ONE

Asymptotically
Getting There

I. THE TEXT OF THE WORLD. In the beginning, Peirce tells us, is the monad. The monad, the dyad, and the triad, corresponding to Firstness, Secondness, and Thirdness, make up what for Peirce are the fundamental categories of signs, of thought, of mind, and by extension, of the universe. I wish to begin by speaking of the monad in the most general sense: the Pure Monad. Accessible by neither analysis nor inferential reasoning, self-contained and self-sufficient, it can be assimilated only in one impossible, monolithic, instantaneous gulp. Peirce’s description of this Monad coincides curiously with orthodox metaphysical accounts of the deity—a nonrelative totality without parts, devoid of spatial and temporal localization, and independent of anything other than itself. The fundamental difference is that the Monad is sheer potentiality, while the deity is usually conceived to be an actuality (Hartshorne 1952:221). The Monad does, however, entail the possibility for actualization of an inexhaustible multiplicity of entities. In other words, the “being” of Monadic quality as mere potentiality is a continuum without existence, for the existence of particulars is dyadic, while consciousness of and perceptual judgment regarding existent particulars is triadic (CP:6.197, 1.328).

Thus we have Peirce’s conception of ongoing interaction between continuity and discontinuity, possibility and actuality, the one and the many, which will be evoked throughout this inquiry. The continuum of possibilities is necessarily nebulous and vague. Actuality, in contrast, is the force of existence which punctuates the undivided to produce discrete units. In this sense, continuity is tantamount to generality, and continuity and generality “are two names for the same absence of distinction of individuals” (CP:4. 172). Peirce’s Monad also bears on what he enigmatically called the “logic of the universe,” or “logic of vagueness,” which consists of the entirety of all possible “logics” as a whole, a continuum—unformalizable, we must presume—of a higher type more rudimentary than classical logic. Upon duly considering the course of this logic. Peirce declares, we cannot but conclude that “it proceeds from the question to the answer—from the vague to the definite” (CP:6.189-92)—i.e., by the undifferentiated differentiating itself, homogeneity taking on heterogeneity, the cloud condensing into myriad ice crystals. Indeed, Peirce’s three categories cannot be adequately considered without an acknowledging nod to the Monad.

But the nuts and bolts of our everyday world are far removed from this ethereal entity, which is the stuff of highest abstractions, mystical insights, and the wildest of dreams. Leaving the Monad as a Grand First aside for the moment, let us step down to the Firsts of ordinary mortals. Peirce, once again, offers a trope, which is, nonetheless, grounded in his “logic of the universe” and hence virtually as abstract as the concept of the Monad. The rise of the most primitive of categories, Firstness, entails the possibility of an initial “cut” in what Peirce called the imaginary “book of assertions,” the domain of “nothingness,” consisting of “separate sheets, tacked together at points, if not otherwise connected” (CP:4.512). The first sheet in this chimerical book, the standard “sheet of assertion,” represents, by virtue of the “cuts” on its pages, “a universe of existent individuals,” the various parts of the surface indicating “facts” about, or propositions asserted of, that universe. Peirce asks us to imagine “cuts” penetrating the surface of this sheet such that what is placed inside each “cut” is severed from the sheet itself (MS 455:10). “Cuts” can be made in the sheet enabling passage into successive sheets and into “areas of conceived propositions which are not [yet] realized” (CP:4.512). Subsequent “cuts” in these successive sheets can then allow entry into worlds which, “in the imaginary worlds of the other cuts, are themselves represented to be imaginary and false, but which may, for all that, be true, and therefore continuous with the sheet of assertion itself, although this [truth] is uncertain” (CP4.512).

Commensurate with Peirce’s thought, a given universe of discourse in the book of assertions is comparable to a particular “semiotically real” world—i.e., the realm, from within that universe of discourse, of “semiotic objects”—which, given Peirce’s “convergence theory” of truth, stands a chance of at least approximating the “actually real.”¹ Thus a universe of discourse can consist of a variety of statements regarded as (1) either true or false in terms of certain relations holding between them, whether they are empirical or not,² (2) patently meaningless or absurd,³ or (3) fictitious and imaginary, whether Meinongian “objects” (i.e., square-circles, gold mountains [Parsons 1980]), Vaihingerian (1935) “as if” hypostats, counterfactuals (Lewis 1973), or everyday figurative language use (see also Merrell 1983).⁴

Peirce invites us to regard his initial blank sheet of assertion as a film upon which there exists the as yet undeveloped photograph of the “facts” (propositions) of the universe. But this cannot be a literal picture, he hastens to point out, for when we consider historically the vast range of partly to wholly incompatible “facts” that have been at one time or another asserted to be true, we must conclude that the book of assertions can be none other than a continuum since these “facts” must somehow “blend into one another” (CP:4.512). This continuum, like a topological space,

must clearly have more dimensions than a surface or even than a solid; and we will suppose it to be plastic, so that it can be deformed in all sorts of ways without the continuity and connection of parts being ever ruptured. Of this continuum the blank [initial] sheet of assertion may be imagined to be a photograph. When we find out that a proposition is true, we can place it wherever we please on the sheet, because we can imagine the original continuum, which is plastic, to be so deformed as to bring any number of propositions to any places on the sheet we may choose. (CP:4.512)

In this manner, each “cut” corresponds to an area on the initial sheet of assertion where the actual state of things is signified. All successive sheets represent an indefinite set of potential “facts,” or propositions, many or most of which can, at an indefinite time and place, become intermittently part of the “semiotically real.” Peirce suggests that “in order to represent to our minds the relation between the universe of possibilities and the universe of existent facts, if we are going to think of the latter as a surface, we must think of the former as three-dimensional space in which any surface would represent all the facts that might exist in one existential universe” (CP:4.512).

This is indeed a significant visual metaphor. It suggests a method by which the evolution of forms, by the succession of “cuts,” can be conceived in timeless fashion as if in bloc. There is a problem, however, regarding the distinction between experience and intellection. For example, according to Newtonian dynamics, to the question, “What is the distance traveled by an object that begins from rest and accelerates to a velocity of 40 ft./sec. in a time of 15 seconds?” the mathematical solution is elementary:

Distance = 1/2 vt = 1/2 × 40 × 15 = 300 feet

FIGURE 1

But what is meant by the resulting statement “The distance the object travels is equal to 300 feet”? The copula, is, implies a timeless state of affairs, a Parmenidean universe in which everything is there simultaneously (Park 1980:22-35). This is a mummified rather than a living image, mechanical rather than organic, static rather than dynamic. It has hardly anything to do with the immediate quality of mind, of feeling, of sentience. We also could have solved the problem geometrically by plotting velocity against time along the Cartesian coordinates and determining the area of the triangle in terms of time and acceleration (see figure 1). But once again we would have a static form on paper far removed from the experience of acceleration. On the contrary, we could assimilate the two-dimensional graph in one perceptual grasp from our three-dimensional vantage point, just as a three-dimensional Spherelander could perceive a vast expanse of a Flatlander’s world in an instant.⁵ In this vein, M. C. Escher (1971:15) writes, regarding his prints, that

Anyone who wishes to create a universe on a two-dimensional surface (he deludes himself, because our three-dimensional world does not permit a reality of two nor of four dimensions) notices that time passes while he is working on his creation. But when he has finished and looks at what he has done, he sees something that is static and timeless; in his picture no clock ticks and there is only a flat, unmoving surface.

Peirce’s book metaphor, in like fashion, presents something akin to the Minkowski “block” interpretation of Einsteinian relativity, a conception, sub specie aeternitatis as it were, of the whole of signs, the Grand Monad—which from a critical vantage could threaten intellectually or conceptually to “spatialize” the ongoing temporal flux of semiosis. This point aside, though I shall return to it later, Peirce’s notion of a cut marring the continuum of possibilities, besides its obvious relevance to a “Dedekind cut” bisecting a line (mathematical continuum), is tantamount to an organism’s initial grasp of some-thing in the flux of experience. This cut entails primordial negativity, since that which is is separated from that which is not. In other words, the cut is a selection; it cuts something (as a First) and distinguishes it (as a Second) from something else. Then, as other cuts are exercised and combined into aggregates, they can become classified in the most general sense—with language in the case of human sign activity—as objects, acts, events, and abstractions of thought. Since the initial flux is a continuum, as with Peirce’s book of assertions, there is potentially an infinity of ways to cut it up at partly to wholly arbitrarily selected joints. Hence any given set of cuts must be construed to be largely relative, for the cuts always could have been made otherwise.

Peirce’s book of assertions evokes four important further considerations. First, the concept of continuity is strictly mathematical, and it bears on Peirce’s doctrine of fallibilism. In fact, the continuum as pure potentiality is fallibilism objectified, for according to fallibilism, “our knowledge is never absolute but always swims, as it were, in a continuum of uncertainty and of indeterminacy” (CP:1.171). Commensurate with Peirce’s fallibilism, his doctrine of tychism entails a universe of absolute chance, a potential containing an “aggregate of all possibilities that are consistent with certain general conditions.” This potential is “greater in multitude than any possible multitude of individuals, but, being a mere potential, it does not contain any individuals at all. It only contains general conditions which permit the determination of all individuals” (CP:6.185).

We have here a distinction between the actualized aggregate of individuals, of Seconds, to consciousness, and the unactualized continuum of possibilities which are not yet available to consciousness as such, for as Firsts they remain unrelated to anything else. These two domains might be termed respectively the selective and the nonselective, and the particulate and the nonparticulate, domains. Given Peirce’s continuity postulate, the nonselective, nonparticulate domain must eventually come under consideration, the former being circumscribed by the latter. It has been an error of traditional cosmological principles to infer that since the universe comprises particularities, it is itself something particular. On the contrary. Each representation of the circumscribed domain implies actuality (selection) only in consequence of the uncircumscribed (non-selected) domain. That which is actualized is necessary because it contains some things and excludes others; but the nonactualized is also necessary because of there being nothing that it does not contain. The actualized exists by virtue of not containing everything that is merely potential, in other words, by virtue of not containing that which might have been actualized but is not. The nonactualized exists by virtue of containing everything that remains or is absent from any given actualized domain. Quite obviously, in the actualized domain there will always be any given number of entities which are not, for without the absence of something, the domain of actuals is precluded from existing as a set of selected entities.

The second consideration is that of the very existence of discontinuity (the coming into existence of cuts, of particulate actuals) which disrupts the continuum. Peirce offers the example of a clean blackboard to represent “the original vague potentiality”—“nothingness” before Firstness (CP:6.203). The blackboard is an indeterminate multitude of possible dimensions, joints, and points, just as the ideal continuum is an indeterminate multitude of possible qualities. If we draw a line on the blackboard a discontinuity is produced, but this discontinuity is itself a continuity complementary with the continuity of the blackboard—that is, it is a line consisting of an infinite continuum of points. But the visible chalk mark is not really the line, for a line is mathematically of infinitesimal thickness. The mark can actually be conceived of as a narrow plane which severs and displaces a segment of the black surface:

Thus the discontinuity can only be produced upon that blackboard by the reaction between two continuous surfaces into which it is separated, the white surface and the black surface. The whiteness is a Firstness—a springing up of something new. But the boundary between the black and white is neither black, nor white, nor neither, nor both. It is the pairedness of the two. It is for the white the active Secondness of the black; for the black the active Secondness of the white. (CP:6.203)

A cut in the continuum, like the initial mark on the blackboard, is the beginning of a series. It might be countered that such a discrete series, in conjunction with the original continuum, creates a problem. If along a line a cut is made equidistant between the beginning and the end—a theoretical impossibility, since ideally a line continues indefinitely—then another can be made halfway between the first cut and the end, and still another bisecting the final segment, and so on. And one of Zeno’s paradoxes ensues, which is the result of contradictorily interjecting discontinuity into continuity.

Peirce, however, sees no problem here. Granted, from the view of the continuum, the road is endless. But from the view of the discontinuous series of events, Achilles suffers no embarrassment: after a few lengthy strides he easily overtakes his plodding counterpart. In fact, Peirce, in elaborating on his doctrine of abduction, argues that consciousness itself overcomes Zeno’s quandary in that, “just as Achilles does not have to make the series of distinct endeavors which he is represented as making,” so abductive inference, as it shades into perceptual judgment, and because it is subconscious, “does not have to make separate acts of inference, but performs its act in one continuous process” (CP:5.131; also 6.177-80). Let it not be assumed, however, that Achilles’ path therefore takes on a “reality” more “real” than the continuum. For, according to Peirce, the continuum of possibilities constitutes the principal character of the universe (CP: 1.62, 6.169-72).

The third consideration bearing on the book of assertions stems from this notion of the “real” and Peirce’s “convergence” theory of knowledge. In Peirce’s view, “truth” and the “real” are most adequately accounted for in scientific discourse. Science focuses on the general, on nature’s regularities. Irregularity and disorder should ideally lie beyond its purview, hence according to Peirce they must be considered “unreal.” Given the accumulation of our knowledge, the domain of the “unreal” is successively becoming smaller, and the “real” (the scientifically accessible) and the natural (the actual world) increasingly tend to coincide.⁶ In other words, although certain problems will always remain to be solved, Peirce noted that the number of problems being put forth was on the increase, but so was the capacity for solving them. If the rate of increase of the latter were greater than that of the former, the probability of a given problem’s eventually being solved would be favorable; otherwise the probability would be zero. Peirce believed the former state of affairs to be the case. In this sense knowledge will continue to increase, now receding, now proceeding, though over the long haul asymptotically approximating the “truth.” One again, Zeno, in another garment.

Finally, the fourth consideration: Peirce’s notion that ideas (another word for signs) spread continuously, which is also related to his law of mind and his conception of consciousness. The continuous spreading of ideas implies that all ideas affect all other ideas. As each idea spreads, though it loses intensity and power, it gains in generality and becomes welded with other ideas (CP:6.104, 6.143). By means of this continuity hypothesis, past ideas are connected to present ideas “by a series of real infinitesimal steps” (CP:6.109). This is ultimately made possible through the mediacy of consciousness. Consciousness must cover a temporal interval, for if not, it is hardly conceivable that we could acquire a sense of time. We are, therefore, forced to admit, Peirce asserts, that “we are immediately conscious through an infinitesimal interval of time” (CP:6.110).⁷ In this infinitesimal interval, “not only is consciousness continuous in a subjective sense, that is, considered as a subject or substance having the attribute of duration, but also, because it is immediate consciousness, its object is ipso facto continuous” (CP:6.111, also 6.227).

However, consciousness of such and such cannot be immediate, a feeling, or Firstness, as it were. It is invariably mediate. For example, “we can never think, ‘this is present to me,’ since, before we have time to make the reflection, the sensation is past, and, on the other hand, when once past, we can never bring back the quality of the feeling as it was in and for itself, or know what it was like in itself” (CP:5.289). There is, consequently, no absolute simultaneity of Firstness, Secondness, or Thirdness to consciousness.

In addition to the continuous extension of mind and consciousness through time, Peirce’s assertion of their extendedness through space paints a strange picture indeed. In fact, the mind’s continuous envelopment of space forms the basis of Peirce’s general law of mind which stipulates that minds, like ideas, “tend to spread continuously and to affect certain others which stand to them in a peculiar relation of affectibility” (CP:6.104). Peirce generalizes this hypothesis to the actors in a human community—and to all communities of organisms, for that matter—holding that there must be a community of feeling not only between parts of the mind but also between minds and minds (CP:6.133, 6.150-52).

Peirce’s belief that consciousness and mind have continuous extension in space, that ideas and minds are connected continuously, and that feelings flow together with ideas and mind, flies in the face of the subjective/objective, sentiment/reason, body/mind, and other such stultifying dichotomies. In fact, if privileging there be in Peirce’s metaphysics, it rests in feeling rather than intellect, in the heart rather than in the mind, so to speak. Peirce’s contention that mental phenomena are governed by law “does not mean merely that they are describable by a general formula; but that there is a living idea, a conscious continuum of feeling, which precedes them, and to which they are docile” (CP:6.152). Consequently, mind is not to be isolated from its place in the totality of nature, and in its behavior clues can be discovered regarding the nature of the world. In fact, as we shall note in chapter 7, Peirce conceives the universe itself as mind in the process of development, and hence the possibility of knowing the universe depends on the notion that “human thought necessarily partakes of whatever character is diffused through the whole universe” (CP: 1.351).

Two problems apparently ensue. In the first place, the “ocular metaphor,” the mind as a “mirror of nature,” which underlies the classical world view, surfaces. This metaphor has of recent been the focus of scathing criticism, and with a considerable degree of reason (see Bernstein 1983; Derrida 1974; Rorty 1979; Tyler 1987). I do not apologize for Peirce, though I must point out that in his anti-Cartesian posture he took a giant step toward deconstructing traditional metaphysics, as Derrida (1974:48) himself acknowledges. In the second place, a paradox becomes imminent, to which in chapter 7 I shall refer as the “Arithmetical Paradox.” The mind, which is immanent, partakes of the universe, which is mind. In another manner of speaking, the set of all sets (universe) is a member (mind) of itself—and thus we appear to violate Bertrand Russell’s (1910) proscription in his Theory of Logical Types. In what sense can the mind make of itself its own object such that it is both the observed and that which observes? Following G. Spencer-Brown (1979:105), this paradox might well be irresolvable, for the very physicist who describes the universe “is, in his own account, himself constructed of it. He is, in short, made of a conglomerate of the very particles he describes, no more, no less, bound together and obeying such general laws as he himself has managed to find and to record.”

The universe sees itself in a mirror, which is it. But in order to do so, it must first cut itself up into at least one state that sees, and another that is seen.⁸ Of course, the universe is inevitably itself, but in any attempt to see itself, it must make itself distinct from itself. In this condition it can be no more than an abstraction, partly false to itself, or as Nietzsche (1968:24) would put it, a fiction, or fable. What is more, when that aspect of the universe which has been cut is combined with what remains uncut, the selected with the nonselected, everything—possibility (Firstness), potentiality (Thirdness), and actuality (Secondness)—is paradoxically there, in the now. It all “exists” in simultaneity in the book of assertions, or, so to speak, in the text of the world.

II. STRIATED OR SMOOTH? This conception of “existence,” though it ultimately and inexorably leaves us with an antinomy of Kantian magnitude, is not the real bugbear in Peirce’s thought according to Charles Hartshorne (1973). He laments the fact that Peirce’s belief in an actual continuum of possibilities as “real” as the actualities drawn from it prevented him from envisioning the importance of discontinuity as it is amply demonstrated in contemporary quantum theory. Hartshorne informs us that the continuous aspects in quantum physics are those of real possibility, while the discrete aspects are those of actual happenings. This fits Peirce’s view, at least “when he was not blinded by his attachment to continuity.” The problem supposedly is that for Peirce the continuous aggregate of possibilities and the set of actuals are coequal in the sense that they are both tantamount to, and modeled by, a mathematically dense continuum. In other words, “Peirce was beautifully clear as to the contingency of actuality as such . . . but somehow he was unclear as to the implied discontinuity of contingent actuality” (Hartshorne 1973:196).

Hartshorne argues that Peirce’s synechism, the doctrine of continuity, erroneously entails a maximum of possibility: it keeps all of them open, hence it must be qualified as continuous rather than discontinuous action. In fact, Peirce pointed out that the possible, the general, and the continuous are ultimately the same (CP:6.189-92). In the spatial continuum (the nonselective domain) every conceivable size and shape can pop into existence, and the space-time continuum can hold every conceivable motion and change. In contrast, discontinuity (the selective domain) implies the exclusion of portions of these infinite possibilities. And such an exclusion is, precisely, actuality. Hartshorne (1973:191-92), however, goes on to argue that

either the actual world is every possible world in one, or it is not. If not, and few besides Spinoza have believed in the exhaustive actualization of possibilities, hence in principle there must be quantization, at least if the exclusion of possibilities is subject to any general rule or regularity. And why should this exclusion affect space but not time? In this, present quantum physics and relativity physics do indeed belong together. It is space-time, not just space, that characterizes reality.

Elsewhere, Hartshorne expresses his dismay that not only was Peirce unable to anticipate quantization in physics, but he believed in the existence of an actual continuity with enthusiasm. He justified his synechism by arguing that we must adopt the postulate that all possibilities are kept open as a continuous whole, otherwise some possibilities would be ruled out a priori. Hartshorne (1973:193) submits that this argument “involves a subtle confusion.” Discontinuity as a principle “rules out nothing except the one infinitely extreme supposedly possible case, continuity.” He then rather condescendingly suggests that perhaps in Peirce’s effort to avoid nominalism he committed the “synechistic error” of affirming the continuity of becoming in which the past is a set of particulars (actuals), the future is potentiality, and the present somehow incorporates both.

For sure, both Peirce and Whitehead are disconcertingly realistic in the Platonic sense, the former regarding his “eternal objects,” the latter his “Firstness.” Despite their differences, however, they agree upon a “full blown” realism, which, William Reese (1952:225) points out, apparently employs the principle: “Whatever is needed to explicate reality must be granted a place within reality.” Peirce and Whitehead’s “reality” includes, in addition to actual existence, the mode of possibility and the mode of process—i.e., the actualization of possibilities, from “eternal objects” or “Firstness.” For the nominalist, the possible cannot be a nonactual manifestation of the actual; in contrast, for Peirce and Whitehead, the actualization of particulars into “reality” implies what might be termed a “background,” or an “underlying order,” which is no less “real” than the existent set of actuals. A fundamental distinction between the possible and the actual rests in that the former is nonlocalized while the latter consists of presumably localizable singularities—with the attendant risk of what Whitehead (1925) calls the “fallacy of simple location.” The former is “mutually implicative” rather than, as is the latter, “mutually exclusive” (Reese 1952:229).

From another perspective, nominalism asserts that the past categorically is no more; it remains outside the present, the product of an incessant passing through the instant of the “now.” The philosphical realist, in contrast, would retort that the past, though past, is conserved in the present—contained within Peirce’s Thirdness. Reese argues, and rightly so, I believe, that the nominalist regress of the discrete series of “nows” into the past implies a potential infinity. Like the number series, in a manner of speaking, it contradictorily implies infinity, for the series cannot be halted. However, supposing one to be capable of reconstructing the series of all past events, like traveling from zero through the negative numbers, or much like Borges’s (1962:59-66) Funes the Memorious remembering his stream of sensory perceptions into the remote past—one still cannot recede further than the actual steps one has taken. In fact, throughout human history the infinity of steps cannot be actualized; there can be neither a conceivable beginning nor an end. According to Peirce, on the other hand, this contradiction of past times is avoided by assuming that all past events up to the present—although this implies an infinite number of events in the nominalist’s view—compose one Cosmic Event. This totality is, we must suppose, continuous, and it can be subdivided indefinitely. Consequently, an infinitely receding series pointing toward an inaccessible first event presents no real problem, for Peirce at least.

Peirce attempted to circumvent the apparent dilemma further by arguing—albeit vaguely—that the possible is somehow “real” and that the past is equally “real.” But this seems to fly in the face of common sense. Peirce’s time and space as continuous and devoid of singularities obviously have no room for Secondness, that is, for actualities. They are exclusively Firstness, existing as a self-referential, self-contained whole. In other words, “a continuum which is without singularities must, in the first place, return into itself” (CP:6.210). Only thus can the series of reiterated cuts be initiated: in the sense of Spencer-Brown, the universe necessarily separates itself from itself in order that part of it may turn inward and distinguish itself from that which it is not.

Peirce (CP:6.211) offers the example of two intersecting lines, each of which is an infinite continuum of points (see figure 2). The intersection of the two lines is a singularity, something akin to a Dedekind “cut,” which mars both continua. Peirce then severs the lines through that point and combines them at their extremity, which can be no more than an imaginary combination, since each line is presumed to continue indefinitely. Finally, an oval—the “ovum” of the universe—can be formed, which is in a sense both finite and unbounded and infinite and bounded; there is no beginning or end, yet one can begin or end anywhere (Boler 1964). The curve doubles back upon itself to describe an unlimited set of possibilia. Peirce reminds us that such a curve, which is theoretically invisible since it is infinitesimally thin, is “merely a Platonic world, of which we are, therefore, to conceive that there are many [worlds], both coordinated and subordinated to one another; until finally out of one of these Platonic worlds is differentiated the particular actual universe of existence in which we happen to be” (CP:6.208).

FIGURE 2

This ethereal realm is, of course, the continuum of possibilities, the undifferentiated (nonselective) domain before it has been mutilated with a cut (selection). Peirce’s notion of a cut in the uncut is remarkably commensurate, as I shall point out repeatedly in this inquiry, with Spencer-Brown’s calculus of indications based on one symbol, , the mark of distinction, especially as it is interpreted by Francisco Varela (1975, 1979), Humberto Maturana and Varela (1980, 1987), and Louis Kauffman and Varela (1980). It is also closely related to what in Semiotic Foundations (1982) I call “boundaried spaces.” We must be mindful, in this light, that Peirce was a mathematician and logician—which is to say, a semiotician—before he was a metaphysicist. Hence his conception of time and space is quite appropriately mathematical, though, given his place in the history of human thought, it remains in part commensurate with classical physics and Euclidean geometry. Consequently it conflicts with Hartshorne’s conception of things, which appears to be more subjective, intuitive, and “commonsensical” than mathematical. Moreover, contrary to Hartshorne, as we shall observe in chapter 2, an important aspect of Peirce’s thought, especially concerning Firstness, the Monad, is not antithetical to quantum theory according to one of its more creative interpretations. But before entering into that dialogue, I will present a few preliminary remarks on a crucial problem arising out of the contemporary scientific world view, especially regarding time, space, change, and objectivity.

III. NARY A GLIMPSE INTO ETERNITY. Against Peirce’s notion of temporal and spatial continua, Hartshorne appears to follow Whitehead and William James according to whom Zeno’s paradox of Achilles and the tortoise succeeds only in demonstrating that an alien property is interjected into the denseness postulate of time and space. A crucial issue of Zeno’s paradox is whether or not the mathematical notion of continuity along a dense string of points is applicable to temporal succession. Whitehead and James, in this respect contrary to Peirce, assert that time does not possess the structure of a mathematical continuum. Their chief reason for this denial is based on their belief that (1) our conscious awareness of temporal order is pulsational not continuous, and (2) the order of the pulses is discrete. Specifically, James (1948:187) remarks that “the mathematical definition of a continuous quantity as ‘that between any two elements or terms of which there is another term’ is directly opposed to the more empirical or perceptual notion that anything is continuous when its parts appear as immediate neighbors, with absolutely nothing between.” And Whitehead (1978:69), after favorably quoting James on Zeno, tells us that

the modification of the “Arrow” paradox, . . . brings out the principle that every act of becoming must have an immediate successor, if we admit that something becomes. For otherwise we cannot point out what creature becomes as we enter upon the second in question. But we cannot, in the absence of some additional premise, infer that every act of becoming must have had an immediate predecessor.

The conclusion is that in every act of becoming there is the becoming of something with temporal extension; but that the act itself is not extensive, in the sense that it is divisible into earlier and later acts of becoming which correspond to the extensive divisibility of what has become.

Adolf Grünbaum (1967:48) raises two questions concerning James’s and Whitehead’s conclusions: (1) Does our awareness actually consist of a discrete series of pulses, of “nows”? and if so, (2) Is their postulated isomorphism of the perceptual and physical orders correct? Grünbaum argues extensively in favor of the first thesis, but he rejects the isomorphism thesis. In brief, he proposes that (1) awareness of the coming into existence of the thingness of the universe is mind-dependent, and (2) time and space in modern quantum theory presupposes a dense continuum—which is contrary, we shall note in chapter 2, to Hartshorne’s interpretation of quantum theory as discretizing time and especially space, which becomes, he believes, grounds for his attack on Peirce. Grünbaum’s first proposal, of course, has nothing new to offer. Its modern embodiment is found most notably in Hermann Minkowski’s mathematical rendition of Einstein’s special theory of relativity into the “block” universe which I mentioned above, a rather uncomfortably Parmenidean formulation. Time, according to this view, loses its movement; it is “spatialized,” much like, on an exceedingly more mundane level, the problem of acceleration in figure 1. Consequently, time plays no part in this instantaneous clash of the universal symphony. Mathematician Olivier Costa de Beauregard (1981: 29) puts it trenchantly:

In Newtonian kinematics the separation between past and future was objective, in the sense that it was determined by a single instant of universal time, the present. This is no longer true in relativistic kinematics: the separation of space-time at each point of space and instant of time is not a dichotomy but a trichotomy (past, future, elsewhere).⁹

In other words, according to this view, there is no knife-edge present moving along the linear race of time from past to future. All “events” in the “block” are at once both past and future—insofar as they are accessible to a given sentient organism along its “world-line”—as well as “elsewhere”—that which lies outside the organism’s peceptual grasp, outside its “light cone,” because of the finite velocity of light. This monolithic four-dimensional manifold is described by Hermann Weyl (1949:116) as a universe which “simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the life line of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time.” Consequently, the perceived universe is distinct, to a greater or lesser degree, for each individual along his “world-line.” And the “elsewhere,” lying outside his purview, bears witness that the universe cannot be accessible to any immanent observer. Time is relative to each observer, and change is dependent upon the mind-dependence of time. In Kurt Gödel’s (1949:557) words:

Change becomes possible only through the lapse of time. The existence of an objective lapse of time, however, means (or at least, is equivalent to the fact) that reality consists of an infinity of layers of “now” which come into existence successively. But, if simultaneity is something relative . . . reality cannot be split into such layers in an objectively determined way. Each observer has his own set of “nows,” and none of these various systems of layers can claim the prerogative of representing the objective lapse of time.

However, Grünbaum (1967:55) takes pains to point out the obvious: Minkowski’s becomingless universe is a view sub specie aeternitatis in the first place, which makes no reference to anyone’s particular “now.” An event is becomingless in the sense that it occurs at a certain time, t, and within a static network of what we would ordinarily perceive to be both earlier and later. Hence becoming is tenselessly in a network of relations of timelike—from our perspective, at least—separation. Quite obviously, becoming as mind-dependent, following Minkowski’s “block” universe and according to Grünbaum’s contention, goes against the grain of James’s and Whitehead’s thesis.

This in an indirect way bears on Peirce’s rather problematic conception of time, which I will treat briefly by way of a digression. Hartshorne points out that Secondness in the absence of Thirdness remains symmetrical and without direction. The asymmetry of time’s arrow is the mere successive actualization of Seconds from the possibilia of Firstness. He argues that if the First, the Monad, were truly independent, then its successors and all other successors could be either First or Second to each other, and time would have no direction; it would be “spatialized,” in bloc (Hartshorne 1964:461). However, Peirce at least hinted that time is the succession of genuine Seconds to Firsts which are actually relative rather than nonrelative. This presents the ambiguous condition of Firsts’ being both relative and nonrelative, depending upon the perspective. But since Seconds are largely directionless, they depend on Firsts conceived to be relative rather than nonrelative in a more primordial sense. It is the Thirdness of consciousness which constructs time’s arrow and hence asymmetrical temporal movement.

In other words, insofar as time is a pure continuum—in conformity with Peirce’s doctrine of synechism—it has neither direction nor metric, and if the principles are to be attached to a definition of time, it must be by way of the content of time: what Peirce calls the law of mind gives the direction and flow of time from past to future. This law entails ideas affecting other ideas through relations of asymmetry, transitivity, and irreflexivity. In this case for two ideas, A and B, if the former is affectible by the latter, then it is later; if vice versa, then it comes before; if neither is affectible by the other, then they are simultaneous. Causality among ideas constitutes the basis for temporal direction (CP:6.127-31). But according to Peirce’s “objective idealism,” since matter is fossilized mind, and temporal order by way of mental processes establishes the direction of time for the physical world, time is mind-dependent. In this fashion, matter, as another form of mind, generates time, and since the individual mind—or minds—is in time, time is, paradoxically, in itself (see Murphey 1961:387-88).

If for Peirce mind gives direction to the physical universe, Einstein’s belief in an objective world and the “block” interpretation of special relativity coupled with Gödel’s idealistic interpretation of the “block” leads to the view that not only are all mechanical processes reversible, but in addition, the direction of time is mental. Or in the terms of this inquiry, we could say that the direction of time is “semiotically real” rather than “actually real”; it is mental in the broadest sense rather than purely physical (see CP:6.554, 6.68ff.).

On the other hand, quite understandably, while Peirce generally conceived space and time as continuous and absolute in the Newtonian sense, he never envisaged them to compose a four-dimensional continuum. Yet he approximated the concept of a space-time continuum in his suggestion that time is akin to a one-dimensional hyperbolic continuum in space. In this manner time and space are not exactly independent of one another (CP:1.273, 6.575). The nature of time thus manifests similarities with the nature of space, but with a major distinction: time, unlike space, cannot be symmetrical, because an entity cannot possess two contradictory properties simultaneously (CP:1.492-95, 1.501). That is to say, something cannot possess both x and non-x. However, it can, over time, first possess x and later non-x. Phillip can be both drunk and sober, provided the two Phillip events are separated by a temporal interval (CP:1.494). Peirce’s more “general” logic, as well as his hypothesis of time, accounts for how an entity becomes. In contrast, that which never changes cannot exist in time.¹⁰

Regarding the second aspect of Grünbaum’s argument—time and space as dense, not discrete—the question is posed as to whether Zeno’s arrow and stadium paradoxes, both of which can be interpreted as directed against the continuity hypothesis of space and time, present the same challenge to quantum theory that Achilles did for the continuity thesis of motion and becoming. Grünbaum suggests that the initial assumption that this is indeed the case appears to be derived from the notion that quantum theory quantized physical properties which were continuous in classical physics. Speculations subsequently arose concerning the possibility of minimal increments of space (“hodons”) and time (“chronons”)—that is, discontinuous space and time in line with Hartshorne’s notion.¹¹ This conjecture eventually led to naught. At best, according to Grünbaum (1967:111), “genuinely atomic space and time is nowadays a gleam in the eyes of hopeful speculative theoreticians.”

In this sense, Peirce, for whom space in and of itself is continuous and without singularities, as is time, appears at the outset to lie midway between the discontinuity thesis and the continuity thesis (CP:6.82, 6.87). In another way of putting it, he is outside classical physics with one foot still caught in the door. Movement of something from one place to another in classical physics was defined as a change of spatial position with respect to a change in time. In relativity theory, in contrast, there is no such rate of change in the objective sense, for the observer is inextricably caught “within” one frame of reference or another, all frames of references being relative to some imaginary observer, or better, no observer at all. The rate of change of a space-time computation from within one reference frame would be available for computation from within another reference frame solely with respect to the relative movement of the second frame to the first one (Gerach, 1978). In other words, use of the term movement in relativity theory depends, like simultaneity, upon a translation from the language of the space-time computations of one reference frame to the language of the space-time computations of another reference frame. There can be no absolute computation or description of any universal phenomena—simultaneous translation of all languages into all other languages—from the vantage of an observer trapped “within” the system.

Yet, the Einsteinian notion of a possible mental instantaneous grasp of the whole as if it were in the eternal “now” testifies to the perseverance of mind constructs pointing toward systems of configurations with possible permutations as if permanence and stasis (monism) were more fundamental than movement and change (pluralism)—a topic that will engage us in chapters 6 and 7. But they are not. With due respect to Hartshorne’s critique of Peirce, the prevalent world image created by the conflicting views of quantum theory and relativity resurrects the very continuity-discontinuity problem presented here, which has actually been with Western thought at least since Greek antiquity.

First and foremost, the continuity-discontinuity problem is germane to the limitations of language. Natural language, when used as if each word carved a granitic referent out of the world, is not only unreliable for a Parmenidean immutable universe, it is also entirely inadequate if reality is construed as Heraclitean flux or Peircean process. Natural language is an instrument primarily for specifying objects and their attributes, and secondarily what they do or what is done to them. Unlike mathematical language, it is obviously an incapable instrument for giving account of the four-dimensional universe of relativity. And natural language is certainly a poor candidate for providing a “description” of the fleeting events or happenings of the quantum world—the term in quotes is actually a misnomer, because it implies something relatively permanent, thus rendering itself to specification at a given point in time. Moreover, Louis de Broglie (1953:219), who is certainly not alone among twentieth-century scientists, tells us that scientific theories and even the most rigorous of formal languages they are embedded in are idealizations. These idealizations

most likely become less applicable to reality as they become more complete, and hence, contrary to Descartes, nothing is more misleading than a clear and distinct idea, for if it is supposedly clear and distinct, it has no necessary bearing on reality, and if it supposedly has no bearing on reality, it is invariably subject to alterations.¹²

Peirce could not have said it better, this semiosic image par excellence. Like the world’s furniture, signs and thought, thought-signs, whether ultimately dressed in natural or formal language, are destined to flow along an ongoing stream of perpetual displacement without the possibility of any given sign’s being at any moment determinately there to be tacked down, contemplated, analyzed, known as it is in the now and as it will be in all future nows. This provokes knee-jerk reactions from even the most stalwart among us. We almost intuitively scream out for the comforting counterimage of hooks connecting signs to things, for determinate joints in the world. Yet, given the presently emerging view of the universe, of the mind, and of the human semiotic animal from a variety of disciplines, there appears to be no alternative on the distant horizon: the cloud of unknowing lingers above more ominously today than when de Broglie put his words on paper.

In addition to the limitations of language which have created innumerable conundrums and conflicts throughout the course of intellectual history, a preference for what Gerald Holton (1973) has called “themata” has also been a source of constant confusion and debate. For a case in point, Einstein and Bohr locked horns for years over something far greater than either of them: a complete and harmonious account of the universe (see Miller 1978, 1986). The contrapuntal give-and-take between these two mental titans was not a mere matter of equations. Rather, it involved the very nature of “reality” at its most basic. Einstein’s world existed “out there”; Bohr’s depended upon the observer’s choice of observing equipment, which brought with it uncertain consequences. Einstein’s ultimate “reality” was the “thema” of a space-time continuum; Bohr’s came in discrete “chunks.” And Einstein’s intangible search for determinacy—”God does not play dice,” he occasionally remarked—was countered by Bohr’s adoption of Heisenberg’s uncertainty principle and his own complementarity.

The notorious double-slit experiment, to be discussed below, uncannily patterns these two antagonistic views. The split beam, like the particle/wave duality itself, is “split-brained.” As a particle, the electron is discrete, a no-nonsense, hard-core realist. It is left-brainy, so to speak. As a wave, it is right-brainy: smooth, continuous, holistic, a range of possibilia, a thought-sign rather than a thing-sign. The electron’s hard-core manifestation generally follows classical mechanics, using variables appropriate for the notion of billiard balls: position, velocity, angular momentum, mass, time. Its soft-core manifestation uses continuous variables somewhat suitable for the effect of dropping a billard ball into a pond of water: wavelength, phase, frequency, amplitude. The real mind-bender is that the schizophrenic “electron” does not stand alone; both of its descriptions can be roughly applied to all “objects.” In other words, it appears that the universe may ultimately be a contradictory collusion of two incompatible “themata,” with no translatability between the languages used to describe each of them.¹³

Moreover, this incompatibility and nontranslatability between languages may well be paralleled by an incompatibility between modes of conception and perception, whether linguistically determined in the Sapir-Whorf sense or the product of everyday language use.¹⁴ Idiosyncratic modes of perception and conception might play a greater role than is generally conceded. Peirce (CP:2.277) was prone to diagrammatic thinking. Popper (1974:182) confesses that he compulsively thinks in schemata. Einstein once claimed he first thought in pictures, thereafter translating his images into their equational form, while Heisenberg, Max Born, and others found concrete imagery disgusting (Hadamard 1945; Miller 1978).¹⁵

Of course such summary comparisons and contrasts are easily abused, and I will attempt to go no further. The point to be made is that, when all is said and done, and “themata” and nontranslatability of languages notwithstanding, we must concede that ultimately we cannot but “see” the world through filtered goggles. That is to say, to borrow German biologist Jakob von Uexküll’s (1957) term, we are locked into our particular species-specific Umwelt (see also Thure von Uexküll 1982, 1985, 1986, 1988). Along a functional cycle or information-conveying loop, external signals enter and become internal signals, having been processed in the transition according to our particular capacities. Consequently, each organism, from the amoeba to Einstein, is limited not only to its unique Umwelt but also to its experientially unique Innenwelt. If an amoeba could write its own physics down on paper, it would differ radically from our physics. Notions of space and time would appear completely bizarre, perhaps entirely unintelligible, from one physics to another. It follows that the “world” of each species is ordinarily taken by that species to be a self-consistent and complete world by way of its particular Umwelt. It is for it simply the way things are. What lies outside its Umwelt does not exist.

In view of Peirce’s continuity of mind through time, and the classical mechanics/relativity/quantum theory clash, J. T. Fraser’s (1979:22-26; 1982: 30-31) formulation, inspired by von Uexküll, is worthy of note. Fraser divides the various Umwelten into atemporal (time cannot be recognized by the human being as such), prototemporal (time and space are distinguishable, but events and things are interchangeable—i.e., time is symmetrical), eotemporal (time is pure succession; it flows, but past-present-future cannot be clearly distinguished), biotemporal (that of sentient organisms), and nootemporal (beginnings and endings are recognized; self-consciousness and self-identity are possible). It hardly needs mentioning that bio-nootemporality and to a degree eotemporality are characteristic of the human Umwelt. The empirical bodies of classical physics belong to the reversible time of eotemporality. At a more basic level, electrons and other kindred entities are confined to prototemporality, and photons to atemporality.

Photons are in constant flux; their world is intrinsically chaotic and restless. Although from the photon’s Umwelt, traveling at 300,000 m./sec., the universe would be bleak and dark, an entirely static state of affairs; it would be like the empty “book of assertions,” the ultimate Monad containing everything as possibility and nothing as actuality. At the prototemporal level of quantum events, in contrast, enduring entities exist, though fleetingly. They resist the idea of permanence and continuity. Of necessity, quantum theoretical formulations generally divide the electron and its immediate cousins into their schizophrenic manifestations in order to make them somewhat intelligible for us from within our eo-bio-nootemporal framework, the wave-particle duality being the product of this split. This is fine insofar as it allows the physicist to get on with her equations and the high-school physics instructor to describe the world of prototemporality to her students. The electron can be either a particle or a wave but not both at the same instant. In this sense, the externally “real” is not the really “real” but the product of an Umwelt: it is a “semiotically real world.” And what would the nimble electron have to say about all this? Most likely little or nothing at all. It is unaware of instants and of linear time. As far as it is concerned from within its prototemporal world, it is both a particle and a wave simultaneously, or neither a particle nor a wave. Take your pick.

Mind, and mind-generated time, in the sense of Peirce, apparently enter the scene at higher levels. Like prototemporality, the most fundamental aspect of the universe of Peirce’s Secondness is motion and change in time—as picked up by the roving, restless mind. However, these particulars, viewed from within our eo-bio-nootemporal Umwelt, are gathered up into a rather dense concoction of differences made into sameness and even identities: tokens are “typized,” individuals become universals, haecceities become quiddities. One must be aware, nonetheless, that in light of semiosis, rest and stasis are not primitive but evolutionary stopping points. They are the temporary repository of matter as effete mind, as habituated mind, for mind, in Peirce’s cosmological framework, is primary.

This assumption, in light of the above discussion on the distinction between relativity and quantum theory, bears further commentary.

IV. ONE WAY OR MANY WAYS?: FROM VAGUENESS TO THE FULLNESS OF THINGS. The relativity concept implies continuous movement of a given entity along its “world-line” within the space-time manifold. The same seems to apply prima facie for movement as change in position during a time interval in classical mechanics. A radical distinction between the classical and the relativist views, however, consists in the latter’s implication—as Whitehead insisted—that there can be no timeless localization of atoms, molecules, or elementary particles. There are no substantive singularities in the space-time continuum. If such material entities existed, they would exhibit the property of being cut off from the remainder of the universe; they would be limited to particular spatial ambulatory prisons.

But they are not. Rather, they are tantamount to the product of Peircean cuts or Spencer-Brownian marks, which are artificially exercised by particular minds, from the human animal to nonhuman organisms, in accord with particular Umwelten—those minds being themselves cuts. From the quantum theoretical view, the Copenhagen interpretation stipulates—though there is controversy in this regard—a wave “collapse” (cut) by an observer-instrument to produce a “particle,” which can then be conceived as a discrete entity. In the John Archibald Wheeler (1977, 1980a, 1980b, 1984) rendition of the quantum world, remarkably in line with Peirce and Spencer-Brown, as we shall note in chapters 4 and 5, a choice is, so to speak, made. A question is then asked of the universe, and an answer distinguished to yield “stuff’ in the space-time field. However, according to the field concept inherent in relativity theory as I understand it, what appears as discrete “stuff’ is actually a “knot” in the continuum which is not an absolute singularity but is connected, hyperbolically or asymptotically as it were, with every other “knot” in the field. That is to say, the “warp” in the field’s embrace of the “knot” curves hyperbolically without ever “cutting” itself off “entirely” from the field, much like the ripples of a pond which are a perturbation yet a manifestation of the pond’s continuity (in general see Capra 1975). Cuts are, in this sense—and as Spencer-Brown takes pains to explain—artificial mutilations of the universe as an interconnected fabric.

Mendel Sachs (1988) presents an intriguing survey of the ongoing controversy surrounding the conflicting relativistic and quantum theoretical conceptions of the world. As I intimated above, through the ages, cosmologies have generally inclined toward one of two “themata”: matter as discrete or as continuous. The most recent scientific revolution, in contrast, has not been able to shake its schizophrenic character. Relativity on the whole implies a continuum view of matter in terms of the field concept, while quantum theory is essentially atomistic, yet of a different sort from classical atomism.¹⁶ In another, rather cryptic, way of putting it, and in light of Fraser’s categories, Einstein’s view is largely eotemporal—though devoid of the infinitely extended continuity of classical time and space. In contrast, Bohr’s quantum theoretical view is largely prototemporal—though devoid of the observer/observed break inherent in classical mechanics (Fraser 1982:93-94). In a loose sense the probabilistic character of the prototemporal world is an evolutionary step between atemporal chaos and eotemporal determinism, the shimmering, scintillating vacillation of the prototemporal domain between particle/wave, this/that, having conjoined otherwise inconsistent antagonists (Fraser 1982:69-70).

Significantly enough, the quantum theoretical view is roughly commensurate with Peirce’s (CP:5.505) concept of vagueness, which has no use for the principle of noncontradiction. In contrast, the continuous field of relativity is a counterpart to Peirce’s generality, which renders the excluded middle impotent, though, from any given reference frame, information and hence knowledge must inextricably remain incomplete—it can be at best only approximated. Vagueness implies a plurality of possibles (Firsts) some of which will eventually be actualized (as Seconds) at different times and places. Generality calls for a synthetic grasp of that which is actualized with respect to those possibles that remain as such, though they could just as well have popped up instead. Were this synthetic whole to be had in one cognitive grasp, the Final Interpretant would be realized. This being impossible, a given sign of generality, an interpretant which is inexorably incomplete, is destined to pass away into its successor just as it emerged from its predecessor.

Now, to place this image within an even broader framework. The ultimate ramifications of the semiosic process—the Final Interpretant, Determinism, Law, Regularity, Habit, Generality—imply a continuous (synechistic) domain. This semiosic goal, the Final Interpretant, is not for the kingdom of this world, however. A perceptual—as well, we must suppose, as a conceptual—grasp of it will always remain unrealized—i.e., incomplete—for the immanent observer-interpreter-knower. On the other hand, pure indeterminism, pure chance, belongs to the tychistic field. It is unbounded; everything can possibly be and anything can presumably go. Consequently, since the noncontradiction principle does not apply, the system remains inconsistent, vague. Yet the immanent, finite observer-interpreter-knower persists in her push toward a modicum of certainty, determinacy, consistency, and clarity. Hence she naturally inclines to flee from vagueness toward generality, but without being able to either fall back on the one or reach the other.

The continuous, synechistic domain of Thirdness evinces symmetry between observer-interpreter-knower and observed-interpretant-known, and non-linearity. Such is comparable to relativity’s four-dimensional space-time domain for the mapping of all components of the semiosic fabric within its enclosure. In contrast, the tychistic domain of Firstness is an open system of possibilities, a portion of which can be actualized into Seconds (elementary “particles”)—recall the selective and nonselective domains, and Peirce’s “book of assertions.” This system entails asymmetry between observer-interpreter-knower and observed-interpretant-known, and linearity in terms of the stream of signs becoming other signs ad infinitum. Such is the separate spatial and temporal actualization of individual signs from within the indeterminate domain of possibilities. Roughly, such also is the universe of complementarity as defined by Copenhagen quantum theory.

The first system is, relatively speaking, objective: sign features are dependent upon their interpreters. Hence a sign, its interpretation remaining, as it were, indeterminate, leaves to the interpreter the right of pushing the determination of its interpretation toward completion for herself. The second system is relatively subjective: sign features depend upon other signs in order that their indeterminate interpretation may progress along the rail leading toward completion (CP:5.505). The objective view is reversible. The interpreter is separated (she separates herself) from the semiosic field, expanding her vision nonlinearly to encompass ever-broader parameters. Ultimately, the mind as sign can theoretically and potentially—though not in actual practice—become coterminous with the field of signs. As such both would have attained the final state of generality—Parmenidean stasis. The subjective view, in contrast, is irreversible. The interpreter is not separated from but is in inextricable interrelationship with signs, and, as it were, she interacts with another utterer or with her other self in an ongoing linear movement. Nothing is at a standstill. There is constant activity on both sides of the equation, between this and that, which are never equal to one another, for the interaction perpetuates itself—Heraclitean flux. But neither state in its totality, to repeat, is of this world of finitude: we are victims of our own immanence. The way of generality presumes some inaccessible “real world” embodying the Grand Truth; the way of vagueness allows for a picking and choosing of many little truths—all of them to a greater or lesser degree fictions, or fables.

Yet the fact remains that Peirce’s doctrine of signs is a rather contradictory fusion of these two extremes. Ontologically he appears to be an idealist, though he holds to his faith in an objective world “out there.” Epistemologically or methodologically he is a realist or objectivist, though his seeker of knowledge could never know the truth if he had it. He could not completely know it, as it were, because of the indeterminacy of meaning and therefore knowledge, which is reflected in the failure of the excluded-middle principle—that is, in the inevitable degree of generality in any and all determinations—and in the breakdown of the noncontradiction principle, which breeds vagueness of different sorts. Through the centuries various thinkers dwelling on human limitations from Heraclitus onward have arrived at the conclusion that the universe is in-determined and indeterminate. It cannot therefore be the object of total and precise knowledge, but only a partial and conjectural one, an image harking back ot Xenophanes and forward to Karl Popper, who occasionally delights in quoting the Greek. An indeterminate universe giving rise to indeterminacy of knowledge entails centerlessness, which has been a bane for those tirelessly in search of certainty and a soporific for those embracing segments of the spectrum, from wide-eyed liberalism to deconstructionism to hard-line relativism to full-throttle nihilism—not to mention Feyerabend’s “epistemological anarchism.” All disciplines have been flooded with what is conceived as this vice, virus, or vigor, depending upon the eye of the beholder. In the words of Lewis Thomas (1980):

We have learned that we do not really understand nature at all, and the more information we receive, the more strange and mystifying is the picture before us. . . . There is no center holding anywhere, as far as we can see, and we can see great distances. What we thought to be the great laws of physics turn out to be local ordinances, subject to revision any day. . . . It is, when you give it a thought, shocking.

Or as Peirce would say, nature is the ultimate “outward clash” which surprises. This “clash” can give way to mystery, or Einstein’s “subtle is the lord.” It has also awakened the West from the epistemological somnambulism Nietzsche (1968:2) anticipated in 1885: “Since Copernicus man has been rolling from the center toward X: the aim is lacking; ‘why’ finds no answer.” Freud found the same “clash” in the unconscious, concluding that the “I” is not the master of its own vessel. Descartes’s certainty becomes for Freud’s unconscious, as for Peirce’s semiosis, a sign perpetually displaced by other signs. Indeed, dreams, slips, lost memory traces, lapses are symptomatic of what Freud described as “overdetermined.” But “determination” as used here has little to do with classical linear causality. Signs in this scheme have no simple one-to-one meaning. Rather, each sign points toward a nonlinear multiplicity of possible pathways, each with the hope of a pot of gold at the end. The sign is “determined” only in the sense that it was drawn from another sign, and that from another sign, and so on. This “determination” is perpetually open-ended, like that of language itself, of thought, of mental “objects,” of the “real.” Just as there are many ways, potentially an infinity of ways, of explaining a relatively rich and complex concept, so there is potentially an infinity of descriptions of both the positive and negative properties of a “semiotic object” “out there.”

Regarding vagueness and generality further insofar as they bear on indeterminacy, consider, for a moment, the Tao. It is notorious that any idea of the Tao can be no more than vague and imprecise. Then how can it be defined? It cannot be, any more than other vague concepts such as mind, beauty, the good, freedom, and so on, which have confounded philosophers for centuries. If such “entities” or “things” or whatever we choose to call them are necessarily vague, then should not our idea of them be correspondingly vague as well? Any feasible response would appear to be affirmative. In such case the character of the idea would be a picture or mirror image of the “thing” of which it is an idea, hence by virtue of the logical positivist’s picture theory of meaning the idea would be accurate rather than vague precisely because of its capacity for mirroring that which is vague.

In other words, the idea would be precise because it is a faithful picture, but imprecise because that which it pictures is vague. Furthermore, according to Peirce, the idea of the originary vague sign is itself a sign, and in such case, it must resemble it (iconically), indicate it (indexically), or say it (symbolically) by an inferentially grounded proposition or argument. If the picture theory is not acceptable—which it is not, in the framework of Peirce’s semiotic—then resemblance is suspect. Indexicality and symbolism are not likely candidates either, for a “quite universal” concept or idea would as such be unrelated to an “other”; it would possess no Secondness or Thirdness. Then how comes it that this orginary universal sign is vague? Insofar as it is vague, it is like the Tao. Any precise idea of the Tao would be imprecise because of its very precision, hence it would be absolutely distinct from the Tao in that it is precise, but the Tao is not precise, hence the idea of it is insufficient. Any idea of the Tao must be as vague as is the Tao itself (Smullyan 1977:11-12).

Now, any respectable logical positivist would declare the proposition “The originary sign (Tao) is vague and general” to be neither true nor false but meaningless. Moreover, she would claim the ability to prove its meaninglessness by irrefutable logic, thus assuring us that it cannot fall into either the category of truth or that of falsity. In other words, following Peirce’s idea that noncontradiction is ipso facto inapplicable to the vague sign and the excluded-middle principle to the general sign, our logical positivist would demonstrate that the proposition “The originary sign (Tao) is vague and general” is vague and general. And she would in a sense be correct. Not only is it vague and general and therefore by and large meaningless—according to her stringently limited notion of meaning—but it is absolutely yet indeterminately vague and general.

The vague sign, then, is inconsistent as well as indeterminate (truth or falsity is undecidable). And the general sign is incomplete as well as indeterminate. Indeterminacy is the character par excellence of the Monad, the unary sign, the icon, Firstness, as it is of the mediary sign, the symbol, Thirdness (CP: 1.468).

V. IT BECOMES A MATTER OF THE SELF AND ITS OTHERS. Interestingly enough, according to currently accepted limitations on knowledge, and especially following Gödel, if a sufficiently rich formal system presents itself as complete and self-sufficient, then it has an inconsistency hidden away somewhere in the closet; and if it is consistent, then it cannot but be incomplete. From the broadest possible view, the inconsistency of vagueness—or by implication, the quantum theoretical formulation—prevents completeness, since something cannot be, to use our above examaple, both a particle and a wave. In contrast, generality in the absolute sense demands completeness or nothing at all. But, as we have noted, there is no absolute generality in a finite world. Of course, from one reference frame an entity can appear as a particle and from another as a wave, but, since perspective is inevitably bound to particular reference frames, there can be no completeness for generality. On the other hand, though vagueness bans noncontradiction from its playground, it cannot entertain the actualization (into Secondness) of two inconsistent entities at the same instant: they are, so to speak, complementary, to be seen as one or the other alternately. Thus the inexorable incompleteness of generality and the inconsistency of vagueness are enforced. Generality and vagueness are in this manner themselves complementary at a more general level.

To take the analogy further, if I may be so permitted, the “quantum theoretical” or vague aspect of semiosis is indeterminate and linear, from First to Second. There must be a separate space-time property of each sign instantiation and its relation to and interaction with its “semiotically real” object. Such interaction implies, given Peirce’s pragmatic maxim, contextualized action regarding the manipulation of signs. That is, before the truth of a given proposition can be established, we must determine its meaning, and to that end Peirce provided a criterion for meaning, his maxim, which states: “Consider what effects that might conceivably have practical bearings, we conceive the object of our conception to have. Then our conception of these effects is the whole of our conception of the object” (5.402).

What Peirce seems to be saying is that the meaning of a proposition is given in another proposition, and that in another, ad infinitum, all of them calling for some conceivable test, as a result of observable properties that one would expect under certain circumstances if the original proposition were true. Since the semiotic web is indefinite and, as far as the finite thinker is concerned, for practical purposes infinite in extension, the meaning of the original proposition, therefore, is destined to remain indeterminate.¹⁷

Moreover, such activity as called for by the proposition is subjective (i.e., mind-dependent). The “semiotically real” can itself be composed of contradictory, or complementary, alternatives, such as the switching of the two possible interpretants of a Necker cube as it flip-flops from face up to face down, and back again. In contrast, the “semiotically real object” and the “real object” will continue to be intransigently incompatible on certain points, for the finite community at least. There is saving grace, however, inasmuch as the sign and its “semiotically real object” can approximate (hopefully at least) the “real.” Pragmatically speaking, then, each proposition can take on a particular context-dependent meaning, which is also dependent upon the context-dependent meanings of all related propositions, which, practically speaking, renders impossible any absolutely determinate meaning for some interpreter in a particular context.¹⁸

On the other hand, the “relativistic” or potentially general aspect of semiosis is, over the theoretical long haul of things, determinate and nonlinear: Thirdness. All sign entities in the semiotic field can theoretically be continuously mapped one onto another at certain points, hence the same space-time framework applies to them as a whole. Relations between signs are also theoretically “real”; they pertain to a given “semiotic reality.” Generality, insofar as it applies to the “real,” should be completely objective. This is, however, the ideal which can be no more than approximated. The “semiotically real” general sign is, and will always be for the finite mind, incomplete.

Thus the “clash” between First and Second, on the one hand, and Third, on the other, to yield “First-Second/Third,” illustrates (1) complementarity relations between the two sides of the virgule, (2) the dual nature of the asymptote metaphor, and (3) inherent inconsistency and incompleteness of all “semiotically real” conceptions of the world. There appears to be a basic distinction, then, between law (Thirdness—and, by extension, the general laws of nature) and rules governing our Umwelt-generated perception and conception of the furniture of the world (items of Secondness as semiotic actualizations of the possibilities of Firstness). Law, that is, law as the final interpretant, the fullness of all things, requires reversibility (time symmetry), continuity, and determinacy regarding the flow of events. In contrast, our perceived and conceived world requires irreversibility, discrete events, and acausality (probability of happenings, natural selection, or, at the highest levels, selection by free will).

Law is the law of generation (of the Final Interpretant), which implies the end of the long trail having been reached, for every would be has been actualized as a this-hereness. The process is a fait accompli. The Minkowski “block” universe this concept evokes entails determinism, “objectivity” (subject/object), and limitlessness. On the other side of the ledger, semiosis is the ongoing flux of sign interaction. It is the locus of information to be processed—the “reading” of signs—during one’s perceiving-conceiving the world, or, at the micro-level, during measurement in quantum theory. It is linear, irreversible, and time-dependent for the finite, limited, immanent observer, the subject-object dichotomy having been dissolved. The whole picture, including process and product, the stream of signs and its congealment, is possible solely by way of this Heraclitean conflictive interplay of opposites: the possible-actual on one side and law on the other, or Firstness-Secondness on one side and Thirdness on the other. In a nutshell, then, the complementary incompatibilities I have alluded to in the immediately preceding discussion are, once again, quite comparably those between quantum theory and relativity (see Sachs 1988; Pattee 1982).¹⁹

It is significant to note that Peirce struggled with a related complementary theme with respect to his two types of “other”: (1) the interaction between the self and the social and physical Other “out there,” and (2) the dialogue between the self and its other inner self. For Peirce the self does not attain a state of self-awareness—awareness of its other self—until it has erred regarding its perception and conception of the Other “out there.” When this occurs, the self artificially extricates itself from everything else in the universe which leads, residually, to what it is not as an objective realm set in the context of time and space—and the Cartesian split suffers its first birth pains. Yet this move is necessary, though it need not necessarily be taken to its extreme manifestation. That is, there can be mindfulness of self and/in the Other rather than self pitted against the Other. Indeed, this relationship, which can be written self/other// Other, is a double binary which generates a triad, and what is most significant, a complementarity relationship.

The self is a curious and unique some- “thing.” One assumes in general that it somehow represents an “object,” “I-me,” which functions in the external world as do all other objects. Yet it is nonempirical. As an “object” it is at most only partly available to any form of sensory experience. It is chiefly the product of imagination: it is inferred. One problem is that the self is not unary but binary in nature, and in addition, like all signs, it manifests a trinary character. For Descartes self-consciousness is immediately intuited, and the self is autonomous: it is primary, existing independently of all external constraints. In contrast, Peirce’s radically anti-Cartesian posture envisages a self which, like all signs, becomes external upon addressing itself to some other (CP:5.253; also Michaels 1977). In fact, it must determine some other, which is the task of all signs. In this sense, the self becomes aware of itself on becoming aware of what it is not, of the nonself, the Other (CP: 1.324).

This becoming of awareness entails action/reaction, Secondness, the “real.” During the rough and tumble of everyday life we are constantly “bumping up against hard fact.” We develop habits and generate expectations, rather mindlessly taking things for granted as simply the way the world works. But the world inevitably sets up a resistance against our imposing ourselves upon it; it forces surprises on us, which constantly remind the self of what it is not. This notion of the “other, of not, becomes a very pivot of thought” (CP:1.324). Secondness is predominant in this scheme because it is essential to the very idea of “reality,” for the “real” is that which “insists upon forcing its way to recognition as something Other than the mind’s creation” (CP:1.325).

The “real” is actual, though of course it is Other. In conjunction with the self it makes up the other pole of a dyad. But the “real” is not immediately present to consciousness as such and such. Consciousness becomes aware of the “real” solely by trinary mediation—Thirdness—between itself and that Other, and by inference, as it were. That is, there is no time in the “present instant” for an inference, least of all for an inference concerning that very instant. An inference is possible only mediately, after the occurrence of a new event. This new event in the form of a surprise shocks one to attention such that one can then generate an inference regarding (1) the break between that which was expected and the event that is perceived to have occurred, and (2) the reason for such a break in the first place. Consciousness of must be mediate, Peirce argues, since the presence of the “real” has no respect for one’s will or wishes; consciousness of is not merely dependent upon an act of volition. This implies that the immediately present is conative rather than perceptive. Consequently, consciousness of the “real” present is a perpetual struggle over what was expected and what is perceived actually to take place, the break between them evoking an incessant call for revamped expectations (CP:5.462).

One is in some sense immediately conscious of one’s feelings, of course, but they are not feelings of a self-conscious ego, for the ego-self is inferred rather than immediate (CP:5.462). Peirce suggests that we are in command of no power by which an intuition can directly and immediately be known. An intuition must exist at some primordial first instant, and for it to be an intuition in the full Cartesian sense, apprehension of it as a cognition must occur at the selfsame instant; that is, it must be an event occupying no time. But since for Peirce, (1) any and all cognitions are always in a process of becoming and passing away, (2) it is impossible to know intuitively if a given cognition is not determined by a previous one, and (3) a cognition, like a perception, is not available to consciousness except mediately, then (4) there can be no apprehension of an intuition in the blink of an instant (CP:5.264-317).

Given that any cognition for Peirce is consciousness of an object as represented to self-consciousness, he means to say the same of one’s knowledge of or consciousness of one’s self. Such self-consciousness is not “a mere feeling of subjective conditions of consciousness, but of our personal selves. Pure apperception is the self-assertion of the ego; the self-consciousness here meant is the recognition of my private self. I know that I (not merely the I) exist” (CP:5.225).

Kant once suggested—and Peirce agrees on this point—that the retarded use of the “I” in children, since they manifest other powers of thought at an earlier age, was evidence of an undeveloped self-consciousness in them. When a child hears, say, a bell, she does not become aware of herself as hearing something which she herself is not; she is merely aware of some object which is making a sound. Or when she “wills” to move a table, she does not become aware of herself as desiring that the table be displaced, only that an object is fit to be moved (CP:5.230). The child soon discovers the relationship between objects “out there” and her own body as another object, which raises it to a level of importance and centrality. And later, after learning a language, the child is subject to sentences such as the warning “The stove is hot.” And so long as she never comes into contact with the stove, she will not know whether it is actually hot or not. If she touches it, on the other hand, the sentence is verified in a striking way:

Thus [s]he becomes aware of ignorance, and it is necessary to suppose a self in which this ignorance can inhere. So testimony gives the first dawning of self-consciousness. . . . In short, error appears, and it can be explained only by supposing a self which is fallible. Ignorance and error are all that distinguish our private selves from the absolute ego of pure apperception. (CP:5.233-35)

On so speaking of the ego, Peirce refers to the triad feeling-volition-cognition. Feeling is Firstness, quality. Volition is dual: force and resistance, agent and patient, self and Other. The shock of an unexpected event is volitional, the result of interaction between the self and the “real,” which gives rise to dyadic consciousness of an ego and a nonego (CP:1.334-35). Cognition, or Thirdness, is the process of mediating between feelings and volitions. Metaphorically put: “Position is first, velocity or the relation of two successive positions is second, acceleration or the relation of three successive positions third” (CP:1.337). Peirce’s analogy is apropos. Velocity is continuous, but there is merely change of position. In contrast, acceleration is continuous change of change; both position and velocity undergo successive alteration.

Quite obviously, feeling, volition, and cognition correspond to Peirce’s tripartite sign. The representamen is immediate. That for which it stands, the object, is other than the self and subject to volition. And the idea to which the representamen gives rise is its interpretant, which entails cognitive activity. The object of representation is not the “actually real” but a “semiotically real object” represented by the sign, so the object of representation can be none other than another representation of which the first representation is the interpretant, and an endless train of representations can be conceived to have the “absolutely real” object behind it as a limit, which can no more than be approximated asymptotically (CP:1.339).²⁰ In brief, every interpretant becomes a sign-representation in the ongoing semiosic process. Like position, the sign stands for the “semiotically real object,” which is moved along by the incessant transmutation of interpretants into sign-representations. And the interpretant, as mover, accelerates the sign, like the force of gravity at 32 ft./sec./sec. or the expanding universe which brings about the “red shift” phenomenon, toward the ideal limit.

This brings up the point, introduced above, that in addition to the external Other, there exists an internal one, the other self created by the passing of a sign into an interpretant, an interpretant into a sign, and the self of one moment into the self of another. The radical absence of the self from its other self, semiotically (i.e., symbolically) evidenced by shifters, creates not a stable but a restless semiosis incessantly sliding along the slope of signification. This renders the timeless identity of the self impossible:the self cannot be itself in the immediate present, but only what it was not during the moment past, and what it not yet is, what it will be, in the future moment.

However, the Other, above all, remains elusive. Regarding the sign triad, the interpretant is acknowledgment of the Other by way of mediation between representamen and object. But since it is itself another representamen whose interpretant is yet another one, ad infinitum, its self-identity incessantly conceals itself, and, as Derrida (1974:49) says of the signified, it is always on the move. The Other to which consciousness points via the interpretant is never fully present, though its presence is always felt, since the “real” represented by signs as the “semiotically real” remains as fugitive alterity and absence: “In the idea of reality, secondness is predominant; for the real is that which insists upon forcing its way to recognition as the mind’s creation” (CP:1.326). The “real” conceived as Other is in this manner resistance, surprise, a subversion of exteriority against the self-conscious self.

But this sense of exteriority, of the presence of a nonego “which accompanies perception generally and helps to distinguish it from dreaming” (CP:1.332), is not merely a sense of the world “out there.” It is the product, rather, of the dialogic self-other, “I-me,” the “me” resting tenuously between the “I” and the “it,” which constitutes the “real.” Freud’s (1925) concept of negation also bears on the “I-it” interaction. The prononoun it marks an irreversible loss of the self’s self-presence, the self’s “me” as absent from its “I.” The child initially uses it to designate what will later correspond to its own displaced self. At this early stage there is no subject/object, no inner/outer. Gradually the “it” transmutes into “me” and becomes “outside” and “alien,” in contrast to the “inside” and generally conceived “presentness” of the “I.”

And thus Peirce’s train of signs embodied in the “I-me” dialogue has lurched from the starting blocks. When one is thinking, the “I” uses signs by means of which to persuade the “me” that something or other is the case. In this activity,

a person is not abolutely an individual. His thoughts are what he is “saying to himself,” that is, saying to that other self that is just coming into life in the flow of time. When one reasons, it is that critical self that one is trying to persuade, and all thought whatsoever is a sign, and is mostly of the nature of language. The second thing to remember is that the man’s circle of society (however widely or narrowly this phrase may be understood), is a sort of loosely compacted person, in some respects of higher rank than the person of an individual organism. (CP:5.421)

In other words, the immediate “I” as First (icon) refers to the object (index) or that which is the object of persuasion, in addressing itself to the other, the interpretant (symbol) in the process of “coming into life in the flow of time,” which in turn becomes itself a sign with its own object (CP:2.274).

This other-Other concept is illustrated by the two “Borgeses” of that remarkable little tale “Borges and I” (1962:246-47). The sentient Borges (Borges₁) is the “subject” whose life “is a flight,” for he loses more and more to the writer Borges (Borges₂), the “predicate.” Borges₂, the fictional being who appears only in the mail, on lists of professors, or in biographical dictionaries, appropriates more and more of Borges₁, who retreats into ever-smaller sub-frames. At the precise moment when Borges₂ completely takes over, specific reference to either of the two disappears. THE BORGES (Borges₁ and Borges₂) now exercises his dominance. There is no longer any line of demarcation between “subject” and “predicate” (“object”). THE BORGES is self-contained and self-sufficient:he is as he is. As such there can be no inconsistency or incompatibility between the two because there is no external reference point from which to make a cut between the one and the other. But as such neither can any fictions—i.e., the fictional Borges₂—be constructed from “within” the system, for fictions require (1) the existence of two partly inconsistent frames, and (2) incompleteness, such that a potentially infinite regress of metaframes can theoretically be constructed (see Merrell 1983). The status of THE BORGES, then, is at the meso-cosmic level equivalent to the cosmic experience of the ineffable totality within which the self is contained. In this state, nothing is marked or cut, there are no frames, and hence fictionality (i.e., what the “real” in part is not) is no longer possible, for all simply is. There is no longer any dialogue.

Thus the self-other relation emerging from the self-Other entails separation and union. Self and other are from one vantage logically distinct entities. The self enters into dialogue with its other and that other with the self, as separate selves in mutual relation. At the same time, from an alternate vantage, self and other are one, continuous and inseparable, a self-referential unity. In this respect the relation is closed rather than open, like the self-Other relation. But the question arises:How can the individual self come into existence if the self-other is an indivisible whole? An answer might be forthcoming, once again, in view of Peirce’s asymptote. Self-other as an indivisible whole appears in the sense of a tenuous approximation to that ideal goal, when all dialogue ceases, when total Oneness prevails, and Truth in all its plenitude is made manifest: the self-other, that is, the interpreter, at this point becomes coterminous with the ultimate interpretant.

In addition, there is an approximation in the opposite direction toward the ideal self-Other relation wherein the self itself has reached finality: it is an absolute absolutely set apart from the absolute Other. Yet, as an approximation, there is no more than an illusion of the categorical split between the one and the other; there is always at least a modicum of oneness in the self-Other interaction (much like the Yin/Yang complementarity, which likely influenced Bohr in the development of his quantum principle). In other words, self-Other does not entail the same ontological status as self-other; rather, the apparent objectivity of the self in this case is over the long haul an idea, an inference, an artificial construct. It is the manifestation of a whole arrived at by reflection on what the that is and why it is other than this. Solely the self-other state is capable of erecting the self-Other construct, but the self-Other must be acknowledged in terms of something other than what it otherwise would have been—i.e., when there is awareness of error. Without the self-Other there can be no scientific discourse. In fact, there can be no dialogue whatsoever from which the self-Other can arise.²¹

The observing self, then, is only apparently an autonomous subject, and the Other only appears to be an object “out there” and ready to be manipulated by the subject. Once more, there is neither exclusively one nor many, continuity nor discontinuity, but perpetual interplay between them. In contrast, there is an ordinarily contradictory yet complementary relation between self-other and self-Other which in turn complements the complementarity between self and other and self and Other. If self-other and self-Other were a continuous, inseparable whole, it would undoubtedly evince qualities comparable to the Tao of Lao Tzu. It would be simply that which is, outside all sensation of time, space, and “stuff.” It would be ontologically prior to all existent things:a closed, recursively autonomous, symmetrical, timeless system—tantamount to the Minkowski “block” or the originary Monad. In contrast, combining the two pairs to form self/other//Other creates a complementary relationship. In one of his earlier formulations of complementarity, Bohr (1934:91) concluded that the concept “bears a deep-going analogy to the general difficulty in the formulation of human ideas, inherent in the distinction between subject and object.” Since that time, there have been few discussions of the problem of measurement or the meaning of quantum explanations without invoking the mind (self-other) of the observer as an irreducible element.

This schizophrenic collusion of many and one, of holism and atomism, that is, this contradictory complementarity, an important aspect of contemporary thought especially regarding both the quantum world and relativity, is, most significantly, patterned in Peirce’s own metaphysical schizophrenia.