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This review article appeared in Volume 7 (2) of the Semiotic Review of Books.

Infinity

R. J A. Kilbourn

Ad Infinitum: The Ghost in Turing's Machine (Taking God out of Mathematics and Putting the Body Back in: An Essay in Corporeal Semiotics). By Brian Rotman. Stanford: Stanford University Press. 1993. 203pp. ISBN 0-8047-2128-9.

There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite.
- Borges, "Avatars of the Tortoise"

Note: Throughout this review article, the term "ellipsis" is represented by "... " instead of a character resembling a rounded upside-down caret symbol, due to the restrictions imposed by internet programming.

Infinity lies at one extremity of Western (and, some would say, Eastern) thought, one end of a metaphorical line which, in conformity with the non-Euclidean geometry of consciousness, bends back upon itself, its ends appearing to converge in a meeting of radical negativity and limitless magnitude, beyond space and time.

In the midst of his demystification of modernity as secularized eschatology, Hans Blumenberg acknowledges that the best evidence for it is Giordano Bruno's application of the attribute of infinity to the world (Blumenberg 1985: 79). This is not to say that the world or the universe had never been conceived in these terms before; Plato, for one, ascribes a conditional infinitude to the universe in the Timaeus (section 38). Under the Christianity of the Renaissance, infinity was the last attribute left to God, besides being. If infinity 'migrated' from God to the world when the former withdrew from the latter (or when God was denied being as well), then now that we no longer hold the universe to be infinite, where does this attribute reside? In other words, in what disciplines or discourses does infinity still have a place or perform some function? Mathematics, for one. In his first book - Signifying Nothing: The Semiotics of Zero (1987), which left off where his new one begins - Brian Rotman recounted the crucial significance of zero in the history of calculation. Zero, of course, has two distinct functions in numerical terms: it is, on the one hand, point of origin and telos, and on the other, a place-holder that also affects the value of adjacent numerals, despite the fact that it has no numerical value in itself (Rotman 1987: 12). The presence of infinity in classically conceived mathematics is tied to that of zero, which had appeared in the West by the twelfth century in the form of the Hindu-Arabic sign, which is also the root of 'cipher.' Rotman maintained that "the mathematical infinite was the fruit of the mathematical nothing: it is only by virtue of zero that infinity comes to be signifiable in mathematics (Rotman 1987: 71). But where theology and, more recently, metaphysics, were displaced as authoritative discursive systems in the post-medieval period, mathematics has retained an authoritative status, not least because, in addition to what Einstein somewhat facetiously called its 'absolute truth' and indisputability' (giving to "the exact sciences a measure of confidence which they could not otherwise attain" [Rosenfeld 1988: 314]), mathematics in its instrumental function undergirds the technological and commercial world that most people inhabit (Rotman 1993: 141). Generally speaking, from a late-twentieth century vantage point, infinity is in no danger of vanishing from the contemporary field of theoretical inquiry - a situation whose apparent irony is subverted by a deeper logic.

Brian Rotman's more recent book, the exhaustively titled Ad Infinitum: The Ghost in Turing's Machine (Taking God out of Mathematics and Putting the Body back in: An Essay in corporeal semiotics) engages with the problem of the infinite in mathematics and semiotics. Rotman's book, as an essay with a relatively narrow set of concerns (including infinity) is a generally coherent, often demanding work. In general, Rotman's systematic exploration of a possible alternative to a metaphysically based, Euclidean mathematics raises a host of other questions revolving around the venerable problem of the representation or signification of something which evades reduction to the finite materiality of this or that signifier. More specifically, Rotman's thesis is a thorough-going attempt to confront what would make most people very bored but which seems to make him very nervous: the possibility of potentially or actually endless counting.

The opening of Rotman's 'abstract' draws attention to the long-standing connection between infinity and the divine (ix), and as such introduces his argument's principal element of contention: as he repeats throughout, 'how to think the infinite?' What is meant by the infinite in this semio-mathematical context? Having introduced this foundational problem, Rotman immediately shifts away from the theo-metaphysical to the mathematical (and, secondarily, semiotic) infinite, where it is found at work in both "the geometrical continuum of points on a line and their integer-based real number descriptions - two linked abstractions which ground all post-Renaissance mathematics. And it is the founding signified, the crucial ontological term, in contemporary mathematics' description of itself as an infinite hierarchy of infinite sets" (ix). That the object of thought is relegated to this specific context does little to alleviate the problem of thinking infinity.

Rotman asks how a set of 'natural' numbers (1, 2, 3... ) can exist, like the symbolic order, not merely independently of, but prior to, the subject who uses them in counting (x). This is a problem because of the ellipsis (... ) that always ends the example: what Rotman calls the fundamental ideogrammatic shorthand for infinity (ix). "How," Rotman asks, "does infinity get to be an exact, rigorously specified mathematical object - an object about which mathematics delivers 'true' and 'objective' knowledge?" (ix). In other words, how does infinity, in itself unthinkable, get to be thought, when only a thing, by definition, is 'thinkable' (can be thought, thematized, comprehended, totalized)? To totalize infinity as something beyond its presence as a sign is a logical contradiction, in that it requires a 'finitization' of infinity, but then the idea of infinity is itself a logical contradiction. So how, as Rotman asks, are we to think infinity? Rotman rightly points out the easily forgotten precept that we "are never presented with the pure idea of infinity as such. How could we be?" (x). The 'idea' of infinity, like the idea of God, is not a thing, properly speaking. But for Rotman "being thought in mathematics always comes woven into and inseparable from being written." In other words, it is precisely as a 'thing,' a sign (word, ideogram) that Rotman chooses to think infinity: if numbers, being written, "can be uncreated, rewritten, deconstructed, altered," then infinity "becomes inseparable from certain effects of the signifier, a phenomenon of mathematical texts, grammar, syntax, notations, and discourse" (35).

Rotman frames his discussion of infinity from the very beginning in terms of mathematics, rather than what he dismisses in the opening paragraph as "its philosophico-theological obscurities and contradictions" (ix). This will prove to be decisive in his subsequent conceptualization of a non-infinitist mathematics according to a restrictive binary logic: either infinity is allowed for in all its intolerable unendingness, or it is excluded, and the system re-calibrated in terms of the comfortingly positive ground of the counting subject's bodily (limited, mortal, finite) presence. For Rotman, "the question of reinstating the body [is] all-pervasive and crucially important" (179 n. 7).

Rotman's focus (as he resolutely maintains), is on the mathematical concept - even the infinite itself - as written : "Thinking in mathematics is always through, by means of, in relation to the manipulation of inscriptions. Mathematics is at the same time a play of imagination and a discourse of written symbols" (x). He then posits that "the question of the mathematical infinite" be posed "as a question of language, as part of an overall study of the nature and practice of mathematical signs - as part, that is, of a semiotics of mathematics" (x).

In developing this semiotics Rotman renounces a "philosophical critique of the metaphysical system, the rampant Platonism, that threads its way through the contemporary interpretation of mathematics" (xi). Rather, he engages in a critique of what seems to him "an altogether more subtle metaphysical principle that permeates the entire subject": "the principle of ad infinitum continuation¨ inseparable from the mathematical community's wholesale acceptance of the view that the numbers are 'natural,' and its failure to ask the question of where these numbers could possibly have come from" (xi). The question of the mathematical infinite is thus from the start tied to the question of counting and therefore to the question 'who counts?'; that is, the question of a mathematical subject. Rotman's investigation of infinity is inextricably bound up with a concern for the counting subject, the one who is faced with the prospect of potentially endless counting so succinctly expressed in that ideogram '¨'. But Rotman's question about 'thinking the infinite' does not disappear so easily, since even the answer of thinking via the written still implies someone or something doing the thinking. Rotman's reiterated concern with reinstating the body in mathematics cannot dispel the insistence of consciousness in any discussion of infinity. It emerges that this rhetoric of the body is a means of providing conceptual or even cognitive limits to an otherwise limitless discussion; i.e. everything corporeal is finite and therefore limited and knowable (temporality, implying change and mortality, are for the moment unmentioned). In other words, to emphasize the centrality of consciousness - of the counting subject as mind rather than body - would be to contaminate the investigation of one instance of radical unknowability (the mathematical infinite) with another, assuming the presence in consciousness of a constitutive and irreducible negative dimension (whether figured as the 'unconscious' or otherwise). Nevertheless, the question of the mathematical subject's relative self-consciousness will prove to be unavoidable.

The three elements in the title of the 'abstract,' "God, Number, The Body," are presented in ascending order of importance to Rotman's thesis. He returns, to begin with, to the topic briefly broached at the outset: the insistence (that is to say, the unavoidability) of God in any discussion of infinity - a topic that appears to cause Rotman anxiety if only because he feels it distracts from the real and rational matter at hand: a body-based mathematics. "[Is] there not in the very idea of their endlessness , their continuation ad infinitum , something strange and other about the whole numbers, the imprint or trace of some disembodied transcendent maker, perhaps?" (3). Rotman provides a brief overview of the history of mathematical thought (3), highlighting moments of 'crisis,' when "the question of the infinite¨pushed itself to the foreground." Here he emphasizes the Aristotelian distinction "between a safe and legitimate potential infinite, and endless coming into being, and a dangerous, paradox-infested completed or actual infinite" (4). This distinction will prove to be of more than historical significance to the subsequent development of Rotman's argument. Rotman considers the potential infinite to be the only conception of infinity that can be "cognized as meaningful and mathematically interpretable" (44).

As in the history of philosophy, mathematical thought emerges out of the two major systems of Aristotle and Plato, and modern mathematics is still thoroughly imbued with elements of both, primarily the latter. "For most mathematicians, mathematics is a Platonic science, the study of timeless entities, pure forms that are somehow or other simply 'out there,' preexistent objects independent of human volition or of any conceivable human activity" (5). In classical terms, then, mathematics is thus conceived, inconceivably, as an infinite set of infinite sets. Overagainst this exists a 'constructivist' alternative, which insists "that any mathematical proof of the existence of an object had to be in the form of¨a finitely specifiable procedure that could 'in principle' be executed in the mind" (5). In this view, the 'progression of integers' is "a potential and not an actual infinity," where the act of counting is "to be performed deep inside our - Kantian - intuition of time" (5). Rotman attempts to position himself as neither a Platonist nor a constructivist (constructivism being immersed in "an unexaminedly ideal mentalism" [6]).

This leads directly into the problem of 'natural' numbers ('natural' "because they are given at the outset, taken for granted as a founding, unanalyzable intuition outside any critique that might demand an account of how they come or came - potentially or actually - to 'be'" [5]) and the even more basic question of number's ultimate provenance: "Where do numbers come from? If not from Kant's transcendental intuition or¨God, then where?" (6). Again Rotman has recourse to writing, inscription, signification, via the inescapable process of counting, since counting "is an activity involving signs" which "works through¨significant repetition" (6). From which follows a question which effectively reframes his initial question regarding infinity in terms more mathematical but no less epistemological: "How are we to imagine a business of repeating the self-same signifying act without end, of iterating for ever ? Or, which will come to the same, what would it mean to deny the possibility of endlessly repeating a signifying act?" (6).

Mathematics is perhaps unique as a language in the number of signifiers it harbours which "seem to require an 'infinity' as their signifieds" (6). For Rotman this is evidence that "our contact with infinity is always and only through writing" (6). The strange thing about mathematical ideograms, as Rotman points out, is that they provide both something to count and the means by which to count (7). Above and beyond the integers themselves, however, Rotman distinguishes a language of 'formal mathematics': what he calls 'the Code,' comprised of terse imperatives devoid of 'indexical expressions such as subject/object pronouns, adverbs, inflected verbs, etc. - words which "tie the meaning of messages to the physical context of their utterance (7)." Given this language, or 'Code,' Rotman then develops a model that reflects what it means to "do mathematics," that situates the mathematical agency implied by the Code.

This model is imagined as a thought experiment in Peircian terms, "played out through written signs" (8) and "organized in terms of three figures" or 'semiotic agencies' operating "simultaneously at different levels of discourse": "the mathematical Subject " - the reader/writer of formal mathematical texts, "who uses the Code but has no access to any description of itself" (8); the "Person ," immersed in history "and in the cultural subjectivity coded by the 'I' of natural language that permeates the metaCode" of 'informal mathematical language.' In other words, the 'Subject' is not a subject properly speaking, whereas the 'Person' is. As if the Subject weren't abstract enough, it has an "Agent," an "idealized simulacrum of itself as its surrogate," an "automaton without the ability to engage with any meanings," operating "only with signifiers at a sub-Coded level." All three of these 'agents' (although the 'Agent' is not an agent properly speaking) are necessary to "enact a single thought experiment narrative. What Rotman has articulated with this triple agency is the inside-out metaphysics he perceives as constitutive of mathematics, which, like many alternative universes and possible worlds posited in fiction, decreases in verisimilitude the 'deeper' the thought-experimenter (the 'Person'?) delves. In fact, the faint echo of a rabbit hole or dim reflection of a looking glass world become more insistent when one notices Rotman's repeated use of the trope of the 'waking dream' for both the practice of mathematics and the (theoretical) thought experiment about it: "I read mathematical signs in terms of a certain written practice, a business of manipulatinginscriptions that characterizes mathematical thought as a kind of waking dream" (xii); or: "[the] thought-experimental model allows us to read mathematics as a business of making certain kinds of 'rigorous fantasies' or waking dreams" (9). The imagined structure of the thought experiment verges on the Borgesian: "The imagining Subject corresponds to the dreamer dreaming the dream, the skeleton Agent to the imago, the figure being dreamed, and the Person to the dreamer awake in the conscious subjectivity of language telling the dream" (9). The potential for this model to turn into an infinitely regressing mise-en-a-dream is thwarted by that third term: the conscious, subjective, embodied, wide-awake 'Person,' recounting his or her dream, not in the mathematical meta-Code but in what is presumably the non-mathematical language of narrative, the telling of a tale which shares its etymology with tallying (both 'tale' and 'tally' are traceable back to the Indo-European *del, to "aim at, calculate, lie in wait"), or potentially endless counting, but has for a variety of reasons succeeded in hypostatizing a form of arbitrary closure within its specific logic (narratology). This is by no means to stray beyond the bounds of Rotman's text: if it is possible to conceive of a discursively-constituted subject that narrates itself into being, it might be possible to conceive of a mathematical subject that counts numbers into being. Assuming the unitary nature of such a subject, there is no need that a limit be imposed from outside on the potential endlessness of counting; the subject's physical death (assuming it exists within time) will guarantee its cessation.

As Rotman remarks in a later chapter: "[counting] presents itself as prototypical of the very business of sign creation itself. We count by repeatedly enacting the elemental process of creating identity by nullifying difference, repeatedly affixing the same sign '1' to individual 'things' - objects, entities - that are manifestly not the same qua individuals in the world-before-counting from which they have been taken" (51). Rotman interprets counting "as a mathematical ur -cognition, as the pure and distilled mode of the production of identity and sameness" (51).

For Rotman the culprit is what he names "the ad infinitum principle - the principle of always one more time (52)," whose mathematical version is the axiom that "for any number x there exists a number y such that y = x + 1," without which it is impossible to conceive of the endlessness of numbers. In considering the implications of rejecting this endlessness, Rotman's focus is once again the counting subject: rather than posit the finitude of the physical universe (a finite quantity of particles to be counted), Rotman suggests the limitation of "the time-span of an individual life (53)," and just as quickly rejects this "move of constraint" as "unacceptably arbitrary."

In the triadic subject of classical mathematics, it is the relation between Subject and Agent, specifically their resemblance, which is crucial, and in this respect "the decisive characteristic is that of physicality" (9). According to Rotman, if it is going to count endlessly for us, as it were in our place, the Agent-imago cannot have a body, it "has to be something transcendental, it has to be a ghost" (9-10). It cannot even be a puppet or machine, for example, since any "scrap of physical being however rarified and idealized" will necessarily sabotage its "efforts to count endlessly," as it will then be subject to spatio-temporal contingencies, energy loss, entropy, and so forth. Rotman focuses on this element of physicality: "[w]hy should an embodied mathematical Subject, whose identity and ability to interpret signs are inseparable from its physical being and contingent presence in the world, create a totally disembodied Agent as its proxy?" (10). The danger of the classical schema is that the "disembodied Agent," "a spirit, a ghost or angel required by classical mathematics to give meaning to 'endless' counting," comes to resemble the God of metaphysics (10). Thus the ghost moves from Turing's machine to "Plato's True Unchanging Heaven." Rotman contends that it is therefore necessary to disbelieve in this disembodied Agent; to "reject not only Platonic orthodoxy, but, more fundamentally, the very idea of disembodiment itself, to refuse altogether the imago of endlessness" (10).

Rotman proceeds to grant this classical mathematical Agent (itself a particularly subtle 'metaphysical principle' [23]) a "suitably idealized but never absent" body (10). "The resulting corporealized mathematics opens out into a new conception of iteration, of counting and therefore of what we might and could mean by 'number.' What emerges is a non-Euclidean arithmetic" (10-11). At this point it is important to read carefully, for what Rotman is instituting is not a denial of infinity per se (the idea of infinity) but of what Hegel called the 'finite' or 'bad infinite' specifically: the endless "iteration of the same" (11; cf. Hegel 137-38 [section 93-95]). Something becomes another; this other is itself something; therefore it likewise becomes an other, and so on ad infinitum." (Hegel 137 [93]; translation slightly modified). Rotman does not deny a "transcendentally mysterious infinitude" so much as set it off absolutely from "our actual experience of iteration" (11; see 180 n. 12). This interrogation of natural number's endlessness (corresponding to a 'givenness') is replaced by a physically limited coming into being, determined by the embodied Agent.

The Russian mathematician Nikolai Lobachevsky published his treatise on Non-Euclidean Geometry in 1826, in what is now seen as a major turning point in the history of modern mathematical thought. But non-Euclidean thinking had already captured the attention of non-mathematicians, writers like Kleist ("On the Marionettetheatre") and, at the end of the century, Dostoevsky (The Brothers Karamazov ). It could be said that non-Euclidean mathematics is in principle virtually as old as Euclidean mathematics, against which it defines itself. In other words, what Rotman describes is a relatively profound epistemological shift that is not contingent so much on the confluence of certain historical factors as on what he sees as the immense existential pressure exerted on the subject by the pre-existent order that is the 'givenness' of number, and the unfathomable prospect of endlessly iterated integers stretching off into infinity. Rotman does not need to disprove Euclid's theory of asymptotic parallels, for example, to postulate a non-Euclidean geometry; this potentially unlimited line of numbers is adequate: rather than stretching infinitely in either direction (i.e. on the 'plus' or 'minus' side of zero), this line (the temporal x-axis on a graph) in Rotman's schema just 'peters out' into non-existence ("an entropic diminuendo"). In this respect, Rotman does not (or chooses not to) see the full implications of a non-Euclidean system.

Rotman's "non-Euclidean arithmetic" purports to be anti-Platonic, anti-metaphysical, a-theistic, non-psychologistic, and non-relativistic. Furthermore, although Rotman admits that his account of mathematics is "undeniablyconstructivist" (22), it also departs from the constructivist line in such important respects as the question of number's coming into being (23). At the same time, Rotman positions his thinking about language in general within the 'conflict' between "the so-called continental outlook dominated by Nietzsche, Husserl, Heidegger, Wittgenstein, and Derrida" and the current 'Anglo-American' "analytic mindset associated with [Gottlob] Frege, Bertrand Russell, and their empiricist forebears" (16). That is, the slogan "Language speaks man into the world" versus "Man speaks language about the world." To say that Rotman leans toward the latter, however, is not to suggest that his re-conceived mathematics is also a straightforward reinstatement of 'man.'

In his second chapter ("Language") Rotman poses the question as to whether mathematics can be considered a language, which, from a conventional semiotic perspective, may be considered rhetorical; that is to say mathematics may not be what Rotman calls a 'natural language,' but it is a signifying practice, a sign system (see 27ff). Rotman's insistent anti-psychologism is predicated on his semiotics: for him mathematics is irreducibly written, a series of marks made by the embodied mathematician, however 'idealized' (Rotman's word): "mathematical language and discourse deal in, are oriented toward, and are 'about' mathematicians' own inscriptional activities; so that, if one insists on using the term, mathematics might be said to 'refer' (like music) to nothing other than itself" (24). On this basis Rotman argues that mathematics, as a signifying practice, "would rapidly become unintelligible" were it not for its as it were parasitic (my word) relation to what he calls a ''natural,' non-mathematical host language."

Rotman's assertion of mathematics' fundamental 'writtenness' has its antecedent of course in Of Grammatology (1976), where Derrida invokes theoretical mathematics as exemplary of a non-phonetic 'language,' one which does not depend and has never depended on the actual or metaphysical presence of a signifying intention (Derrida 1976: 10). In a long note, Rotman makes explicit the connection between his Code/metaCode opposition and Derrida's characterization of the relation of writing to speech as 'secondary' and 'supplementary,' "inside a logocentricized Western thought," to use Rotman's formulation. However, Rotman claims that his analysis bears an inverse relation to the precepts of grammatology: "[what] operates in mathematics is not logocentrism, not the privileging of speech over writing, of primary self-presence over a despised secondarity, but the reverse: a form of graphocentrism, the privileging of the formal writing of the Code over an eliminable, theoretically unnecessary - epiphenomenal - metaCode" (185 n. 28). Rotman appears to overlook the fact that Derrida was talking about not just "alphabetic writing" when he singled out mathematics as the model of a non-phono-logocentric, language, "untainted by the metaphysics of presence" (185 n. 28). In a long note, Rotman maintains that the thrust of his essay is ultimately against Derrida's conception of mathematics as quintessentially written, toward the locating of "a deeply metaphysical principle at work within mathematics' current conception of 'number' but also to reinstate the body and the subject with their talk, noise, and physics of presence onto the mathematical scene" (185 n. 28). He would root out metaphysical presence in order to better reinstate physical presence in mathematics. The precise nature of the proposed subject's ideal, non-metaphysical, embodiedpresence is never adequately explained.

In discussing mathematical 'language' or signs, Rotman explains that he is referring to "ideograms in the usual sense of written characters conveying, invoking, or denoting conceptual content - signifying - through their graphic identity, as visually presented marks" (26). Rotman singles out the ideograms '0' and '... ' as occupying a "more primitive and originary signifying level" and as therefore underpinning arithmetic counting (0, 1, 2, 3... ). It is common knowledge however, as Rotman himself explains in his earlier book, Signifying Nothing: The Semiotics of Zero, that there was counting in this sense of arithmetic progression or 'tallying,' long before the introduction of zero, which served to open up whole new vistas of calculation and numerical representation. The point that Rotman does not make is that it was the inception of zero that permitted the representation, and thus in a certain sense the conceptualization, of positive infinity. That is, the introduction of '0' was a prerequisite for the introduction of '... ' as a mathematical ideogram. Rotman does not pursue this causal relation in the new book.

Via summaries of the theories of Saussure, Peirce and, to a lesser extent Benveniste, Rotman fleshes out his earlier line about "language speaking man into the world" (16), in so far as subjectivity is constituted in the individual language-user's appropriation of pre-existent indexical forms (most notably 'I') to define him/herself in relation to other users ('you'), within a specific spatio-temporal physical context ('here,' 'now,' 'this') (30). In Rotman's reading, the semiotic subject is still an abstract 'type,' and subjectivity therefore something 'there' prior to the situated physicality of the speaker (his 'person,' as Benveniste says [30]).

Rotman's third chapter considers the infinite in relation to the finite, moving away in his usage temporarily from the mathematical infinite to something more metaphysical (39). In a note, however, he seeks to distance his usage of the infinite from Hegel's insistence on a 'bad' infinite that is "conditioned, conceptually limited," and by definition finite (180 n. 12). Rotman's protestations aside, it remains unclear how the mathematical infinite he elaborates differs from Hegel's bad infinite, particularly in light of his obsession with the ideogrammatic infinite ('... '), and with its limitation or 'finitization' by the 'presence' of the body of the counting subject. Rotman rejects the good/bad distinction (and its echoes in contemporary mathematics) on the grounds that it represents "the inevitable return of an unacknowledged and buried theism" (180 n. 12). This 'theism' is most obviously 'buried' beneath the corollary to the bad infinite; the absolute infinite which Hegel apophatically but rather mean-spiritedly describes as "a wretched neither-one-thing-nor-another" (Hegel 138 [94]).

Rotman gives away his position vis-a-vis the infinite as source of "inconsistencies, contradictions, paradoxes, antinomies, and other productions of discourse intolerable to mathematical reasoning" (39): it plainly makes him nervous. In this vein, Rotman traces the impact of Zeno's paradoxes on subsequent mathematical thought. On the one hand, there is a distrust of motion, based on infinite divisibility and infinitesimal quantities - in other words the familiar stadium and arrow paradoxes - and on the other hand a distrust of the very opposite, "an infinitely straight line," which gave rise to Euclid's parallel line axiom, wherein "parallel straight lines are straight lines which, being in the same plane and beingproduced indefinitely in both directions, do not meet one another in either direction" (Rosenfeld 35) (40).

Rotman forges ahead in his quest to formulate a critique of the endless iteration of counting which remains within "a quite narrowly drawn conception of the rational" (54), and thus paves the way for the emergence of non-Euclidean arithmetics which allows for a kind of closure (56). Non-Euclidean numbers are themselves, in Rotman's view, a function of the entropic physical universe (rather than the other way around), and as such do not behave in the predictable and 'pure' manner of the 'natural' numbers of Euclidean arithmetic. Rotman allows for the possibility however that his establishing the possibility of denying the ad infinitum principle is by no means "to be taken as a repudiation of the classically conceived Euclidean infinite as such" (58). He readily admits that, from the contemporary, Platonist or constructivist viewpoint, the adoption of a non-Euclidean mathematics presents no "real challenge to the idea of infinity" (58). A 'Euclidean' system contains within itself the ground for the representation of the unrepresentable (infinity, God, Truth - an aesthetic of the sublime), whereas a 'non-Euclidean' system like Rotman's provides the language, the metaphorics, for the representation only of a finite, experiential reality of a unitary, embodied consciousness within the theoretically knowable physical universe; in short, a phenomenology of a counting Subject without the ideal term of an 'absolute spirit.' (60).

Following Rotman's reasoning, time is to arithmetic (counting and recounting) what space is to geometry and figuration (60) (thus the further analogy could be made - as it in fact is by Rotman - of arithmetic to logic as geometry to rhetoric [e.g. 68]). This stark distinction blurs, however: for counting to unfold, the spatial operation of differential disruption must occur; the iteration of numbers, as in verbal language, depends on difference, which is, as Derrida pointed out, spatial in its functioning (Derrida 1976). Rotman is therefore too dogmatic in his contention that "counting, however idealized, is a temporal process." It was to escape such strictly binary thinking that Derrida coined différance , whose effects are neither exclusively temporal nor spatial, predicated on an absence which has no corresponding presence to negate it. The absence, the effect of the trace, in différance is irreducible, but Rotman's refashioned mathematics is predicated on what might be called an irreducible presence (cf. 185 n. 28). As Rotman remarks in the next chapter, "mathematical logic is inseparable from a species of rhetoric" (68), and rhetoric, with the 'persuasional' emphasis Rotman gives it here, suggests a relatively traditional model of speaker and audience in a relationship of mutual presence, however 'attenuated' or idealized. In this sense, the one 'doing' mathematics is analogous to the speaker, using his or her whole body in the repertoire of 'communicative acts' that constitutes rhetoric - and a very strange way of 'doing' mathematics.

Rotman finally poses the question at the crux of his thesis: what is the nature of the relation between "the empirically constituted, corporeal mathematical subject who sits down to read, write, and count mathematical signs," and the fictive being, the "imagined simulacrum," Rotman posits who is the one who actually performs the endless counting? (62). This question never receives a satisfactorily clear answer, which hampers Rotman's own persuasiveness at key points.

The foregoing question leads, in the next chapter ("Experimental Thought"), directly into Rotman's advocation of the mathematical thought experiment as the most viable means of exploring this relation between subject and agent (66). Rotman sees thought experiments as germane to the practice of mathematics (67). Their most significant function is to effect a shift from the actual to the virtual, from experience to the imagining of experience (68), recalling the earlier analogy of the tripartite dreaming subject. The absence of "familiar indexical signs" in the Code is evidence, for Rotman, of a centuries-old, formalist, apotropaic effacing of any spatio-temporally locatable subject (73). Rotman sees this as a concerted effort to construct an I-less, metaphysical mathematical subject, in an "always-already there present: a timeless voice from no one and from nowhere" (74). In effect, the Person's task is to report on and interpret the relation between Subject and Agent ("imaginer and its imago"); the relative degree of 'similitude' between these two on which the Subject, as "idealization of the Person," cannot comment, having no access to the "indexical self-description" provided by the metaCode (78). Rotman would have us believe in a subject that is already there, already constituted and embodied, and yet unable to refer to itself reflexively (if it were 'merely' an abstract mathematical subject, this would not be a problem). Once again, Rotman notes the isomorphic resemblance this model bears to that of the dream, with the single proviso that "mathematics deals in waking dreams," for the likely reason that 'real' dreams are not the direct product of rational consciousness. To use the analogy of the dream qua dream would be to allow for the possibility (however figurative) of an unconscious function, and therefore of a 'ground' as abyssal as the zero-infinite differential is for mathematics.

The irony of his triadic model, which Rotman seems to appreciate, is that the Subject's corporeal 'presence' within the circuit is entirely metaphorical, even 'ideal,' only in terms of a different ideal order than its idealized proxy, the Agent (86). The Agent is more ideal, the Person less so: the circuit in its entirety is an abstraction, a fiction (91). This raises a number of interesting questions, such as what is the nature of idealized corporeality? (the Subject has "an idealized but not nonexistent body [100]") Why should the subject be mortal? Rotman asks (92). The Agent, "if it is to perform the operations imagined for it by the Subject, will not merely exhibit some idealized version of the Subject's corporeality, but will possess no physical presence whatsoever. It will be a ghost" (93). It seems that the only reason this Subject needs to be embodied at all is to set it off from the Agent it dreams up to perform tasks the Subject is prevented from performing because it is embodied and therefore mortal. This, at least, might be the obvious conclusion. Rotman, though, is determined to use the Subject's corporeality to limit the Agent's ideally attenuated (i.e. metaphorical) corporeality (94), such that the Agent "cannot be allowed to perform any action that is not capable - potentially - of being realized, of being materially instantiated and made actual within the physical universe inhabited by the subject" (94). This is because "asking an Agent to perform inherently nonrealizable actions, is to invoke a being who moves in a universe other than the one we - and all conceivable mathematical Subjects - occupy" (94): a being able to act according to Euclidean physical laws. Hence the attention paid to the degree of similarity in the relation between Subject and Agent. This is a crucial moment in Rotman'sargument: "the envelope of the Subject's attenuated corporeality is simply every imagined action for which it is not impossible that it be instantiated and become actual" (95). In other words, any process of potentially infinite duration, such as endless counting, would be ruled out.

Rotman's reinterpretation of iteration demands that "the ideogram of indefinite continuation - the '... ' - " be rewritten to signal the new limit associated with it: '... $'. The latter is the sign for the limit of 'mechanical dissipation' (109), which applies to the automaton-Agent; Rotman also provides another sign '... @,' for the Subject's "cognitive fade-out into unintelligibility" (109). Rotman's justification for imposing this seemingly arbitrary limit is that, unlike the Agent (not a true agent at all) "what the Subject does must be intelligible, intersubjectively interpretable in terms of signs" (105). Rotman's model makes no provision for the unintelligible, irrational, unthinkable, impossible, or unknowable. Thus he imposes this 'limit of intelligibility,' "the principle of this-universe realizability" (106), the '$', which, he maintains, is very familiar in "nonmathematical situations" (105). Here, Rotman has recourse to "perceptual psychology," in a slight departure from his otherwise unwavering focus on the body.

For Rotman, then, mathematics and infinity become incompatible because phenomenology and infinity are already in a sense mutually exclusive; that is to say, the prospect of indefinite iteration on the 'meta-' model (stories-within-stories; plays-within-plays; exponents-within-exponents, adding up to hyperexponents; in short, any manner of code) soon breaks down into unintelligibility, uncognizability (108). Rotman sums up the inherently rhetorical nature of this sort of speculation: "one can no more exhibit or make manifest such a limit than think the unthinkable or utter the ineffable" (108). Few sentences in the book better express Rotman's staunchly rationalist attitude vis-a-vis the negative, which might have been somewhat disappointing, had his real interest actually been infinity. As he states at the end of this chapter: "[the] whole account here springs from a semiotically based refusal to accept the currently available explanations - in fact, lack of explanations - as to how the natural numbers come into the world to be humanly observed and manipulated" (113).

This is by no means to suggest that Rotman is 'wrong' in what he says about cognition's 'resistance' to meta-iteration past a certain level (109). Rotman insists that he is not in these examples rehearsing the neo-heraldic mise-en-abime , which he wants to restrict to the domain of visual representation; a move which is not only pre-empted by the widespread application of this term throughout literary studies (for example), but it remains unclear why the examples he provides from other 'codes' (speech, writing, arithmetic [107]) are substantively different from that of a painting-within-a-painting, and so on. Even more than this, mise-en-abime's vertiginous structure implies a potentially endless iterating continuation in either direction: a classic instance of (b)ad infinitum in practice (109).

Rotman is also adamant that his cognitive limit-designation ('@') not be taken for a reinstatement of the Kantian transcendental limit. The rationale behind this denial offers insight into the ostensibly anti-metaphysical quality of Rotman's model: "for the Code of mathematics the cognizable is neither more nor less than the symbolizable, since the inseparability of ideas from their inscription, of signifiedsfrom signifiers, inherent in mathematical activity, forces one to couple what is imaginable with the intersubjective production and exchange of written signs" (110). An embracing of the most literal sense of Derrida's "il n'y a pas d'hors-texte ." Even the imagination is subject to a limit determined by what is imaginable for the Subject as the idealization of the Person (110-11).

In exploring the possibility of a non-Euclidean arithmetic, Rotman uses the obvious analogy of non-Euclidean geometry, whose provenance is discussed above. He wonders "whether one can treat geometry's relation to its object as a paradigm for arithmetic's relation to its object" (118). This is highly problematic: first of all, Rotman flatly states that geometry's 'object' is "extension in space," whereas arithmetic's is "passage through time." If the first were acceptable under certain circumstances, the second is simply unacceptable. Perhaps Rotman intends something else by 'object'; otherwise it would seem that "passage through time" has to do rather with the subject's experience of using arithmetic, where counting, say, transpires over time, in a manner directly analogous to reading. The latter, in so far as it depends on a materially present text, has its own built-in limit function: reading stops either artificially, when the page or book or particular story comes to an end, or when the reader gets tired or has to do something else. This would correspond to Rotman's realizability limit. On the macro- level, the fact that there is a finite number of legible texts in the world (relative to the life-span of a single reader) determines the equivalent of the cognitive limit function. But perhaps reading is not the best analogy for counting, at least not from Rotman's point-of-view: perhaps writing is better; but even writing (as narrating, re-counting), as suggested above, has its internally-instituted, conventionalized means of coming to an end, however arbitrary. And of course it is only fair to recall that Rotman's counting subject is specific to mathematics, and could not be expected to have anything to say to a writing or reading subject at a party - let alone behave like one - particularly not in the absence of indexical expressions, like 'I.'

In discussing Euclid's unease about his own axiom of parallels, Rotman chooses to ignore the 'troublesome' "idea of a straight line being prolonged infinitely far" (119). He refuses to acknowledge that this 'unease' may have been the result of the possibility, contrary to reason and logic, of the two asymptotic lines eventually meeting, at a point that cannot be formulated in the terms of either Euclidean geometry or the Kantian categories, or whatever; a point that is neither spatial nor temporal; that is not a point, properly speaking; that is impossible, inconceivable, and so forth. Rotman contends that the arithmetical 'cognate' to the parallel axiom is the ad infinitum principle itself (120). There is no reason to disagree with this conclusion, although in his explanation Rotman does not clarify the most obvious basis for the comparison: unending counting, like an endless straight line, is unthinkable; both are in effect metaphors for a metaphysical 'reality' inaccessible to thought. And, like the line, the iterating series of numbers will end up curving back upon itself, in defiance of Euclidean precepts as much as the physical laws that govern the universe: what Rotman is arguing for here is, after all, a phenomenology as it is a model for objective 'reality.'

What Rotman fails to mention is that Aristotle and Hegel are not the only philosophers to have theorized differing orders of infinity; an excusable omission,given the proclaimed mathematico-semiotic focus of his discussion. In the period of epistemological upheaval that resulted in what Blumenberg has termed 'the modern age,' there was a trend of thought that had numerous points of contact (Descartes, Pascal) with the philosophical 'mainstream' but remained for a variety of reasons on the margins. Blumenberg himself singles out two thinkers, Giordano Bruno and Nicholas Cusanus, as being of signal importance in the transition from the premodern to the modern. Cusanus, the more 'medieval' of the two, was a bishop, bureaucrat, theologian, and mathematician, with a 'scientific' cast of mind that makes him one of the first 'modern' thinkers, at once seminal and liminal. But it is Cusanus's place in the history of mathematics that justifies his mention here. What keeps him in the eyes of some from fully crossing the threshold into modernity is, among other things, his use of mathematics - specifically geometry - to render an approximation of what is otherwise absolutely unrepresentable: God's infinite being.

Arguably, and perhaps paradoxically, one of the most compelling aspects of Cusanus's thought is his sympathy with negative theology, and the manner in which he combines its language and logic with a geometrical metaphorics. Unlike the advocates of conventional, affirmative theology, Cusanus's use of geometry is predicated on a relation of absolute non-resemblance between the form and what it 'represents.' There is an unbridgeable distance between, say, a straight line and the perfect, infinitely straight line that is God. For Cusanus, every finite line partakes of curvature, since if it were really, 'maximally' straight it would be the "maximum, infinite line" (there can be only one), a change which, on the rhetorical level (as opposed to the conceptual) amounts to a leap from the as it were 'concretely symbolic' into pure metaphor (Docta Ignorantia 63). The straight line in geometry, for that matter, is never actually straight, as it is inescapably determined by what might be called the non-Euclidean universe of fallen creation. In the created world, all lines are crooked. The same holds for more complex geometrical figures: one that is in fact used in the Docta Ignorantia is the polygon inscribed within a circle (Docta Ignorantia 1. 3; 3. 1; 3. 4). For Cusanus, this compound figure represents the incommensurable difference between the intellect and truth; as the number of the polygon's sides increases, it grows increasingly similar to the circle, without ever achieving identity with it - without, in short, ever becoming absolutely circular. And this irreducible difference, as that which separates the intellect and truth, remains infinite, no matter how small the increments separating the two appear to become (see Docta Ignorantia 1. 3).

Like Plato and many others, Cusanus is "careful to distinguish the infinity of the cosmos from that of God" (Harries 6), but unlike Plato (and like Gregory of Nyssa; see below), Cusanus recognizes a radical discontinuity between the two: the cosmos is 'privatively infinite;' that is, "it lacks limits in which it can be enclosed, while the infinity of God precludes all indeterminacy" (ibid.). This is by no means to suggest that Cusanus articulated a vision of a universe cut off irremediably from God - this is a much more recent development. The infinity of the universe, in Cusanus's words, "contrasts with the infinity of God because it is due to a lack, whereas [the infinity] of God is due to an abundance Thus, the infinity of matter is privative, [but the infinity] of God is negative" (Docta Ignorantia 2. 8).(Note 1)

Rotman's crucial mistake is in not recognizing (or at least not acknowledging) that geometry, especially certain forms of anticipatory non-Euclidean geometry, employed geometrical forms for a purpose completely other than the representation of what Rotman blithely calls "the structure of actual externally presented space" (120). And this is not to have recourse to a variation on classical infinitist mathematics; rather, it is to draw attention to an apophatic trend of thinking, grounded in the unthinkability of the relation between being and non-being, created and uncreated (self and other?) - the proto-différance that Gregory of Nyssa in the fourth century named 'diastema ': (Note 2) the irreducible gap or interval between the reach of human cognition (what can be known) and God's absolute unknowability as what Levinas has called "Being beyond being" (Levinas 1981). The diastema's determination is radically negative; it describes neither one term or the other but the differential relation between them, grounded in unknowability.(Note 3) As a name for "the gap that separates creation and Creator" (Gregorios 67), which is both ontological and epistemological, diastema is unambiguously theological, but this does not prevent its logic from being appropriated and applied in other contexts in which is faced the problem of thinking something unthinkable (where thinking, in Rotman's terms, is representing). In other words, where there is no longer any question of denying ontological continuity, the epistemological discontinuity remains. Thus in terms of different orders of infinity, there is no basis of resemblance as there is for, say, Plato's universe of conditioned infinitude, created in imitation of the eternal pattern.

What we receive from this persistent strain of apophatic thought, then, is a means of thinking about not merely two different orders of infinity, but about two different orders of things - mutually contradictory, incommensurable - at the same time; that is, of thinking about their relation, the irreducible space between the finite and the infinite (or finitely infinite and negatively infinite), since consciousness, being finite, remains necessarily on one side. The great insight of this apophatically-inflected strain of thought is that, if the absolute cannot be thought in itself (i.e. if it is no longer an option for thought), then the relation can, in the positive form of a figure for an irreducible gap, which, no matter how close the mind seems to come to an apprehension of the absolute, remains uncrossable because it is finite and bounded on one side, and unbounded and infinite on the other. In other words, as an alternative to Rotman's alternative, it is not necessary to worry about the infinite at all.

Towards the end of the book Rotman seems to come almost full-circle, in an acknowledgement of the inescapability of Euclidean concepts on the grounds of an admittedly dubious 'intuitive obviousness' (127): "evidently, realizable arithmetic is radically non -Euclidean. But against this difference there is also an identity: any acceptable understanding of number has to be locally Euclidean" (127). Rotman suggests that therefore there are two orders of number, one 'ideal' (the 'classical ordinals') and one 'realizable,' but then wonders how one would be able to tell the 'difference' between them: "does it not seem that counting to ten on our fingers is counting to ten - whether the counting is prolonged beyond 10$ or 10 by a realizable Agent or by a classical ad infinitum agent able to count forever?" (128) Rotman identifies a 'difference' or 'divergence' here, between "classical andrealizable" arithmetical laws (128).

The ideogram '@' denotes the limit of what makes sense on a universal level; '$' denotes the corresponding limit on a subjective level. Can the limit symbolized by @ be thought of as cognate with the Cusan notion of a 'privative infinite'? (i.e. not as a limit-function of the physical universe but of what is 'possible' within that universe). A principle (if unstated) goal of Rotman's thesis seems to be the elimination of contradiction and paradox, of undecidability in any guise (eg. 130-31). In other words, Rotman's notion of a "realizable arithmetic" is "locally Euclidean," and "radically non-Euclidean" only on a 'global scale' (121-30). Rotman's mathematical model, as fundamentally 'written,' conforms with neither the phono-logocentric nor the grammatological (where each 'iteration' of an utterance, written or spoken, is a new signifying event conditioned by its context and independent of a present signifying intention as proof of origin); rather, Rotman argues for a mathematics in which writing and thinking are interdependent to the extent that the Subject's 'active presence' - not merely the intention but the body itself - is required for any mathematical act to take place (143). That is, for Rotman, each implementation of mathematical signs is a unique and originary event, regardless of the iterative, conventional nature of these signs, which Rotman in fact acknowledges (140-41). On the other hand, Rotman does not allow for the primacy of either the world or mathematics as absolute origin; the world, always already 'mathematized,' provides mathematics with its spatio-temporal model (142-44). Rotman identifies an 'oscillation' ("not confined to mathematical signs, but having a particular force for them") between the signifier's coexistence as material mark and as "general, idealized, non-materially presented type " (144). Neither precedes the other, nor have they existed in this state of mutual dependence forever, since that would contravene the non-Euclidean nature of the system. This is the only concession Rotman makes to any sort of 'irrationality' or irresolvability, preferring to ground his model in what is effectively the phenomenology of the singular mathematical Subject, "a semiotic agency made available by the code - engaged in the dreaming of its own numerical boundaries" (145). Rotman makes explicit the connection between this Subject and phenomenological experience of the world in terms of his discussion of time: the time that the Subject inhabits, he states, is the same as that inhabited by "any reader of this text" (147).

Rotman admits to having deliberately excluded from his model of the (dreaming) mathematical Subject any level or dimension of unconsciousness, and that this is an attempt to repress (consciously?) what is potentially one of the most interesting aspects of a project of denying infinity. In admitting this, Rotman only draws attention to what amounts to the (unintentional) dialectical negation of infinity: as he admits without wanting to, the object of repression emerges as what is most interesting (193 n. 52).

Notes

1) For a further explanation as to why Cusanus's universe is not really infinite (and a valiant attempt at a clarification of Cusanus's complex and often confusing use of terms like 'privatively' and 'negatively' infinite), see Hopkins 1978: 30-32. Cf. Descartes's letter to Chanut of 6 June 1647, in which he cites Cusanus on the issue of the "indefinite extension" of the universe over against the infinity of God (Descartes 221).
Back to where you left off.

2) "We have no English word by which to translate diastema. To translate it as gap or interval could be to miss out its meaning of extendedness" (Gregorios 75). Gregorios points out that diastema has been translated into French as 'espacement ,' suggestive at once of spacing, difference and movement (ibid.).
Back to where you left off.

3) I owe this observation to Charles Lock.
Back to where you left off.

References

Blumenberg, Hans (1985) The Legitimacy of the Modern Age. Robert M Wallace (trans.) Cambridge, Mass. and London: MIT Press.

Borges, Jorge Luis (1964) "Avatars of the Tortoise." Labyrinths. Donald A. Yates and James E. Irby (eds.). New York: New Directions: 202-208.

Derrida, Jacques (1989) Edmund Husserl's "Origin of Geometry": An Introduction. John P. Leavey, Jr. (trans.). Lincoln and London: University of Nebraska, 1989.

--- (1976) Of Grammatology. Gayatri Spivak (trans.). Baltimore and London: Johns Hopkins.

Descartes, Réne (1970) Philosophical Letters. Anthony Kenny (trans. and ed.). Oxford: Clarendon.

Gregorios, Paulos Mar (1988) Cosmic Man: The Divine Presence - The Theology of St. Gregory of Nyssa. New York: Paragon House.

Harries, Karsten (1975) "The Infinite Sphere: Comments on the History of a Metaphor." Journal of the History of Philosophy 13. 1 (January): 5-15.

Hegel, G. W. F. Hegel's Logic (Part I of the Encyclopedia of 1830). William Wallace (trans.). Oxford: Clarendon, 1975.

Hopkins, Jasper (1978) A Concise Introduction to the Philosophy of Nicholas of Cusa. Minneapolis: University of Minnesota.

--- (trans.) (1990) Nicholas of Cusa On Learned Ignorance: A Translation and Appraisal of De Docta Ignorantia. Minneapolis: Arthur J. Banning.

Levinas, Emmanuel (1981) Otherwise than Being, or Beyond Essence. Alphonso Lingis (trans). Hingham, Mass.: Kluwer Academic.

Lock, C. J. S. (1992) "Texts of the Body and the Mind: Semiotics and the Face." [Unpublished].

Plato. Timaeus and Critias. London: Penguin, 1971.

Rosenfeld, B. A. (1988) A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. Abe Shenitzer (trans.). New York, Berlin, etc.:Springer Verlag.

Rotman, Brian (1987) Signifying Nothing: The Semiotics of Zero London: Macmillan.

Russell J. A. Kilbourn is a graduate student in the Centre for Comparative Literature at the University of Toronto. His thesis deals with the relationship between negative theology and the modern novel, and he also works on film and literature. Russell is currently teaching in the Literary Studies program at Victoria College, U. of T.


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