Barbara Herrnstein Smith is Professor of Comparative Literature and English and Director of the Center for Interdisciplinary Studies in Science and Cultural Theory at Duke University. Arkady Plotnitsky is Visiting Scholar at Duke University’s Center for Interdisciplinary Studies in Science and Cultural Theory.

Mathematics, Science, and Postclassical Theory

By Barbara Herrnstein Smith, Arkady Plotnitsky

Duke University Press

Copyright © 1997 Duke University Press
All rights reserved.
ISBN: 978-0-8223-1857-6

Contents

Introduction: Networks and Symmetries, Decidable and Undecidable,
Thinking Dia-Grams: Mathematics, Writing, and Virtual Reality,
Concepts and the Mangle of Practice: Constructing Quaternions,
The Moment of Truth on Dublin Bridge: A Response to Andrew Pickering,
Explanation, Agency, and Metaphysics: A Reply to Owen Flanagan,
Agency and the Hybrid,
The Accidental Chordate: Contingency in Developmental Systems,
Complementarity, Idealization, and the Limits of Classical Conceptions of Reality,
Is “Is a Precursor of” a Transitive Relation?,
Fraud by Numbers: Quantitative Rhetoric in the Piltdown Forgery Discovery,
A Glance at SunSet: Numerical Fundaments in Frege, Wittgenstein, Shakespeare, Beckett,
Microdynamics of Incommensurability: Philosophy of Science Meets Science Studies,
Notes on Contributors,

CHAPTER 1

Thinking Dia-Grams: Mathematics, Writing, and Virtual Reality

Brian Rotman

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 counterpropaganda 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 alphabeticism: the insistence that we interpret all writing — understood for the moment as any systematized graphic activity that creates sites of interpretation 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 imprisonments 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 speechless characterization of itself is impossible 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 alphabeticism — 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, philosophizing, 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 system 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 mathematics 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 presumably 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 difficulties put in the way of any proper examination of mathematical writing by the traditional characterizations of mathematics — Platonic 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 mathematical 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 communication, significance, and semiosis are brought about. In other words, the constitutive nature of mathematical writing is invisibilized, 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 internal 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 geometrical 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?” 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 (semiotically unmediated) 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 interplay of imagining and symbolizing, familiar on an everyday basis to mathematicians within their practice, goes unseen.

Nowhere is the documentist understanding of mathematical language more profoundly embraced than in the foundations of mathematics, 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 procedure is twofold. First, vernacular mathematical usage is made informally rigorous by having all of its terms translated into the language of sets. Second, these informal translations are completely formalized, 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 (see diagram at right). 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 language 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 translated 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 converts the asserted relationship between these abstracted but still visualizable sets into the de-physicalized and decontextualized 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, metamathematics 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, Gödel, Turing, and Cohen, among many, many others). Philosophically, however, the original purpose of the whole foundational enterprise was to illuminate the nature of mathematics by explaining the emergence of paradox, clarifying the horizons of mathematical reasoning, and revealing the status of mathematical objects. In relation to these aims the set theoretization of mathematics and the technical results of metamathematics are unimpressive: not only have they resulted in what is generally acknowledged 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, particularly 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 practitioners. 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 of it.” He understood signs accordingly 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 modification of our habitual responses to the world. We acquire new habits in order to minimize the unexpected and the unforeseen, to defend ourselves “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 instances 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 behavior,” where the fear is that of ambiguity in the form of cognitive oscillation or irresolution, blurring or shifting of boundaries, imprecision, or any departure from the clarity and determinateness of either/or logic.

Now, it is precisely the elimination of these phobia-inducing features 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 — himself 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 applying a phobically derived apparatus, as it were, to explicate an unrepentant 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. Indeed, 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 sustained attention they deserve. Doubtless, part of the explanation for this comparative neglect of experimental reasoning lies in the systematizing approach to the philosophy of science that has foregrounded questions of rigor (certitude, epistemological hygiene, formal foundations, 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 mathematics, 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 following diagram summarizes these actors and the arenas in relation to which they 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 preparatory and epiphenomenal to the real — proper, rigorous — business of doing mathematics. Included in the MetaCode’s resources would be the stories, motives, pictures, diagrams, and other socalled heuristics which introduce, explain, naturalize, legitimate, clarify, and furnish the point of the notations and logical moves that control the operations 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 domain of all legitimately imaginable operations, that is, as signifying possibilities available to an idealization of the Subject. This idealization, the one-who-executes these activities, the Agent, is envisaged as a surrogate or proxy of the Subject, imagined into being precisely in order to act on the purely formal, mechanically specifiable correlates — signifiers — of what for the Subject is meaningful via signs. In unison, these three agencies make up what we ordinarily call “the mathematician.”

(Continues…)Excerpted from Mathematics, Science, and Postclassical Theory by Barbara Herrnstein Smith, Arkady Plotnitsky. Copyright © 1997 Duke University Press. Excerpted by permission of Duke University Press.
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