Pondering Bohmian Mechanics

3 July 2019

Wolfgang Smith

As if the bewilderment over quantum theory were not formidable enough in itself, the Bohmian Ansatz — which purportedly returns us almost to classical clarity — has added what may prove to be the most perplexing element of all to the mix.

Let us start at the beginning: with a statement, namely, of what exactly the “hidden variables” account of quantum theory is meant to accomplish. What motivated David Bohm to radically restructure the mathematical formalism of what might well be termed the most accurate physics the world has ever seen? The answer may in fact be gleaned from the preface to his last book,1D. Bohm & B. J. Hiley, The Undivided Universe  (Routledge, 1993). Bohm himself died unexpectedly “just as the final touches were being put to the manuscript.” and is actually manifest in the subtitle itself, which reads: “An Ontological  Interpretation of Quantum Theory” [my emphasis]. So far from being a positivist, or an aficionado of some kindred school, it appears that Bohm was actually a metaphysician at heart, and that it is this propensity — so very rare in our day — that sets him apart. What he had his eye upon was not primarily how things appear, but what in truth they are.  And therein apparently lies the motivation behind his “hidden variables” approach, which was designed — neither to simplify, nor to extend the range of quantum mechanics — but to manifest its ontological  significance. As his co-author B. J. Hiley explains:

Indeed the most radical view to emerge from our deliberations was the concept of wholeness, a notion in which a system formed a totality whose overall behavior was richer than could be obtained from the sum of its parts. In the ontological theory that we present here, this wholeness is made manifest through the notion of nonlocality, a notion that is seemingly denied by relativity.2Op. cit., p. xi.

For my part, I find these observations profoundly significant. There are three assertions here:  (1) that the authentic system constitutes a whole which is more than the sum of its parts;  (2) that it is precisely this wholeness  that gives rise to the phenomenon of nonlocality; and (3) that the question remains whether Einsteinian relativity and nonlocality are actually incompatible (as appears indeed to be the case). Leaving aside the third — with which we shall not be concerned in the present article3For my part, I maintain (in company with Albert Einstein himself!)  that nonlocality and Einsteinian relativity are, most definitely, not  compatible. — let me note, first of all, that I concur wholeheartedly with the first two affirmations, and actually perceive in them the key to an ontological comprehension of quantum mechanics.

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What then has, in Bohm’s view, gone awry in the standard4I.e., quantum theory as formulated by its founders, beginning with Werner Heisenberg in 1925. formulation? The error resides, supposedly, in the omission of an essential element from the quantum-mechanical description: i.e., the apparatus of measurement. The “wholeness” of the measuring process, Bohm insists, has thus been compromised: what has been jettisoned — what the standard wave function fails to describe — stands in fact at the very center of what physics is about: “Indeed, without the measuring instruments in which the predicted results appear, the equations of the quantum theory would be just pure mathematics that would have no physical meaning at all.”5Op. cit., p. 2. True enough: physics is after all “the science of measurement,” as Lord Kelvin has observed, which is to say that it is in the act of measurement, precisely, that physics comes into its own. Now, for Bohm this means apparently that the formalism of physics — its actual equations — must somehow incorporate the measuring process itself. It is hardly surprising, therefore, that he perceives the standard formulation of quantum mechanics — which conceives of measurement simply as a “collapse” of the wave function, a discontinuity which cannot be bridged by any differential equation — to be no more than a means of prediction which sheds no light on the underlying process. And, clearly, this omission of what he takes to be ontologically  essential constitutes, for Bohm, the fundamental defect he wishes to correct through an extension and refinement of the mathematical formalism. It was not mere mathematical bravado, thus, but rather a profound metaphysical striving that led Bohm to his astounding breakthrough, and explains why he regarded “the most accurate physics the world had ever seen” to be woefully deficient. As Bohm and Hiley observe dismissively: “All that is clear about the [standard] quantum theory is that it contains an algorithm for computing the probabilities of experimental results.”6Ibid., p. 1.

According to the dominant strains in the philosophy of science, this is of course precisely what a quantum mechanics is meant to do. What primarily interests Bohm, on the other hand, as we have seen, is not the prediction of measurable effects, but the ontology of the cosmos, its very wholeness. As Bohm and Hiley explain: “We have chosen as the subtitle of our book ‘An Ontological Interpretation of Quantum Theory’ because it gives the clearest and most accurate description of what the book is about”:  of what in fact Bohmian mechanics as such is about. “The question of determinism is therefore a secondary one, while the primary question is whether we can have an adequate conception of the reality of a quantum system.”7Ibid., p. 2. It is to be noted that in the standard formulation, the quantum system as such is ultimately conceived as something “less than real”:  as in a way “reminiscent of Aristotelian potentiae,” to quote Heisenberg. And this brings us back to the question of measurement:  that’s where the mystery resides.

This explains the radical departure of Bohmian mechanics from standard quantum theory: the fact that the “hidden variables” approach brings the process of measurement itself into the quantum-mechanical description. In place of a two-tier process consisting of a “Schrödinger evolution” described by the wave equation, followed by a discontinuous and mathematically unpredictable “projection” effected by an act of measurement, Bohm conceives of a continuous evolution of the combined system — the measured plus the measuring — in which all is governed by a single system of differential equations, without “inexplicable” discontinuities.

To a non-Bohmian quantum physicist this is apt to sound miraculous, and it is safe to say that prior to 1952, when David Bohm accomplished his feat, not a single quantum theorist of repute thought it could be done. In light of quantum mechanics as envisaged in its standard formulation, the ultimate components were conceived to be “half particle, half wave,” which is to say that they could in fact be neither. The notion that there exist bona fide  particles, moving under the guidance of a “pilot wave” in well-defined orbits calculable by means of differential equations descriptive of the integral system,8I.e., quantum system plus measuring apparatus. seemed thus to have been rigorously excluded. And as a matter of historical fact, when the notion of “hidden variables” was first enunciated by Louis de Broglie at the 1927 Solvay Conference, it was quickly disposed of — eviscerated in fact — by the presiding heavyweights.

Yet the founding premise of de Broglie-Bohm physics is simple in the extreme, and by no means unattractive to physicists: it affirms that real  particles interact with a real  wave to define real  trajectories determined by a system of differential equations — the trick is to discover these equations, to write them down and apply the resultant formalism to the solution of actual problems. And to the amazement of just about every quantum physicist — including possibly Bohm himself — it turns out that this can actually be done, and that the resultant mechanics yields in fact the same results as the standard theory.9To which one should add: at least most of the time. The matter is currently in dispute among quantum theorists.

As could have been predicted, the reaction to Bohm’s epochal discovery has been mixed. On the opposing side I would, above all, cite Werner Heisenberg, the first to write down — in a form of his own — the famous equation definitive of quantum mechanics, which in a thousand experiments has never yet yielded a false result. In Physics and Philosophy,  published six years after Bohm’s breakthrough, he gives a summary critique of “hidden variables” theory, which at the very least is dismissive of Bohm’s ontological claims. Heisenberg’s central objection is that “the hidden parameters of Bohm’s interpretation can never occur in the description of real processes if quantum theory remains unchanged,”10Werner Heisenberg, Physics and Philosophy  (Harper & Row, 1958), p. 132. in support of which he refers to physical processes on an atomic scale, such as those underlying the Compton effect. In a word, Heisenberg perceives the “hidden variables” of Bohmian mechanics as inherently speculative conceptions that prove to be de facto  meaningless on the level of physics properly so called, which he evidently conceives as an interplay of theory and experiment. In this optic Bohmian mechanics reduces to an unnecessary and potentially misleading means to arrive at results which can be obtained far more directly by way of the standard theory.

Bohm, to be sure, perceives the matter differently: he is, after all, a “metaphysician at heart,” as we have noted. What interests him primarily is the ontology, the underlying “wholeness” of which the measurable properties are the effects, and one almost senses a note of sarcasm when he laments that, in the standard or so-called Copenhagen interpretation, all that remains is “an algorithm for computing the probabilities of experimental results.”

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It is that Bohmian conception of “wholeness” thus — the notion of a “totality whose overall behavior is richer than could be obtained from the sum of its parts” — that calls for reflection. I too am persuaded that authentic ontology demands such a “wholeness,” and concur that this entails — and in a way explains — the phenomenon of nonlocality. What interests me above all, however, in the aforesaid declaration, is something Bohm and his followers seem not to perceive: an implication, namely, discernable by way of Aristotle’s conception of “quantity” as that which “admits mutually external parts.” For this entails a recognition, the significance of which could hardly be overstated: the fact, namely, that a whole which does not reduce to “the sum of its parts” cannot be fully conceived in quantitative terms.

One more step is needed to reveal, in full, what has thus come to light. For if such be the case, what else, besides quantities, does that wholeness comprise? The question is ontological:  what else, besides quantities, can there be? And here too the Aristotelian categories prove to hold the key: the complement of quantities, namely, are qualities, which unlike the former, do not  “admit mutually external parts,” and cannot therefore be quantified.11The notion, for example, that the color red can be identified with a corresponding light frequency is hardly worthy of a response. Red — or redness, if you prefer — is a quality, and anyone who takes it to be a quantity (a frequency, say) is simply confused. That there is a correlation between color and light frequency cannot be denied, but to say that the two are the same is to speak nonsense. Aristotle had it right: there is  a category distinction between quantities  and  qualities.  And this fact proves to be critical to cosmology.

We arrive thus — by means of rudimentary conceptions tracing back over two thousand years — at a conclusion which appears to have become virtually unthinkable in our day. My claim is that inasmuch as the cosmos constitutes an “unbroken wholeness,” it does not reduce to quantity, and consequently contains qualities as well. It cannot, therefore, be fully described or understood in terms of physics alone. In other words: no formulation of physics — not even Bohmian mechanics — can give us more than a partial or limited comprehension of the universe, inasmuch as it leaves out of account the qualitative dimension of reality. And one might add, in regard to this “qualitative dimension,” that in light of traditional ontology this proves in fact to be the “higher” of the two, inasmuch as it is expressive of morphe  or form  as opposed to hyle  or materia.  For the “quantities” to which physics reduces the universe correspond to numerus, which according to a well-known Scholastic dictum, “stat ex parte materiae”  (stems from the side of materia).12To be precise, numerus  as distinguished from the integers and their ratios, which according to Platonic and Aristotelian philosophy, do not  “stem from materia,”  but derive actually from the opposite pole: from what Pythagoreans term the monad. And these are what, to this day, are termed rational  numbers. It was Descartes, moreover, who once again “muddied the waters” by subsuming the latter under the category of numerus  by way of his “analytic” geometry: by means of “Cartesian coordinate systems” he obliterated the categorical distinction between the “rational” numbers and numerus. But whereas this assimilation may be expedient for physics — the “science of measurement” — it utterly obliterates arithmetic  as understood in the Pythagorean tradition, which was far more than a “mathematics” in the contemporary sense. It was in fact a science upon which arts — from music to architecture — could be based, and which holds relevance to a myriad phenomena in the organic sphere: from the anatomic proportions of the anthropos  to the shape of leaves. We will need, sooner or later, to rethink whether all this ancient lore constitutes in truth no more than a “prescientific superstition.” An authentic introduction to that long-forgotten science may be found in Thomas Taylor’s The Theoretic Arithmetic of the Pythagoreans  (Phoenix, 1934). See also the monumental treatise entitled Die Harmonikale Symbolik des Alterthums  by Albert Freiherr von Thimus, published at Cologne in 1868, which stands alone as a source.

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To the neutral observer it cannot but seem remarkable that even the most brilliant physicists — including those who espouse “unbroken wholeness” — have apparently failed to grasp that the world does not in fact consist of quantities alone: it makes one realize how powerful a force the Zeitgeist  actually proves to be. Since the onslaught of the Enlightenment in the seventeenth century, Western intellectuals have been hypnotized, as it were, by the Cartesian dichotomy — the notion that reality splits neatly into an “objective” world of res extensae, definable in purely quantitative terms, plus a “subjective” realm of so-called res cogitantes — upon which, to this day, our scientistic Weltanschauung  is based. We have thus come to regard the external world as a realm bereft of qualities, which can therefore be described without residue in purely mathematical terms. And that conviction, I would point out, has actually cast us into what may quite properly be termed a state of schizophrenia: one moment the grass is green, and the next — when our “higher” education kicks in — it is not.  The problem is that Descartes has cut asunder what in the act of knowing is conjoined:  we have forgotten what it is “to know.”

It thus behooves us to reflect anew upon the “act of knowing” that connects us to the “external” world, beginning with visual perception, upon which that knowing is in fact mainly based. I would point out, first of all, that the subjectivist interpretation of visual perception is not — and in fact, cannot be — based on empirical grounds: no amount of experimentation can establish that what we “actually perceive” when we look at an external object is simply “in our head.” The point is that perception  is something radically different from sensation. To be sure, the current theory of visual perception, espoused almost universally by cognitive psychologists, is indeed subjectivist to the core; it needs however to be realized that this subjectivism is actually implied by the presiding “retinal image” or “camera” paradigm itself, which has dominated the discipline since the Enlightenment. It happens however that the tide began to turn in the 1940’s when a young researcher at Cornell named James J. Gibson, commissioned under a government contract to discern how pilots perceive a landing point, arrived at the recognition that such a perception cannot in fact be achieved on the basis of “retinal image” psychology! We will presently have more to say concerning this key discovery.13Following a long series of journal articles, Gibson published a definitive formulation of his theory under the title The Ecological Approach to Visual Perception  (Lawrence Erlbaum Associates, 1986). On this subject I refer to my article, “Do We Perceive the Corporeal World?”, where additional references are given.

As should by now be apparent, we have broached the subject of perception precisely because measurement  is consummated, after all, in such an act. If it be true, therefore, that colors  for instance prove not  to be mere res cogitantes, this would imply that the instrument of measurement does not in fact reduce to res extensa. And this in turn would of course entail that standard quantum mechanics is right, after all, in distinguishing categorically between the physical system and the measuring apparatus, and that, by the same token, Bohmian mechanics is mistaken — not, to be sure, in an operational or pragmatic sense — but from an ontological  point of view, precisely. And this is what I now affirm on empirical grounds, based squarely upon Gibson’s refutation of “retinal image” psychology.

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It follows that the Copenhagen formulation of quantum theory — the one that reduces supposedly to a mere algorithm for computing probabilities — had it right after all. For if it be indeed the case that what we perceive in the act of visual perception is not, after all, “in our head,” but pertains actually to the “external” domain, this enables us — forces  us,  in fact — to distinguish ontologically between the physical system, characterized by quantitative attributes, and the measuring instrument, which consequently owns qualitative  attributes, such as color, as well. What confronts us in the measurement configuration is thus a meeting of two distinct ontological planes — what I term the physical  and the corporeal — followed by a resultant actualization of a physical state, which as Heisenberg surmised, constitutes indeed a transition from potency  to act. One thus arrives, quite naturally, at an ontological  understanding of quantum mechanics.14As first enunciated in The Quantum Enigma: Finding the Hidden Key  (republished by Angelico Press in 2005). For a summary account I refer to my monograph, Physics and Vertical Causation: The End of Quantum Reality  (Angelico Press, 2019). A brief introduction may also be found in my article, “From Schrödinger’s Cat to Thomistic Ontology.”

What I find objectionable in “hidden variables” theory is its very structure: by incorporating the measuring apparatus, namely, into the physical system, it renders invisible the ontological dichotomy upon which quantum mechanics as such is based. It thus implicitly negates the actual ontology, the very thing Bohmian mechanics was meant to discern, and instead perpetuates the myth of Democritus: the misbegotten notion that qualities — “color, the sweet and the bitter” — constitute illusory superimpositions, that at bottom reality reduces to “atoms and the void.”  Thus, inasmuch as the distinction between the quantum system and the measuring apparatus proves to be metaphysical, it turns out that by his very Ansatz,  Bohm has closed the door to an ontological comprehension of quantum mechanics. I find it tragic that the endeavor to bring to light the ontology of quantum physics — so costly and so brilliant — should have led to the denial of the very distinction by which that ontology can be discerned.

The failure, moreover, to distinguish ontologically between the physical system and the instrument of measurement likewise impedes a second major recognition: the discernment, namely, of a hitherto unknown mode of causation, which I refer to as vertical.15See Physics and Vertical Causation  (Angelico Press, 2019). What distinguishes this mode is the fact that it does not operate “in time” by way of a causal chain, and cannot therefore be expressed in terms of differential equations; and what moreover renders that causation detectable in the act of measurement is the self-evident fact that a transition between two distinct ontological planes cannot but be instantaneous.

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It behooves us, finally, to reflect somewhat upon “the mystery of knowing” as such: the relation, that is, of a knower  to the known. If we say that knowledge constitutes a kind of union, what is it, then, that connects the two? And if that union proves indeed to be “a certain oneness” as Aristotle claims, how is it possible for two things thus to “become one”?  I see no other way to deal with this question effectively than that of hylomorphism. By distinguishing between what the Greeks termed  hyle  and  morphe,  and the Romans  materia  and  forma,  conceived to be related as “recipient” to that which is “received” — somewhat as clay is receptive to the likeness of Socrates — it becomes possible for a second “receptor” to receive the very same form.16The italics are meant to indicate that the word is used in the specific sense of forma. For inasmuch as a form  has no corporeal being of its own — is not itself a “thing” — it is not separated, spatially or otherwise, from a potential knower. It is possible then, for a subject with a “materialike” receptivity, to “receive” that form, even as clay can receive “the form of Socrates.” And thus — “in a certain sense” as Aristotle has it — the knower and the known “become one.”

Returning to the case of visual perception, it thus appears that the connecting link uniting the percipient to the perceived can indeed be none other than form. Since the onset of the Enlightenment, on the other hand, we have been led to believe that the link consists simply in a physical process, answering in fact to the “retinal image” or “camera” paradigm — which however entails that in truth there exists no “link” and indeed no “objective” knowing at all. What we in truth perceive then is an “image” of some kind, a phantom-like thing “inside the head,” which would mean that we do not actually perceive the “external” world. Predictably so: because as we have noted before, nothing other than form  permits the miracle of knowing  to take place.17One might add that the reason resides in the fact — recognized by philosophers from Plato to Alfred North Whitehead — that to know  is something sui generis:  is not, in other words, the same as to be. It should come as no surprise, therefore, that whereas no trace of forms  is to be found in the “visual image” theory of perception, forms  do in fact play the key role in the previously mentioned alternative: for it happens that Gibson’s theory — which he refers to as the “ecological theory of visual perception” because the object of perception is what he terms the “environment” — hinges upon a “pickup of invariants from the ambient optic array,” in which the “invariants” prove in fact to be forms.18It should be noted that Gibson himself does not make this identification, and it is unclear to what extent he was philosophically inclined. He comes across as a “hard headed” empiricist, howbeit with powers of discernment rarely to be found. It is thus indeed by way of forms  that the “environment” — corresponding to what I term the corporeal  domain — can be visually perceived.

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Getting back to David Bohm, I wish now to show that, in the course of his ontological reflections, he has come within a hair’s breadth of making essentially the same discovery as James Gibson: the realization, namely, that visual perception hinges upon a pickup of what turns out in the end to be — yes, a form. Yet whereas, at first glance, this may seem incongruous, it is scarcely surprising in view of Bohm’s fascination with the idea of “wholeness”:  after all, the “invariants” by which we perceive what Gibson terms the “environment,” and which are said to be given at every location within that ambient optic array, exemplify in their own way the idea of “the part containing the whole” at which Bohm arrived by way of quantum theory.

Let us recall two quantum-mechanical findings, in particular, which suggested that conception to Bohm: first, the phenomenon of nonlocality, an inalienable characteristic of quantum theory which affirms that particles half a world distant may be so connected as not in truth to be separated at all, an effect that cannot actually be understood in mechanistic terms;19And may consequently be recognized as an effect of vertical causation. and secondly, that according to quantum theory “the whole may actually organize the parts, not merely through the strong connection of very distant elements, but also because the state of the whole is such that it organizes parts”20David Bohm, Unfolding Meaning  (Ark Paperbacks, 1987), p. 7. [my emphasis]. Gone is the key principle of classical physics: the idea that the universe can be understood in terms of its “atomic” parts. Not only does the whole fail to reduce to the sum of its parts, but it has actually the capacity to act upon the parts, to “organize” them.

But even this proves ultimately to be insufficient: from these initial reflections Bohm appears to have advanced to the full-blown realization that, in some exceedingly recondite sense, the whole is actually present to or “contained in” every part. Thus, alluding to a holograph, Bohm notes that “each part is an image of the whole object,” which leads him to the pivotal conception of enfoldment:   “Therefore every part contains information about the whole object … information about the whole is enfolded  in each part.”

My point, now, is that it was this crucial insight that enabled Bohm to rediscover in essence the salient feature of Gibson’s theory:  this “enfoldment,”  namely, proves to be the key to the enigma of visual perception. Bohm tells us so himself with the utmost clarity:

The light from all parts of the room contains information about the whole room and, in a way, enfolds it in this tiny region going through the pupil of your eye, and is unfolded by the lens, and the nervous system — the brain — and somehow consciousness produces a sense of the whole room unfolded in a way which we don’t really understand. But the entire room is enfolded in each part. This is crucial, because otherwise we wouldn’t be able to understand what the room was — the fact is that there is a whole room, and we see the room from each part.21Ibid., p. 10.

This is simply brilliant, and is tantamount in fact to Gibson’s notion of “invariants in the ambient optic array.” Bohm too, in his own way, has in effect rediscovered “forms.” Their respective accounts of visual perception concur, moreover, right up to the point where the light enters “the pupil of your eye,” following which they become antithetical. The fact is that “inside the head” Bohm’s description is based upon the “camera” paradigm, in keeping with what the “experts” have been telling us for the past so many centuries; and here Gibson obviously holds the advantage. Having found “what the experts say” to be fallacious, and arrived by way of empirical inquiry at a new paradigm, he envisages visual perception in a radically different way. The point, however, is that this new approach proves to be none other, inherently, than what Bohm himself was seeking: for it happens that the Gibsonian “pickup of invariants from the ambient optic array,” so far from reducing to a mechanical process, hinges precisely upon the idea — yes, of enfoldment. Gibson’s account of visual perception proves thus to be in truth as “Bohmian” as a theory could possibly be, and would doubtless have delighted the master himself.

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Let us not fail to note, finally, that the rediscovery of realism by way of Gibson’s “enfoldment” theory of visual perception confirms what we have deduced directly from Aristotle’s categories: the fact, namely, that so far from reducing to the world as conceived by the physicist, the universe proves to be incomparably richer and more sublime than we have been taught to believe. It turns out that our terrestrial environment, in particular, is endowed with a plethora of forms, which not only combine with materia  to produce corporeal entities, but enable us also to know these corporeal entities by way of perception. Yet ever since the Enlightenment, Western civilization has warred against form  in a Promethean endeavor to reduce the world to quantity.  And whereas the universe as such remains evidently what it is, our conception thereof has changed to the point that an objective ontology has become de facto  unthinkable. As Pavel Florensky, the venerable and pansophic priest (executed, significantly enough, by the Soviets in 1937) observed a century ago: “On the basis of positivism and materialism and, in general, of trends of thought that reject the essential reality of forms, there is no place for realism.”

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1. D. Bohm & B. J. Hiley, The Undivided Universe  (Routledge, 1993). Bohm himself died unexpectedly “just as the final touches were being put to the manuscript.”
2. Op. cit., p. xi.
3. For my part, I maintain (in company with Albert Einstein himself!)  that nonlocality and Einsteinian relativity are, most definitely, not  compatible.
4. I.e., quantum theory as formulated by its founders, beginning with Werner Heisenberg in 1925.
5. Op. cit., p. 2.
6. Ibid., p. 1.
7. Ibid., p. 2. It is to be noted that in the standard formulation, the quantum system as such is ultimately conceived as something “less than real”:  as in a way “reminiscent of Aristotelian potentiae,” to quote Heisenberg.
8. I.e., quantum system plus measuring apparatus.
9. To which one should add: at least most of the time. The matter is currently in dispute among quantum theorists.
10. Werner Heisenberg, Physics and Philosophy  (Harper & Row, 1958), p. 132.
11. The notion, for example, that the color red can be identified with a corresponding light frequency is hardly worthy of a response. Red — or redness, if you prefer — is a quality, and anyone who takes it to be a quantity (a frequency, say) is simply confused. That there is a correlation between color and light frequency cannot be denied, but to say that the two are the same is to speak nonsense. Aristotle had it right: there is  a category distinction between quantities  and  qualities.  And this fact proves to be critical to cosmology.
12. To be precise, numerus  as distinguished from the integers and their ratios, which according to Platonic and Aristotelian philosophy, do not  “stem from materia,”  but derive actually from the opposite pole: from what Pythagoreans term the monad. And these are what, to this day, are termed rational  numbers. It was Descartes, moreover, who once again “muddied the waters” by subsuming the latter under the category of numerus  by way of his “analytic” geometry: by means of “Cartesian coordinate systems” he obliterated the categorical distinction between the “rational” numbers and numerus. But whereas this assimilation may be expedient for physics — the “science of measurement” — it utterly obliterates arithmetic  as understood in the Pythagorean tradition, which was far more than a “mathematics” in the contemporary sense. It was in fact a science upon which arts — from music to architecture — could be based, and which holds relevance to a myriad phenomena in the organic sphere: from the anatomic proportions of the anthropos  to the shape of leaves. We will need, sooner or later, to rethink whether all this ancient lore constitutes in truth no more than a “prescientific superstition.” An authentic introduction to that long-forgotten science may be found in Thomas Taylor’s The Theoretic Arithmetic of the Pythagoreans  (Phoenix, 1934). See also the monumental treatise entitled Die Harmonikale Symbolik des Alterthums  by Albert Freiherr von Thimus, published at Cologne in 1868, which stands alone as a source.
13. Following a long series of journal articles, Gibson published a definitive formulation of his theory under the title The Ecological Approach to Visual Perception  (Lawrence Erlbaum Associates, 1986). On this subject I refer to my article, “Do We Perceive the Corporeal World?”, where additional references are given.
14. As first enunciated in The Quantum Enigma: Finding the Hidden Key  (republished by Angelico Press in 2005). For a summary account I refer to my monograph, Physics and Vertical Causation: The End of Quantum Reality  (Angelico Press, 2019). A brief introduction may also be found in my article, “From Schrödinger’s Cat to Thomistic Ontology.”
15. See Physics and Vertical Causation  (Angelico Press, 2019).
16. The italics are meant to indicate that the word is used in the specific sense of forma.
17. One might add that the reason resides in the fact — recognized by philosophers from Plato to Alfred North Whitehead — that to know  is something sui generis:  is not, in other words, the same as to be.
18. It should be noted that Gibson himself does not make this identification, and it is unclear to what extent he was philosophically inclined. He comes across as a “hard headed” empiricist, howbeit with powers of discernment rarely to be found.
19. And may consequently be recognized as an effect of vertical causation.
20. David Bohm, Unfolding Meaning  (Ark Paperbacks, 1987), p. 7.
21. Ibid., p. 10.