1999 Technology of Mathematical Persuasion

Inscribing Science Edited Tim Lenoir, Stanford U. Press. On line

My aim here is to outline a conception of mathematics-as language and indicate, contrary to the received view, how mathematical reasoning is inseparable from persuasion and the exercise/creation of a certain mode of subjecthood. Let me frame my approach to the question of language and mathematics by means of two historically separated and very different citations.

“Philosophy is written in this grand book – the universe, which stands continually open before our gaze. But the book cannot be understood unless  one first learns to comprehend the language and to read the alphabet in which it is composed. It is written in the language of mathematics … ”   Galileo[1]

Against Galileo’s confident invocation of language there is its contemporary problematization.

“However the topic is considered, the problem of language has never been simply one problem among others. But never as much as at present has it invaded, as such, the global horizon of the most diverse researches and the most heterogeneous discourses, diverse in their intention, method, and ideology.”   Derrida [2]

Where to begin? One could start from the outside, from mathematics’ evident and inescapable immersion in the sociocultural matrix, and track its historical production as a form of instrumental reason and source of so-called objective, true, value-free knowledge. The last two decades has seen much work of this kind — heterogeneous but operating under the unifying premise that all forms of knowledge together with their legitimating claims to ‘truth’ and the like are social constructions (mathematical no less than any other). [3] In relation to the present aim the hope would be to close in on mathematical practice and, by focusing in on its communicational and symbolic functioning, produce an understanding of mathematics-as-language. Or one could start from the inside, which is what I shall do. This means assuming (and bracketting its complexities) the ultimate success of an external characterization of mathematics as a communicational activity, and attempting to complement this description by giving a quasi-phenomenological, internal account that might reflect what it means to do mathematics — write its symbols and think its thoughts — from within. And since there is virtually nothing in the way of a history to such a project one starts from the present.

 A present in which the cultural scene of the humanities is being shaped by an engagement with a certain linked set of contentiously theorized problems and intellectual moves that goes under the mark — simplifying but nonetheless useful for our purposes — of the Post (post-structuralist, post-modern, post- patriarchal, post-humanist, post-industrial, postEnlightenment, post-realist …). Though no single issue, discursive move or programme captures the boundaries of the Post, a key site of conflict and innovation is the nature and business of language. By this I mean “language” in the large, to include natural speech, but also all forms of writing, recording, communicating, representing, cyphering and notating through the use of spoken words, written and gestured signs, iconographic devices, ideograms, symbol systems, and the like.

A discourse within the humanities resistant and unsympathetic to the de- stabilizing subversions of the Post, is mainstream Anglo-American analytic philosophy, whose focus on logic and the nature of mathematical and scientific knowledge makes it naturally relevant to any examination of mathematics.  Here the refusal of the Post (essentially of post-structuralist, deconstructive theorizing) takes the form of a deep, entrenched and perhaps irreconcilable, conflict between an old and much worked through orthodox tradition regarding language, and a newer philosophical outlook opposed and alien to it: a conflict between the so-called continental outlook whose conception of language — dominated by Nietzsche, Husserl, Heidegger, Wittgenstein and Derrida — assigns it a constitutive role and whose slogan might run “Language speaks man into the world,” and the current analytic mindset which understands language as an inert, transmissive medium and their empiricist forebears — whose banner might read “Man speaks language about the world.”

To say, then, of some human endeavor, in our case the practice of mathematics, that it is a “language” or symbol system or mode of discourse, is to confront  this conflict; something that seems inevitable if one is to place mathematics, at this late point in the century, in relation to the current of late twentieth century thought. But such an engagement brings with it certain risks and difficulties since, as is generally acknowledged, the status of “language” (and not just within philosophy) is in flux — contentiously theorized, open-ended, problematic. Evidently, any description of the linguistic and signifying capacity of mathematics is likely to raise issues that will appear artificial or foreign to mathematics’ previous un-languaged or inadequately languaged image of itself.

 Leaving such problematics aside, there is, on the face of it, little need to insist that mathematics is a language: who after all among those familiar with it  would deny the proposition? Certainly not those users – accountants, engineers, economists, actuaries, statisticians, cliometricians, meteorologists and the like – who have no choice but to translate in and out of mathematical expressions and terminology on their way to other interests and forms of engagement; and certainly not the professional practitioners of the physical and life sciences and technologies who have always appealed to and depended on mathematics as an essential linguistico-cognitive resource. Is not the contemporary techno- scientific picture of physical reality literally unthinkable outside the apparatus  of mathematical notations and terms used to articulate it? For Galileo, reading these terms was not so much articulating the language of science as deciphering the writing of God; and if in the four centuries since his proclamation science has progressively sought by a series of disclaimers to distance itself from the more blatant aspects of Galileo’s theism, it has left his understanding of mathematics as the language and alphabet of the physical universe very much  in place, adhered to, with remarkably little questioning, by innumerable scientists from the seventeenth century to the present-day.

Yet, despite the many much repeated recognitions of mathematics’ linguistic nature, there has been little sustained attempt (from either side of the continental/analytic philosophical divide) to develop the theoretical and conceptual consequences of saying what it means for mathematics to be a language or be practised as a mode of discourse. A notable, important  exception would be Wittgenstein’s extended remarks on the foundations of mathematics with their intent to characterize the doing of mathematics (at least in its elementary computational aspects) as a motley of certain kinds of language games. But, though canny, interesting and provocative, Wittgenstein’s fragmentary, deliberately unsystematic dicta do not address the theoretical question of mathematical language and discourse in any way. Nor do they indicate how one might do so. Notwithstanding this, however, the viewpoint here is sympathetic to his radical nominalism which sees any mathematical “object” as an effect of the notation system that supposedly describes it.

Galileo’s amalgamation of language, writing, books and alphabetic letters raises certain crucial issues: what do languages –divine, natural or man-made –have to do with alphabets? Is mathematics a “language” or a form of writing? What is the difference? And, related to these: is there a distinction here between “depth” and “surface,” between, for example, a project for a “grammar” of science and the apparently more subjective, impressionistic study suggested by “language” of science? And why the genitive here? Why of science? Why not the grammar, the language, the code, the symbolic system, or whatever, of mathematics itself–cut free from all questions of its instrumentality?

Large scale questions. In the present context we need to be more specific and focused. Thus, one might ask whether mathematical texts produce meanings that differ fundamentally, in kind, from those of speech and its alphabetic inscription: what is the relation, for example, of mathematical proofs to written narratives or to the arguments and dialogues of everyday discourse? And one wants to know what it means linguistically –to do mathematics: what manner of symbolic activity, of imagining and scribbling, are mathematicians engaged in when they think/write mathematical signs?

Semiotics – professing itself to be the study of any kind of sign, language, or communicational system and with an ongoing finger in every kind of signifying pie imaginable– is the obvious place to look for a means of addressing the question of mathematics-as-language. Now Semiotics, stemming separately from the work of Saussure, a structural linguist, and Peirce, a metaphysically minded pragmatist, is a two-headed affair: both a kind of abstract formalism  and a philosphico-taxonomic study of what it means to signify. From Saussure comes the definition of a sign as a coupling — said to be unmotivated and arbitrary — of signifier (“hundt,” “chien”) and signified (doghood); and the insistence on downgrading referential meaning in favour of a sign’s structural relations to other signs. Peirce held that anything whatsoever that stands to somebody for something was a sign, and that any sign always created or involved an “interpretant” — variously another sign or interpreting agency — as an essential part of its action.

These definitions have been much discussed. This is not the place to elaborate them or to attempt a comparison between the formalist/structuralist and interpretive/discursive semiotics they give rise to. Instead, I shall sketch two very different ways of approaching the question of mathematical signs that I’ve pursued over the past few years which can be seen — very crudely and schematically — as instances of these two semiotics. Significantly, though employing divergent methods, aims and starting places, each ends up by foregrounding the role of the corporeal, sign writing/reading subject.

The first, which I shall be very brief about, concerns the analysis of one particular and rather special mathematical sign — namely zero. As is well known, neither Greek nor Roman mathematics had aconcept of zero. In the form we have it it is a Hindu invention transmitted into Europe via the Arab Mediterranean. Its introduction marking a discontinuity in western thought concerning the concept, use, theorization, metaphysics and writing of number. At the end of the 16th century, some three centuries after its introduction in northern Italy, the Dutch engineer and mathematician Simon Stevin, engaged in the project of extending the place notation from finite to infinite decimals in his treatise The Dime, was moved to a kind of wonder at the creative power of zero — “the true and natural beginning” of the numbers he called it. As the point was to the geometrical line so the nought — “poinct de nombre” he wanted to call it was to the arithmetical progression. What are we to make of his perception of a natural beginning and the analogy it suggested to him? What is it about zero as a sign that might give it this singular generative quality? For Stevin if not for us?

My approach to this question in Signifying Nothing [4] was to give a sufficiently abstract, generic characterization of zero and then construct a kind of archeology, a la Foucault, of the zero sign in relation to other, similarly conceived, non-mathematical signs. Starting from a formulation of particular kind of meta-sign — specifically a sign for the absence of certain other signs –I identified a series of semiotic relations between zero in the code of arithmetic, the vanishing point in the code of perspectival painting, and socalled imaginary money, or Bank money as Adam Smith called it, in the code of economic exchange. These relations being built on parallels between the generative, originating roles played by the meta-signs in their respective codes: zero engendering an infinity of new number signs, the vanishing point delivering an endless supply of a new type of visual image, and imaginary money making possible an unlimited range of previously unavailable transactions.

On the basis of the resulting isomorphisms between the codes of arithmetic, painting and money it becomes possible to articulate certain features shared by the semiotic actors who operate within these codes; between, that is, the subject of arithmetic — the one-who-counts– and the other subjects involved–the one- who-paints and the one-who-transacts. In particular, it becomes possible to apprehend the dual status of these subjects as certain kinds of discursive agencies, that is, semiotic subjects who both use and are created by their respective codes.

One consequence of this, radically antagonistic to prevailing Platonistic interpretation of mathematics (and to cognate realist or objectivist accounts of scientific knowledge), but familiar within Post (particularly post-structuralist) thinking about language, is that the “objects” concerned here numbers, visual scenes, values — far from belonging to prior, pre-semiotic worlds waiting to be signified, are inseparable from, and in a radically uneliminable sense owe their being to, the very codes thought to be referring, notating and describing them.

The comparative insights offered by such an archeology, the sorts of homologies between mathematical counting, visual depicting and economic transacting it makes available (which extend further than my brief summary indicates) are novel and suggestive. But its capacity to illuminate the overall nature of mathematical discourse is evidently limited. After all, there are mathematical signs other than zero, and mathematical activities — giving definitions, executing operations, introducing notations, making assertions, following proofs and arguments — that are not reducible to acts of counting.

Another approach to mathematics, employing a discourse-based semiotic rather than an archelogical/structuralist one, is called for; one that would respect the fact that “The world of rigorous fantasy we call mathematics,” as Gregory Bateson put it,[5] is imagined and thought into existence. The thought-experiment (Gedankenexperiment) has been a widely used  element of scientific practice since Galileo. Only fairly recently, however, have these scenarios and narratives of imagined activities– actions conceivable in principle but difficult or impractical to realize — received any serious attention. Being used as devices of illumination, explanation and persuasion they were seen in relation to proper, theoretically grounded explanation and real experiments as merely rhetorical. The putting into question of this “merely,” as part of a growing recognition of the role of persuasion in the constitution of scientific practice, is an historiographical achievement of the last two decades. The historian Thomas Kuhn, for example, has posed a series of provocative questions about thought-experiments and their constitutive function, observing how they have “more than once,” in the mid 17th and early 20th centuries, “played a critically important role in the development of physical science.”[6]

Proposed here is a semiotic model of mathematical activity fabricated around the idea of a thought-experiment. Such a model would identify all mathematical reasoning — proof, justification, validation, demonstration, verification — with chains of imagined actions that detail the step-by-step realization of a certain kind of symbolically instituted, mentally experienced narrative. Thus, unlike physical science where thought-experiments are contrasted with real ones as one ratiocinative and persuasional device among others, mathematics, as presented here, will be exclusively founded on them.

Peirce seems to have been the first to suggest that mathematical reasoning was akin to the making of thoughtexperiments, “abstractive observations,” he called them. While Peirce’s contemporary, the physicist Ernst Mach, saw in them not only essential tools of physics but also, like Peirce, understood them as the basis for any kind of planning and forward-directed rational thought. Now the relation between real experiments and thought-experiments is, I would argue, less obvious than it looks. To see this it will be helpful, before introducing Peirce’s characterization, to go back historically to the point when the category “experimental” was itself being legitimated as the foundation of empirical science.

In their study of the origins of English experimental science as instituted by Robert Boyle in the late 17th century, the historians Stevin Shapin and Simon Schaffer set out to demonstrate that “The foundational item of experimental knowledge”, that is to say the very category of scientific “fact”, “and of what counted as properly grounded knowledge generally, was an artifact of communication and whatever social forms were deemed necessary to sustain and enhance communication.”[7] To this end they identify several technologies — “knowledge-producing tools” — employed by Boyle to establish the categorial identity and legitimacy of scientific “facts” and their separability from purely theoretical observations. Chief among these being what they call the “technology of virtual witnessing.” This amounted to a rhetorical and iconographical apparatus, a carefully controlled manner of writing and appealing to pictures and diagrams which, by causing “the production in the reader’s mind of such an image of an experimental scene as obviates the necessity for either direct witness or replication,” [8] allowed the “facts” to speak for themselves and be disseminated over the widest possible arena of belief. Behind the employment of this apparatus was Boyle’s anxiety to  distance his method of empirical verification, and with it the claim that such a method produced and dealt in the natural category of “facts,” from the kind of experiments involving only pure “ratiocination,” unconnected to the rigors of actual experimentation, that he attributed to others, notably Pascal. Now Pascal was of course a mathematician and making experiments out of a process of pure ratiocination — thought experiments — is, I contend, precisely how mathematicians manufacture mathematics.

The proposal, then, is that Boyle’s rhetorically accomplished replacement of the actual by the virtual, the shift from doing/watching to the imagining of doing/watching, from real, witnessed and executed, experiments to their virtual, reproduced-in-the-mind versions, charted by Shapin and Schaffer, points, contrary to Boyle’s efforts to separate them, to a certain duplication of persuasive technique between the establishment of empirical factuality and mathematical certainty. Each comes about by the use of an elaborately designed apparatus able to mask its rhetorical features under the guise of a neutral  method for discovering pre-existing “facts” of nature or, in the case of classically conceived arithmetic, objective “truths” about the so-called natural numbers. Identifying the thought-experimental mechanism for mathematics might, then, provide a way of explicating the persuasional and rhetorical basis for mathematical reasoning and logic. Of course, there are differences between real experiments — however their meanings and outcomes are rhetorically processed, negotiated and disseminated – and imagined ones. In particular, what corresponds in mathematics to an empirical world studied by science – mathematics’ external “reality” as it were — is already a symbolic domain, a  vast field of ideal, semiotic objects. As a result the distinction between direct manipulation of such symbolic objects and their virtual or thought-  experimental manipulation is less obvious and identifiable. To avoid all sorts of confusion we need to pay attention to the language of mathematical texts. More specifically, since the issue concerns the imagining of actions rather than their direct execution, we need to focus on the way texts invoke agencies and instruct them to manipulate mathematical signs. We can can start from a division of mathematical discourse into two distinct modes; a division valorized by the mathematical community  as a proper, formal mode and an improper, supplemental and extra- mathematical one.

The formal or rigorous mode — mathematics “proper” consisting of all those written texts — on paper, blackboards, screens — presented according to very precise, unambiguous rules, conventions and protocols regulating what is a permitted and acceptable sign-use with respect to making and proving assertions, giving definitions, manipulating symbols, specifying notations, etc..  I call the sum total of all such rules, conventions and associated linguistic devices accepted and sanctioned as formally correct by the mathematical community the CODE. The informal or unrigorous mode consisting of the mass of signifying activities which — in all but the most austere and artificial instances accompanies the first mode: drawing diagrams, providing examples, underlying motivations, narratives, intuitions, applications and intended interpretations, and generally using natural language (figuratively, ostensively, descriptively) together with iconographical devices to convey all manner of mathematical sense. I call this heterogeneous collection of semiotic means the METACODE.

 As indicated, the standard account — enshrined for example in the writing of purely internalist histories of the subject –would be that the metaCode is epiphenomenal, a matter of mere affect, psychology, heuristics and handwaving, subordinate to the “real” business of doing mathematics taking place in the Code, and in principle completely eliminable. I shall indicate how such a view of the metaCode, as a practically necessary but theoretically dispensable supplement, a tool or prop to be discarded once the passage into pure reason and formal truth has been accomplished, is entirely misconceived.

Mathematical signs, formal and informal, are of course communal and intersubjective. They are transmitted by, rely on and are addressed to subjects. In the case of the Code I call the common, idealized sign-using agency – the universal reader/writer addressee of the Code — the mathematical SUBJECT. An examination of the signs addressed to such a Subject reveals two principal features of formal mathematical discourse. The first is the total absence of any expression connecting the Subject to the inhabited world: no mention of his/her immersion in culture, history, or society nor any reference to psychological, physical, temporal or spatial characteristics. Crucially, the user of the Code is never asked to interpret a message which makes reference to the Subject’s embodied presence. This elimination of the corporeal from what is allowed as correct mathematical language has a many-sided history related to effects of idealization and demands of reductive abstraction to deliberately prosecuted programmes of rigor and the need to deliver to the physical sciences a suitably “objective” formalism. But, the complexities of the history of mathematical rigor aside, the result is that a group of fundamental devices, present it seems in all natural languages–what Peirce called “indexical” signs, Jakobson “shifters”, Quine “token-dependent” signs, and others “deictic” elements–are absent from the mathematical Code. Such terms as “I,” “you,” “here,” “now,” “this,” “was,” “will be” which depend for their interpretation on the located or locatable presence of their utterers, are simply unavailable to the mathematical Subject.

According to the linguist Emile Benveniste the indexicals, chiefly the first person “I”, are the vehicle through which subjectivity itself — via discourse — gets produced:

Language is … the possibility of subjectivity because it always contains the linguistic forms appropriate to the expression of subjectivity, and discourse provokes the emergence of subjectivity because it consists of discrete instances. In some way, language puts forth “empty” forms which each speaker, in the exercise of discourse, appropriates to himself and which relates to his “person” at the same time defining himself as I and a partner as you.[9]

Evidently, the mathematical Subject — though an embodied sign writing and reading user of the Code — is an I-less subject, unable to articulate her or his embodied subjectivity within the Code. It is rather in the metaCode, permeated by ostensive gesture and the indexicals of natural language, that this articulation can take place. In contrast to the Subject let us call the sign user of the metaCode, the one who has access to the “I” of the metaCode and the subjectivity it enjoys and makes available, the PERSON.

The second principal feature of the language of mathematical texts is the giving of commands. Mathematical discourse is permeated with injunctions: define A, compute B, consider C, prove D, integrate E, construct F, iterate G, and so on. In fact, so dense is the network of injunctions that mathematical texts that mathematics can appear to be entirely an operative discourse, its communications little more than ensembles of orders to be carried out. Who issues these exhortations and to whom are they variously addressed? And,  given that mathematicians do no more than hang around thinking and scribbling, what sort of actions are demanded of them?

Structural linguists distinguish two sorts of imperative: the speaker-inclusive (“Let’s go!”) and the speaker-exclusive (“Go!”). In mathematics this division corresponds to a radical split between two types of mathematical command: inclusive ones such as “let us consider a Hausdorff space,” “define a mapping m,” “show that alpha is the case,” ask that speaker and hearer institute and inhabit a common — imagined — world; they are issued by and addressed to the mathematical Subject. Exclusive ones such as “invert the matrix M,” “iterate  S,” “integrate the function g,” require that certain actions meaningful in some already imagined world be executed. But who carries out such actions? Clearly, it is not the Subject who can be asked to iterate indefinitely or well-order the continuum or invert an arbitrary matrix. Rather it is an idealized and truncated version, a model or simulacrum of the Subject, which I shall simply call the AGENT, and Peirce, in his description of a thought-experiment, calls a “skeleton diagram of the self,” that is dispatched to perform these activities:

It is a familiar experience to every human being to wish for something beyond his present means, and to follow that wish by a question, “Should I wish for that thing just the same, if I had ample means to gratify it?” To answer that question, he searches his heart, and in so doing makes what I term an abstractive observation. He makes in his imagination a sort of skeleton diagram, or outline sketch of himself, considers what modification the hypothetical state of things would require to be made in that picture, and then examines it, that is, observes what he has imagined, to see whether the same ardent desire is there to be discerned. By such a process, which is at bottom very much like mathematical reasoning, we can reach conclusions as to what would be true of signs in all cases …[10]

I have introduced three agencies — Person, Subject, Agent –which I claim operate in any mathematical thought-experiment. The roles of Subject – imaginer – and Agent – imagined – are evident enough, but what of the Person? Pierce’s description makes no mention of a third figure or agency. Ought he to have done so? There is an analogy here between mathematics and waking dreams. The Agent is the imago, the figure dreamed about; the Subject is the imaginer, the dreamer dreaming the dream; and the Person the dreamer awake in language  consciously observing, articulating and interpreting the dream. The necessity  for this third level occurs because the dream-code is restricted, making it impossible to articulate the meaning of the dream – as a dream – within the frame available to the dreamer. In just the same way the language in which the mathematical Subject operates — the Code — is restricted by its lack of indexicality. And what is unavailable — the Subject — precisely as a result of this lack — is, as we shall, is how  the  technology of thought- experiments achieves persuasion.

How do thought-experiments persuade? They furnish the Subject with a scenario enacted by the Subject’s proxy, the Agent, of what he/she would experience. This can only impinge and have persuasional relevance for the Subject by virtue of the similarity between Subject and proxy: only on the basis of the affirmation “it is like me” is the Subject persuaded that what happens to the Agent mimics what would happen to him or herself. And it is just this affirmation – which rests on a recognition of a sufficient likeness between imaginer and imago – that the Subject cannot articulate, since to do so would require access to an indexical self-description denied it. Only in the metaCode, the domain of the Person, can such a description can be given. Of course, if as Peirce seems to, one takes the imaginer to be carrying out his reflections within natural language, then language and metalanguage coincide, and the need for a Person distinct from a Subject doesn’t arise. That being so it is Peirce — the author advocating the thought-experiment as the means of selfpersuasion —  who is the missing person/Person.

The logician’s picture of a mathematical proof is a step-bystep assent to a sequence of logical moves ending in the assertion being validated. But, as all mathematicians know, it is perfectly possible to agree with (fail to fault) every step of a proof without experiencing any conviction; and without such experience, a sequence of steps fails to be a proof. Presented with a new proof, mathematicians will seek the idea behind it, the principle or story that organizes the logical moves into a coherent whole: as soon as one understands that the persuasional structure of the proof can emerge. Proofs embody arguments — discursive semiotic patterns — that work over and above – before – the  individual steps and which are not reducible to these steps; indeed, it is by  virtue of the underlying story or idea or argument that the sequence of steps is the sort of intentional thing called a proof and not merely an inert string of formally correct inferences. The point of this in relation to thought-experiments is that such underlying stories are not available to the Subject confined to the Code; they can only be told by the Person from within the metaCode.

Implicit in this characterization of proof as thought-experiment is the idea that mathematical assertions, statements of content, are to be seen as predictions; specifically, predictions about the Subject’s future encounters with signs. They foretell what will happen if he/she does something. Thus, the assertion “2+3=3+2″ predicts that if the Subject joins 11 to 111 the result will be the same as if he/she had joined 111 to 11. The assertion ” 2 is irrational” predicts that no matter what particular integers the Subject substitutes for p and q in the expression p2 – 2q2 the result will not be zero. Not all assertions yield so directly to a construal in terms of predictions as these examples, and one needs more than this, some kind of overall argument based on the recursive construction of mathematical assertions to secure the point in any general way.

Let me summarize the tripartite structure of the technology of mathematical persuasion sketched here. There are three semiotic figures. The Agent, an automaton with no capacity to imagine, who performs imaginary acts on ideal marks, on signifiers; the Subject who manipulates not signifiers but signs interpreted in terms of the Agent’s activities; the Person who uses metasigns to observe and interpret the Subject’s on-going engagement with signs. In terms of these agencies any piece of mathematical reasoning is organized into three simultaneous narratives. In the metaCode the underlying story organizing the proof-steps is related by the Person (the dream is told); in the Code the formal deductive correctness of these steps is worked through by the Subject (the dream is dreamed); and in what we might call the subCode the mathematical operations witnessing these steps are executed (the dream is enacted) by the Agent.

It is possible, as I’ve shown elsewhere,[11] to use this tripartite scheme to give  a unified critique of the three standard accounts — Hilbert’s formalism, Brouwer’s intuitionistic constructivism, Fregean Platonism — of mathematics. Briefly, the move one makes is to consider the triad of signifier, signified, Subject and show how each of the standard accounts systematically occludes one of the three elements. Thus, intuitionism, relying on a idealized mentalism, denies any but an epiphenomenal role to signifiers in the construction of mathematical objects; formalism, fixated on external marks, has no truck with meanings or signifieds of any kind; Platonism (the current orthodoxy), dedicated to discovering eternal, transhistorical truths, repudiates outright any conception of the (in fact, any humanly occupiable) Subject position in mathematics. Plainly, the valorization of a proper, formally sanctioned Code over an improper and merely supplemental metaCode deeply misperceives how mathematics traffics with signs. A misperception intrinsic to and formative of Platonism, since in order to deny the presence of persuasion within mathematical reasoning it has to understand the language of mathematics as a transparent, inert medium which manages (somehow) to express adequations between human description and heavenly truth. On the contrary, only by understanding language as constitutive of that which it “describes” — only through such a post-realist reversal of mathematical “things” and signs in which, for example, numbers are as much the result of numeral systems as numerals are the names of numbers which antedate them — can one make sense of an historically produced apparatus of persuasion and an historically conditioned account of the — human — engenderment of the numbers. But this  is in the future: the history of the Subject, Agent, Person no less than the  history of mathematics as a sign practice of which these semiotic agencies would be a part has yet to be written.

Such a reversal works to dissolve a split in “number,” a hierarchical division characteristic of western thought and operative since the beginning of the classical period, which valorized arithmetica (numbers contemplated philosophically; ideal, perfect objects) over logistica (numbers as empirical objects calculated with by slaves). The hegemony of arithmetica now looks to be over. Computer technology, the contemporary manifestation of Logistica, by insisting on the materiality of counting/calculating as a business of real time  and memory cannot but re-instate the material, (electronically) embodied slave upon the mathematical scene. Computer technology’s effect on and interaction with mathematics is already so massive that it has become difficult to think the fate of mathematics separate from that of logistica: certainly it seems strange and increasingly artificial (outside Platonism/realism) to talk about our relation to number without invoking some conception of agency, some suitably idealized version, that is, of an embodied mathematical sign-using subject. In the wake of this destablizing of arithmetica, that is, the putting into question of  a prior, theoretically given “number,” one can rephrase Galileo’s challenge to the Schoolmen. Instead of asking: Why should the physical world conform to a prior, Aristotelean metaphysical theorization of it? One asks: Why should counting and hence numbers be imagined within a prior metaphysical scheme that knows nothing of their materiality and is silent about how they come to be?

Why indeed? To illustrate what is it stake here let me, finally and very briefly, indicate a recent application.[12] I have made of the semiotic model outlined here, namely: a putting into question and making problematic the idea of the mathematical infinite – that simple and all pervasive idea we are asked to  invoke when we interpret the ideogram “…” in the expression 1, 2, 3, … to mean go on counting without end, forever.

The persuasive force of a thought-experiment works, as I have indicated, through the resemblance between ourself as imaginer and the imago we conjure as our proxy; between the mathematical Subject and his/her Agent. This  relation between imaginer and imago lies between two limits: that of total identity (which if reached would mean that we perform real not imagined actions) and total difference (which would destroy any sense to the idea of a resemblance being in play). Suppose now that the thought-experimental activity is that of counting, and we ask the question: Is the Agent imagined to be corporeal in any way whatsoever? Since the Subject, as a sign reading/writing agency, is clearly and irremediably embodied the question is manifestly one of resemblance between imaginer and imago. If we answer yes, and maintain that our Agent has some — however idealized, vestigial or attenuated — immersion  in the material world, then its action will not escape the regimes of space, time, energy use, decay and so on, that govern all physical process, and it will not, as a result, be able to go on endlessly. If we answer no, as we must if we are to cognize infinity and counting without end, then our Agent will be a  disembodied ghost, and we are confronted by the question of persuasion. Why should a Subject whose embodiment is inseparable from its engagement with signs create an incorporeal phantom as its proxy? In what way can the   imagined manipulation of signifiers by such a transcendental imago persuasively image the Subject’s manipulation of signs? The negative answer would be that of current infinitistic mathematics were it to admit to an analysis in terms of Subjects and Agents; but short of so admitting neither the answer  nor the difficulty it leads to is likely to make a direct impact on such mathematics. The positive answer has radical, large scale and interesting consequences: it would eliminate all talk of infinity from mathematics and put in place of classical arithmetic with its endless progression of transhistorical, transcendental integers — the so-called natural numbers — a non-Euclidean arithmetic of realizable, this-universe constructible numbers; numbers that fade into indeterminacy and non-existence as counting them into being is ever further prolonged.

NOTES

[1]Stilman Drake, Discoveries and Opinions of Galileo (New York: Doubleday, 1957), p. 238.

[2]Jacques Derrida, Of Grammatoloqy, tr., Gayatry Spivak, (Baltimore: The Johns Hopkins University Press, 1976), p. 6.

[3]For example, and by no means comprehensively: David Bloor, Knowledge and Social Imagery (London: Routledge, 1976). David Bloor and Barry Barnes, “Relativism, Rationalism and the Sociology of Knowledge,” in Martin Hollis and Steven Lukes, eds., Rationality and Relativism (Cambridge, Mass.: M.I.T. Press, 1982). [Get Pages] Sal Restivo The Social Relations of Physics, Mysticism, and Materialism (Boston: D. Reidel, 1983). Sal Restivo, “The  Social Roots of Pure Mathematics,” in Susan Cozzens and Thomas Gieryn,  eds., Theories of Science in Society (Bloomington: Indiana University Press, 1990) [Get Pages] . Theodore Porter, “Quantification and the Accounting Ideal in Science,” Social Studies of Science (1992), 22, pp. 633-652. Paul ErnestThe Philosophy of Mathematics Education (New YorkPalmer Press, , 1991).

[4]Brian Rotman Signifying Nothing: the Semiotics of Zero (London: Macmillan Press, London, 1987).

[5]Gregory Bateson Steps Toward an Ecology of the Mind (New York: Ballantine Books, 1972), p. 422.

[6]Thomas S. Kuhn, The Essential Tension (Chicago:University of Chicago Press, 1977), p. 240.

[7]Steven Shapin and Simon Schaffer, Leviathan and the Air-Pump (Princeton: Princeton University Press, 1985), p. 25.

[8]Ibid. p. 60.

[9]Emile Benveniste, Problems in General Linguistics, tr., Meek, (Coral Gables: University of Miami Press, 1971), p. 227.

[10]J. Buchler, ed., The Philosophy of Peirce: Selected Writings, (Routledge: London, 1940), p. 98.

[11] Brian Rotman, “Toward a Semiotics of Mathematics,” Semiotica, 72, pp. 1- 35, 1988.

[12] Brian Rotman Ad Infinitum … the Ghost ln Turing’s Machine: Taking God Out Of Mathematics And Putting the Body Back In (Stanford: Stanford University Press, 1993).

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