Emergence of a Meta-Subject

 

Brian Rotman, Signifying Nothing: The Semiotics of Zero (New York; St. Martin's, 1987), Chapter 2, pp. 27-56

ANTERIORITY OF THINGS TO SIGNS

 

Each of the transformations described so far, in arithmetical signs, picture signs, and money signs can be thought of as a different model of the following abstract scheme:

 

There is a system (Hindu decimal place notation, principles of linear perspective, mechanism of capitalist exchange) which provides a means of producing infinitely many signs (numerals, pictures, transactions). These signs re-present (name, depict, price) items in what is taken to be a prior reality (numbers, visual scenes, goods) for an active human subject (one-who-counts, one-who-sees, one-who-buys-and-sells). The system allows the subject to enact a thought-experiment (calculating, viewing, dealing) about this reality through the agency of a meta-sign (zero, vanishing point, imaginary money) which initiates the system and affects a change of codes (gestural/graphic, iconic/perspectival, product/commodity).

 

I want now to deconstruct this scheme. What lies at its centre, explicit in the talk of 'prior' reality, is some supposed movement into signification, some shift from object to sign, from presentation to representation, from a primary given existence to a secondary manufactured description. In each of the cases of number, vision, and money a field of entities is assumed to exist anterior to the process of assignation performed by the system. What are taken to be pre-existing numbers are given names, scenes from some supposedly pre-existing visible world are depicted, goods conceived as existing independently of, and prior to, the agency of money are assigned a price. In each case this process of assignation hinges on the meta-sign which both initiates the signifying system and participates within it as a constituent sign. And it is this double ambiguous role played by zero, by the vanishing point, by imaginary money, which ultimately destabilises the scheme presented here and deconstructs the anteriority to signs this reality is supposed to enjoy.

 

In other words, the simple picture of an independent reality of objects providing a pre-existing field of referents for signs conceived after them, in a naming, pointing, ostending, or referring relation to them, cannot be sus-tained. What gives this picture credence is a certain highly convincing illusion. Once the system is accepted, on the basis of a perfectly plausible original fiction, as a mechanism for representing some actuality, it will

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continue to claim this role however far removed its signs are from this putative reality; so that, for example, numerals can be written which name 'numbers' that are unrealisable by any conceivable process of human counting or enumeration, pictures can be painted that depict purely imaginary, non-existent, or visually impossible 'scenes', transactions can be drawn up that price humanly unachievable relations between 'goods'.

 

The result is a reversal of the original movement from object to sign. The signs of the system become creative and autonomous. The things that are ultimately 'real', that is numbers, visual scenes, and goods, are precisely what the system allows to be presented as such. The system becomes both the source of reality, it articulates what is real, and provides the means of 'describing' this reality as if it were some domain external and prior to itself; as if, that is, there were a timeless, 'objective' difference, a transcendental opposition, between presentation and representation.

 

From one direction this deconstruction of the anteriority of object to sign can be construed, as we shall see, as the path by which the meta-sign engenders a secondary formation of itself. Each of the signs zero, vanishing point, imaginary money, will now be shown to have, in relation to the whole original system of signs, a natural closure which emerges as a new, highly potent meta-sign, important in its own right, in the codes of number, vision and money. A meta-sign whose action, at one remove from the original 'anterior'field of entities, accompanies a radically different, self-conscious form of subjectivity which I call the mesa-subject.

 

CLOSURE OF ZERO: ALGEBRAIC VARIABLE

 

At the end of the sixteenth century the Dutch mathematician Stevin advocating in his treatise, The Dime (Stevin, 1958) the extension of the Hindu system of numeration from finite to infinite decimals, expressed great wonderment at the creative power of zero, at the ability it gave the principle of place notation to manufacture an infinity of number signs.

 

Stevin rejected the classical notion of number, arithmos, as a fundamental misunderstanding of the nature of numbers. For both Plato and Aristotle arithmos always, as Klein (1968) in his investigation Greek Mathematical Thought and the Origins of Algebra expresses it, 'indicates a definite number of definite things. It proclaims that there are precisely so and so many of these things'. And though they differed on a crucial point of interpretation, 'Plato speaks of numbers which have "visible and tangible bodies". . . so that in counting dogs, horses, and sheep these processes yield dog-, horse-, and sheep-numbers' (Klein, 1968, pp. 46-7), while Aristotle, concerned to clarify what it means to say that two numbers were equal, saw them as abstractions from particular concrete collections. Both would have assented to the formula 'number as an assemblage of "units"'.

 

For Stevin the source of error here is not so much the formula itself but the quasi-geometrical interpretation of a 'unit' contained in the classical view that underlies it, the notion that units were abstracted, individualised 'things' that were not (obviously) pluralities and not in fact numbers at all. On the contrary, Stevin argued, the unit was a number like any other; it was not the unit that was the arche of number, but the nought. Zero was the proper origin of number,'The true and natural beginning' (Steven, 1958, p.499); and just as the point in geometry generates the line, so zero, which he wanted to call poinct de nombre gives rise to numbers.

 

To make zero the origin of number is to claim for all numbers, including the unit, the status of free, unreferenced signs. Not signs of something, not arithmoi, certainly not real collections, and not abstractions of 'units' con-sidered somehow as external and prior to numbers, but signs produced by and within arithmetical notation. In the language of Saussure's distinction Stevin rejected numbers as signs conceived in terms of positive content, as names for 'things' supposedly prior to the process of sign)fication, in favour of signs understood structurally, as having meaning only in relation to other signs within the sign system of mathematics. In effect, Stevin was insisting on a semiotic account of number, on an account which transferred zero's lack of referentiality, its lack of 'positive content', to all numbers. In so doing he overturned the belief in the anteriority of 'things' to signs that that classical formulation of arithmoi depends on.

 

Stevin's primary interest, however, was not semiotic but mathematical. He proposed to extend the principles of the Hindu place notation from the whole numbers to all possible real magnitudes. This meant the creation of a system of infinitely long signifiers which relied for their interpretation on infinite summation, that is, on the process of an infinity of numbers being added together. Thus, just as 333 means 3(1) + 3(10) + 3(100), so 0.333 . . . ad infinitum means 3(1) + 3(/10) + 3(1/100) + 3(1/1000) + . . . ad infinitum. Such non-finite addition requires an infinite limiting process; a phenomenon visualisable in geometric terms as an asymptotic approach of a line to a curve. Stevin's extension of Hindu numerals to an infinite format, which resulted in a numeralisation of the real one-dimensional continuum, had far-reaching consequences for seventeenth century, and indeed for all subsequent, mathematics. Semiotically, it is the language, algebra, in which these consequences were developed, and the opposition such language offers between determinate, possibly unknown, but fixed numbers (constants) and the indeterminate non-numerical entities (variables), that is of primary interest.

 

Algebra, the art of manipulating formal mathematical expressions such as equations, formulas, inequalities, and identities, is co-extensive with the idea of a variable. In any algebraic expression, such as for example the identity (x - y) (x + y) = x2 -y2, the letters x and y are said to be variables, by which is meant that they denote arbitrary individual numbers in the

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sense that any particular number may be substituted for x and y. A variable is thus a sign whose meaning within an algebraic expression lies in certain other, necessarily absent, signs. In relation to these other signs, which con-stitute its 'range', a variable is a meta-sign, that is one which indicates the virtual - the potential, possible, but not actual - presence of any one of the particular signs within its range.

 

The essential mathematical idea of a variable, that is as an indeterminate that can be calculated with as if it were a determinate number, is due to Stevin's French contemporary, Vieta. Unlike Stevin, Vieta did not challenge the classical picture of number. His notion of variable which he called species was, therefore, formulated, as Klein observes, in referentialist terms:

 

For practical mathematical purposes Vieta's referentialism (repeated, as we shall see, in contemporary accounts) is of no importance. It is only significant, when one asks the question what a number sign is and what it means semiotically for a variable to range over numbers. For such questions, the referentialism behind explanations of indeterminateness in terms of scholastic distinctions between first and second intentions, essentially the opposition between things and signs of things, is the point at issue: for it is precisely this opposition that mathematical indeterminacy, semiotically perceived, requires to be unmasked.

 

The point is central. I shall elaborate it by taking issue with the current orthodox, that is realist, explanation of a variable; a simple, purely syntactical definition within set theory which runs as follows: to say 'x is a variable ranging over the numbers N' means simply that 'x denotes an arbitrary member of the set N of numbers', where to denote a number means to point to the set constituting it. According to the set-theoretical realism behind this definition, numbers - and indeed all mathematical entities such as points, lines, invariants, geometrical figures, ratios, constructions, functions, spaces, predicates, relations, and so on - are 'things' celled sets. So that mathematical discourse is entirely descriptive and referential: its signs are names, more or less explicit, more or less contained within other names, of determinate set-

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theoretical objects, its assertions are propositions, unambiguously'true'or 'false' statements about some prior, extra-linguistic, universe of sets.

 

What, then, are sets as conceived by set theory? They are ensembles, collections, classes and aggregates, of members which are themselves ensembles, collections, classes and aggregates of members: where the regression of membership is imagined to end either in some arbitrarily introduced atomic entities - ur-elements - or, as is actually the case in current practice, in the set-theoretical first cause, the equivalent of zero for collections, namely the empty set {0}. In this conception sets have precise, unambiguous criteria for membership governed by axioms; they are static, unchanging, determinate things. Theorems proved 'true' about them remain true, and so must always have been true; they are in no sense invented but discovered as timeless objects that simply exist in the Platonic (Parmenidean, in fact) universe of pure motionless being. Sets are in short the 'things', the ultimate Dinge, required by a nineteenth century realism immersed in the anteriority of things to signs (of things).

 

Conceived as sets in this way, the objects of mathematics are subjectless, unauthored referents that exist independently of human agency. By excising the subject from its conception of mathematical activity such realism offers an epistemology as well as an ontology that is semiotically incoherent: it misrepresents proof, for example, as validation of eternal truths about 'things', and misdescribes a variable as a place-holder for the members of some infinite set of such things. As signs variables are those which can be replaced by any sign of a given kind. In the paradigm case, that of number variables, the 'given kind' are the number signs, and since these cannot be specified except through the process of counting, it follows that a semiotic characterisation of a variable can only be given in terms of the active, constructing, counting subject. Or to make the comparison in historical terms, the algebraic variable, semiotically perceived, amounts to a re-description of Vieta's species: one which reinstitutes the subject - the one-who-counts - into Vieta's discussion, replacing the arithmoi there by the purely semiotic picture of number that Stevin so marvelled at.

 

What is the connection between zero and a variable as meta-signs? Zero functions dually: it moves between its internal role as a number among numbers and its external role as a meta-sign initiating the activity of the counting subject. So it is with the algebraic variable, except that the internal/ external duality is enacted at one remove from the one-who-counts. Perceived internally, variables present their familiar appearance as manipulable algebraic objects, as signs among signs within formulas. In expressions like x + 1 = y, 2x + 3y-z = 0, and so on, the letter signs enter into full arithmetical relations with number signs: multiplying them, being added to them, being numerically compared to them, being substituted for them, and generally being treated as if they were number signs according to a common syntax. Perceived externally, this unitary syntax comes apart, since

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variables are not of course number signs; on the contrary, they are signs which meta-linguistically indicate the possible, but not actual, presence of number signs.

 

This duality is mediated by a new mathematical subject, the algebraic subject, whose relation to the one-who-counts mirrors the relation between a variable and a number. Thus, the algebraic subject has the capacity to signify the absence of the counting subject, the displacement of the one-who-counts from an actual to a virtual presence. Now at cer,tain points, when variables are instantiated by numbers, this displacement ceases to operate - the two subjects coalesce. So that, for example, the algebraic subject who reads the identity x2 - 1 = (x-1) (x + 1) becomes the subject who checks the result of writing, say, x = 10; who, in other words, computesultimately by counting - the arithmetical validity of 100 - 1 = 9 x 11. But this sort of arithmetical localisation is extraneous to the difference between the two subjects: when variables are manipulated as algebraic objects within formal calculations any such fusion between the counting and algebraic subject is precluded; and the algebraic subject performing these calculations remains as an autonomous, arithmetically self-conscious, agent whose meta-linguistic distance from the one-who-counts is the source of the difference, when they are considered as mathematical discourses, between algebra and elementary arithmetic.

 

The semiotic connection between zero and the variable thus emerges as one of symbolic completion: by ranging over all number signs, that is over all possible records that can be left by the subject whose sole capacity is to repeat, the algebraic subject performs an operation of closure on the infinite proliferation of number signs that come into being with zero. In effect, the algebraic number variable is a sign for the signs that can, at least in principle, be produced by the one-who-counts.

 

CLOSURE OF THE VANISHING POINT: PUNCTUM

 

In his Vision and Painting: The Logic of the Gaze Bryson (1983) draws a fundamental distinction between two stages of perspectival painting in terms of the requirements of physicality, of spatial locatedness, that the system imposes on the spectator:

 

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In terms of the semiotic activeness charted so far, this transition from what Bryson calls the 'Glance to the Gaze', from the vanishing point to the punctum, from a corporeal spectator, object)fied by the deictic insistence of the 'pure' perspectival image, to one who is disembodied, who views the painting from an unoccupiable, purely notational, point in space, will correspond to a shift in subjectivity: from the subject created by one meta-sign to the subject of a secondary meta-sign which forms the semiotic closure of, what it renders as the previous, system of images.

 

To see how this comes about, how the first epoch of perspective in the fifteenth century gives way to its second culminating form two centuries later, we shall look at two paintings, The Artist in his Studio by Vermeer, and Las Meninas by Velasquez, each of which, in very different ways, exemplifies the illusion of vanishing point perspective through an image which unmasks it, and denies it in favour of a radically more complex, self-conscious, form of viewing.

 

If there is a single vanishing point in Vermeer's painting it is certainly not evident. On the one hand, the usual indication (a pair of orthogonals that can be seen to intersect) is evaded: the foreground and background chairs and the artist's stool, all suitably right-angled, are placed obliquely to the frame, the table which appears square to the frame displays only one orthogonal, and so on. On the other hand, the usual indication is curtailed: the flagstone floor, though it implies the conventional continuity between spectator and scene, and in earlier perspective paintings would have given an immediate determination of the vanishing point, is here so flattened, so reduced in scope and deliberately masked by objects, that it determines nothing beyond its declaration 'floor'.

 

Clearly, then, the viewer is not being asked to enter the painting by focusing on and occupying the central geometric absence created by a vanishing point. Instead, there is a more pressing, visually immediate absence, put there by Vermeer and not as a result of perspectival requirements, marked by the back of the artist's head - that of his face, which positions the spectator. From this absence of vision one is impelled to another, the face of the model, whose downcast if not closed eyes point to the final absence of sight, the white death-mask staring blindly upwards from the table. This chain of absences, at first direct but unseen, then oblique, downward, and masked, and ultimately open but sightless, vividly concretises Vermeer's ghostly and enigmatic celebration of seeing, or rather (though he would perhaps have appreciated the elision) his celebration of the painting of seeing.

 

Vermeer's title, Di Schilderkonst, understandably but incorrectly translated

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Illustration 7 Jan Vermeer: The Artist in His Studio.

 

as The Artist in His Studio, means the art of painting, and his depiction of an artist (perhaps Vermeer) making a depiction is precisely about the process of picture making itself As a sign depicting the making of perspectival images, Vermeer's painting embodies a visual meta-sign, a sign about picture signs. And the artist responsible for this meta-sign, the one who has painted the scene, is nowhere to be seen. Not only can he not be traced through any identifiable vanishing point, with the result that the painting is

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seen in multi-perspective, it can be viewed £rom an arbitrary position, but he has expunged any trace of his visual presence from the act of painting it. The convex mirror at the base of the chandelier (unlike the mirror in van Eyck's Arnolfini painting two centuries earlier reflecting back van Eyck to the spectator) gives back no image of the painter. On one side it shows the light by which the scene is rendered visible to the internal artist painted by Vermeer, and on the other an indistinct reflection of this same artist; in the middle, instead of the painter, it is the chandelier, prosaically inserting itself, that is reflected (see Illustrations 8 and 9).

 

Illustration 8 Jan Vermeer: detail of chandelier from The Artist in His Studio.

 

If, as seems reasonable, the seated figure can be taken to depict Vermeer, then, because Vermeer could not have seen himself in this way, his painting represents a scene he has imagined. The scene was nowhere but in his head. But if a scene can be literally nowhere, have no 'real' location, what then is the difference between presenting and representing? The priority of things to signs which sustains the distinction seems to disappear. As if to counter this merging of the literal and the metaphorical, of 'things' and signs (of things), of the real and the depicted, but in fact compounding it, Vermeer emphasises the fiction of the thing/sign opposition by painting the same leaves three times: first as posed or advertised "reality" on the model's head, then, at the moment of being painted by the figure, as a represented 'sign' of this so-called reality. And then again as, unobtrusively signified back-ground'reality'within the curtain on the left of the painting. Thus visual perception and the painting of this perception cease to be opposed categories; Vermeer no longer records perceptions but, as Bryson (1983, p. 116) puts it, 'notates them, and then demonstrates (paints) the act of notation'.

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Illustration 9 Jan van Eyck: detail of mirror from The Arnolfini Wedding.

 

Vermeer's image, then, like all the meta-constructs encountered so far, operates ambiguously between its lingual and meta-lingual roles. Internally, as a sign within the code of perspectived paintings, it is a single image, a picture sign among picture signs, depending on other paintings for the interpretation of its narrative, its structure, the meaning of its constituent signs, and so on. Externally, the meta-sign calls attention to this very system of images and to the account of perception it promotes; and in so doing offers the spectator an unlocated subjectivity, inviting him to occupy the

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place of the unsituated artist, the absent, invisible, notional subject who painted the image; the one without whom Vermeer's picturing of the act of painting would be merely another depictive painting - one which accepted, rather than put into enigmatic question, the illusion of representation offered by perspectival images.

 

As the north European algebraic variable is to the zero of the Italian Renaissance, so Vermeer's painting, one of the culminating images of the Northern Renaissance separated by more than two centuries from the 'first geological age of perspective' of Brunelleschi and Alberti, is a continent apart from its Italian forebears. It makes sense to ask, therefore, whether there are semiotic features of Northern, essentially Dutch, painting which might have facilitated the emergence of such a meta-sign.

 

S. Alpers (1983), in her persuasively illustrated book, The Art of Describing, provides an analysis of Dutch seventeenth-century art and the theories of vision inseparable from it, which yields a response to this question. One which starts from the difference between two ways of drawing the transversal and orthogonal grid-lines of perspective: the Italian method of construction, and the distance-point method recommended by Viator in the first North European treatise on perspective at the beginning of the sixteenth century. Her illustrations of the difference, taken from Giacomo Vignola, are shown in Illustrations 10 and 11.

 

Illustration 10 Giacomo Vignola: the first 'regola' or the 'construzione legittima'.

 

Now it is geometrically obvious and well known that these two rules can be made to yield identical systems of perspective lines; making it easy to appropriate the second, later rule to the first (seeing it, for example, as nothing more than a practical Northern device, a draftsman's rule of thumb, for the swift production of transversals). Such an understanding of distance-point construction would, Alpers argues, violate a fundamental distinction between the two rules. What is significant about Viator's method, at least in

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Illustration 11 Giacomo Vignola: the second 'regola' or distance point method.

relation to Dutch art, is the emphasis it places on the existence of an internal viewer. So that in Saenredam's image, for example:

 

 

In an argument, too historically textured and ram)fied to summarise here, Alpers offers the presence of such an internal viewer - a depicted looker as she says - as a paradigmatic feature of Northern art, one intimately bound up with the essentially descriptive mode of Dutch visualisation with its attachment to the visual representation of knowledge through maps, plans, diagrams, and drawings of the world as seen through the microscope. As Alpers points out, the figure gazing up at the organ in Saenredam's interior is not, as might appear, merely a device of scaling inserted to underline the grandeur of the church's architecture; but, on the contrary, is the organising visual focus, the point of visual entry into Saenredam's image (see Illustration 12). The spectator and the 'looker' fuse. What is emphasised and made available to the viewer is an ocular, ambulatory presence inside the image, and not a view from without determined by external design as required by Italian perspective. Alpers' account makes the uncannily life-like 'realism', the strange hyper-verisimilitude found in so much of Dutch art, a comprehensible effect of this internalisation of the vanishing point. Saenredam's drawings of churches, Vermeer's Delft, and so on, persuade the spectator of their visual immediacy, the physical occupiability of what they depict, precisely because in front of them there is 'no prior view to establish a position . . . from which, as we say, to take in the work' (Alpers, 1983, p. 41).

Alpers' concept of the 'looker' suggests a mediating term between two visual subjects: a subjectivity situated between the perspectival subject,

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Illustration 12 Pieter Saenredam: Interior of the Church of St Bavo in Haarlem, 1636.

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associated with the primary meta-sign of the Italian vanishing point, and the meta-subject associated with its closure, with the punctum found, so far, in Vermeer's painting.

 

One can thus articulate a sequence of four visual subjects that span the movement from Gothic art to; the culmination of Northern painting in Vermeer's Art of Painting:

 

(1) the Gothic subject, the one who uses Roman numerals, whose mode of signifying is dominated by the iconic, made up of what Bryson (1983, p. 116) cells 'diffuse non-localised nebula of imaginary definitions', (2) the perspectival subject, coded by the vanishing point, situated outside the frame in a relation of imagined identification with the artist's viewpoint, the result of the Gothic subject having been objectified mirror-like into a visual pronoun; (3) the 'looker', the figure of internal vision in Dutch art, an internalisation of the perspectival subject, whose interior presence calls into question, and so suggests the absence of, any exterior point of view; (4) the meta-subject, engendered by the punctum, able to signify what the presence of the 'looker' can only raise as an interpretive possibility, namely the necessary absence of any externally situated, perspectival seeing.

 

Corresponding to these four phases of subjectivity one has four types of signification:

(1) no meta-sign; (2) meta-sign = vanishing point; (3) internalised meta-sign; (4) meta-sign = punctum.

And, in turn, four forms of anteriority - the presumed primacy of 'things' to signs which refer to them:

 

(1) Anteriority is non-existent. Medieval image makers did not categorise their efforts in terms of referentiality. They neither claimed nor denied a posterior relation to some real, waiting to be depicted, physical world. The pictorial values they prized were the mythic, the numinous, the spiritually potent, not those of accuracy or depictive 'truth', the space of Gothic images was not representational, but conceptual and iconic (recall de Cusa's interest in the portrait of a face in terms of its capacity to iconise God's vision.) (2) Anteriority is present. Italian perspectival painting was founded on the fiction of a framed portion of nature, a detached fragment of some prior visual 'reality', being represented with truth and accuracy, to the viewer; its image makers prized copying and perfect mimicry. To be praised as an imitatore della natura or, more hubristically, a surpasser of nature, was the greatest compliment they sought. (3) Anteriority is in abeyance. The visual world does not so much precede the viewer as exist alongside him, coter-minous with his viewing presence; the internal 'looker' end the reality he

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sees are simultaneously presented artefacts - a phenomenon which leads Alpers to observe, about Saenredam's images, that 'such pictures are not properly architecture viewed, in which an external viewer is presented with a view of architecture, but rather views of architecture viewed' (Alpers, 1983, p. 52). (4) Anteriority is deconstructed. The Art of Painting renders perspectival viewing, the framed picture as a geometrised simulacrum of a prior world, impossible. There is no unsignified 'real' which can be, accurately or other-wise, simply recorded and depicted. All that is seen is already a pictured universe. The world and visual signs for the world cannot be separated: 'Observation is not distinguished from what is observed. That is the grand illusion that the picture . . . creates' (Alpers, 1983, p. 168)

 

Despite its manifold differences - in scope, iconography, style, mode of composition, pictorial setting, cultural assumptions - Velasquez' image delivers, and seems intended to deliver, a semiotic lesson cognate to Vermeer's on the illusive difference between presenting and representing (see Illustration 13). Certainly, in terms of content, the two images present a striking set of common features and preoccupations with respect to which they exhibit parallel or polar opposite responses.

 

Both paintings are signs that contain, as essential elements within their schemes, already signified 'realities', constructed images, designs, depictions - the pattern of leaves reproduced in Vermeer's curtain, his scrupulously wrought map of Holland dominating the background, Velasquez'accumula-tion of studio canvases, barely discernible through the gloom or the flattening of perspective - which call attention to the paintings' preoccupations with the nature of visual representation, with the illusions attendant on the business of painting signs to stand for things.

 

Both are paintings of artists painting: one seated, absorbed in the process of seeing, motionless, faceless, private, caught from behind; the other standing, dynamic, frontally and publicly present, interrupting his painting with a look indicating his awareness of being seen.

 

Both depict an artist painting a portrait of a figure from life, physically present to them as models. In Vermeer's image the representation of the model, a girl who, idealised as the historical muse Clio, is already a representation, is visible to us, the viewers. In Velasquez' painting the model is the Infanta, the royal Margarita, present to us in unidealised form, but her representation is invisible. The opposition between the paintings these doubles of Vermeer and Velasquez are producing reflects this alternation of absence and presence: Vermeer's internal painting looks to be a small and delicate portrayal of his model, open and declarative, against the absent and hidden vision of its producer. Velasquez' internal canvas, enormous and mysterious, mocking through its revealed back the pretensions of pictorial depth, has an image on it as obscured and absent as its producer is defiantly present.

 

Both paintings feature mirrors that conceal as much as they reflect.

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Illustration 13: Diego Velasquez: Las Meninas

Vermeer's, withdrawn from prominence, and naturalised, as if it were there accidently as part of a contingent background, nevertheless calls attention, by a comparison with earlier convex mirrors in Northern painting, to what it fails to reflect, showing only the painter's double and the light he paints by but no painter. Velasquez' mirror, placed so advertently at the back of his image at its visual centre, likewise shows no painter but reflects, and insists that we notice the fact, two ghostly figures.

 

It seems to be generally accepted that the figures are the king and queen of Spain in whose service Velasquez' picture is being produced. Any inter-

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pretation of Las Meninas has, then, to explain their mirrored status within the intentional structure of the painting. Foucault, in an intricate reading of the subjectivities organised within the painting, claims that this mirror, by containing the vanishing point of the entire picture, occupies a pivotal and ambiguous role:

According to this we become aware of the place outside the space of the painting, directly opposite the mirror, from which we, the spectators, organise these subjectivities: the persons reflected in the mirror, for whom the foreground figures have been arranged, are standing where we stand. Velasquez' double is looking at them, at us and at himself; or rather, at the absent Velasquez who has painted this whole scene.

 

In a recent essay, Las Meninas and the Mirror of the Prince, which offers a definitive account of the painting Snyder (1985) dismisses Foucault's reading on the grounds that it is based on a fundamental error: the vanishing point is not, as Foucault (and others) have maintained, within the mirror but is located well to the right of it near the elbow of the figure in the doorway. This being so the mirror cannot contain the reflections of the real, corporeal king and queen standing in the foreground opposite the vanishing point; rather it reflects a 'reflection'of them, an ideal of their royalty which Velasquez' vvhole picture is to be taken as manifesting and celebrating. Snyder's reading which is embedded in a persuasively detailed appreciation of the role played by mirrors, as metaphors for the 'ideal' in seventeenth century Spanish thought, is convincing. Whether it displaces Foucault's quite different interest in the exchange of subjectivities is open to doubt. Certainly Foucault's error about the vanishing point removes the grounds for his reading, but not necessarily the insights that the reading makes available. In fact, the readings are, in the end, as much complimentary as mutually excluding. Where Foucault (1973, p. 16) describes Velasquez' painting as the place where 'representation undertakes to represent itself' and emphasises its organisation around 'an essential void: the necessary dis-appearance of that which is its foundation - of the person it resembles and the person in whose eyes it is only a resemblance', Snyder (1985, p. 564) insists not on means but ends, on the function of the painting to externalise the very idea of the 'ideal', so that 'Las Meninas is a speculation about speculation, a reflection by an exemplary artist of an ideal image that engenders images'.

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From the formal semiotic point of view adopted here, both Velasquez' and Vermeer's paintings can be seen to contain, rest on, and be entered through, an absent subject - the painter/viewer - which the mirror must fail to reveal. The subject we are being asked to con£ront, through a chain of absences in Vermeer, a void in Foucault's reading of Velasquez, or a non-corporeal 'idealised' reflection in Snyder's reading, is a meta-lingual one, a visual mesa-subject at once notional, invisible, and absent, who by making evident the false anteriority of visual things to visual signs, allows depiction to become, for the first time, self-conscious. Moreover, like Vermeer's image, Las Meninas operates as a sign about signs, a meta-sign whose domain is the mode of depiction itsel£ Its offer to the spectator, an invitation to occupy the place of the one-who-paints-the-image, is addressed to the meta-subject; a subject who, through the ambiguous action of the painting, sees - through what is after all a perspectivally-based image -the limits of perspectived seeing.

 

This meta-subject within the code of visual signs, who appeared with the variable as the algebraic subject within the signs of mathematics, will appear again, below, in relation to a new form of money sign. Before this I want to give another version of self-consciousness emerging from the activity of a subject depicting himself depicting; one which occurred at the same time as the construction of the algebraic subject by Stevin and Vieta.

 

Montaigne in the medium of words, written but fundamentally different from the written media of mathematical, visual, or monetary signs, not only saw himself making precisely such a self-depiction in his autobiographical Essays, but also, characteristically, perceived the illusion behind it:

 

 

Where Vermeer and Velasquez painted images of themselves painting, Montaigne wrote himself writing, inscribing an image of himself within the text; an image busily writing ('scribbling' he called it) about the imaginary scenes Montaigne chose to lay before him. When Montaigne affirms that the self he is depicting cannot be separated £rom the process of depiction, that it is 'in some sort formed' by the very activity of framing its portrait, he is articulating in the written words of his Essays what Las Meninas and The Artist in His Studio render through visual signs: namely, the illusion of anteriority, the fantasy of a world of the 'thing' - in Montaigne's case some unitary, pre-

45 

existent, waiting to be written, 'true' self - prior to a domain of the 'sign of the thing' which represents it.

 

What are the imaginary scenes Montaigne puts in front of his double in the text? What, in the Essays, corresponds to the object of internal depiction, to the girl-as-Clio Vermeer's double is painting or the royal Infanta being painted in Velasquez' image? Montaigne conceived himself to be a creature in motion, to be identified as a self only through activity, through the pattern of his responses to the world. This being so, Montaigne depicts his double - the one who says 1' n the Essays - as writing down his (the double's) reactions, his responses to this or that circumstances imagined for him, from a place outside the text, by Montaigne.

 

It is natural to perceive the mirror reflexivity here in terms of painting and to see the Essays as giving 'not just a painter painting a picture of himself but a painter painting a picture of himself painting a picture and so on' (Sayce, 1972, p. 73); a metaphor which leads to a self situated at some infinite vanishing point. But the image of painting, unless it is cognisant of the deconstruction of object and sign accomplished for example in the images of Vermeer and Velasquez, rests on precisely the false anteriority they reveal. Another metaphor, non-inscriptional and closer perhaps to Montaigne's sense of himself as a being known only through activity, is that of theatre. For once Montaigne admits that his 'serf' is a performance, that what he observes has been posed for the observer, his supposedly autobiographical text cannot but reflect this duplicity: 'his medium, in order to transmit that aspect of his self, must reveal not the role that is played but the actor in the process of performing' (Rider, 1973, p. 78).

 

Such a self-conscious staging of the act of writing about writing makes the Essays into a certain kind of meta-theatre in which the actor, having perpetually to divulge the lie of acting, and moving ambiguously between the depicted self who says 'I' end the absent unknowable author, is a textual meta-subject: an agency able to convey, via a written text, the illusion that there exists a self, a prior 'thing', anterior to written depictions of that self.

 

To say this is in no sense to deny the materiality of the world, to pretend that there is no corporeal presence answering to the historical figure called Montaigne whose 'serf' is the subject-matter of the Essays, and in relation to which one recognises his text as autobiographical. Rather, when Montaigne (1889,'The Affection of Fathers') writes that 'Je me suis presente moy-mesme a moy, pour argument et pour subjet', he must be interpreted as denying the existence of some pristine, pre-textual self, some original and given 'me' (moy, moy-mesmes) being referred to by an independent and posterior 'I' (Je) in favour of a written construct that precisely transcends the I/me opposition. Thus, the sense in which the Essays are autobiographical, that is the nature of the semiotic relation between the meta-subject - the self thus constructed - and the self of the historical 'Montaigne' is not one of descrip-

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tion or reference (transparent, abstract, neutral language pointing to a pre-linguistic thing), but a relation between signs that rests on an iconic and motivated resemblance between an embodied and a textual self.

In an exactly parallel way, behind the abstract and unsituated character of the visual punctum lies a material world of sight, perception, and physical apprehension which is not described by the images of Vermeer and Velasquez as if it were there independently of these images. The obvious 'life-likeness' of such paintings, their recognisability as visual evocations, testifies to an iconism between signs that offer themselves as seeing (depictions) and the seeing of seeing (meta-signs that depict such embodied depictions).

CLOSURE OF IMAGINARY MONEY: PAPER MONEY

In the sign systems of mathematics and painting, zero and the vanishing point give rise to secondary meta-signs which form their semiotic closures. We shall now witness the same phenomenon occurring within the field of money signs.

We saw how bank money, the imaginary ideal money of the Marc Banco, by assigning a price via the agio to gold money, produced a certain kind of written credit, a monetary promise made by a bank to a named, individual merchant. And though such a note could be passed from hand to hand, and could, therefore, in some sense circulate freely, its link to the original merchant, no matter how complex the ,circuit of exchange, was never erased. The secondary meta-sign which forms he closure of imaginary money, arises from the complete severing of tlis attachment, the separation of money signs from traceable named owners, and the creation of a new, depersonalised and anonymous, form of money sign--that of paper money.

This attachment of bank money to particular persons arose from the fact that it, like any promissory note (as well as bill of exchange, receipt, IOU, note of indemnity, cheque, and so on) it, was an inherently deictic, that is to say, indexical sign. Its meaning as a money sign pointed to and was inseparable from the physical circumstanes of its use. One can say that its utterance as a sign was governed by a demonstrative personal pronoun tying it to the concrete particulars of a temporally located, named individual, since in order to circulate as money it needs to be turned over and endorsed, that is written to a payee by its owner through a reference, a date, and a signature. The presence of such a pronoun sharply separates imaginary money from money in currency. Indeed, it was in relation to the signs of currency--the circulating gold coins--that imaginary money appeared as a meta-sign indicating, through this pronoun, a mercantile subject who was

47

not present, and indeed could not be mentioned, within the code of exchange determined by circulating gold money.

The deixis inherent in imaginary money is the monetary analogue of the pronoun of place and point of view particularised by the vanishing point. And, as with the vanishing point, to emerge from it and form its closure, to become de-deictified, depersonalised, freed from the attachment to a spatially particularised viewer or owner, imaginary money has to be replaced by a different sort of sign presentation; a presentation that makes a new type of monetary promise--the kind of promise, for example, printed on English and Scottish bank notes (see Illustrations 14 and 15).

Illustration 14: Scottish twelve pound note, 1716.

At the end of the seventeenth century the Bank of England introduced, and the Bank of Scotland popularised, the printed bank note. The notes illustrated here display two versions of the promissory formula. In the

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Illustration 15 English one pound note, 1955.

 

Scottish note there is the older deictic promise to an individual whose name has been written in, together with an alternative general promise (which in fact renders the older form redundant) to an unnamed bearer. The English note dispenses with the first pronominalised, indexed alternative, and simply addresses its printed declaration to 'the bearer'.

 

The move from a deictic promise, with its embodiment in dates, individual names, and apparatus of witness, to one promising to pay a de-personalised anonymous bearer is highly significant; and it was achieved, in terms of historical process, against a particular semiotically revealing form of legal resistance concerning the signing of monetary notes within English Law. Neither the resistance (nor what it reveals) would have been evident if bank notes, with this same promise to an unnamed bearer, had been first introduced in Scotland or France, since the law of these countries required no written discharge of a monetary promise made out to a 'bearer':

 

 

The legal point at issue was whether the common law of England (for which a 'contract' was always between named individuals) should be inter-preted as allowing 'things-in-action' (which included bills of exchange as well as promissory notes) to be assignable to strangers. Whether or not, in

49 

other words, the practice of 'making bills (payable to bearer) transferable without a slow and expensive assignation, or even any endorsement' should be permitted. No final judgement was ever given. Initially the practice was allowed for bills of exchange but disallowed for promissory notes; a judge-ment which was then overturned to include promissory notes. This inclusion was then almost immediately reversed with the decision that 'all writs being promissory notes and not bills, were illegal, and could not be assigned or transferred; and finally this last judgement was ignored, and notes payable to a 'bearer', requiring no endorsement or validating signature, enjoyed a free de facto circulation.

 

The oscillation in law here between signing and not signing a note, between a named and an unnamed individual, between a particularised payee and generalised unlocated bearer, was the reaction of a judiciary reluctant to abandon the familiar localised subject addressed by a deictic promise - the subject of a contract- for the global, anonymous subject indicated on paper money by 'The Bearer'. The bearer is a variable subject, a subject in meta-lingual relation to any particular named and dated individual (the temporary owner of the note at a particular time) able to instantiate it, a meta-subject inscribed in the meta-sign 'paper money', whose analogue, in the signs of mathematics, is the algebraic subject.

 

Zero and the vanishing point gave rise to their closures through the process of a certain deconstruction. The illusion of anteriority, that is the belief that numbers and visual scenes were prior to the arithmetical signs that named or the signs of painting that depicted them, was exposed by the construction of meta-signs that insisted on 'things' and signs-for-things being coexistent. These were meta-signs, moreover, for which the categorial opposition between anteriority and posteriority was dissolved - it being as 'true' to hold that numbers and scenes were posterior, creations of the very system of signs invoked in order to name them, as to believe the opposite.

 

For money signs the illusion of anteriority collapses at exactly the point when the printed bank note is recognised as an instrument for creating money. Unlike metallic

 

 

The scandal of paper-created money, that is money written or printed into existence, was well appreciated by those who lost their 'real' money to Law's

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financial wizardry. And it has haunted the users of paper money and all the financial instruments descended from it ever since. The scandal is the loss of anteriority: paper money, instead of being a representation of some prior wealth, of some anterior pre-existing quantity of real gold or silver specie becomes the creator, guarantor and sole evidence for this wealth (see Illustration 16). And yet paper money still offered to the bearer of a note, the exchange: palpable specie, money in the hand, for an anonymous paper sign that can go and come from anywhere (the Chinese called it'flying money'), thereby insisting on the existence of anterior wealth waiting to be redeemed. The scandal is that of self-creation:paper money, as a circulating money sign, appeals to the very anteriority that as a creative meta-sign it has the capacity to deconstruct.

 

Illustration 16: US dollar bill silver certificate, 1957. A silver certificate which unlike the standard dollar bill explicitly promises redemption in specie.

 

Unsurprisingly, paper money was viewed with hostile suspicionexchanging real gold or silver for a flying paper note seemed lunatic - and enthusiastic greed- the bank note's ability to create wealth seemed unlimited. In the United States, where the widespread use of paper money was pioneered, these responses crystallised into a debate in the early nineteenth century between rival monetary philosophies.

 

Marc Shell (1980) sees in this 'debate about coined money and paper money [which] dominated American political discourse from 1825 to 1845', a wider, far-reaching conflict about the nature of signs - aesthetic and literary as well as monetary - about the nature, that is, of the difference between'reality' (things) end 'appearance' (signs-of-things). Certainly, the issue between the 'gold bugs' (for whom money meant gold, paper money being merely a licensed fraud) and the 'paper money men' (whose notes promised absent specie stored in a bank) was that of monetary reality and appearance (see Illustrations 17 and 18). And as with the vacillation of the English judiciary over signing or not signing a promissory note, the conflict arose from the refusal to surrender a palpable reality to an invisible

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appearance which the new paper sign required of its users. For the 'gold buys' money was solid, physical and real, a precious commodity which was owned by individuals, a'thing'as opposed to a sign; and they refused the offer made by the 'paper money men' to become 'bearers'- anonymous globalised versions of themselves - of notes that were mere insubstantial signs for money.

 

Shell, who is interested in the effect of this conflict on aesthetic representation, shows how Poe, in his story 'The Gold Bug', creates images that display the suspicious, uncertain, mysterious, fraudulent, and ghostly nature of the promise of gold on a piece of paper.

 

Illustration 17 'A shadow is not a substance', 1870. Specie reality and 'greenback' appearance.

 

In the cartoon 'A Shadow is not a Substance' for example, the specie can be viewed as one cause of the shadow. The specie and the sun are two links in the chain of cause and effect which a detective might say

 

Illustration 18 Milk tickets for babies. 'In Place of Milk', 1870.

 

produces the shadow. In 'The Gold Bug' the events in the plot and the eventual discovery of gold are not connected by this kind of necessity. To say that the design of the gold bug on the paper is a link in a chain of symbols or events which leads inevitably to the gold is, adopting the terms of the caricature, to say that the shadow is the cause of the sub-stance rather that vice versa. According to the Gold Bugs, this is the ostensibly absurd position of the Paper Money Men, of whom Legrand

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seems to be one as he marches through the dark forest with the designed

 paper clutched in his hand. (Shell, 1980, p. 21)

 

Shell illuminates Poe's story as a detective fiction whose mysteryunravelled through a melodrama of subliminal clues, cyphers, invisible writing, hidden treasure - is paper money's scandalous capacity to create gold. In the story, Poe makes the written instructions to the buried treasure (his narrative proxy for the promissory formula of the bank note) appear unnaturally, out of nowhere, as letters cryptically emerging from a blank sheet. He plays with and mocks paper money's ambiguous duality, its ability to appeal to the anterior existence of gold at the same time as it deconstructs, via its capacity to manufacture money, the very possibility of this anteriority. Ultimately in the story symbols - the signs-for-things - and'things'co-exist and create each other.

 

Later we shall see that in the end, within the present-day world monetary order, paper money's promise of payment, its promise to deliver some absent gold or silver, goes by default: on presenting a dollar bill to the US Treasury or a pound note to the Bank of England for redemption, the bearer would not be given any specie but would simply be offered a copy of the note being redeemed. Within such a tautological model of exchange, paper money becomes flying money that can never land (see Illustration 19).

 

Illustration 19 German billion mark note, 1923. Hyper-inflated paper money.

 

NATURALISING META-SIGNS INTO SIGNS

 

The movement from the primary meta-signs of zero, vanishing point, and imaginary money through a characteristic deconstruction to their closures produces a new signifying capacity which, in each case, assumes the priority

54

of objects to signs in order to deny it; a semiosis in which signs are seen to create the very objects they are taken to be depicting, naming, and representing.

 

And if the source of this denial, the meta-subject who exposes the illus-tion that objects are anterior to signs, appears in its different manifestations within mathematical, visual, textual, and monetary signs as a variable, invisible, unsituated, anonymous and absent agency, then the meta-sign which constitutes the means through which it operates requires a certain oblitera-tion: namely, the null)fication as a mesa-sign of the original and primary sign whose closure it is.

 

There is a moment in each of the codes that I have discussed when the fact that this change has happened becomes visible, when the meta-sign in question appears in an already naturalised form as a mere sign: when zero is accepted as just a number among numbers, its meta-lingual role at the origin of these numbers forgotten; when the vanishing point is absorbed into the painting as a location among others depicted there, its geometric initiation of the painting's design ignored; and when imaginary money is internalised as just more money, its meta-lingual distance from gold money suppressed.

 

What is forgotten, ignored, suppressed - in fact repressed - here is the original subject, or rather, the agency of this subject, the activity of the one responsible for originating, through the primary meta-sign, the entire system of signs. And it is in the place left empty by the primary subject that the meta-subject occurs. Its relation to the absented subject being that of temporary fictional identification, as when the subject of algebra imagines himself to be the one who makes the material marks of counting, the visual meta-subject sees himself embodied and situated in the place of the artist's double, the bearer of a paper note believes himself to be the individual, deictically present 'owner' of specie.

 

This move which naturalises meta-sign to sign in the codes of mathematics, vision, texts, and money, can be mapped onto a familiar and banal phenomenon in the code of spoken language, namely the literalisation of figures of speech, the death of metaphors, metonyms and the like. However, the question: what, literally, is a dead metaphor and how and at what point do metaphors die? seems problematic and unanswerable. For Nietzche, for example, whose attack on'truth' initialed the sense of the dif-ficulty Iying behind any simple interpretation of 'literal', it was a question of forgetting:

 

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If we attempt to avoid the circularity of Nietzche's use of metaphor (a monetary one at that) to locate the metaphorical death of metaphor, and speak from within a purely formal semiotics, we can say merely that semiotically a particular meta-sign is alive, its figurative and meta-lingual content still vital, as long as the primary subject, for whom the distinction between literal and figurative is itself still alive and uproblematic, is not, in the ways we have seen, forgotten, ignored, or suppressed; as long, that is, as the meta-subject, for whom presentations and representations coincide, has not yet made an explicit appearance within the code in question. In other words, the literality which sustains a metaphor is put into question at some point in the passage from a fully present primary subject to the sign)fied absence of that subject. For Western depictive painting such a point is almost visible: between the presence of the perspectival subject (unquestioned acceptance of perspectival literalism) and the explicit denial of the possibility of this literalism in the meta-paintings of Vermeer and Velasquez there occurs, as we saw, the internal depicted figure of the looker. By offering an internalised image of the perspectival subject such a figure neither embraces such a subject nor takes any part in signifying its absence; rather the figure is a sign which stands in the code at a place between the two: by challenging, on the one hand, the idea of literality, of a clear-cut distinction between prior reality and a subsequent figurative depiction of that reality, without on the other hand, in any way suggesting how such an idea might be denied or shown to be illusory.

 

It is a commonplace to describe the sorts of sign transformations that have occupied us as part of a general increase in conceptual abstraction that occurred with and through the emergence of Renaissance science. Certainly, algebra is a more abstract, less concrete, art than arithmetic, written calculations are more abstract than abacus manipulations, paper money is more abstract than imaginary money, the use of gold money is a more abstract activity than exchange by barter, and so on. But this sort of simple linear ascent through levels of abstraction, though adequate perhaps for certain purely internal histories of codes, conceals the agent responsible for these changes, the subjectivity in relation to which (and only so) they can be made intelligible. In the codes of mathematics, vision, text, and money, it is the active constructing subject who, by taking part in a thought experiment, makes an abstraction; an experiment in which the subject is enabled to occupy a new semiotic space, one which relies essentially on a reference to the absence of signs that were previously - before the experiment - conceived in terms of a positive, always present, content.

 

Of the signs I have talked about those of mathematics are generally held to be the most abstract. On the view presented here - that abstraction as an attribute of signs is essentially the result of a subject absenting himself-such is not the case. Zero is neither more nor less abstract than the vanishing

 56

point or imaginary money; all arise from and indeed facilitate the same sort of self-absenting move.

 

But zero is, as we observed earlier, a more universal and conceptually naked sign for absence, whose connection to the idea of 'nothing' creases an altogether more primitive and elemental source of abstraction. And it is this coupling between zero and the paradoxical idea of 'nothing' that will occupy us next.