1995 Thinking Dia-Grams

Mathematics, Writing, and Virtual Reality Mathematics, Science and Postclassical Theory. Ed Smith and Plotnitsky, 389-415

In the epilogue to his essay on the develop­ment of writing systems, Roy Harris declares:

It says a great deal about Western culture that the question of the origin of writing could be posed clearly for the first time only after the traditional dogmas about the relationship between speech and writing had been subjected both to the brash coun­ terpropaganda of a McLuhan and to the inquisitorial scepticism of a Derrida. But it says even more that the question could not be posed clearly until writing itself had dwindled to microchip dimensions. Only with this … did it become obvious that the origin of writing must be linked to the future of writing in ways which bypass speech altogether.’

Harris’s intent is programmatic. The passage continues with the injunction not to “re­-plough McLuhan’s field, or Derrida’s either,” but sow them, so as to produce eventually a “history of writing as writing .”

Preeminent among dogmas that block such a history is alphabeti­ cism : the insistence that we interpret all writing-understood for the moment as any systematized graphic activity that creates sites of in­ terpretation and facilitates communication and sense making-along the lines of alphabetic writing, as if it were the inscription of prior speech (“prior” in an ontogenetic sense as well as the more immedi­ate sense of speech first uttered and then written down and recorded). Harris’s own writings in linguistics as well as Derrida’s program of deconstruction, McLuhan’s efforts to dramatize the cultural impris­ onments of typography, and Walter Ong’s long-standing theorization of the orality/writing disjunction in relation to consciousness, among others, have all demonstrated the distorting and reductive effects of the subordination of graphics to phonetics and have made it their business to move beyond this dogma. Whether, as Harris intimates, writing will one day find a sp.eechless characterization of itself is im­ possible to know, but these displacements of the alphabet’s hegemony have already resulted in an open-ended and more complex articu­lation of the writing/speech couple, especially in relation to human consciousness, than was thinkable before the microchip.

A written symbol long recognized as operating nonalphabetically­ even by those deeply and quite unconsciously committed to alpha­ beticism-is that of number, the familiar and simple other half, as it were, of the alphanumeric keyboard . But, despite this recognition, there has been no sustained attention to mathematical writing even remotely matching the enormous outpouring of analysis, philosophiz­ ing, and deconstructive opening up of what those in the humanities have come simply to call “texts.”

Why, one might ask, should this be so? Why should the sign sys­ tem long acknowledged as the paradigm of abstract rational thought and the without-which-nothing of Western technoscience have been so unexamined, let alone analyzed, theorized, or deconstructed,  as a mode of writing? One answer might be a second-order or reflexive version of Harris’s point about the microchip dwindling of writing, since the very emergence of the microchip is inseparable from the action and character of mathematical writing. Not only would the entire computer revolution have been impossible without mathemat-

Thinkin9 Dia-Grams    391

ics as the enabling conceptual technology (the same could be said in one way or another of all technoscience), but, more crucially, the computer’s mathematical lineage and intended application as a calculating/reasoning machine hinges on its autological relation to mathematical practice. Given this autology, mathematics would pre­ sumably be the last to reveal itself and declare its origins in writing. (I shall return to this later.)

A quite different and more immediate answer stems from the dif­ ficulties put in the way of any proper examination of mathematical writing by the traditional characterizations of mathematics-Pla­ tonic realism or various intuitionisms-and by the moves they have legitimated within the mathematical community. Platonism is the contemporary orthodoxy. In its standard version it holds that mathe­ matical objects are mentally apprehensible and yet owe nothing to human culture; they exist, are real, objective, and “out there,” yet are without material, empirical, embodied, or sensory dimension. Besides making an enigma out of mathematics’ usefulness, this has the consequence of denying or marginalizing to the point of travesty the ways in which mathematical signs are the means by which com­ munication, significance, and semiosis are brought about. In other words, the constitutive nature of mathematical writing is invisi­ bilized, mathematical language in general being seen as a neutral and inert medium for describing a given prior reality-such as that of number-to which it is essentially and irremediably posterior.

With intuitionist viewpoints such as those of Brouwer and Husserl, the source of the difficulty is not understood in terms of some external metaphysical reality, but rather as the nature of our supposed inter­ nal intuition of mathematical objects. In Brouwer’s case this is settled at the outset: numbers are nothing other than ideal objects formed within the inner Kantian intuition  of time that is the condition  for the possibility of our cognition, which leads Brouwer into the quasi­ solipsistic position that mathematics is an essentially “languageless activity.” With Husserl, whose account of intuition, language, and ideality is a great deal more elaborated than Brouwer’s, the end result is nonetheless a complete blindness to the creative and generative role played by mathematical writing. Thus, in “The Origin of Geometry,”the central puzzle on which Husserl meditates is “How does geomet­ rical ideality … proceed from its primary intrapersonal origin, where it is a structure within the conscious space of the first inventor’s soul, to its ideal objectivity? “2 It must be said that Husserl doesn’t, in this essay or anywhere else, settle his question. And one suspects that it is incapable of solution. Rather, it is the premise itself that has to be denied: that is, it is the coherence of the idea of primal (semi­ otically unmmediated) intuition lodged originally in any individual consciousness that has to be rejected. On the contrary, does not all mathematical intuition – geometrical or otherwise- come  into being in relation to mathematical signs, making it both external/intersubjective and internal/private from the start? But to pursue such a line one has to credit writing with more than a capacity to, as Husserl has it, “document,” “record,” and “awaken” a prior and necessarily prelinguistic mathematical meaning. And this is precisely what his whole understanding of language and his picture of the “objectivity” of the ideal prevents him from doing. One consequence of what we might call the documentist view of mathematical writing, whether Husserl’s or the standard Platonic version, is that the intricate inter­ play of imagining and symbolizing, familiar on an everyday basis to mathematicians within their practice, goes unseen.

Nowhere is the documentist understanding of mathematical lan­ guage more profoundly embraced than in the foundations of mathe­ matics, specifically, in the Platonist program of rigor instituted by twentieth-century mathematical logic. Here the aim has been to show how all of mathematics can be construed as being about sets and, further, can be translated into axiomatic set theory. The proce­ dure is twofold. First, vernacular mathematical usage is made infor­ mally rigorous by having all of its terms translated into the language of sets. Second, these informal translations are completely formal­ ized, that is, further translated into an axiomatic system consisting of a Fregean first-order logic supplemented with the extralogical symbol for set membership.

To illustrate, let the vernacular item be the theorem of Euclidean geometry which asserts that, given any triangle in the plane, one can draw a unique inscribed circle.The first translation removes all reference to agency, modality, and physical activity, signaled here in the expression “one can draw.” In their place are constructs written in the timeless and agentless lan­ guage of sets. Thus, first the plane is identified with the set of all ordered pairs (x, y) of real numbers and a line and circle are trans­ lated into certain unambiguously determined subsets of these ordered pairs through their standard Cartesian equations; then “triangle” is rendered as a triple of nonparallel lines and “inscribed” is given in terms of a “tangent,” which is explicated as a line intersecting that which it “touches” in exactly one point. The second translation con­ verts the asserted relationship between these abstracted but still visu­ alizable sets into the de-physicalized and de-contextualized logico­ syntactical form known as the first-order language of set theory. This will employ no linguistic resources whatsoever other than variables ranging over real numbers, the membership relation between sets, the signs for an ordered pair and for equality, the quantifiers “for all x” and “there exists x,” and the sentential connectives “or,” “and,” “not,” and so on.

Once such a double translation of mathematics is effected, meta­ mathematics becomes possible, since one can arrive at results about the whole of vernacular  mathematics by proving theorems about the formal (i.e., mathematized) axiomatic system. The outcome has been an influential and rich corpus of metamathematical theorems (associated with Skolem, Godel, Turing, and Cohen, among many. many others). Philosophically, however, the original purpose of the whole foundational enterprise was to illuminate the nature of mathe­ matics by explaining the emergence of paradox, clarifying the hori-


 zons of mathematical reasoning, and revealing the status of mathe­ matical objects. In relation to these aims the set th”eoretization of mathematics and the technical results of metamathematics are unim­ pressive: not only have they resulted in what is generally acknowl­ edged to be a barren and uninformative philosophy of mathematics, but (not independently) they have failed to shed any light whatsoever on mathematics as a signifying practice. We need, then, to explain the reason for this impoverishment.

Elsewhere, I’ve spelled out a semiotic account of mathematics, par­ ticularly the interplay of writing and thinking, by developing a model of mathematical activity-what it means to make the signs and think the thoughts of mathematics-intended to be recognizable to its prac­ titioners.1 The model is based on the semiotics of Charles Sanders Peirce, which grew out of his program of pragmaticism, the general insistence that “the meaning and essence of every conception lies in the application that is to be made ofit.” 1 He understood signs accord­ ingly in terms of the uses we make of them, a sign being something always involving another-interpreting-sign in a process that leads back eventually to its application in our lives by way of a modifica­ tion of our habitual responses to the world. We acquire new habits in order to minimize the unexpected and the unforeseen, to defend our­ selves “from the angles of hard fact” that reality and brute experience are so adept at providing. Thought, at least in its empirically useful form, thus becomes a kind of mental experimentation, the perpetual imagining and rehearsal of unforeseen circumstances and situations of possible danger. Peirce’s notion of habit and his definition of a sign are rich, productive,  and capable of much interpretation. They have also been much criticized; his insistence on portraying all in­ stances of reasoning as so many different forms of disaster avoidance is obviously unacceptably limiting. In this connection, Samuel Weber has made the suggestion that Peirce’s “attempt to construe thinking and meaning in terms of ‘conditional possibility,’ and thus to extend controlled laboratory experimentation into a model of thinking in general,” should be seen as an articulation of a “phobic mode of be­ havior,” where the fear is that of ambiguity in the form of cognitive oscillation or irresolution, blurring or shifting of boundaries, impreci-


 sion, or any departure from the clarity and determinateness of either/ or logic.5

Now, it is precisely the elimination of these phobia-inducing fea­ tures that reigns supreme within mathematics. Unashamedly so: mathematicians would deny that their fears were pathologies, but would, on the contrary, see them as producing what is cognitively and aesthetically attractive about mathematical practice as well as being the source of its utility and transcultural stability. This being so, a model of mathematics utilizing the semiotic insights of Peirce-him­ self a mathematician-might indeed deliver something recognizable to those who practice mathematics. The procedure is not, however, without risks. There is evidently a self-confirmatory loop at work in the idea of using such a theory to illuminate mathematics, in apply­ ing a phobically derived apparatus, as it were, to explicate an unre­ pentant instance of itself. In relation to this, it is worth remarking that Peirce’s contemporary, Ernst Mach, argued for the importance of thought-experimental reasoning to science from a viewpoint quite different from Peirce’s semiotics, namely, that of the physicist. In­ deed, thought experiments have been central to scientific persuasion and explication from Galileo to the present, figuring decisively in this century, for example, in the original presentation of relativity theory as well as in the Einstein-Bohr debate about the nature of quantum physics. They have, however, only recently been given the sort of sus­ tained attention they deserve. Doubtless, part of the explanation for this comparative neglect of experimental reasoning lies in the system­ atizing approach to the philosophy of science that has foregrounded questions of rigor (certitude, epistemological hygiene, formal founda­ tions, exact knowledge, and so on) at the expense of everything else, and in particular at the expense of any account of the all-important persuasive, rhetorical, and semiotic content of scientific practice.

In any event, the model I propose theorizes mathematical reason­ing and persuasion in terms of the performing of thought experiments or waking dreams: one does mathematics by propelling an imago­ an idealized version of oneself that Peirce called the “skeleton self”- around an imagined landscape of signs. This model depicts mathemat­ ics, by which I mean here the everyday doing of mathematics, as a certain kind of traffic with symbols, a written discourse in other words, as follows: All mathematical activity takes place in relation to three interlinked arenas-Code, MetaCode, Virtual Code. These represent three complementary facets of mathematical discourse; each is asso­ciated with a semiotically defined abstraction, or linguistic actor­ Subject, Person, Agent, respectively-that “speaks,” or uses, it. The actors (Agents, Subjects,Person), their arenas (virtual Code, Code, metaCode) and their semiotic means (signifiers, signs, metasigns)  operate as an interlinked triad.

The Code embraces the total of all rigorous sign practices-defining, proving,  notating,  and  manipulating  symbols-sanctioned  by  the mathematical community. The Code’s user, the one-who-speaks it, is the mathematical Subject. The Subject is the agency who reads/writes mathematical  texts and has access to all and only those linguistic means allowed by the Code. The MetaCode is the entire matrix of unrigorous mathematical procedures normally thought of as prepara­ tory  and  epiphenomena!  to the  real-proper,  rigorous-business  of doing mathematics. Included in the MetaCode’s resources would be the stories, motives, pictures, diagrams, and other so-called heuristics which introduce, explain, naturalize, legitimate, clarify, and furnish the point of the notations and logical moves that control the opera­ tions of the Code. The one-who-speaks the MetaCode, the Person, is envisaged as being immersed in natural language, with access to its metasigns and constituted thereby as a self-conscious subjectivity in history and culture. Lastly, the Virtual Code is understood as the do­ main of all legitimately imaginable operations, that is, as signifying

Thinking Dia-Grams    397

possibilities available to an idealization of the Subject. This idealiza­ tion, the one-who-executes these activities, the A9ent, is envisaged as a surrogate or proxy of the Subject, imagined into being precisely in order to act on the purely formal, mechanically specifiable corre­ lates-signifiers-of what for the Subject is meaningful via signs. Jn unison, these three agencies make up what we ordinarily call “the mathematician.”

Mathematical reasoning is thus an irreducibly tripartite activity in which the Person (Dreamer awake) observes the Subject (Dreamer) imagining a proxy-the Agent (Imago )-of him/herself, and, on the basis of the likeness between Subject and Agent, comes to be per­ suaded that what  the Agent experiences is what the Subject would experience were he or she to carry out the unidealized versions of the activities in question. We might observe in passing that the three-way process at work here is the logico-mathematical correlate of a more general and originating triangularity inherent to the usual divisions invoked to articulate self-consciousness: the self-as-object instanti­ ated here by the Agent, the self-as-subject by the Subject, and the sociocultural other, through which any such circuit of selves passes, by the Person.

Two features of this way of understanding  mathematical  activity are relevant  here:  First, mathematical  assertions  are to be seen,  as Peirce insisted, as foretellings, predictions made by the Person about the Subject’s future engagement  with signs, with the result that the process of persuasion is impossible to comprehend if the role of the Person as observer of the Subject/Agent relation is omitted. Second, mathematical thinking and writing are folded into each other and are inseparable not only in an obvious practical sense, but also theoreti- . cally,  in relation  to the cognitive possibilities  that  are mathemati­ cally available. This is because the Agent’s activities exist and make narrative  logical sense  (for the Subject)  only through  the Subject’s manipulation of signs in the Code.

The second feature of this model, the thinking/writing  nexus, will occupy us below. On the first, however, observe that there is an evi­ dent relation between the triad of Code/MetaCode/Virtual Code here and the three levels-rigorous/vernacular/formal-of  the Platonistic


 reduction illustrated above. Indeed, in terms of external attributes, the difference drawn by the mathematical community between un­ rigorous/vernacular and rigorous/set-theoretical mathematics seems to map onto that between the Meta Code and the Code. This is indeed the case, but the status of this difference is here inverted and dis­ placed. On the present account, belief in the validity of reasoning, or acquiescence in proof, takes place only when the Person is persuaded, a process that hinges on a judgment-available only to the Person­ that the likeness between the Subject and the Agent justifies replac­ ing the former with the latter. In terms of our example, the proof of the theorem in question lies in the relationship among the Person, who can draw a triangle and see it as a drawn triangle; the Subject, who can replace this triangle with a set-theoretical description; and the Agent, who can act upon an imagined version of this triangle. By removing all reference to agency, the Platonistic account renders this triple relation invisible. Put differently, in the absence of the Person’s role, no explication of conviction-without which proofs are not proofs-can be given. Instead, all one can say about a sup­ posed proof is that its steps, as performed by the “mathematician,” are logically correct, a truncated and wholly unilluminating descrip­ tion of mathematical reasoning found and uncritically repeated in most contemporary mathematical and philosophical accounts. But once the Person is acknowledged as vital to the mathematical activi­ ties of making and proving assertions, it becomes impossible to see the MetaCode as a supplement to the Code, as a domain of mere psychological/motivational affect, to be jettisoned as soon as the real, proper, rigorous mathematics of the Code has been formulated.

If we grant this, we are faced with a crucial difference operat­ ing within what are normally and uncritically called mathematical “symbols,” a difference whose status is not only misperceived within the contemporary Platonistic program of rigor, but, beyond this, is treated within that program to a reductive alphabeticization. The first split in the following diagram, the division of writing into the alpha and the numeric, is simply the standard recognition of the non­ alphabetic character of numeric, that is, mathematical, writing. The failure to make this distinction, or rather making it but subsuming all writing under the rubric of the alphabet, is merely an instance of what we earlier called the alphabetic dogma. The point of the latter diagram is to indicate the replication of this split, via a transposed version of itself, within the contemporary Platonistic understanding of mathe­ matics. Thus, on one hand, there are ideograms, such as ‘+’, ‘x ‘, ‘1’, ‘2’, ‘3’, ‘=’, ‘>’, ·.. .’, ‘sinz’, ‘logz’, and so on, whose introduction and interaction are controlled by rigorously specified rules and syn­ tactic conventions. On the other hand, there are diagrams, visually presented semiotic devices, such as the familiar lines, axes, points, circles, and triangles, as well as all manner of figures, markers, graphs, charts, commuting circuits, and iconically intended shapes. On the orthodox view, the difference is akin to that between the rigorously literal. clear, and unambiguous ideograms and the metaphorically unrigorous diagrams. The transposition in question is evident once one puts this ranking of the literal over the metaphorical into play: as soon, that is, as one accepts the idea that diagrams, however useful and apparently essential for the actual doing of mathematics, are nonetheless merely figurative and eliminable and that mathematics, in its proper rigorous formulation, has no need of them. Within the Platonist program, this alphabetic prejudice is given a literal manifes­ tation: linear strings of symbols in the form of normalized sequences of variables and logical connectives drawn from a short, preset list determine the resting place for mathematical language in its purest, most rigorously grounded form.


There is a philosophical connection between this transposed alpha­ beticism and classical ontology. The alphabetic dogma rides on and promotes an essential secondarity. In its original form, this meant the priority of speech to writing, that is, the insistence that writ­ ing is the transcription of an always preceding speech (and, taking the dogma further back, that speech is the expression of a prior thought, which in turn is the mirror of a prior realm, … ). Cur­ rent Platonistic interpretations of mathematical signs replay this sec­ ondarity by insisting that signs are always signs of or about some preexisting domain of objects. Thus the time-honored distinction be­ tween numerals and numbers rests on just such an insistence that numerals are mere notations-names-subsequent and posterior to numbers which exist prior to and independent of them. According to this understanding of signs, it’s easy to concede that numerals are historically invented, changeable, contingent, and very much a human product, while maintaining a total and well-defended refusal to allow any of these characteristics to apply to numbers. And what goes for numbers goes for all mathematical objects. In short, contem­ porary Platonism’s interpretation of rigor and its ontology-despite appearances, manner of presentation, and declared motive-go hand in glove: both relying on and indeed constituted by the twin poles of an assumed and never-questioned secondarity.

Similar considerations are at work within Husserl’s phenomeno­ logical project and its problematic of geometrical origins. Only there, the pre-semiotic-that which is supposed to precede all mathematical language-is not a domain of external Platonic objects subsequently described by mathematical signs, but a field of intuition. The prior scene is one of “primal intrapersonal intuition” that is somehow­ and this is Husserl’s insoluble problem-“awakened” and “reacti­ vated” by mathematical writing in order to become available to all men at all times as an objective, unchanging ideality. The problem evaporates as a mere misperception, however, if mathematical writ­ ing is seen not as secondary and posterior to a privately engendered intuition, but as constitutive of and folded into the mathematical meaning attached to such a notion. What was private and intraper­ sonal is revealed as already intersubjective and public.

Nowhere is this more so than in the case of diagrams . The in­ trinsic  difficulty  diagrams  pose  to  Platonistic  rigor-their  essentialThinking Dia-Grams    


 difference from abstractly conceived sets and the consequent need to replace them with ideogrammatic representations-results in their elimination in the passage from vernacular to formalized mathemat­ ics. And indeed, set-theoretical rewritings of mathematics, notably, Bourbaki’s, but in truth almost all contemporary rigorous presenta­ tions of the subject, avoid diagrams like the plague. And Husserl, for all his critique of Platonistic metaphysics, is no different: not only are diagrams absent from his discussion of the nature of geometry, a strange omission in itself, but they don’t even figure as an important item to thematize.

Why should this be so? Why. from divergent perspectives and aims, should both Platonism and Husserlian phenomenology avoid all fig­ ures, pictures, and visual inscriptions in this way? One answer is that diagrams-whether actual figures drawn on the page or their imag­ ined versions-are the work of the body; they are created and main­ tained as entities and attain significance only in relation to human visual-kinetic presence, only in relation to our experience of the culturally inflected world. As such, they not only introduce the his­ torical contingency inherent to all cultural activity, but, more to the present point, they call attention to the materiality of all signs and of the corporeality of those who manipulate them in a way that ideograms-which appear to denote purely “mental” entities-do not. And neither Platonism’s belief in timeless transcendental truth nor phenomenology’s search for ideal objectivity, both irremediably mentalistic, can survive such an incursion of physicality. In other words, diagrams are inseparable from perception: only on the basis of our encounters with actual figures can we have any cognitive or mathematical relation to their idealized forms. The triangle-as­ geometrical-object that Husserl would ignore, or Platonists eliminate from mathematics proper, is not only what makes it possible to think that there could be a purely abstract or formal or mental triangle, but is also an always available point of return for geometrical abstraction that ensures its never being abstracted out of the frame of mathe­ matical discourse. For Merleau-Ponty the necessity of this encounter is the essence of a diagram.

“I believe that the triangle has always had, and always will have, angles the sum of  which  equals two right  angles  … because


I have had the experience of a real triangle, and because, as a physical thing. it necessarily has within itself everything it has ever been able, or ever will be able, to display…. What I call the essence  of  the  triangle  is nothing  but  this  presumption  of a completed synthesis, in terms of which we have defined the thing.”6

In fact, the triangle and its generalizations constitute geometry as means as well as object of investigation: geometry is a mode of imag­ining with and about diagrams.

And indeed, one finds recent commentaries on mathematics indi­cating a recognition of the diagrammatic as opposed to the purely formal nature of mathematical intuition. Thus Philip Davis urges a reinterpretation of “theorem” which would include visual aspects of mathematical thought occluded by the prevailing set-theoretical rigor, and he cites V. I. Arnold’s repudiation of the scorn with which the Bourbaki collective proclaims that, unlike earlier mathematical works, its thousands of pages contain not a single diagram.7

Let us return to our starting point, the question of writing, or, as Harris puts it, “writing as writing.” One of the consequences of my model is to open up mathematical writing in a direction familiar to those in the humanities. As soon as it becomes clear that diagrams (and indeed all the semiotic devices and sources of intuition mobi­ lized by the MetaCode) can no longer be thought of as the unrigorous penumbra  of  proper-Coded-mathematics,  as  so  many   ladders  to be kicked away once the ascent to pure, perceptionless Platonic form has been realized, then all manner  of possibilities  can  emerge. All we need to do to facilitate them is to accept a revaluation of basic terms. Thus, there is nothing intrinsically wrong with or undesirable about “rigor” in mathematics. Far from it: without rigor,  mathemat­ ics would vanish; the question is how one interprets its scope  and purpose. On the present account, rigor is not an externally enforced program of foundational hygiene, but an intrinsic and inescapable de­ mand proceeding from writing: it lies within the rules, conventions, dictates, protocols, and such that control mathematical imagination and transform mathematical intuition into an intersubjective writing/ thinking practice.  It is in this sense that, for example, Gregory Bate-


 son’s tag line that mathematics is a world of”rigorous fantasy” should be read. Likewise, one can grant that the MetaCode/Code division is akin to the metaphor/literal opposition, but refuse the pejorative sense that set-theoretical rigor has assigned to the term “metaphor.” Of course, there is a price to pay. Discussions of tropes in the humani­ ties have revealed that no simple or final solution to the “problem” of metaphor is possible; there is an always uneliminable reflexivity, since it proves impossible, in fact and in principle, to find a trope-free metalanguage in which to discuss tropes and so to explain metaphor in terms of something nonmetaphorical. For mathematics the price­ if such it be-is the end of the foundational ambition, the desire to ground mathematics, once and for all, in something fixed, totally cer­ tain, timeless, and prelinguistic. Mathematics is not a building-an edifice of knowledge whose truth and certainty is guaranteed by an ultimate and unshakable support-but a process: an ongoing, open­ ended, highly controlled, and specific form of written intersubjec­ tivity.

What, then, are we to make of mathematical diagrams, of their status as writing? How are they to be characterized vis-a-vis mathe­ matical ideograms , on the one hand, and the words of nonmathe­ matical texts, on the other? It would be tempting to invoke Peirce’s celebrated trichotomy of signs-symbol, icon, and index-at this point. One could ignore indexicals and regard ideograms as symbols (signs resting on an arbitrary relation between signifier and signified) and diagrams as icons (signs resting on a motivated connection be­ tween the two). Although there is truth in such a division, it is a misleading simplification: the ideogram/diagram split maps only with great artificiality onto these two terms of Peirce’s triad. In addition, there is a terminological difficulty: Peirce restricted the term “dia­ gram” to one of three kinds of icon (the others being “image” and “metaphor “), which makes his usage too narrow for what we here, and mathematicians generally, call diagrams. The artificiality arises from the fact that the ideogrammatic cannot be separated from the iconic nor the diagrammatic from the indexical. Thus, not only are ideograms often enmeshed in iconic sign use at the level of algebraic schemata, but, more crucially, diagrams, though iconic, are also, less obviously, indexical to varying degrees. Indeed, the very fact of


their being physically experienced shapes, of their having an opera­ tive meaning inseparable from an embodied and therefore situated gesture, will ensure that this is the case.

But this is a very generic source of indexicality, and some diagrams exhibit much stronger instances. Thus, consider the diagram, funda­ mental to post-Renaissance  mathematics,  of a coordinate axis


which consists of an extended, directed line and an origin denoted by zero. Let’s ignore the important but diffuse indexicality brought into play by the idea of directed extension and focus on that of the origin. Clearly, the function  of  the ideogram  ‘o’ in this diagram is to establish an arbitrarily chosen but fixed and distinguished “here” within the  undifferentiated  linear continuum.  The ideogram  marks a “this” with respect to which all positions on the line can be ori­ ented; such is what it means for a sign to function as an origin of coordinates. Indexicality, interpreted in the usual way as a coupling of utterance and physical  circumstance,  and recognized  as present in the use of shifters like “this,” “here,” “now,” and so on, within ordinary language,  is thus  unambiguously  present  in  our diagram. It does not, however, declare itself as such: its presence is the re­ sult of a choice and a determination made in the MetaCode,  that is, outside the various uses of the diagram sanctioned within mathemat­ ics proper. It is, in other words, the written evidence or trace of an originating act by the Person. Thus, zero, when symbolized by ‘o’, is an ideogrammatic  sign for mathematical ‘nothing’ at the same time that it performs a quasi-indexical function within the diagram of a coordinate  axis.8

It would follow from considerations like these that any investiga­ tion of the status of diagrams has to go beyond attempts at classifying them as sign-types and confront the question of their necessity. Why does one need them? What essential function-if any-do they serve? Could one do without them? Any answer depends, it seems, on who “one” is: mathematicians and scientists use them as abundantly and with as much abandon as those in the humanities avoid them. In fact, diagrams of any kind are so rare in the texts produced by historians, philosophers, and literary theorists, among others, that any instance sticks out like a sore thumb. An immediate response is to find this avoidance of visual devices totally unsurprising. Would not their em­ brace be stigmatized as scientism? Indeed, isn’t the refusal to use figures, arrows, vectors, and so forth, as modes of explication part of the very basis on which the humanities define themselves as different from the technosciences? Why should texts committed-on whatever grounds-to communicating through words and not primarily inter­ested in the sort of subject matter that lends itself to schematic visual representation make use of it? But this only pushes the question a little further out: What allows this prior commitment to words to be so self-sufficient, and what determines that certain topics or subjects, but not others, should lend themselves so readily to diagrammatic commentary and exegesis? Moreover, this separation by content isn’t very convincing: philosophers are no less interested in space, time, and physical process than scientists; literary theorists occupy them­ selves quite as intensely as mathematicians with questions of pattern, analogy, opposition, and structure. Furthermore, whatever its value, such a response gives us no handle on the exceptions, the rare re­ course to diagrams, that do occur in humanities texts.

To take a single example, how should we respond to the fact that in Husserl’s entire oeuvre there is but one diagram (in his exegesis of temporality), a diagram that, interestingly enough, few commen­ tators seem to make any satisfactory sense of? Are we to think that Husserl, trained as a mathematician, nodded-momentarily slipping from the philosophical into a more mathematical idiom? Or, unable to convey what he meant through words alone, did he resort, reluc­ tantly perhaps but inescapably, to a picture? If the latter, then this fact-the possible inexplicability in words of his account of time­ would surely be of interest in any overall analysis of Husserl’s philo­ sophical ideas. It would, after all, be an admission-highly significant in the present context-that the humanities’ restriction to picture­ less texts may be a warding off of uncongenial means of expression rather than any natural or intrinsic self-sufficiency in the face of its subject matter. But then would not this denial, or at any rate avoid-


ance, of diagrams result in texts that were never free (at least never demonstrably so) of a willful inadequacy to their chosen exegetical and interpretive tasks; texts whose wordy opacity, hyper-elaboration, and frequent straining of written expression to the edge of sense were the reciprocal cost of this very avoidance?

And what goes for diagrams goes (with one exception) for ideo­ grams: their absence is as graphically obvious as that of diagrams; the same texts in the humanities that avoid one avoid the other. And the result is an adherence to texts written wholly within the typo­ graphical medium of the alphabet. The exception is, of course, the writing of numbers: nobody, it seems, is prepared to dump the system that writes 7,654,321 in place of seven million, six hundred fifty-four thousand, three hundred twenty-one; the unwieldy prolixity here is too obvious to ignore. But why stop at numbers? Mathematics has many other ideograms and systems of writing-some of extraordi­ nary richness and subtlety besides  the number notation based on zero.

What holds philosophers and textual theorists back? Although it doesn’t answer this question, we can observe that the place-notation writing of numbers is in a sense a minimal departure from alphabetic typography: an ideogram like 7,654,321 being akin to a word spelled from the ‘alphabet’ 0, 1,2,3,4,5,6,7,8,9 of ‘letters’, where to secure the analogy one would have to map the mathematical letter o onto some­ thing like a hyphen denoting the principled absence of any of the other given letters.

I alluded earlier to Weber’s characterization of Peirce’s semiotics as founded on a fear of ambiguity and the like. It’s hard to resist seeing a reverse phobia in operation here: a recoil from ideograms (and, of course, diagrams) in the face of their potential to disrupt the famil­ iar authority of the alphabetic text, an authority not captured but certainly anchored in writing’s interpretability as the inscription of real or realizable speech. The apprehension and anxiety in the face of mathematical grams, which appear here in the form of writing as such-not as a recording of something prior to itself-are that they will always lead outside the arena of the speakable; one cannot, after all, say a triangle. If this is so, then the issue becomes the general relation among the thinkable, the writable, and the sayable, that is, what and how we imagine through different kinds of sign manipu­ lations, and the question of their mutual translatability. In the case

Thinkin9 Dia-Grams   407

of mathematics, writing and thinking are cocreative and, outside the purposes of analysis and the like, impossible to separate.

Transferring the import of this from mathematics to spoken lan­ guage allows one to see that speech, no less than mathematics, mis­ understands its relation to the thinkable if it attempts a separation between the two into prior substance and posterior re-presentation; if, in other words, the form of an always re-presentational alphabetic writing is the medium through which speech articulates how and what it is. By withdrawing from the gram in this way, alphabetic writ­ ing achieves the closure of a false completeness, a self-sufficiency in which the fear of mathematical signs that motivates it is rendered as invisibly as the grams themselves. The idea of invisibility here, how­ ever, needs qualifying. Derrida’s texts, for example, though written within and confined to a pure, diagramless and ideogramless format, nevertheless subvert the resulting alphabetic format and its automatic interpretation in terms of a vocalizable text through the use of vari­ ous devices: thus a double text such as “Glas,” which cannot be the inscription of any single or indeed dialogized speech, and his use of a neologism such as differance, which depends on and performs its meaning by being written and not said. But, all this notwithstanding, any attempt to pursue Harris’s notion of a speechless link between the origin and the future of writing could hardly avoid facing the ques­ tion of the meaning and use of diagrams. Certainly, Harris himself is alive to the importance and dangers of diagrams, as is evident from his witty taking apart of the particular diagram-a circuit of two heads speaking and hearing each other’s thoughts-used by Saussure to illustrate his model of speech.9

But perhaps such a formulation, though it points in the right direc­tion, is already-in light of contemporary developments-becoming inadequate. Might not the very seeing of mathematics in terms of a writing/thinking couple have become possible because writing is now-post-microchip-no longer what it was? I suggested above that the reflexivity of the relation between computing and mathematics­ whereby the computer, having issued from mathematics, impinges on and ultimately transforms its originating matrix-might be the crux of the explanation for our late recognition of mathematics’ status as writing.

To open up the point, I turn to a phenomenon  within the ongoing


microchip revolution, namely, the creation and implementation of what has come to be known as virtual reality. Although this might seem remote from the nature and practice of mathematics and from the issues that have so far concerned us, it is not, I hasten to add, that remote. In addition to many implicit connections to mathemati­ cal ideas and mathematically inspired syntax via computer program­ ming, there are explicit links: thus, for example, Michael Benedikt, in his introductory survey of the historical and conceptual context of virtual reality, includes mathematics and its notations as an impor­ tant thread running through the concept.10

An extrapolation of current practices more heralded, projected, and promised than as yet effectively realized, virtual reality com­ prises a range of effects and projects in which certain themes and practices recur. Thus, one always starts from the given world-the shared, intersubjective, everyday reality each of us inhabits. Within this reality is constructed a subworld, a space of virtual reality that we-or rather certain cyberneticized versions of ourselves-can, in some sense, enter and interact with. The construction of this virtu­ ality, how it is realized-its parameters, horizons, possibilities, and manifestations-varies greatly from case to case. Likewise, what is entailed by a “version” of ourselves, and hence the sense in which “we” can be said to be “in” such virtual arenas, varies, since it will de­ pend on what counts, for the purposes at hand, as physical immersion and interaction, and on how these are connected and eventually im­ plemented. In all cases, however, virtual arenas are brought into exis­ tence inside computers and are entered and interacted with through appropriate interface devices and prosthetic extensions, such as spe­ cially adapted pointers, goggles, gloves, helmets, body sensors, and the like. Perhaps the most familiar example is dipping a single finger into a computer environment via the point-and-click operation of a computer mouse. But a mouse is a very rudimentary interface device, one which gives rise to a minimal interpretation, both in what the in­ ternalized finger can achieve as a finger and because a finger is, after all, only a metonym of a body: all current proposals call for more comprehensive prosthetics and richer, more fully integrated modes of interaction with/within these realities once they are entered.

Let’s call the self in the world the default or real-I; the cyberneti-

Thinkina Dia-Grams    409

cized self we propel around a virtual world, the surrogate or virtual-I; and the self mediating between these, as the enabling site and means of their difference, the jacked-in or 9099led-J. Operating a virtually real environment involves an interplay or circulation among these three agencies which ultimately changes the nature of the original, default reality, that is, of what it means to be a real-I inhabiting a/the given world. This circulation and especially its effect on, ultimately its transformation of, the given world motivate a great deal of virtual­ reality thinking. To fix the point I’ll mention two recent, differently conceived proposals, a social-engineering project and a fantasy ex­trapolation, which explore the possibilities offered by a virtualization of reality. The first, Mirror Worlds, is part propaganda, part blueprint for a vast series of public software projects by computer scientist David Gelernter, and the second, Snow Crash. is a science fiction epic of  the  near-cybered-future   by  novelist  Neal  Stephenson.11

Mirror Worlds sets itself the task of mapping out, more or less in terms of existing software technology. a way of virtualizing a public entity, such as a hospital or university or city (more ambitiously. an entire country, ultimately the world). Its aim is to create a virtual space, a computer simulation of, say. the city-what Gelernter calls the “agent space”-which each citizen could enter through various interface tools and engage in activities (education, shopping. infor­ mation gathering, witnessing public events, monitoring and partici­ pating in cultural and political activity, meeting other citizens, and so on) in virtual form. The idea is that the results of such virtual-I activities would reflect back on society and effect changes in what it means to be a citizen within a community-to be a real-I-changes in previously unattainable and, given Gelernter’s downbeat take on con­ temporary fragmentation and anomie, sorely needed ways. In Snow Crash Stephenson posits an America whose more computer-savvy denizens can move between a dystopian reality (panoptic surveil­ lance and Mafia-franchised suburban enclaves) and a freely created, utopian computer space, the “metaverse,” where their virtual-I’s, or “avatars,” can access the information net and converse and interact with each other in various virtual ways. Crucial to the plotting and thematics of Stephenson’s narrative is the interplay between the in­ side of the metaverse and the all-too-real outside; the circulation, in


other words, of affect and effects between virtual-I’s and real-I’s as the characters put on and take off their goggles. Although they move in opposite directions-in Stephenson’s fantasy the virtual world in the end reflects the intrigue and violence of the real, while for Gelern­ ter the virtual world is precisely the means of eliminating the anomic violence of the contemporary world-they share the idea of opposed worlds separated, joined, and mutually transformed by an interface.

A certain homology between virtual reality and mathematical thought, each organized around an analogous triad of agencies, should by now be evident. The virtual-I maps onto the mathematical Agent, the real-I onto the Person, and the goggled-I onto the Subject. In accordance with this mapping, both virtual reality and mathemat­ ics involve phenomenologically meaningful narratives of propelling a puppet-agent – asimulacrum, surrogate, avatar, doppelganger, proxy (Peirce’s “skeleton self”) – of oneself around a virtual space. Both require a technology which gives real-I’s access to this space and which controls the capabilities and characteristics of the skeleton­ self agent. In both, this technology is structured and defined in terms of an operator, a figure with very particular and necessary features of its own, distinct from the puppet it controls and from the figure­ the Person or real-I-occupying the default reality, able to put on goggles and operate in this way. And both are interactive in a material. embodied sense. In this they differ from the practices made possible by literature, which (like mathematics) conjures invisible proxies and identificatory surrogates of ourselves out of writing, and they differ from the media of theatre, film, and TV, where (like vir­ tual reality) proxies are not purely imagined, but have a visual presence. The difference arises from the fact that although these media allow, and indeed require of, their recipients/participants an active interpretive role, this doesn’t and cannot extend to any real-materi­ ally effective-participation: mathematicians manipulate signs, and virtual realists act out journeys.

Mathematics, then, appears to be not only an enabling technology. but a template and precursor, perhaps the oldest one there is, of the current scenarios of virtual reality. But since something new is enabled here, what then (apart from obvious practical differences) distinguishes them? Surely, a principal and, in the present context,

Thinking Dia-Grams    411

quite crucial difference lies in the instrumental means available to the operator-participants: the mathematical Subject’s reliance on the writing technologies of ink and chalk inscriptions versus the pros­ thetic extensions available to the virtual reality operator. Therefore, what separates them is the degree of palpability they facilitate: the gap between the virtuality of a proxy whose repertoire (in the more ambitious projections) spans the entire sensorimotor range of mo­dalities – ambulatory, auditory. proprioceptive, tactile,  kinetic – -and the invisible, disembodied Agent of mathematics. The virtual space entered by mathematicians’ proxies is, in other words, entirely imag­ined, and the objects, points, functions, numbers, and so on, in it are without sensible form; percepts in the mind’s eye rather than in the real eye necessary for virtual participation. Of course, the journeys that mathematical Agents perform, the narratives that can be told about them, the objects with which they react, and the regularities they encounter are strictly controlled by mathematical signs. Con­necting these orders of signification, recreating the writing/thinking nexus through the interactive manipulation of visible diagrams and ideograms and the imagined, invisible states of affairs they signify and answer to, determines what it means to do mathematics.

We are thus led to the question: What if writing is no longer con­ fined to inscriptions on paper and chalkboards, but becomes instead the creation of pixel arrangements on a computer screen? Wouldn’t such a mutation in the material medium of mathematical writing effect a fundamental shift in what it means to think, and do, mathe­ matics? One has only to bear in mind the changes in consciousness brought into play by the introduction of printing-surely a less radi­ cal conceptual and semiotic innovation than the shift from paper to screen-to think that indeed it would. The impact of screen-based visualization techniques on current scientific research and on the status of the theory/experiment opposition, as this has been tradi­ tionally formulated in the philosophy and history of science, already seems far-reaching. Thus, although primarily concerned with certain aspects of the recent computerization and mathematization of bi­ ology. the conclusion of Tim Lenoir and Christophe Lecuyer’s inves­ tigation, namely. that “visualization is the theory,” is suggestive far outside this domain.12


New type of mathematics – ways of thinking mathematically ­ have already come into existence precisely within the field of this mutation. Witness chaos theory and fractal geometry, with  their essential reliance on computer-generated images (attractors in phase space, self-similar sets in the complex plane, and so on) which are nothing less than new, previously undrawable kinds of diagrams. And, somewhat differently, witness proofs (the four-color problem, classi­ fication of finite groups) that exist only as computer-generated enti­ ties. Moreover, there’s no reason to suppose that this feedback from computer-created imagery and cognitive representations-in effect, a vector from an abstract, imagination-based technology to a concrete, image-based one-to the conceptual technology of mathematics will stop at the creation of new modes for drawing diagrams and notating arguments. But diagrams, because their meanings and possibilities stem from their genesis as physically drawn, bodily perceived objects, are already quasi-kinematic. In light of this, it’s necessary to ask why such a process should be confined to the visual mode, to the creation of graphics and imagery, and not extended to the other sense mo­ dalities? What is to stop mathematics from appropriating the various computer-created ambulatory, kinesthetic, and tactile features made freely available within the currently proposed schemes for virtual reality? Is it unnatural or deviant to suggest that immersion in a vir­ tually realized mathematical structure-walking around it, listening to it, moving and rearranging its parts, altering its shape, dismantling it, feeling it, and even smelling it, perhaps-be the basis for mathe­ matical proofs? Would not such proofs, by using virtual experience as the basis for persuasion, add to, but go far beyond, the presently ac­ cepted practice of manipulating ideograms and diagrams in relation to an always invisible and impalpable structure? The understanding of writing appropriate to this conception of doing mathematics, what we might call virtual writina, would thus go beyond the “archewriting” set out by Derrida, since it could no longer be conceived in terms of the “gram” without wrenching that term out of all continuity with itself.

On at least one understanding of the genesis of thinking, nothing could be more natural and less deviant than using structures outside ourselves in order to think mathematics. Thus,  according to Merlin

Thinking Dia-Grams   413

Donald’s recent account, the principal vector underlying the evolu­ tion of cognition and, ultimately, consciousness  is the development and utilization of external forms of memory: our neuronal connec­ tions and hence our cognitive and imaginative capacities resulting, on this view, from forms of storage and organization outside our heads rather  than  the reverse. 13

Evidently, natural or unnatural, such a transformation of mathe­ matical practice would have a revolutionary impact on how we con­ ceptualize mathematics, on what we imagine a mathematical object to be, on what we consider ourselves to be doing when we carry out mathematical investigations and persuade ourselves that certain as­ sertions, certain properties and features of mathematical objects, are to be accepted as ‘true’. Indeed, the very rules and protocols that con­ trol what is and isn’t mathematically meaningful.  what constitutes a ‘theorem’, for example, would undergo a sea change. An assertion would no longer have to be something capturable in a sentencelike piece of-presently conceived-writing, but could  be  a  configura­ tion that is meaningful only within a specifically presented virtual reality. Correspondingly, a proof would no longer have to be an argu­ ment organized around  a written-as presently conceived-sequence of logically connected symbols, but could take on the character of an external, empirical verification. Mathematics would thus become what it has long denied being: an experimental subject; one which, though quite different from biology or physics in ways yet to be for­ mulated, would be organized nonetheless around an independently existing, computer-created and -reproduced empirical reality.

This union, or rather this mutually reactive  merging,  of mathe­matics and virtual reality-a  coming together of a rudimentary  and yet-to-be consummated  technology of manufactured  presence with its ancient,  highly developed  precursor-would  take the form of a double-sided process. As we’ve seen, from outside and independent of any mathematical desiderata, the goal of this technology is to achieve nothing less than the virtualization of the real, a process that will en­ gender irreversible changes in what for us constitutes the given world, the default domain of the real-I. From the other side, in relation to a mathematics whose objects and structures have a wholly virtual, nonmaterial existence already, the process appears as the reverse, as


effectively a realization of the virtual, whereby mathematical objects, by being constructed inside a computer, reveal themselves as ma­ terially presented and embodied, a process that will likewise cause irreversible and unexpected changes in the meaning we can attach to the idealized real. One could give a more specific content to this by looking at the most extravagantly virtual concept of contemporary mathematics, that of infinity; a concept so inherently metaphysical and sp ctral as to be unrealizable-actually or even in principle­ within the universe we inhabit . Or so I have argued at length else­ where.’

Of course, these remarks on the joint future of mathematics and vir­ tual reality, although they could be supported by looking at computer­ inflected practices within the current mathematical scene, are little more than highly speculative extrapolation here. I have included them in order to get a fresh purchase on the notion of a diagram: by overtly generalizing the notion into a virtually realized (i.e., physically presented) mathematical structure, one can see how the question of diagrams is really a question of the body. To exclude diagrams-either deliberately as part of the imposition of rigor in mathematics or less explicitly as an element in a general and unex­ amined refusal to move beyond alphabetic texts and the linear strings of ideograms that mimic them within mathematics-is to occlude materiality, embodiment, and corporeality, and hence the immersion in history and the social that is both the condition for the possibility of signifying and its (moving) horizon.


This article was written during the period when I was supported by a fellowship from the National  Endowment for the Humanities.

1   Roy Harris, The Oriyin of Writing  (London, 1986), 158.

2 Edmund Husserl. “The Origin of Geometry,” trans . David Carr, in Husserl: Shorter Works, ed . P. McCormick and F. Ellison (Notre Dame, 1981), 257.

3         Brian Rotman , “Toward a Semiotics of Mathematics,” Semiotica 72 ( 1988): 1-35; and Ad Infinitum … The Ghost in Turing’s Machine : Takiny God Out of Mathematics and Putting the Body Back In (Stanford, 1993), 63-113.

4         Charles Sanders Peirce , Collected Writinys, ed . Philip Weiner (New York, 1958), 5:332 .

Thinking Dia-Grams    415

5    Samuel Weber, Institution  and Interpretation  (Minneapolis,  1987), 30.

6   Maurice Merleau-Ponty,  The Phenomenolo9y  of Perception,  trans. Colin Smith (London, 1962), 388.

7     Philip Davis, “Visual Theorems,” Educational Studies in Mathematics 24 ( 1993): 333-44.

8     See Brian Rotman, Si9nifyin9  Nothin9: The Semiotics of Zero  (Stanford,  1993).

9     Roy Harris, The Lan9ua9e Machine (Ithaca, 1987), 149-52.

IO     Michael  Benedikt, Cyberspace: First Steps (Cambridge, 1992), 18-22.

11      David Gelernter, Mirror Worlds (New York,  1992); Neal Stephenson, Snow Crash

(New York, 1992).

12         Timothy Lenoir and Christophe Lecuyer, “Visions of Theory” (to appear).

13          Merlin Donald, Ori9ins of the Modern Mind  (Cambridge, MA, 1991).

14         Rotman, Ad Infinitum.

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