TWO

Bohm’s Tropology and
Peirce’s Typology

I. OF THE BUDDHIST SELF-REFLECTING STRING OF PEARLS. I engaged in the above digressions before returning to the thread of the Peirce-Hartshorne debate for a specific purpose. David Bohm’s metaphysics, which I shall in this chapter relate to Peirce’s doctrine of signs, cannot be divorced from the concepts of continuity and mind (or self-other). Nor does the mind/matter dichotomy apply. To put it rather cryptically, just as Einstein introduced us to the space-time continuum, so Peirce and Bohm, among others, and from within rather distinct conceptual frameworks, ushered in the mind-matter continuum.

At the outset I should point out that although quantum theory and relativity are incompatible on many counts, they share the fundamental premises that (I) the universe is an interconnected whole, a Monad; dichotomies of space and time, matter-energy and information, gravity and inertia, are actually different aspects of the same phenomena; and (2) there is no such thing as observing—setting oneself apart from—this interactive whole with respect to a neutral frame of reference; necessarily, irrevocably, and paradoxically, we are inside the fabric (Sachs 1988).¹ In other words, the selected cannot be coterminous with the nonselected. Since the early days of the Copenhagen interpretation, it has become evident that modern physics must divide the world not into things but into connections, that is, relations. The world consequently begins to appear “as a complicated tissue of events, in which connections of different kinds alternate or overlap or combine and thereby determine the texture of the whole” (Heisenberg 1958a:107).

Bohm (1957, 1977, 1980) has extrapolated this concept to the extreme. He reminds us that the classical notion of independent atoms making up the universe has been completely overthrown by modern science, hence it should now be asserted that an inseparable “quantum interconnectedness” constitutes the fundamental “reality,” and that to conceive of independently behaving parts is erroneous, for they are merely individual and contingent. This unbroken wholeness cannot be subject to localization, nor can it be explicitly describable: it is, comparable to the pure Monad, simply that which is. Bohm painstakingly argues that the undivided whole is—much like Peirce’s initial (uncut) plastic book of assertions—n-dimensional and atemporal, and it cannot be handled in any way whatever by three-dimensional thought. Indeed, Bohm maintains that we must now develop an entirely new conception of things more distinct from Descartes than Descartes was from the Greeks.

The postulate of undivided wholeness is rooted in a strange feature of quantum theory sometimes called “phase entanglement,” which arises when two or more entities capable of manifesting both wavelike and particlelike characteristics interact (see especially Herbert 1985). Whenever two such entities meet, so also do their representative “proxy waves,” which merge into a whole. Then, after their momentum has carried them great distances apart, their wave phases remain entangled in such a way that an interference effect on one entity will instantaneously be transmitted to the other.² Ordinary waves, say, those of the surface of a pond, do not become phase entangled because, unlike quantum entities, they do not occur in what is called “configuration space.” This space of n-dimensions—rather than one, two, or three dimensions, such as waves along a taut rope, on the surface of the pond, or sound waves, respectively—requires three dimensions for each entity. Two entities reside in six-dimensional space, three in nine-dimensional space, and so on. There is no upper limit. The disconcerting feature of phase entanglement is that, on paper at least, there appears to be action at a distance: in multidimensional space, action on one entity can have an instantaneous effect on another, though information cannot pass between them in the conventional form because it is, according to relativity theory, limited by the speed of light. This apparently unexplainable phenomenon is, precisely, the essence of Bohm’s cosmology (Herbert 1985:168-72).³

Bohm’s undifferentiated whole, which he terms the enfolded, or implicate, order, is comparable to Peirce’s ultimate Firstness or the Monad as what might be before there is any originary cut and mediary consciousness of it. It is continuous and self-contained; it entails the process of coming into being of the thisness of things. On the other hand, the actualized world of particulars, the unfolded, or explicate, order, is secondary to the primary implicate order, as is Peirce’s Secondness to Firstness.⁴ The implicate order, Bohm tells us, indicates an entire realm not hitherto afforded serious attention in physics, chiefly because it demands an entirely new mindset. The equivalent of the unfolded or explicate order was the focus of classical mechanics, whose analysis of phenomena into separate points along the Cartesian coordinates constituted the general understanding of the universe. Relativity and quantum theory have not entirely departed from these classical methods. In contrast, Bohm proposes that in the most general sense, quantum theory calls for an exceedingly broader and more fundamental conception than the classical paradigm: his implicate order. The essential notion of movement, implicit in quantum theory, is, Bohm (1977:39) argues, “not that of an object translating itself from one place to another, but rather, it is a folding and unfolding, in which the object is continually being created again, in a form generally similar to what it was, though different in detail.” Since the infinity of factors determining what an entity is at a given instant changes in the next instant, though ever so slightly, everything is in constant change, and nothing can remain identical with itself over time (e.g., Shiva’s dance) (Bohm 1957:155). The explicate order, on the other hand, is mere appearance (e.g., Maya). It is abstracted from the implicate order on which it depends and from which it derives its whole form and set of characteristic relationships (Bohm 1977:39).

Bohm illustrates the relationship between the two orders with a metaphor. If we put some viscous fluid such as glycerine in a container, place a drop of insoluble dye close to one edge, and stir the mixture slowly with a mechanical device, the dye is “stretched” out in a circle until it seems to disappear. It is still there, but it is now enfolded, implicate. Merely because an entity is not empirically available to us—i.e., nonselected—Bohm hastens to add, is no guarantee that it is not “real”—and here the “realism” of Whitehead and Peirce, discussed in chapter 1, comes into view. Now if we slowly reverse the rotation of the stirring device, the drop reappears in its previous form. It has become unfolded, explicate. The important point is that during the entire experiment, the dye was part of the whole of things. While in its enfolded (nonselected) condition, it existed in an unactualized state. It was simply there potentially to become explicated (selected) (Bohm 1980:149-56). The combination of the implicate order (possibilities) and the explicate order (actualities) is what Bohm (1957:133-35) terms the qualitative infinity of nature. Much like Whitehead, he argues that there can be no end to the levels of interconnected networks, from the infinitely great to the infinitely small, such that there is no end to the number of perspectives. In fact, Bohm’s implicate might conceivably be an explicate from somewhere else in another dimension.

In his early work, Bohm relates the qualitative infinity of nature to a background and substrate (recall Peirce’s and Whitehead’s concepts of “background” and “mutually implicative”). Every entity, however fundamental, depends for its existence on the maintenance of appropriate conditions in its infinite background and substructure. These conditions, in turn, are affected by their mutual interconnections and interactions with the entities under study, hence the explicate and the implicate cannot be separated. In this manner, the seamless fabric of the implicate exercises an influence on a set of explicate entities, and vice versa. This interaction Bohm (1957:144) calls the reciprocal relationship. Such a replete universe reminds one of the rabbinical idea of a scroll where the blank spaces take on importance equal to that of the actual ciphers.

Bohm (1980:186-89) also speaks of the implicate order as a higher-dimensional reality which projects into lower-dimensional elements to endow them with nonlocal and noncausal relationships. This higher-dimensional order obviously approaches randomness. But it is not absolutely random, because it possesses regularities capable of generating a myriad diversity of entities as space-time events. Nonetheless, as Bohm takes pains to argue, it is n-dimensional and atemporal, hence it cannot be cognized or handled in any fashion whatsoever by conventional thought and by the languages, both formal and natural, now used in physics. It is available solely to intuitive faculties, if at all. But, of course, we have already heard about such a higher-dimensional reality: Peirce’s logic of vagueness, the logic of the universe, and Firstness as a continuous cosmic whole, all of which lie outside our own logic, analysis, and reason (CP:5.505-508, 6.185-213; also Smullyan 1952; McKeon 1952; Weiss 1952; Nadin 1982, 1983).

Bohm (1980:186-89) offers another visual trope for his nonvisual, and rather ineffable, higher dimension. Imagine two TV screens in a room showing a fish in a tank, one projected from the side of the fish and the other head-on. As we stare at the screens, we notice that when the fish on one screen moves slightly, the fish in the other instantaneously does the same, but at a 90° orthogonal to the first fish. Were we to be unaware that the two cameras were trained on the same fish from different angles, this behavior might be rather disconcerting. Obviously, there is no temporal cause-and-effect sequence, but some simultaneous interaction between the two bodies bringing about reciprocating movements. The problem is that we are viewing the TV screens as if each were a two-dimensional world, while the cameras, from a higher dimension, have created what appears to be an unexplainable phenomenon. This higher dimension, like the background and the implicate order, is inaccessible to linear reasoning.

Recall the above discussion of the graph constructed in figure 1 from the elementary problem of an object’s acceleration in time. Spatially the lines on the sheet are two-dimensional, but they imply, along one of the coordinates, a third dimension: time. From our three-dimensional perspective this time dimension has no movement, it is entirely static. It is comparable to our own three-spatial plus one-temporal world conceived as the timeless Minkowski “block” lying outside our perceptual grasp. Or, to use Peirce’s trope, it is as if a three-dimensional Spherelander were viewing the cuts on a given sheet from some point in the book of assertions, while a Flatlander, confined to the two-dimensional world of one of the book’s sheets, would be incapable of such a perceptual grasp.⁵

Bohm elaborates on an even more intriguing trope, relating the combination of his implicate and explicate orders to a hologram. The hologram operates on the principle of interfering wave patterns. A laser beam is directed toward a half-silvered mirror in such a way that part of the beam passes through, striking a photographic plate, and part is reflected onto the object to be photographed. The light reflected from the object strikes the photographic plate at an angle with that part of the beam which passed directly through the mirror to produce a complex array of interference patterns. When the plate is developed and then illuminated, again with laser light, the object reproduced on the two-dimensional surface appears to possess the three-dimensional characteristics of the original object. The intriguing feature of laser photography is that there is no one-to-one correspondence between the parts of the photographed object and the parts of the photographic plate. Rather, the interference pattern of each region of the plate is equal to the whole. If the corner of a plate is broken off and illuminated, it will reproduce the entire object, though the smaller the piece the more vague the reproduced image becomes.

Actually, Bohm’s hologram metaphor is a fixed image of the state of an electromagnetic field—a set of interacting waves—in the flat space containing the photograph. But this fixed state is not static. It is a shimmering, dancing bundle of standing waves in incessant movement. The sum of its myriad moves can be described as atemporal, yet there is pure change. In order that the hologram model be true to the combined implicate and explicate orders, which Bohm labels the holomovement, it must be conceived as both a static set of interference patterns and the unfolded three-dimensional image jumping in and out of the two-dimensional plane: it is perpetually enfolding and unfolding. The quantum formulation, Bohm argues, is precisely the mathematical description of this holomovement. But in quantum theoretical terms, there is no physical notion of movement in the ordinary sense. The physicist uses his mathematical metaphor because it produces results, but there is no meaning outside the mathematical symbols. In Bohm’s words, on constructing a language with which to describe the implicate order, the holomovement must be

considered as the totality in which all that is to be discussed is ultimately to be [foregrounded]. Similarly, in the algebraic mathematization of this general language, we consider as a totality an undefinable algebra in which the primary meaning of each term is that it signifies a “whole movement” in all the terms of the algebra. Through this key similarity there arises the possibility of a coherent mathematization of the sort of general description that takes the totality to be the undefinable and immeasurable holomovement. (Bohm 1980:164)

The language of the holomovement, then, can be none other than algebraic, which is to say that it is imageless and by and large incompatible with natural language: as a whole it is a pure metaphor, or allegory if you will, of that which it signifies. Bohm discusses the possibility of “relatively autonomous sub-algebras” which entail aspects of the very general and undefinable “whole algebra.” As each aspect of the holomovement enjoys only limited autonomy, so the sub-algebras are dependent upon the larger formulation. It is interesting to speculate that, just as Bohm’s sub-algebras take on meaning in the context of research programs and only in regard to the general and undefinable whole algebra, so also what Peirce calls subjective (classical) logic exists only with respect to, and is dependent upon, the exceedingly more general and more complex logic of the universe (CP: 6.189). With the holomovement, like the logic of the universe, a form of “objective reality” is in a sense restored: what is not (Firstness or Thirdness, both in their pure state continuous) is as “real” as what is (Secondness, actuality, discontinuity). The ongoing schizophrenia between particle/wave, matter/mind, continuity/discontinuity no longer exists in the classical sense.

In order to qualify this assertion, we must return briefly to Hartshorne’s critique of Peirce.

II. OUR RESTLESS WORLD. For Peirce, as we noted in chapter I, time and space are continuous because they embody conditions of possibility, and since the possible is general, and generality is continuity, they are two names for the same absence of distinction. Peirce’s failure to anticipate quantization in physics was a grave error, Hartshorne tells us—though this should actually be no surprise, for after all, the most eminent physicists of the nineteenth century failed in this respect as well. Hartshorne argues, as we have noted, that Peirce believed the past to be particular, the future irreducibly general or potential, and the present somehow both. Given, possibility entails futurity as such, which is commensurate with Peirce’s tychism (the doctrine of chance). But Hartshorne (1973:193) observes that Peirce’s indeterminism or tychism, rather than an arbitrary feature of his thought, was “central and essential to it.” That might be fine and dandy, Hartshorne suggests. But, he warns, to say with Peirce “that possibility as well as actuality is real . . . is not to say that possibility is exhaustively actualized.” Hartshorne goes on to state that Peirce’s real problem is that “Peirce was beautifully clear as to the contingency of actuality as such and the continuity of possibility as such, but somehow he was unclear as to the implied discontinuity of contingent actuality” (Hartshorne 1973:196).

Contrary to Hartshorne, and in light of the above discussion of quantum theory according to Bohm’s interpretation, space and time can quite legitimately be conceived as continuous: the former embodies the potential position of an entity and the latter potentially the instant of an event. The future in this regard is a continuous, general set of possibilia for contingent actualization in the present. Moreover, Bohm’s qualitative infinity of nature represents a potentiality enfolded within the implicate order for an unlimited set of actuals, given an indeterminate lapse of time. In this regard, when Peirce is placed alongside Bohm, there is no call for claiming he proposed that the actualized world of empirical entities is at any given point in time continuous.

For Peirce, the coming into being of actual entities is contingent: after having become, they are discontinuous, but as contingent possibilities they were-are-will be continuous. And the conjunction of continuity and generality implies the absence of discrete actuals. Yet, over unlimited time, the becoming of actuals—in order to realize the Final Interpretant—would certainly constitute a continuous domain. The problem stems from Hartshorne’s contention that Peirce affirmed (1) an actual continuity engulfing an infinity of entities, and at the same time (2) the continuous becoming or actualization of entities.

Statement (1) is in reality not the case, as Hartshorne poses it at least. Actualized entities constitute a discrete order whose existence is dependent upon a higher realm, the potential for future actualization. I speak here of fundamentally the equivalent of Bohm’s qualitative infinity of nature, a composite of the implicate (nonselective) and explicate (selective) orders, of a continuous and discontinuous domain. Peirce’s continuous logic of the universe is copresent with entities actualized in the world at a given moment, and like Bohm’s implicate order, it is infinite, and in a manner of speaking “actual,” though not empirical. The vast majority of the entities in the potentia remain implicate at any given point in time, yet the possibility exists, however remote, for their being explicated somewhere at another time, hence the conjunction of potentia and indefinite time is unlimited (the term potentia is, as pointed out in chapter I, section I, the Aristotelian term applied by Heisenberg [1958a] to the quantum world).

That is, the explication or actualization of entities is for Peirce a process of continuous becoming. Once having become, a given set of entities at a particular point in time and space can be no more than discrete. Yet over unlimited time and space, the total set of actualized entities must make up none other than a multitudinous whole—though whether or not it is dense might remain an open question. Hence (2) is somewhat deceptive, for though becoming, over the indefinite long haul of things, is continuous, that continuity, like the implicate order, is not empirically available.

Hartshorne (1973:194) argues further that the Heisenberg uncertainty principle cannot be effectively explained away by postulating “hidden variables.”⁶ At most, Hartshorne claims, the hidden-variables interpretation reveals the nonempirical characteristic of actualized entities and the statistical characteristic of possibilities. The only valid reason for accepting this interpretation is that otherwise there is no coherent logic for the notions of possibility and actuality. Peirce, we are told, argued for the impossibility of exact measurement, given continuous statistical variables, hence there could be no determinism. However, what he failed to recognize is that “if the world has both continuous and discontinuous aspects then, a fortiori, exact causal laws cannot apply. For the interplay of continuous and discontinuous aspects forbids such laws” (Hartshorne 1973:194). In support of his critique of Peirce, Hartshorne turns to the example of photons of light partly reflected and partly refracted when passing through a medium such as glass. A given photon’s being either reflected or refracted is determined probabilistically by the angle of incidence, but since this angle can vary continuously, the discontinuity between reflection and refraction is explainable solely by a probabilistic account.

However, consider an alternative combination of continuity and discreteness: the double-slit experiment. A stream of electrons is fired at a thin sheet of metal punctured by two minuscule holes with a screen behind the sheet to detect the resultant pattern. If one hole is closed, some of the electrons pass through the other hole and strike the screen in a manner comparable to bullets from a machine gun passing through a hole in a barrier and striking a thick plank of wood behind it. They behave as particles. Now imagine that both holes are left open. We normally would expect there to be two patterns, one behind each hole, duplicating each other. But this is not the case. An interference pattern is set up in the space immediately behind the orifices which, on the screen, appears as if water waves had been passing through two slits in a barrier. The electrons, formerly behaving as particles passing through one hole, now act as waves interfering with one another after passing through two holes. The first trial demonstrated the electrons’ discreteness, the second their continuity. And the mixture of the two entails the essence of Bohr’s complementarity principle. In the first case classical laws inhere; in the second they do not. The first evinces the image of actualized entities in the present which make up a determinate past; the second, as waves, consists of a continuous set of statistical possibilities until they become actualized and take on their particle personality upon reaching the screen.

Bohm insists that this combination of classical and postclassical characteristics, a contradictory combination of causal and statistical phenomena, testifies to the incomplete nature of Copenhagen quantum theory. It is, he argues, the surface manifestation of a deeper, subquantum level: his implicate order of incessant flux, the laws of which are qualitatively different from those of the quantum level. Hence there is no reason why quantum equations describing either the wave characteristics or the particle characteristics should have any relevance at the lower level (Bohm 1957:111-16). If Bohm is correct, then this deeper level is every bit as “real” as the quantum level, though it contains no existing (that is, actualized) entities.⁷ And, to extend our parallel, commensurate with Bohm’s subquantum level, Peirce’s logic of the universe as a continuous infinite or quasi-infinite realm of possibilities exists in the present as well as the future and the past.

Significantly, in this regard, like Bohm’s holomovement, an important facet of this deeper logic is not formalizable with conventional algebras and calculi. And neither is the generality of Peirce’s conditions of possibility in space and through time, which exclude exhaustively describable distinctions. I see no reason, then, for Hartshorne’s somewhat disparaging remarks on Peirce. Quite frankly, either to criticize Peirce for his failure to divine the essence of Copenhagen quantum weirdness or to denounce Hartshorne for his apparent ignorance of Bohm’s anti-Copenhagen interpretation would be baseless—though actually, Hartshorne’s paper was published when Bohm was still relatively unrecognized outside a small group of practicing physicists. The fact is that Peirce’s concept of continuity and originary Firstness is strikingly compatible, though he is not clear on some points, with the particular aspect of Bohm’s quantum cosmology being discussed here.

Another aspect of Peirce’s notion of continuity sheds broader light on the issue at hand. Peirce believed that the most convincing reason for his continuity postulate rests in its accounting for how an entity, be it mind or matter, can act continuously upon another entity (CP: 1.170). Hartshorne retorts that the postulate of a finite number of events between two entities (or minds) does not discount the existence of internal relations connecting these entities, hence Peirce’s notion of a continuous infinity of infinitesimals connecting them is cumbersome, contradictory, and unnecessary. The potentiality of events is continuous in space and time, not their actuality. Following Whitehead, Hartshorne tells us that if consideration is limited to the future potentiality of an entity, then it spreads out finally to pervade the universe; its actuality, on the other hand, is discrete. Although Peirce is vague on the spatio-temporal atomicity of actual events, the problem seems to be that he speaks of continuity at one level while Hartshorne refers to it at another level.

Regarding what Hartshorne conceives as Peirce’s illegitimate collusion of temporal continuity and discontinuity, he elsewhere alludes to the absurdity of an infinity of experiences between two events, say, having breakfast and having lunch (Hartshorne 1964:469). I believe it is now quite apparent that this “dilemma” is also common to contemporary physics, and as far as the physicist knows at present, it might be endemic to the very nature of reality. For Planck light was of discrete and continuous character. Combining the wave mechanics of de Broglie and Schrödinger with Heisenberg’s matrix mechanics, somehow electrons also seemed to be both continuous and discrete. Bohr’s complementarity metaphysically brought the incompatible descriptions together, and P. A. C. Dirac even combined them mathematically (Sachs 1988).

But problems remained. In fact, Zeno’s paradoxes, we have observed, are germane to the very issue. His immobile arrow in flight is predicated upon the idea that just as space can be conceived as consisting of an infinity of contiguous points, so time can be an infinite collection of contiguous instants. True, our experience of the arrow in flight appears to us to be strictly continuous, but perhaps it actually consists of William James’s “drops of experience” which merge into one another to present the sensation of continuity. Then how can Zeno legitimately arrest the arrow in flight and then proclaim at the end of his argument that it never could have moved from one point to another in the first place? Physically the transition in question may be possible, but there is nothing to prevent Zeno’s rendering it impossible by a sheer act of mind. On his so doing, the continuum is discretized by a contradictory mixing of two incompatibles, which is no problem for mind, though experience and intuition rebel against it (Dantzig 1930:124-27).

For example, in modern times the mind, via the calculus, can easily describe a continuous curve as a series of infinitesimal straight segments, an abstraction which is also no more than a skeleton of the world as perceived by our senses. And fortunately for us, a host of “real world” problems are solved by such fictions, which the mathematician has been trained to use for so long that he may come at times to prefer the substitute to the genuine article. Even the practical person who ordinarily demands at least the appearance of “reality” tends to regard mathematical terms not as symbols or thought—and much less fictions—but as images validly representing “reality.”

Yet, Peirce’s contradictory mixing of continuity and discontinuity, of infinity and finiteness, is, I would submit, quite appropriate to the task at hand. In the first place, independently of Richard Dedekind, and anticipating somewhat the work of Georg Cantor, Peirce established a logical distinction between finite and infinite collections, which was later to become the basis of set theory. The breakthrough was that an infinite set is equivalent to a proper subset of itself, whereas a finite set is not. For example, the set of natural numbers is equal to its many subsets, which can be illustrated by setting up correspondences:

These series are paradoxical if one reflects on the meaning of the statment “Two (infinite) sets have the same number of elements.” However, Cantor proposed that two sets could be considered to have the same (cardinal) number of elements if they are equivalent, which follows the ordinary usage of the term set quite closely. If a hotel has rooms with single beds only, and if there is an occupant for each room, one can say that there are the same number of rooms as people; whatever number is assigned to the collection of rooms must also be assigned to the collection of people.⁸ By the same token, it follows that the set of even numbers is equivalent to the set of natural numbers, which goes against common sense as well as traditional mathematical conceptions.

Nonetheless, Cantor’s contribution—which in part can be attributed to Peirce, though much of the latter’s work in this regard awaited posthumous publication—was to illustrate that the statement “The whole is greater than any of its parts” is false when applied to infinite sets (for example, CP:6.115). This concept is embodied in the Tristram Shandy Paradox. Shandy labored for two years chronicling the first two days of his life, lamenting that at this rate no matter how long he continued to write, he would be further and further from the completion of his autobiography. To this Russell (1957:85-86) retorts: “Now I maintain that, if he had lived forever, and had not wearied of his task, then, even if his life had continued as eventfully as it began, no part of his biography would have remained unwritten.” Finite and infinite sets, so it appears, are absolutely incompatible, which adds more brush to Hartshorne’s bonfire.

However, we must note that, mathematically speaking, there are two kinds of infinity: actual and potential. And there are three basic kinds of series: nondense and denumerable, dense and nondenumerable, and neither dense nor denumerable (see Huntington 1929). Peirce’s possibilia, like Bohm’s implicate order, is an infinitely extended, nonselective domain each subset of which is equivalent to the whole. Peirce’s actuals, like Bohm’s explicit order, constitute a finite selective domain each subset of which is not equal to the whole, but which, given unlimited time, can in asymptotic fashion approximate the equivalent of the implicate totality. The first order, it would appear, is tantamount to the actual infinite; the second is potentially infinite in extension. These two domains also pattern the conflict, discussed above, between relativistic continuity and quantum theoretical discontinuity, or in another way of putting it, between the dense and the nondense, and the nondenumerable and the denumerable. The series of whole integers and a collection of nonoverlapping three-dimensional regions of space are dense and at most denumerably infinite. An example of a dense and nondenumerable series is the class of all nonterminating decimal fractions. A series which is neither dense nor denumerable is that of a line bisected ad infinitum in the impossible Zeno fashion to yield an infinite series of infinitesimal segments (however, Peirce was not in agreement with Cantor on all points [see Eisele 1979; Murphey 1961]).

The type of dense, nondenumerable series to which I wish to refer here is that of Dedekind continuity. Dedekind noted that a straight line consisting of an infinity of points is infinitely richer than the domain of rational numbers: between each set of points there is an infinity of points, whereas there are gaps between rational numbers. The line is complete and continuous, while the series of numbers always remains incomplete and discontinuous. Dedekind then posited a “cut” in the line which severs it into two portions or classes, one to the left of the “cut” and the other to the right of it. It follows, he argued, that just as any point on the line severs it into two contiguous, nonoverlapping regions, so every real number constitutes a means for dividing all rational numbers into two classes which have no element in common, but which, in combination, exhaust the entire domain of rational numbers (Dantzig 1930:164-78).

Cantor’s concept of infinity has its counterpart in the Dedekind theory. But there is a fundamental difference. Cantor discovered a property Tobias Dantzig (1930:148) calls the “self-asymptotic nature” of a convergent sequence, which can be illustrated by the following:

1/2, 3/4, 7/8, 15/16, 31/32, 63/64, 127/128, . . . . . . . .

3/4, 7/8, 15/16, 31/32, 63/64, 127/128, 255/256, . . . . . . . .

7/8, 15/16, 31/32, 63/64, 127/128, 255/256, 511/512, . . . . . . . .

Generation of this collection can obviously be continued indefinitely, with all sequences being asymptotic to one another. Noting that these sequences are asymptotic to the series of rational numbers, Cantor identified them with the real number series. In this light, Cantor’s generation ad infinitum is dynamic. Numbers tend to grow beyond measure toward a limit. In contrast, Dedekind’s line is static; it is there, in bloc for all time, and when punctuated with a “cut,” though having suffered a mutilation, the left side and right side remain eternally the same, a continuum of points smoother than the richest vanilla ice cream. Cantor’s set of convergent sequences, it hardly needs mentioning, is also directly relevant to Peirce’s notion of the potential completion of the Final Interpretant and the approximation of knowledge to truth (see Rescher 1978).

In Peirce’s conception of continuity, time eventually enters the scene. He believed that multitudes correspond to linear series, and that the potential aggregate of elements is infinitely greater than any aggregate of actualized elements. Yet, as a potential, it does not “contain” any elements at all. It only “contains general conditions which permit the determination” of elements (CP:6.185). This potential constitutes Peirce’s notion of a “true continuum,” which “is something whose possibilities of determination no multitude of individuals can exhaust” (CP:6.170). Drawing a set of individual elements from the potential can never exhaust it, though it can successively approximate it. Peirce adds that this nature of infinity does not spell out any categorical realm of ignorance on our part. Rather, we can have a vague idea of the totality. Take, for instance, the series of whole numbers. The aggregate of all such numbers cannot be counted within a given person’s lifetime, nor can it be counted by an entire community of knowers short of infinite in number. But “though the aggregate of all whole numbers cannot be completely counted, that does not prevent our having a distinct idea of the entire collection of whole numbers. It is a potential collection, indeterminate yet determinable” (CP:6.186).

In this fashion, Peirce’s concept of knowledge cannot but be very imperfectly realized so long as it is confined to actual entities. In contrast, an infinite community of knowers pools its resources into the infinitely distant future containing all possibilities theoretically to arrive, at last, at Truth. Ultimately the focus rests not on existent individuals but on classes of possibilities and potentialities, not on what is but on what might be and would be. In this sense also, and contrary to Hartshorne, possibility, as well as actuality, enjoys certain citizen’s rights in the “real” world.

Further to demonstrate how mathematics pervaded Peirce’s thought, this general notion also bears on Peirce with respect to his concept of personality and consciousness. Personality is like any general idea or an infinitely extended potential. It is not something to be apprehended in an instant; it must be lived in time. Nor can any finite duration of time embrace it. Nonetheless, in each infinitesimal interval of time “it is present and living, though specially colored by the immediate feelings of the moment” (CP:6.155). It is incessantly growing, developing toward some end. Hence there is reference to the future, but that future in terms of the end never arrives. Were it to be explicit in the here-now, “there would be no room for development, for growth, for life; and consequently there would be no personality” (CP:6.157).

Regarding consciousness, Peirce also evokes the mathematical notion of continuity.⁹ He reasons that though a past idea cannot be in present consciousness but only in past consciousness, the idea can nonetheless be present insofar as it is always in a process of becoming past; it is always less past than any assignable date in the past, that is, if it is still in recollective consciousness (memory). This implies, Peirce concludes, that “the present is connected with the past by a series of real infinitesimal steps” (CP:6.109). Now this and other equally enigmatic statements are the prime focus of Hartshorne’s critical remarks. How can we account for an infinite number of infinitesimal steps between breakfast and lunch? How can we even speak of infinitesimals and the nuts and bolts of the physical world all in one breath?

But Peirce continues undaunted. He points out the rather obvious: consciousness necessarily embraces a certain interval of time. If not, we could gain no knowledge of time, “and not merely no veracious cognition of it, but no conception whatever. We are, therefore, forced to say that we are immediately conscious through an infinitesimal interval of time” (CP:6.110). In this infinitesimal interval, not only is consciousness continuous in a subjective sense, but the object of consciousness, the idea in question, is also continuous, just as all signs (i.e., in their fulfillment as interpretants) and minds are continuous in the sense that they merge into one another and are thus welded together. In fact, this spread-out consciousness is patterned by the spread of its contents—in somewhat comparable fashion to the contemporary concept of space expanding as the universe, which contains space and is in turn contained by it, expands.

Peirce attempts to illustrate his heady idea of signs welded to signs, minds to minds, and consciousness to its contents with what are today known as the topological concepts of boundaries and neighborhoods. He asks the reader to imagine a surface bounded by a curve, say, a circle, to be half red and half blue. The red and blue surfaces, to exist as such, must have red and blue spread over them such that the color of any part of the surface is the color of its immediately neighboring parts. What color, then, is the boundary? Peirce responds that since “the parts of the surface in the immediate neighborhood of any ordinary point upon a curved boundary are half of them red and half blue, it follows that the boundary is half red and half blue” (CP:6.126). That is to say, to integrate Peirce’s example with another of his thought experiments (CP:6.203-206), it can be said that the boundary, a line separating blueness (B) from redness (R), is actually neither B nor R: it is BR-lessness. However, this BR-lessness is precisely the property that unites B and R. So there must be some commonality shared by both B and R embodied in BR-lessness, which implies that it is somehow in some sense both B and R, which Peirce would deem “half B and half R.”

This is a difficult pill to swallow, though it must be admitted that BR-lessness welds B and R together by some property, however vague, which they share. Such a property is metaphorically comparable to the Sheffer (1913) “stroke” function, to be discussed in more detail in chapter 4, the essence of which was anticipated by Peirce (Berry 1952; also Merrell 1985a, 1985b, 1987). The “stroke,” “l”, would indicate that BR-lessness denotes “not both not B and not R” or “neither B nor R.” In such case BR-lessness would either oscillate between B and R—hence as Peirce says it would be half B and half R—or it would disclaim membership in either of the two surfaces—hence it would not evince any commonality between B and R and therefore could not be in possession of any power to combine them. It appears that if the latter were true, then there could be no meld of the two and they would stand as incompatible, with an insurmountable barrier between them (see appendix 1).

However, if we avail ourselves of Peirce’s idea that consciousness has the capacity naturally and intuitively to overcome Zeno’s Achilles and the tortoise paradox, this barrier is no real problem. Peirce observes that many scholars had thought it inconceivable that Achilles could pass through an infinity of points (or, following our example, lines separating a series of shades of blue and red). This, he remarks, “does not embarrass Achilles the least in the world, for his final effort carries him through a whole infinity of the points” (CP:2.27). In other words, during the temporal increment required for Achilles’ final lurch, there is, according to Zeno’s iterative slicing of time, an infinity of infinitesimal instants. And space, having been sliced accordingly, evinces an equal infinity of infinitesimal points. Achilles’ final bound can therefore take him as easily through an infinity of space barriers as through an infinity of time barriers.

This introduces us directly to Peirce’s exemplification of continuity in general by way of the model of time as continuous. Peirce frequently engaged in analysis in terms of time series. He also alluded to something like Cantor correspondences in reference to his idea that all signs, minds, and “semiotic objects” are welded together such that they lose their autonomy and identity. These series and correspondences include multitudinous collections that are transfinite, indeed, even beyond Cantor’s Alephs, Peirce claims, in character. For example, the horizontal and vertical series

increasingly becomes larger and larger in one direction and smaller and smaller in the other. There is no end. The series simultaneously shades into the infinitely great and the infinitely small. This principle of merging appears in many guises in Peirce’s thought: growth, evolution, semiosis, consciousness, mind, color and sound intensity, instinct and reason, nature and culture, and above all, time. The successive variation (differentiation) is continuous, not discrete. Herein lies the essence of Peirce’s “real” continuum of possibilia (the implicate) from which all things can be explicated. And herein Peirce’s convergence theory of truth comes fully into view: the continuum (nonselective) domain plus the aggregate of all (selected) particulars, like Bohm’s holomovement, is Peirce’s true universal.

Peirce (CP:6.164-84) curiously defines his continuum as consisting in Kanticity and Aristotelicity, the first positing that there is at least one point between any two points, and the second that the infinite series contains every denumerable point that is a limit to the nondenumerable point in the system. Kanticity is proper to the Dedekind line; it is essentially static, all there all at once, to be cut according to the wishes and whims of the geometer. No other principle is necessary except the power of the mind to categorize entities along a definite scheme which is totally divorced from time. In contrast, Aristotelicity is Cantorian: dynamic. It entails a tendency to grow, to generate, to converge toward something. It necessarily moves within time. The first is time frozen (the Final Interpretant or Third), the second is pure chance, pure spontaneity (the Monad or First), which lends itself to the actualization of particulars in the “real” world. The first implies legato, a continuous harmonic unfolding (real numbers). The second implies a symphony of staccato, represented by successive punctuations in the continuum (natural numbers). Peirce at times appears to have hoped to wed the two by accelerating staccato until it merged into legato, thus explaining both away by their resolution into their opposites: a conjunctions oppositorum. Whether he succeeded or not is, as far as I know, still an open question, Hartshorne’s critique notwithstanding. The fact is, to repeat, that the continuity-discontinuity conflict with which Peirce and others struggled during the latter quarter of the nineteenth century and the beginning of this one has been abruptly foregrounded in the relativity theory-quantum theory logjam. To this theme I now return, as the dialogue between Peirce and Bohm continues to unfold.

III. THE PRIMACY OF MIND. To repeat, though quantum theory and relativity theory differ vastly on many accounts, in a deeper sense they share the implication of undivided wholeness. The Einsteinian space-time field is continuous, indivisible, particles being regarded as an abstraction from the field. They are singularities, or space-time “knots.” A description of the relativity universe entails the structure of such singularities in the continuum field. In terms of the observer-observed relationship, the field of a particular singularity, constituting an object, merges, but not in simultaneity, with the field of the observer-as-singularity. This is actually a different sort of wholeness from that of quantum theory—e.g., the instantaneity of the “phase entanglement” interpretation discussed above in contrast to the nonsimultaneity of relativity. Consequently, in spite of their deep compatibility, relativity and quantum theory have not yet been effectively united to the general satisfaction of the scientific community, the chief reason being that extended structure has not been consistently introduced into relativity (Bohm 1980:111-39). Yet the concept of space inherent in relativity, as I understand it, is not entirely alien to Peirce’s continuum of mind, especially in the sense that minds can influence one another mediately, not immediately, through their overlapping extendedness. And since the character of sign is that of the character of mind, mind itself being a sign, Peirce’s notion of signs merging into one another becomes equally plausible (Hartshorne 1973:199).

A distinct relatedness between minds and the phenomena of the physical world exists in quantum theory. Recalling the double-slit experiment, if the electrons passing through one hole behave as particles, and if passing through both holes they take on wave characteristics, then how could an individual electron at the point of penetrating the barrier know that another electron was at the other hole in order to change itself into a wave? Somehow there must be an interconnection (e.g., “phase entanglement”) between the particles approaching the metal sheet at different points. Quantum weirdness reaches a climax here. It marks the demise of classical objectivity: what happens to the electrons in the double-slit experiment is simply nonobservable in the ordinary sense. The electrons approaching the sheet appear to behave as if they were particles, but upon entering and leaving the holes they are like waves.

But actually they were waves all along, that is, if they went unobserved. For example, assume we are reduced in size sufficiently so as to be able to see individual electrons. Walking along in front of the sheet, we can see the swarm of particles obediently headed toward their destination. We reach the barrier and peer on the other side, expecting to see nothing—for, we assume, there is nothing but waves. To our surprise, an apparently chaotic array of particles heading for the screen meets our eye. What has happened? Our observing the “wave packets” has “collapsed” them into particles. In other words, the “electrons” were never electrons in the particle sense until they struck the screen, the interaction of which actualized them—i.e., the “wave packets” were “collapsed.” What we thought were “particles” in front of the sheet were viewed as such only when we observed (interacted with) them. And when we assumed we would see “nothing” between sheet and screen, there was in reality “nothing” to see, but when we took a peek, electrons as particles jumped up to greet us. Quantum reality demonstrates that there is no objective world “out there”: any world we can know is partly created by the observer.

The double-slit experiment illustrates quantum interconnectedness according to the original Copenhagen interpretation. Bohm, however, carried the interconnectedness idea a giant step further with his more general cosmology, which also, he believes, can incorporate the universe of relativity. From Bohm’s holographic perspective, and with Hartshorne’s critique in mind, it becomes apparent that there is nothing unbearably strange in Peirce’s contention that minds “spread” and act on one another instantaneously. His law of mind stipulates “that ideas tend to spread continuously and to affect certain others which stand to them in a peculiar relation of affectibility. In this spreading they lose intensity, and especially the power of affecting others, but gain generality and become welded with other ideas” (CP:6.104). From what Peirce believes to be a spread of feeling found to occur in protoplasm, he generalized to the human semiotic animal, holding that feeling “has a subjective, or substantial, spatial extension” (CP:6.133). Further, “since space is continuous, it follows that there must be an immediate community of feeling between parts of mind infinitesimally near together” (CP:6.134). This explanation, Peirce asserts, coordinates the action of the nerve-matter of the individual brain with that between minds external to one another. Just as there is action between parts of the same mind that are continuous with one another, so one mind can act upon another, “because it is in a measure immediately present to that other; just as we suppose that the infinitesimally past is in a measure present” (CP: 1.170).

A problem, however, rests in Peirce’s belief that like signs, when thoughts, memories, and minds spread, they suffer a loss of intensity. Quantum wholeness is not a rerun of the Newtonian drama in which everything is instantaneously connected to everything else by means of a mysterious force—gravity—across distances. Gravitational connections diminish with distance according to the inverse square law, thus endowing nearby connections with overwhelming importance while distant connections become relatively insignificant. In contrast, quantum interconnectedness, undiminished by spatial and temporal separation, is altogether different. Phase entanglement indicates that an electron from a bench in Central Park can intermingle with another in Gorky Park as forcefully as it can with one of its neighbors from the same bench. In this respect alone, Peirce’s action of one sign or mind on another appears to be strictly Newtonian. If Peirce was incapable of foreseeing the consequences of contemporary physics, it is principally in this regard, rather than his synechistic doctrine, as Hartshorne argues.

On the other hand, in light of Peirce’s “law of mind,” Bohm, drawing support from Karl Pribram, also suggests that the mind (consciousness) and physical reality are of one whole, and at the implicate level they are governed by a comparable “algebra.”¹⁰ Bohm notes that matter and mind (consciousness) evince certain commonalities, chief of which is their holographic character: both entail implicate and explicate orders.¹¹ First, various forms of energy, such as gravity, magnetism, light, and sound, constantly enfold information regarding the material world into particular regions of space. These energies are felt (Firstness) as the pull of gravity, the attraction of a magnet, light reflected from a stop sign which impinges on the retina, and an auto horn when one steps from the curb onto a street before an oncoming vehicle. Such information enters the sense organs and passes through as such and such (Secondness), which is then cognized (Thirdness). The structure of the information, in other words, is enfolded into the brain—somewhat comparable to music being enfolded into the grooves of a disk, later to be retrieved and reproduced by the speaker system. Hence, is not all matter, Bohm asks, tantamount to enfolded information from various energy sources? In Peirce’s terminology, this is rather comparable to asking: “Is matter not truly effete (enfolded) mind?”

It is certainly plausible that the processing of sensory input resembles an image-constructing device analogous to holographic processes. Moreover, memory, according to Pribram’s thesis, appears to be enfolded in comparable fashion, all over the brain rather than localized.¹² Bohm (1980:198) goes even further to suggest that “when the ‘holographic’ record in the brain is suitably activated, the response is to create a pattern of nervous energy constituting a partial experience similar to that which produced the ‘hologram’ in the first place.”¹³ If Bohm is correct, not only is the brain a repository of “mind-stuff,” but so also is the entire body. Much in the Peircean sense, when one instinctively jumps back onto the curb upon hearing the auto horn, one has resorted to the reflexes of habituated mind. The body acted without there being any conscious and intentional will on the part of the mind to cause the body to so act. From human to ape, from horse to aardvark to amoeba and the plant kingdom, in the Peircean sense all such action on the environment is ultimately the product of jelled mind.

Peirce was critical of the notion, common during his day, that “an idea has to be connected with a brain, or has to inhere in a ‘soul.’ “This is preposterous, he charges. An idea does not belong to the soul; it is the soul that belongs to the idea. The soul does for the idea just what cellulose does for the beauty of the rose; that is to say, it affords it the opportunity to do its thing (CP:1.216). Thought, Peirce argues further, is not necessarily connected with the brain. Rather, it exists “in the work of bees, of crystals, and throughout the purely physical world; and one can no more deny that it is really there, than that the colors, the shapes, etc., of objects are really there” (CP:4.551). The lower organisms, as well as the physical world, as we have observed, are in one form or another different manifestations of mind. But Peirce goes even further still. When, say, a warm solution containing suspended Na⁺ and Cl^- ions is allowed to cool and a seed crystal is placed in it, crystallization occurs, which is “mindlike” inasmuch as the ions are programmed with a “memory” to so behave.¹⁴ All “physical stuff” is in this manner the conglomeration of “mind-stuff.” Although Peirce generally admitted to the dependence of mental action on the brain, or on substance of some sort, that out of which mind arises and can manifest itself (CP:6.559, 7.586), he nevertheless always held that thought in general has “no existence except in the mind,” and only as it is so regarded does it really exist (CP:5.288).

In short, Peirce’s conception of things is actually less bizarre than the proposition that the mind actively explicates matter (i.e., the “collapse” of waves into particles), an idea propagated by the most speculative of the “new physicists.”¹⁵ It is also somewhat less bold than Bohm’s notion that mind is an explicate parallel with that of matter. On the other hand, it bears on James Jeans’s (1958:186) transformation of things into ideas when remarking that the universe is more akin to a Great Thought than the Newtonian-Cartesian Machine. At the same time it seems to hint at some sort of transformation of ideas into things in the grand style of Bishop Berkeley.

IV. MERELY AN ILLUSION? Further to qualify Bohm’s quantum interpretation for the purpose of relating it to the Peircean framework, let us return to his dye drop and glycerine trope. Suppose we put a drop of dye in the container and turn the stirring mechanism n times. We then place another drop nearby and stir n times once again, and we continue the process until a number of drops are enfolded into the viscous liquid. If we slowly reverse the turn of the mechanism, the drops will explicate as a succession of dots. Now suppose we turn the mechanism in the reverse direction, but so rapidly that the drops merge into one another to form what appears to be a solid object, a curved line, moving continuously through space. The development of this line appears to immediate perception to be the equivalent of a growing entity because the eye is not sensitive to concentrations of dye lower than a certain minimum: the observer is incapable of seeing the entire picture. Such perception, in Bohm’s words, “relevates” (makes relevant, lifts up) a certain aspect of the apparent “reality.” That is to say, it foregrounds the line while the rest of the fluid is seen only as a grey background against which the “relevated” object seems to be moving.

Bohm concludes that when his model is placed in a broad theoretical context, it is comparable to that which is relevated (lifted up, explicated) in our immediate perception. We begin with the holomovement, patterned by the whole movement of the dye and glycerine, which enables us to perceive the train of events. Then another system, the eye plus nervous system, including the limitation wrought by our persistence of vison, enables us to “see” an unbroken line when the individual drops are relevated in rapid succession such that they merge into one another. Otherwise, we would “see” only a series of dots when the rate of relevation is lowered considerably—bearing in mind that the holomovement is undefinable and unquantifiable, the dye and glycerine being merely a metaphor.

Bohm then compares his dye-and-glycerine narrative to a quantum context, an observation of tracks left by elementary particles in detecting devices such as a photographic emulsion or a bubble chamber. The tracks, which are the result of the particles’ passage through the medium somewhat like the vapor trail left by a jet airplane, give the appearance of a continuous line rather than a series of dots, as did the dye drops. And, like the dye drops, the tracks are no more than an aspect appearing in immediate perception. In other words, the “track” as a collection of dots is not a line depicting any inherent movement of the electron. The electron does not move as such but is relevated (unfolded) and enfolded at successive spots along its apparent trajectory. The movement or line, as the Tantric Buddhists tell us, is a product of the mind; the mind makes the connections. Bohm warns us, however, that this is an artificial representation of movement described discontinuously. It is not a faithful replica of quantum jumps, since they tend to lead one erroneously to the conclusion that localized particles are capable of manifesting autonomous motion, which is irrelevant to the quantum universe as undivided wholeness.

The implicate coupled with the explicate order, moreover, bears further on Zeno’s paradoxes of motion. In the implicate order, some ensembles unfold into the explicate as continuity (the dye drops emerging rapidly), and the unfoldment of others appears to be discontinuous (when the stirring device is turned slowly). Unfolding in this sense entails both continuity and discontinuity, such a combination apparently providing fodder for Zeno’s arguments. But the problem, it would appear, is dissolved in Bohm’s interpretation of the quantum world as unceasing flow, the holomovement. In illustrating his point, Bohm (1965) refers to Jean Piaget’s studies in child psychology which, he suggests, demonstrate that our earliest experiences involve at least in part the implicate—an incessant, fluctuating enfoldment unfolding. After some time the child learns to mark out the equivalent of distinctions or cuts in the flux. Successive ruptures of the continuous whole made by her interaction with the world eventually serve to construct a model, a world map, following three-dimensional Cartesian space and time supposedly broken by a point, the “now.” This becomes the nuts-and-bolts explicate world of everyday living. It is necessary for the socially sanctioned construction of one’s “semiotically real” world, of course, but a stark abstraction from the whole, that is, the holomovement, of which the child is now generally oblivious.

Yet perplexing questions remain: If in the beginning everything was flow, then how was it possible to enact the first cut? This is the inverse of the arrow paradox: not only from within Zeno’s venerable framework cannot the arrow shift from one temporal increment to the other, it never could have left the bow in the first place. Conversely, if everything is always already in flux, then from what position, point, or particular vantage can it be temporarily halted by a cut, since that position, point, or vantage could itself have been none other than a cut? Zeno, of course, fused continuity and discontinuity in order to carry out his proof, arguing that continuous motion perceived is illusory, and if discontinuity prevails in the absolute sense, then there can be no motion.

Recalling the above problem and its corresponding graph on the distance traversed by an accelerating object, science has traditionally viewed motion conceptually in this and all such problems as if the entire trip were present all at once. More complex scientific and engineering problems involve a more subtle calculus to differentiate, say, a curve into increasingly smaller units to approach an infinite limit: zero. By so fusing discreteness and continuity, it was assumed that Zeno’s puzzles were laid to rest once and for all. In Dantzig’s (1930:138) words, having determined to cling to the notion of infinitesimals in the calculus, “we have no other alternative than to regard the ‘curved’ reality of our senses as the ultra-ultimate step in an infinite sequence of flat worlds which exist only in our imagination.” Quantum jumps reopened this can of worms, however, and the Eleatic paradoxes were once again foregrounded (Grünbaum 1967).

Bohm, unlike Peirce at times, and perhaps more than any other contemporary scientist, seems to appreciate the subtlety of Zeno’s arguments—in fact, to an extent, in his denial of motion, Zeno approaches the implicate/explicate model. Bohm draws, in addition to his series-of-dye-drops model, from stroboscopic and tachistoscopic experiments such as two spots of light flashing on a screen in rapid succession, thus appearing to the observer as a continuous stream of light moving from the point of the first spot to the second one.¹⁶ He then suggests that there is a similarity between the order of such immediate experience and the implicate order as it is apprehended by an abstractive act of mind. This reveals the possibility of a coherent mode of understanding the immediate experience of nature in terms of thought. For example, consider how motion is reduced to a set of points along a line. At a certain time, t₁ a particle is at position x₁, and at a later time, t₂, it is at position x₂. As pointed out in chapter 1, a formula for expressing the velocity of the particle between the two points, or a graph representing it, will be static, as if to say “now it is here, now it is there.” There is no real sense of unbroken wholeness, of the experience of movement (Bohm 1980:198-201).

The calculus solves the problem differently. The time interval, t₁–t₂ and the change in position, x₁–x₂, become infinitesimal, and the velocity of the particles is defined as the limit of the ratio of the change in position divided by the change in time (Δ x/Δ t), as the latter approaches zero. Some reflection reveals, however, that this procedure is ultimately as abstract as the previous one, for neither is there any immediate experience of a time interval of zero duration, nor is it possible to see in terms of reflective thought what this could mean. Moreover, the calculus entails the notion of continuous movement, but quantum-level movement is discontinuous, so its application is limited to classical concepts (i.e., Bohm’s explicate order, such as the movement of billiard balls), which provides at least an adequate approximation in the macroscopic world. Movement in the implicate order, on the other hand, does not involve this problem; it consists of nonlinear series of “inter-penetrating and intermingling elements in different degrees of unfoldment all present together ” (Bohm 1980:203). This activity depends upon the whole enfolded order, which is continuous and determined by the relationships of copresent elements.

Bohm concludes that when we consider movement purely in terms of the implicate order, problems of the Zeno variety do not arise, because it is an outcome of this whole enfolded order determined by relationships of copresent elements rather than by the relationships of elements that exist to others that no longer exist. Through thinking in terms of the implicate order, one arrives at a notion of movement that is logically coherent and more properly represents our immediate experience of movement. Thus the sharp break between abstract logical thought and concrete immediate experience no longer needs to be maintained. Rather, the possibility is created for an unbroken flowing movement from immediate experience to logical thought and back again, and thus for an end to that obstinate, successive fragmentation (Bohm 1980:203).

Of course Peirce, unlike Bohm, did not enjoy access to contemporary logical and mathematical tools. Neither did he live during those years of turmoil when quantum theory was still in conceptual quicksand, which undoubtedly would have stimulated his thought immensely. This is not to say that had Peirce been born a century later the scientific community would have yet another maverick quantum physicist to contend with. Simply stated, Peirce, given not only his pioneer work in logic, mathematics, and the sciences but especially his trailblazing efforts on probability coupled with his infatuation with the problem of the infinite and the finite, continuity and discontinuity, foreshadowed certain key aspects of twentieth-century thought.

I refer most specifically in this context once again to Peirce’s notion of continuity. The problem with Zeno’s argument, Peirce (CP:5.335) tells us, is a self-contradictory supposition: that a continuum has parts. Yet Peirce uses this same contradictory combine repeatedly. A typical example is found in his comments on the succession of ideas (or thoughts). That one idea succeeds another is obvious, but we cannot be consciously aware of this succession. Why not? To understand the problem, Peirce asks how one Lockean idea can resemble another (CP:7.349). An idea contains nothing more than what is present to the mind in that idea. Two ideas cannot exist in simultaneity; hence what is present to the mind in one idea is present only at that particular moment, and it is pushed out of consciousness when another idea is present. They are, Peirce claims, mutually exclusive. Consequently, neither idea, “when it is in the mind, is thought to resemble the other which is not present in the mind. And an idea can not be thought, except when it is present in the mind. And, therefore, one idea can not be thought to resemble another, strictly speaking” (CP:7.349; also 5.289).

Is this mere sophistry? It might appear so at the outset. But if we conceive of time as a succession of instants, like the tick-tock of a clock or the jerks of the second hand on a watch, then Zeno lurks in the shadows, for how is it possible to go from the tick to the tock? On the other hand, if time is considered to be undivided flow, then how can a specific thought (idea, sign) jump into the mind and out again? In order to escape this apparent dilemma, Peirce proposes that if we assume what is present to the mind at one moment as distinct from what is present at another moment, then there must be a process present to the mind, and consciousness must run through the two moments, “perhaps indefinitely.” In this manner, consciousness welds distinct increments together, each of them containing a definite thought (idea, sign). Hence consciousness must possess a duration, and if so, then there can be no instantaneous consciousness. All consciousness mediately relates to process. It follows, Peirce tells us, that

no thought, however simple, is at any instant present to the mind in its entirety, but it is something which we live through or experience as we do the events of a day. And as the experiences of a day are made up of the experiences of shorter spaces of time so any thought whatever is made up of more special thoughts which in their turn are themselves made up by others and so on indefinitely. (CP:7.351)

And synechism, Hartshorne’s nemesis, resurfaces. Peirce’s “indefinite continuity of consciousness” implies the perplexing notion that what was in the mind during the whole of an interval consists of what was in common in the contents of the mind during the composite parts of that interval, and so on, down to infinitesimals. Elsewhere, Peirce develops the same idea, but he enters from another angle:

We must have an immediate consciousness of the past. But if we have an immediate consciousness of a state of consciousness past by one unit of time and if that past state involved an immediate consciousness of a state then past by one unit, we now have an immediate consciousness of a state past by two units; and as this is equally true of all states, we have an immediate consciousness of a state past by four units, by eight units, by sixteen units, etc.; in short we must have an immediate consciousness of every state of mind that is past by any finite number of units of time. But we certainly have not an immediate consciousness of our state of mind a year ago. So a year is more than any finite number of units of time in this system of measurement; or, in other words, there is a measure of time infinitely less than a year. Now, this is only true if the series be continuous. Here, then, it seems to me, we have positive and tremendously strong reason for believing that time really is continuous. (CP:7.348)

Here we once again confront Peirce’s problem. Each state of consciousness is discrete, we would ordinarily assume, but in a year or less the number of states must be infinite in magnitude, hence they compose a continuum. But if this is the case, then how is it possible to get from one state of consciousness to another?—i.e., the impossibility of Zeno’s arrow to proceed from point A to point B. To make matters worse, there is a Peircean counterpart to this dilemma in memory. Peirce (CP:7.674-76) tells us that there are no aggregate parts in memory, no absolute instants. All is flow; everything merges into everything else. There can be neither a first nor a last, for between each two potential moments of memory there exists an infinity of others. Hence memory entails an unlimited number of operations of the mind which lie below consciousness—in fact, the entirety of our past experience is, somehow, always in consciousness (CP:7.547, 7:674-76). But if this is the case, then once again, how is it possible for an item of memory to be abstracted? How can the first cut be made?

On this point Peirce is subjected to his most devastating attack from Hartshorne’s bunker, and perhaps not without a grain of reason. As one incredulously reads Peirce’s words, they can hardly appear to be anything but absurd. Infinity is not of this world, we tend to retort, it is merely a mathematical ideal. This was undoubtedly the reaction of Hartshorne, who, with Whitehead and others, contends that experience must come in least units, in packets. There is no need here to reiterate Grünbaum’s (1967) “refutation” of the “bundle” theory of experience, since neither Peirce nor Whitehead nor Hartshorne offers direct empirical evidence for his hypothesis. It is consequently not wise to attempt to judge whether Peirce and Bohm are right and Hartshorne and Bohr wrong. Rather, a mapping between Peirce and Bohm merely demonstrates that in human thought throughout the ages, comparable ideas (i.e., “themata”) recur.

I now turn to the most intriguing parallel between Peirce and Bohm in their attempts more adequately to account for the phenomenon of consciousness.

V. INTO THE INTERSTICES. As a result of some strange and rather bizarre “experiments” Peirce conducted on consciousness, and commensurate with his continuity postulate, he declared that consciousness is like a “bottomless lake” (CP:7.547; 7.553-54). Percepts are like rain on the continuous surface of the lake which habits (by contiguity) and dispositions (by resemblance) serve to authenticate. Those are suspended at various depths in the lake along a continuum. Ideas near the surface are readily available to the active mind, while those that lie at great depth are discernible only at the expense of great effort. A force comparable to gravitation—roughly following the inverse square law—indicates that the attraction of deep ideas is relatively slight, and hence with greater effort brought toward the surface. In addition, the mind can exercise control over no more than a limited area at each level; raising a group of ideas from the depths requires that others be pushed down. Related ideas exercise a selective attraction on one another. When one idea is at the surface, it creates a certain degree of buoyance in other ideas. However, all ideas have a tendency to sink, to gravitate—i.e., become embedded—into oblivion. The aptness of this lake metaphor, Peirce concludes, “is very great.”

Bohm’s speculation on consciousness is relatively cautious and tentative, as it should be. At the outset, it must be mentioned that Bohm is well aware of the paradox inherent in his notion of consciousness: the very idea or intuition, and hence consciousness, of the holomovement is immanent; it is part of that very holomovement. Consciousness in this sense seems an impossibility, yet it exists. However, in Bohm’s favor, this fact is no more or less enigmatic than Spencer-Brown’s universe—as well as that of much Eastern philosophy—looking at itself in the mirror, which is it (perhaps the best we can do at this ground level is embrace the paradox and get on with the game).

For Bohm the flux of the holomovement is the implicate source of all forms, both physical and mental. The totality of existence, inanimate matter, living organisms, and mind, emerges from the enfolded. Consciousness, which includes thought, feeling, desire, volition, and so on, is thus placed, much in the sense of Peirce, on equal footing with matter. Consequently there is no fundamental distinction between Descartes’s “thinking substance” and “extended substance” (Bohm 1980:196-208). “Extended substance” is ordinarily considered to be discontinuous, but the same cannot be so easily said of the former. A particular thought seems to be discontinuous, but thought itself functions as process. Bohm suggests that though the world “out there” can be most appropriately conceived almost exclusively with regard to its explicate qualities, the implicate should enjoy equal time in any discussion of the totality. And it is precisely mind, or consciousness, that gives us a conceptual handle with which to approach the implicate. If thoughts emerging into consciousness are construed in the same fashion as particles actualized from the implicate, we have an approximation to what Bohm is talking about. Moreover, memory, in line with Pribram’s holographic model of the brain, remains enfolded, but an actualized remembrance, like a thought in consciousness, is an unfolding from the implicate (Bohm 1980:208).

Bohm is not clear on whether consciousness is a derivative directly from the ground, the implicate, or whether it is an epiphenomenon of matter, the brain. In either case, his consciousness as implicate order plus explicated thoughts bears likeness to Peirce’s “bottomless lake” metaphor, especially in view of the fact that the implicate is a higher-dimensional reality than our three-dimensional physical world, and the lake (the implicate) as three-dimensional is a higher-dimensional source for derivatives at the two-dimensional surface.

Yet another commonality between Peirce and Bohm is found in their use of a music metaphor to account for consciousness. But first a word on music regarded as movement, or flux. Consider a polyphonic musical phrase, for example. It is a successive unfolding which must necessarily remain incomplete. At each particular stage a new moment is constituted by the addition of a new musical quality. But this is not mere addition ab initio without modification of what is already there. Rather, the quality of a newly arrived tone, in spite of its particularity, cannot but be tinged with the totality of the antecedent musical context which, in turn, is retroactively changed by the emergence of a new musical quality. The individual notes are not externally related to the whole as if it were a static block, nor does their particularity disappear in the undifferentiated continuum of the musical totality. The musical phrase is a successively differentiated whole which remains integrated in spite of its successive character; yet it remains differentiated, with its marked individuality, in spite of its characteristic dynamic wholeness. It is at once one and many, analog and digital, continuous and discontinuous. Musical experience perhaps comes as close to Heraclitus’s “unity of opposites” as is possible from within the confines of human linear perception.

More specifically, Bohm, by reflecting on and giving careful attention to what happens in certain experiences, avails himself of music perception to illustrate his holographic trope. When one is listening to music, at a given moment a certain note is being played, but a number of previous notes are still “reverberating” in consciousness. The simultaneous presence of these reverberations is responsible for the directly felt sense of movement, flow, continuity. The notes must be of close proximity; listening to a set of them removed by a substantial time increment would destroy altogether the sense of a whole unbroken, living movement providing for aesthetic appeal. Bohm suggests that one does not experience the entirety of this movement by retention of notes during past moments and comparing them with the present. The reverberations are not memories but rather “active transformations” of what came earlier. Such transformations consist of a generally diffused sense of the earlier sounds with an intensity that gradually diminishes, as well as varied emotional responses, bodily sensations, incipient muscular movements, and the evocation of a wide variety of subtle meanings.

One can in this manner experience the direct flow of a sequence of sounds as they are enfolded into many levels of consciousness; the sounds interact to give rise to an immediate feeling of movement. That is to say, the notes, as unfolded Seconds in the air, return to their quality of Firstness in consciousness during the moment of aesthetic apperception. This activity in consciousness, Bohm (1980:199-200) continues,

evidently constitutes a striking parallel to the activity that we have proposed for the implicate order in general. . . . [W]e have given a model of an electron in which, at any instant, there is a co-present set of differently transformed ensembles which inter-penetrate and intermingle in their various degrees of enfoldment. In such enfoldment, there is a radical change, not only of form but also of structure, in the entire set of ensembles . . . and yet, a certain totality of order in the ensembles remains invariant, in the sense that in all these changes a subtle but fundamental similarity of order is preserved.

In the music, there is, . . . a basically similar transformation (of notes) in which certain order can also be seen to be preserved. The key difference in these two cases is that for our model of the electron an enfolded order is grasped in thought, as the presence together of many different but interrelated degrees of transformations of ensembles, while for the music, it is sensed immediately as the presence together of many different but interrelated degrees of transformations of tones and sounds. In the latter, there is a feeling of both tension and harmony between the various co-present transformations, and this feeling is indeed what is primary in the apprehension of the music in its undivided state of flowing movement.

When experiencing a succession of aesthetically pleasing sounds, Bohm (1980:200) concludes, one is “directly perceiving an implicate order.” This order is dynamical in the sense that it is a continual flow into emotional, physical, and other responses that remain inseparable from the transformations out of which it is constituted.¹⁷ Bohm compares his notion with a similar phenomenon concerning visual perception which offers the advantage of its being empirical. This entails studies of the effect of stroboscopic devices, mentioned above, giving the illusion of flowing movement between two light flashes as if they were analog rather than digital. In Bohm’s terminology, the two visual images undergo active transformation as they are enfolded into the brain and nervous system, giving rise to emotional, physical, and other responses of which one may be only at most dimly conscious. These stroboscope experiments are comparable to the reverberation of musical notes in consciousness, the chief difference being that the visual images, unlike their auditory counterpart, cannot be resolved in consciousness to provide an aesthetically satisfying pattern. This suggests for Bohm that

quite generally (and not merely for the special case of listening to music), there is a basic similarity between the order of our immediate experience of movement and the implicate order as expressed in terms of our thought. We have in this way been brought to the possibility of a coherent mode of understanding the immediate experience of motion in terms of our thought (in effect thus resolving Zeno’s paradox concerning motion). (Bohm 1980:201)

In Peirce’s conception of things, space being continuous (the Newtonian view) in contrast to Zeno’s construction of an infinitely converging discrete series, our experience dictates that Achilles encounters no barrier against his traversing an infinity of real infinitesimal steps. This, in light of Bohm and Peirce’s auditory model, suggests that individual “events” (musical notes) from within the continuity of becoming can be cut out of the continuum, though their character is necessarily fictitious: their juxtaposition or coinstantaneity with other “point-events” along the line of becoming is an intellectual abstraction from the concrete world of experience. In contrast, the flow of musical experience is an unfolding of the enfolded, the becoming of a potentiality, a continuity which, unlike the mathematical continuum, cannot be abstracted without being mutilated. In Whitehead’s (1925:54) words, “a note of music is nothing at an instant, but also requires its whole period to manifest itself.” Durationless instants are mere ideal limits, arbitrary cuts in the dynamic continuity of becoming.

More specifically regarding Peirce’s music trope, separate notes are for practical purposes considered with respect to their digital properties, while the melody is continuous. A single tone may be prolonged for an hour, a day, a week, and it exists as it is during each second of that time, present to the senses as if everything in the past were as completely absent as the future, since there would be no background against which the tone could be properly differentiated. In contrast, a melody consists of a succession of sounds striking the ear at different moments, and to perceive it “there must be some continuity of consciousness which makes the events of a lapse of time present to us” (CP:5.395). Of course we perceive the melody by hearing the separate notes, yet we do not directly hear it, for we hear only what appears to be present at a given moment, the remainder of the entire succession remaining unavailable to us at that particular moment. Peirce is addressing himself to two sorts of awareness, that is,

what we are immediately conscious of and what we are mediately conscious of, are found in all consciousness. Some elements (the sensations) are completely present at every instant so long as they last, while others (like thought) are actions having beginning, middle, and end, and consist in a congruence in the succession of sensations which flow through the mind. They cannot be immediately present to us, but must cover some portion of the past or future. Thought is a thread of melody running through the succession of our sensations. (CP:5.395)

In sum, just as the musical score exists in parts as marks on paper, each segment having its own submelody, so in the auditory perception of the piece various systems of relationship of succession subsist together between the same sensations. But once again, physical or spatial imagery unfortunately threatens to take over the trope. This is much the problem of describing quantum events in natural language. When is a “particle-event” a particle, and when is it a probability wave amplitude? The first is physical, spatial, digital, and “visualizable” in the imagination; the second is a continuous whole without predefined parts. The first is an individual in the conventional sense; the second is not: it is of “undulatory” or “pulsational” character, an imageless frequency of potentiality associated with a resultant mass once the wave has been “collapsed” into a “particle.” The wave as wave manifestation is a “pattern.” As Whitehead (1925:193) expressed it, such a “pattern”

need not endure in undifferentiated sameness through time. The pattern may be essentially one of aesthetic contrasts requiring a lapse of time for its unfolding. A tune is an example of such a pattern. Thus the endurance of the pattern now means the reiteration of its succession of contrasts. This is obviously the most general notion of endurance . . . and “reiteration” is perhaps the word which expresses it with most directness. But when we translate this notion into the abstractions of physics, it at once becomes the technical notion of “vibration.” This vibration is not the vibratory locomotion: it is the vibration of organic deformation.

Whitehead’s terms vibrational and pulsational perhaps most adequately describe the sort of synthesis he strives to bring about between the particle/wave antithesis. Rather than movement in space as the displacement of one particle by another, Whitehead prefers alteration of pattern. Instead of the term particle, he prefers event. Pulsational becoming thus supersedes the matter/energy dichotomy.

All this is analogous—alas, analogies are the most we have to go on—to the dynamic structure of polyphony. In a contrapuntal fugue, two or more melodically independent movements, whether harmonious or dissonant, occur simultaneously. In this sense, and to appropriate Peirce’s image, each component unfolds successively alongside the others, and all of them proceed toward the future while “overlapping,” or “merging” into one another, but without sacrificing individuality and autonomy. These “alongside” and “overlapping” relations immediately evoke spatial connotations, though spatiality really has nothing to do with it. The dynamic union of the intertwined successivities is distinct from static spatial juxtapositions. It is cobecoming rather than coexistence, cofluidity rather than correlation. It is n-dimensionality packed into one-dimensional becoming rather than one-dimensional relata spread over three-dimensional space.

This processual and relational conception of semiosis as revealed in Peirce’s river metaphor in conjunction with the music trope is apropos (CP:6.325).¹⁸ Ever-changing, the river flows on, gradually carving out a channel for itself along the most economical route, settling down into habit, but a habit that is itself incapable of rest, for it is invariably subject to variation. An isolated part of this entire flow is meaningless: it must be considered as a whole movement, comparable to the movements of two accomplished pianists playing a fugue. They compose a coordinated whole; their fingers flow along the keyboard, their minds attending not to particular notes, fingers, or keys but to the whole. If a particular key on the piano is out of tune and one of the musicians for an instant focuses on the sound it discordantly produces, the flow may be disrupted, and the entire process is thrown into disarray. While within the flow, in contrast, the pianists’ creative input can bring about variations, some of which improve the general rendition, but some perhaps not, and the process continues, though it is never the same as it was in previous recitals: there is no absolute repetition.

This flow of the fugue, this whole, is not describable as a mere concoction of atomistic events—i.e., pressure applied on certain keys at certain points along the linear stream as the piece is played in order to produce particular sounds. Nor is the piece itself merely a conglomerate of individual tones: it is a whole. To repeat Bohm’s conclusion, when experiencing a succession of aesthetically pleasing sounds, one is “directly perceiving an implicate order” (i.e., Firstness, quality, feeling, tone), the essence of the Monad. It seems that, to paraphrase Leibniz, whether speaking of the holomovement or music, we are speaking of unconscious algebra.

Let us, then, probe deeper into this whole, this bottomless lake.