Irreducible Wholeness and Dembski’s Theorem

We need first of all to be apprised of the fact that not only in the days of Pythagoras and Plato, but in the premodern world at large, a triangle or a circle, for instance, was by no means viewed as a “point set,” but was in effect conceived as an irreducible whole. To regard these geometric entities as mere aggregates would have been seen as a denial of their essence, their very “being.” The same can be said of non-geometric entities such as integers or ratios thereof, which were likewise conceived in premodern times as IW’s—even as they are in the context of musical scales, where one speaks, for instance, of the “octave.”¹  By the time one arrives at what we term “real” numbers, on the other hand—the kind represented by nonterminating decimals—the picture has changed. To refer to these indiscriminately as “real” is to ignore the time-honored distinction between rational and irrational numbers: those which constitute integer ratios and those which do not. But why does that matter? It matters because therein, I surmise, resides the distinction between irreducible and reducible wholeness in the numerical domain.

Of course this calls for a good deal of explanation, which we shall attempt to suggest, at least, in the following section. But granting that it is true: what difference does it make? The answer to this question, I believe, is simple: to amalgamate “rational” and “irrational” numbers into a single category of so-called “real” numbers is in effect to amalgamate irreducible and reducible wholeness, thereby depriving arithmetic of its innate ontological significance. This step alone bears witness to the fact that, in the post-medieval world, the traditional ontological wisdom of mankind has become a closed book: a “brave new intellectuality” has taken its place. Mathematics had by then become disassociated from its “irreducible” content to the point that Principia Mathematica could be seen as definitive of what in truth it is. The very notion of IW having faded into oblivion, the way was open for the reduction of mathematics to formal logic and set theory, thereby diminishing its content ontologically to the status of a nonentity. One might say that set theory is to mathematics what quantum theory is to physics: it is what remains when every last vestige of “being” has been exorcised.

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Yet whether we recognize the fact or not, now as before mathematics deals essentially with irreducible wholes. Descartes notwithstanding, first of all, the ancient distinction between geometry and arithmetic still carries weight: qua IW, a triangle or a circle, for example, does not in truth reduce to the domain of numbers. Whatever computational benefits the amalgamation of these disparate mathematical realms may have achieved, we have surrendered the ontological insights their separation entails. It is one thing to “do” mathematics, and quite another to understand what, from a metaphysical point of vantage, mathematics is “about.” My point is that when it comes to that other side of the enterprise, the recognition of geometry as an irreducible discipline proves to be a sine qua non. We have been far too quick to stigmatize the ancient savants as “primitive” and the like, when in fact there are grounds to wonder whether, metaphysically speaking, the shoe may not actually be on the other foot. For my part, I am persuaded that when it comes to the ontology of mathematics—to the recognition of IW, namely, as the essential—the ancient savants apparently discerned what we no longer comprehend.

Not only, thus, did they recognize the ontological distinction between an irreducible whole and a mere aggregate, but what strikes me as even more significant: they realized apparently that it matters! This is, after all, why the Pythagoreans were visibly impacted by their discovery that the ratio of a side to the hypotenuse of an isosceles triangle is what we term an “irrational” number. Might there be then, conceivably, a connection between “irrational” in this mathematical sense and its broader connotation as a deficiency or lack of some kind, be it cognitive or ontological? Could the copresence of the irrational signify perhaps that the sensible world consists, in the final count, not only of irreducible wholes—of being—but of “nonbeing” as well?

Granting that the Christian ontology does not coincide with the Platonist but ultimately transcends it,²  I would point out that the aforesaid surmise accords with Christian sources. Recall, for instance, the words of St. Augustine in the Confessions, where—addressing himself to God—he declares:

I see these others beneath thee: an existence they have, because they are from thee; yet no existence, because they are not what thou art.

I am not suggesting, of course, that St. Augustine is speaking of irrational ratios: what he affirms unequivocally, however, is that the cosmos is made up not simply of “being,” but entails perforce an element of “nonbeing” as well. On the yet higher level of Scripture, moreover, there is a well-known citation which makes the very same point: I am referring to “the ontology of the Burning Bush” enunciated in Exodus 3:14. Recall the scene: Moses asks God to declare to him His name, and is told in response “Ego sum qui sum”—which can be rendered “I am that which is.” It is not hard to see that this concurs essentially with the words of St. Augustine.

Getting back to the Pythagoreans: I surmise the disciples of Pythagoras did in fact understand the rational/irrational dichotomy in a cognate key—which explains, among other things, why the words “Let no one ignorant of geometry enter here” were reputedly inscribed over the portal of Plato’s Academy.³

There can be no doubt, on the other hand, that the modern world is decidedly—and one might say, blissfully—“ignorant of geometry.” In obliterating the ancient distinction between geometry and arithmetic, and compounding rational and irrational numbers into a single category—labelled “real” no less—we have in effect formally eradicated the distinction between irreducible and reducible wholeness: between “being” and “nonbeing” in the mathematical realm. We have thereby closed the door to an ontological comprehension of mathematical science. Furthermore, having excluded “being” from the realm of mathematics, we have excluded it ipso facto from the physical sciences, and thus from the resultant Weltanschauung. Think of it: what has in effect been jettisoned is the very conception of IW: the very conception of being itself!

I find it ironic that the so-called Enlightenment has, in effect, replaced the rational by the irrational. And let me add, parenthetically, that this explains why “evolution” has become the dominant myth of our age: where being reduces namely to the sum of its parts, it originates evidently through an aggregation of particles—which is, after all, precisely what the tenet of “evolution” reduces to ontologically. A treatise chronicling the transition—the descent, actually—from Pythagoras and Plato to Darwin and the Principia Mathematica, conceived in light of the rational/irrational dichotomy, might prove to be enlightening. Suffice it to note that the decisive shift began apparently near the end of the Middle Ages with a radical denial of IW in the form of an enchantment with nominalism. In the wake of the Enlightenment, moreover, this negation—which initially was opposed to the intellectual and cultural mainstream—became the implicit credo of the modern age—its religion almost, one might say—at least in the Western world.⁴

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The idea of irreducible wholeness is closely associated with the ontological notion of the “tripartite cosmos,” which I take to be not only normative but “perennial” in the sense that, in a way, it has always been known.⁵ According to this cosmography the integral cosmos divides “vertically” into three ontological domains: the corporeal, subject to both space and time; the so-called intermediary, subject to time alone; and ultimately the aeviternal, subject to neither.⁶  One may conceive of this tripartite ontology in terms of a circle in which the center represents the aeviternal domain, the circumference the corporeal, and the interior the intermediary. I am persuaded that this representation is genuinely iconic, and can in truth be employed—somewhat like a mathematical formula—to draw conclusions: ontological conclusions to be precise.⁷  And I regard this “icon” as an invaluable key to many questions, including the enigma of irreducible wholeness: a way of “seeing” what stands at issue.

The icon bears witness, first of all, to what may be termed the ontological primacy of irreducible wholeness. One “sees” that inasmuch as an IW does not reduce to the sum of its spatio-temporal components, it preexists on the aeviternal plane: one might say that it manifests an aeviternal prototype. This conclusion cuts of course against the very grain of contemporary thought, which locates being—or better said, what is left thereof—on the spatio-temporal plane. To be sure, to minds steeped in the Zeitgeist of our age, the very idea that there may be something “beyond” the spatio-temporal smacks of the unbelievable, the utterly fantastic. Add the notion that this “transcendent” and indeed “aeviternal” element constitutes the core reality supportive of spatio-temporal phenomena as such—and chances are not many will be left standing to hear you out.

Yet to the extent one is willing and able to assimilate this ancient—and as I surmise—perennial teaching, one begins to realize not only that it actually makes sense, but that it may well be in essence the only ontology that does. The problem is that we tend, almost irresistibly, to “spatio-temporalize” whatever presents itself to our view; in its present state, at least, our mind appears to be incapable of apprehending unmediated wholeness. It is hardly surprising, thus, that philosophy in the authentic “pre-Enlightenment” sense demanded a stringent discipline: that there was perforce a “yogic” side to the enterprise. We read in the ancient books that the disciples of Pythagoras, for instance, were obliged to observe a vow of celibacy, and that Socrates characterized philosophy as “the practice of death”: can you imagine a contemporary professor of philosophy so much as utter such words? In those pre-Enlightenment times, philosophy—so far from reducing to mere “theory” or “speculation”—was in essence a matter of sight, of seeing—not “as through a glass, darkly”—but actually “face to face”: without a spatio-temporal intermediary, that is.

For readers respectful of the Judeo-Christian tradition let me note that, given what Genesis has to say regarding the Fall of Adam, this should actually come as no surprise. The point is that this prehistoric Fall has in effect reduced human nature to what St. Paul terms a “psychikos anthropos” who “knoweth not the things of God.”⁸  We need to recall that man is traditionally conceived as a corpus-anima-spiritus ternary, corresponding to the Pauline soma-psyche-pneuma. What is missing in the psychikos anthropos is thus the highest component of his tripartite being: pneuma or spiritus, namely. But this is precisely the faculty that enables man to know “the things of God,” which in this context we may take to be the “things” pertaining to the aeviternal realm. This then, I surmise, is the Eden from which Adam was banished, an exclusion which evidently has been passed on to his progeny.⁹

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Irreducible wholeness, then, is not an abstraction—not just somebody’s theory—but the very being, rather, of existent things. In corporeal entities it is the being proclaimed by the spatio-temporal parts: the one within the many. The fact is that being proves to be inseparable from unity or oneness: “Ens et unum convertuntur” say the Scholastics. But how do we know that there is such an ens, that there is such an unus? The fact is we do; we know it as the “whatness” of the thing: as the what that it is.

It has thus become apparent that, strictly speaking, our physical science does not deal with being at all, that in fact it can have no inkling either of ens or unus. It cannot therefore comprehend even the corporeal, but is constrained to deal with mere potentiae actualizable in principle on the corporeal plane. But if, by way of physics, one is unable to conceive even the lowest tier pertaining to the tripartite cosmos, what to speak of the aeviternal plane, the source of IW!

A number of basic ontological recognitions are implicit in what has been said. It is to be noted, first of all, that the causality accessible to physics—which breaks naturally into a deterministic, a random, and a stochastic kind¹⁰ —is based upon spatio-temporal parts, and acts upon spatio-temporal parts in turn. To the extent that it may be productive of wholeness, this part-based causality—which I designate by the adjective horizontal—gives rise evidently to a wholeness which itself reduces ipso facto to the sum of its spatio-temporal components. One arrives thus at what might be termed an ontological theorem:

(*)  Horizontal causation cannot give rise to irreducible wholeness.

Let me interrupt this train of thought to point out once again how utterly incongruous, from a Platonist point of vantage, the contemporary dogma of “evolution” proves in fact to be: it presupposes, after all, that organic wholes arise from a reordering of parts in antecedent organic wholes. It postulates, therefore, that wholes derive from parts and pieces by way of a causality itself derived from parts and pieces—which is either to deny the irreducible wholeness of living organisms, or to suppose that this wholeness can be produced by horizontal causation. It is to be noted thus that (*) suffices to disprove Darwinian evolution.

The question presents itself whether there exists a causation that can produce IW; and it is not hard to see that in fact there must—given that horizontal causality comes into play only on the corporeal plane, and therefore constitutes evidently a secondary mode. What, then, is the primary? What else can it be than a causation arising—not from parts or from a wholeness reducible to parts—but from the primary wholeness that antecedes both spatial and temporal division! It is perforce this causality originating on the aeviternal plane—which I designate by the adjective vertical—that bestows the being and oneness upon a corporeal entity which render its wholeness irreducible. One arrives thus at a second conclusion:

(**)  It is vertical causation that engenders irreducible wholeness.

It is to be noted that whereas VC originates on the aeviternal plane, it acts perforce upon the two lower strata, beginning with the intermediary, where it gives rise to what Aristotelians term substantial forms. What proves to be of capital importance is the fact that these substantial forms are endowed with a capacity to exercise a VC of their own, which they can do by virtue of the fact that, as an IW, they preexist on the aeviternal plane. One consequently arrives at a third recognition:

(***)  Vertical causation not only gives rise to substantial forms,
            but can originate from substantial forms as well.

Let us not fail to observe the radical departure of this Platonist ontology from our current “scientific” outlook: the fact, first of all, that instead of the world being spatio-temporal, it is “time and space” that constitute a fragmentation of an inherently aeviternal world. And needless to say, that Platonist reversal cannot but impact our understanding of science to the point of negating our post-Enlightenment Weltanschauung in its entirety.

Finally, it behooves us to recognize the remarkable asymmetry between that ancient outlook and the contemporary. Fundamental questions, first of all, which appear to be virtually insoluble in contemporary terms, turn tractable in a trice when viewed in Platonist terms: just think of the multitudinous phenomena associated with VC—nonlocality for instance—which to the physicist prove incomprehensible! Notwithstanding its vaunted immensity, moreover, there is yet something “puny” about our “brave new universe,” in which there are in fine finali no frontiers worthy of human conquest. The very opposite can be said of the Platonist cosmos, a distant glimpse of which suffices to ennoble our life! What indeed are billions of years and light-years—supposing they exist—beside the tripartite cosmos, which dwarfs the entire spatio-temporal world, first by its “intermediary” realm, and ultimately by the “aeviternal,” which transcends the normal compass of the human mind.¹¹

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I wish now to point out that—in 1998 to be exact—IW has made its appearance in the form of a mathematical theorem of truly epochal significance. It has however done so incognito inasmuch as the theorem is formulated in the context of information theory, a discipline which conceives of “information” in inherently set-theoretic terms. Yet, implicit though it be, the idea of IW enters as the very crux of the theorem. This fact, however, becomes manifest—not, to be sure, in terms of information theory—but on ontological grounds.

I am referring of course to the famous theorem discovered by the mathematician and information theorist William Dembski,¹²  and associated from the start with the notion of “intelligent design.” I shall argue that Dembski’s theorem can in fact be seen as a special case of the ontological proposition (*), which, as you recall, simply states that horizontal causation cannot give rise to irreducible wholeness. What Dembski’s theorem actually deals with, on the other hand, is not IW as such, but complex specified information or CSI, which can be no more than an information-theoretic instantiation of IW.

Yet the fact remains that Dembski’s theorem is epochal in its significance: it suffices, after all, to invalidate—in a single mathematical stroke—the mechanistic worldview that has dominated Western civilization since the Enlightenment. But in so doing, it raises a crucial question of its own: if horizontal causation cannot produce CSI, what is it, then, that can? Dembski and his colleagues seem to have opted for the notion of “intelligent design.” Yet one sees, on the basis of (*), that this does not get to the heart of the matter. The fact is that Dembski’s theorem demonstrates the existence of a causation hitherto unsurmised by the scientific establishment, which in fact does not fit into our “flat” cosmology. The fact is that this hitherto unsurmised causation proves to be none other than what I had termed vertical causality in the context of the quantum measurement problem.¹³

The recognition implicit in Dembski’s theorem turns out thus to be—not information-theoretic—but primarily etiological. It proves that wherever you encounter CSI—which is just about everywhere—you are confronted by an effect of VC. What ultimately stands at issue, however, is not CSI, but IW: irreducible wholeness namely, which is something incomparably more general. In the final count it is IW that testifies to VC. CSI enters the picture only by virtue of the fact that, in the context of information theory, it exemplifies IW. It needs to be understood that not only is the conception of IW ontological, but that it pertains in truth to the deepest ontology of all: i.e., the Platonist, which ultimately situates that primary wholeness on the aeviternal plane. But there is, strictly speaking, no common measure between CSI and IW.

From a metaphysical point of vantage then, Dembski’s theorem affirms precisely that it takes VC to produce IW; the fact that horizontal causes cannot produce CSI constitutes a drastically special case.

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This brings us to a crucial question: what is it, in Dembski’s formulation, that can be identified as the information-theoretic counterpart of “irreducibility”? We must remember that what the ontological theorem excludes is not the production of a wholeness, but of an irreducible wholeness, precisely. How, then, can the notion of “irreducibility” be formulated in information-theoretic terms? It is here that a touch of genius comes perforce into play.

In keeping with the principles of information theory, Dembski configures his theorem in probabilistic terms. He envisages an “event space” Ω endowed with a so-called probability measure P, which to every measurable subset E of Ω assigns a real number P(E) between 0 and 1. The subset E is termed an “event,” and P(E) is its probability. What is needed now is some additional stipulation which elevates E above the status of an “ordinary” event by rendering it—in Dembski’s terminology—detachable. The very word itself suggests that a “detachable” target must be “more” than a mere point set: how otherwise could it be “detached”! What is it, then, that can render a subset to be “more” than a subset: that is the question, which tends evidently to be disheartening—until, that is, the idea strikes that what is actually “more” than just a subset is in fact an intelligible subset, to put it in Platonist terms! What could it be, then, that can render a subset “intelligible”? Well: a pattern, for example, which can be specified by a rule of some kind—a specification, Dembski calls it. And as it turns out: this works!

But not until a second condition is imposed: for it is evident that any nonempty target can in fact be “hit by chance.” To rule out this possibility requires therefore another condition: what Dembski terms complexity. One needs to assume that the probability of our detachable event E is small enough to preclude “accidental hits.” But whereas this stipulation is both necessary and sufficient to validate Dembski’s theorem, it serves only to neutralize the probabilistic setting of information theory. Inasmuch as that probabilistic setting is itself extraneous from an ontological point of view, the function of complexity, if you will, is simply to return the issue to ontological ground. In this optic Dembski’s theorem is seen to be inherently ontological, as we have claimed.

What horizontal causation cannot produce, it thus turns out, is an event that is both complex and specified. An example may suffice to convey the idea in nontechnical terms: if Ω consists, say, of sequences of length 1000, made up of H’s and T’s—the outcome, say, of tossing a coin 1000 times—the stipulation that H’s and T’s alternate would constitute a specification: an exceedingly simple one, to be sure. What is actually “detachable,” moreover, are not the H’s and T’s, which evidently are elements of Ω, but the pattern, precisely. And let us not fail to discern the fact—as subtle as it is crucial—that the pattern as such is indeed “more than the sum of its parts,” and thus constitutes in truth an irreducible whole. Everything hinges upon this razor-sharp point!

It is to be noted that whereas the “parts”—the H’s and T’s—are visually perceptible, the “pattern,” strictly speaking, is not. Beyond a certain threshold of complexity, at least, it is evidently possible to perceive the H’s and T’s in the given order without recognizing the pattern, which in Platonist parlance is termed an idea. There is consequently a fundamental distinction between the “sensible” and the “intelligible,” the “seeing” of which calls in fact for different faculties, corresponding to the distinction between psyche and pneuma.¹⁴ The crucial point is that, in light of the Platonist ontology, the “intelligible” is actually situated “above space and time,” and thus on the aeviternal plane itself. Unbelievable as it may strike the contemporary mind, the efficacy of what Dembski terms a “specification” resides in the fact that its ultimate referent proves to be aeviternal.

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It may be worth noting again that I originally came upon “vertical” causation in the context of the quantum measurement problem, where it plays the decisive role. What renders the so-called “collapse of the wave function” mystifying, it turns out, is the fact that horizontal modes of causality are simply not up to the task: it requires VC to effect this “collapse,” which in truth it accomplishes, not over an interval of time, however short, but instantaneously.¹⁵  This very instantaneity—the fact that the action of VC turns out to be indecomposable—can itself be seen as a signature of irreducible wholeness. Following this initial recognition of VC in the context of quantum measurement, moreover, the action of VC could be detected in various domains of scientific interest, ranging from what physicists term nonlocality to biological phenomena which are likewise inexplicable in “horizontal” terms.

But whereas VC was initially identified by the fact that it acts—not “in time”—but “instantaneously,” it has since become apparent that this “instantaneity” is the result of a still more basic fact, which we henceforth take to be the defining characteristic of VC: i.e., that even as horizontal causation is the causality emanating from parts, so VC is the causality that emanates from aeviternal wholes.¹⁶  Our theorems (*) and (**) are reflective of this fact.

A question obtrudes at this point: having characterized, in (**), the action of VC as productive of irreducible wholeness, one is hard pressed to conceive of “wave function collapse” as an effect thereof. Let us, then, consider this issue. A corporeal measuring instrument M acts upon a quantum system described by a wave function Ψ. What the act of VC accomplishes on the corporeal plane is evidently a change of state in M: a pointer, say, is moved to a certain position.¹⁷  But this evidently does not affect the IW of the instrument, which remains unchanged. How, then, can it be said that the action of VC is productive of IW? To recognize this, one needs to look at what happens to the wave function Ψ, which as one knows—or, in any case, should know—represents an ensemble of potentiae. And now we see the picture: the vertical act in question has been “productive of irreducible wholeness” in the sense of bringing a given potentia associated with Ψ into an IW, an irreducible wholeness—that namely of the instrument M!

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The fact is that, near the end of the twentieth century, VC has made its appearance as a hitherto unrecognized scientific reality—and this unquestionably signals the end of the present era and the imminent commencement of another. To what extent VC will permit itself to be “domesticated” through incorporation into a scientific framework of some kind, time will tell. One thing, at least, we know beyond any doubt: the hegemony of horizontal causation—and thus of physical science—has been broken irretrievably. It strikes me as a safe prognosis, moreover, that our outlook regarding the “ancient superstitions” of mankind may undergo a drastic revision. I find it not inconceivable, even, that somewhere down the road there may be a resurgence of interest in Pythagorean geometry!

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