At the first beginning of a world like this, we have to place not so much a force being exercised as an infinite goodness communicated. Love is the unfathomable source of all causality.”9The Christian Philosophy of St. Thomas Aquinas  (University of Notre Dame Press, 1994), p. 183.

We are beginning, perhaps, to catch a glimpse of the Thomistic ontology; but let us continue. Not only is God’s love the unfathomable source of all causality, but all causation, as we know it, imitates  that love. To quote Gilson once more: “Beneath each natural form lies hidden a desire to imitate by means of action the creative fecundity and pure actuality of God. This desire is quite unconscious in the domain of bodies; but it is that same straining towards God which, with intelligence and will, will blossom forth into human morality. Thus, if a physics of bodies exists, it is because there exists first a mystical theology of the divine life. The natural laws of motion, and its communication from being to being, imitate the primitive creative effusion from God. The efficacy of second causes is but the counterpart of His fecundity.”10Ibid., p. 184.

This same Thomistic vision of Nature has been expressed by Meister Eckhart in a passage of rare beauty which I would like also to share with you, where he writes: “You must understand that all creatures are by nature endeavoring to be like God. The heavens would not revolve unless they followed on the track of God or of his likeness. If God were not in all things, Nature would stop dead, not working and not wanting; for whether thou like it or not, whether thou know it or not, Nature fundamentally is seeking, though obscurely, and tending towards God. No man in his extremity of thirst but would refuse the proffered draught in which there was no God. Nature’s quarry is not meat or drink … nor any things at all wherein is naught of God, but overtly she seeks and ever more hotly she pursues the trail of God therein.”11Meister Eckhart  (C. de B. Evans, trans., London: Eatkins, 1925), vol. I, p. 115.

Here we have it: a vision of Nature which penetrates to the very heart of things, to that “most profound element” in fact, which St. Thomas has identified as its act-of-being. And to be sure, this is no longer an Aristotelian, but an authentically Christian Weltanschauung. I propose to show next how the findings of quantum theory fit into that Christian worldview.

*   *   *

It needs to be pointed out, first of all, that the Thomistic philosophy, no less than the Aristotelian, is unequivocally nonbifurcationist. There is not the slightest trace of “Cartesian doubt” to be found in either philosophy. What we know by way of sense perception are external objects, period; and these are the objects that enter into the Thomistic ontology. It follows that the findings of physics (our  physics, that is) can be assimilated into the Thomistic worldview only on condition that they be first interpreted in nonbifurcationist terms.

The fundamental problem, clearly, is to situate the physical domain ontologically in relation to the corporeal. Now, we know that transitions from the physical to the corporeal are effected by acts of measurement in which a certain possibility inherent in a given physical system is actualized; and this constitutes, Thomistically speaking, a passage from potency to act. A physical system as such may consequently be conceived as a potency in relation to the corporeal domain. And I might add that this point has in fact been made forcefully by Heisenberg with reference to fundamental particles: “a strange kind of physical entity just in the middle between possibility and reality”12Physics and Philosophy  (New York: Harper & Row, 1962), p. 41. he calls these entities, and goes on to observe that in certain respects they are reminiscent of what he terms “Aristotelian potentiae.” However, when it comes to the macroscopic domain, that is to say, to aggregates of fundamental particles that constitute the SX of a corporeal object X, Heisenberg identifies X and SX without the least scruple — as if the mere aggregation of atoms could effect a transition from potency to act! Nonbifurcation, on the other hand, implies, as we have seen, an ontological distinction between X and SX, which is to say that SX, no less than the quantum particles out of which it is composed, constitutes itself “a strange kind of physical entity just in the middle between possibility and reality.” To be precise, fundamental particles and their aggregates — be they ever so macroscopic! — occupy a position, ontologically speaking, between primary matter and the corporeal domain. It appears that contemporary physics has discovered an intermediary level of existence unknown and undreamt of in pre-modern times: and this is what I term the “physical universe.”

What is it, then, that differentiates the two ontological planes? From an Aristotelian or Thomistic point of view the answer is clear: what distinguishes a corporeal object X from SX is precisely its substantial form. It is this form that bestows upon X its corporeal nature and specific essence, its “whatness” or Sosein, as we have said. And it is important to emphasize that this substantial form is perforce something other than a mathematical structure; for indeed, if it were, X and SX would in fact coincide. One might say that SX itself comprises everything in X that is “quantitative” or reducible to mathematical structure in the widest sense. Substantial forms, therefore, are not amenable to a quantitative or rigorously mathematical science. It is to be noted, moreover, that this fact was clearly recognized by Descartes himself: “We can easily conceive,” he writes, “how the motion of one body can be caused by that of another, and diversified by the size, figure and situation of its parts, but we are wholly unable to conceive how these same things can produce something else of a nature entirely different from themselves, as for example, those substantial forms and real qualities which many philosophers suppose to be in bodies.”13See E. A. Burtt, The Metaphysical Foundations of Modern Physical Science  (New York: Humanities Press, 1951), p. 112. It needs however to be noted that this is precisely the reason why the protagonists of universal mechanism, headed by Galileo and Descartes, rejected substantial forms and banished sensible qualities from the external world: substantial forms as well as sensible qualities had to be excluded because neither could be reduced to mechanical terms. In so doing, however, Galileo and Descartes have cast out the very essence of corporeal being; one is left with a de-essentialized universe, a world emptied of reality.

We need today to free ourselves from the iron grip of this spurious and dehumanizing dogma. We need to rediscover the fullness of the corporeal world, replete with substantial forms and real qualities, which moreover enshrines at its very core the mystery of what St. Thomas calls “the most profound element in all things.” We have need of this discovery in every domain of life, including the scientific: even when it comes to the philosophic or “more-than-operational” understanding of quantum theory, as we have seen. But we have need of a sound ontology above all in the spiritual domain: authentic Christianity, in particular, demands a sacramental capacity on the part of matter which is categorically inconceivable in Cartesian terms. It hardly needs pointing out that in a universe comprised of quantum particles — in which not even a red apple can exist! — the Christic mysteries have absolutely no place. Now, I surmise that of all the true philosophies — and I believe there may be more than one — the Thomistic is for us the safest and most efficacious means by which to effect the liberating intellectual rectification. Whosoever has sensed that “love is the unfathomable source of all causation” has already broken the chains; and whoever has grasped, even dimly, what St. Thomas terms the “act-of-being” is well on his way.

Read Less...

   [ + ]

1. Nick Herbert, Quantum Reality  (Garden City: Doubleday, 1985), p. 15.
2. Nature and Life  (New York: Greenwood Press, 1968), p. 15.
3. The Emperor’s New Mind  (Oxford University Press, 1989), p. 256.
4. Ibid., p. 296.
5. The Concept of Nature  (Cambridge University Press, 1964), p. 30.
6. Nature and Life, op. cit., p. 6.
7. The Quantum Enigma  (Tacoma, WA: Angelico Press/Sophia Perennis, 2005).  A useful summary with helpful commentary has been given by William A. Wallace in “Thomism and the Quantum Enigma,” The Thomist, vol. 61 (1997), pp. 455-467.
8. Summa Theologiae, I, Quest. 8, Art. 1.
9. The Christian Philosophy of St. Thomas Aquinas  (University of Notre Dame Press, 1994), p. 183.
10. Ibid., p. 184.
11. Meister Eckhart  (C. de B. Evans, trans., London: Eatkins, 1925), vol. I, p. 115.
12. Physics and Philosophy  (New York: Harper & Row, 1962), p. 41.
13. See E. A. Burtt, The Metaphysical Foundations of Modern Physical Science  (New York: Humanities Press, 1951), p. 112.