1994 Exuberant Materiality

  Configurations 2, 257-74

 

I take part of my title from a fine book, The Origins of the Modem Mind by Merlin Donald. In the concluding section, called “Exuber­ant Materialism,” Donald places his account of the evolution of culture and cognition within a burgeoning and newly self-confident materialism, part of what he calls a “neuroscientific apoca­lypse.” The aim is to construct a biological morphology of intelligence able to lift itself free from the “tortured logic and compromises” that have characterized discussion of the mind/brain relation in this century, and to reach toward a scientific under­ standing of human consciousness-an understanding, Donald em­phasizes, going beyond the familiar Darwinian naturalism to depict humans as “symbol-using, networked creatures, unlike any that went before us.” “We act,” as he puts it, “in cognitive collectivities, in symbiosis with external memory systems. As we develop new ex­ ternal symbolic configurations and modalities, we reconfigure our own mental architecture in nontrivial ways.” 1 The claim is striking. We are, it seems, constantly rewiring our brains as we go along, physically manufacturing our mentality via the symbols, cognitive tools, and techno-semiotic apparatuses that are the cultural-men­ tal, ideational-achievements of these very brains.

For some years I have been concerned with one particular external    memory    system  or  symbolic configuration, namely mathematics:

  1. Merlin Donald, Origins of tile Modem Mind: Tilree Stages in tile Evolution of Culture and  Cognition (Cambridge,  Mass.:  Harvard  University  Press,  1991), p. 382.

mathematics understood as a techno-semiotic  apparatus, that is, as a discourse with ideograms and diagrams, as a technology (tool for reasoning, thinking, predicting, imagining), and as an apparatus (system of writing and manipulating material symbols). In address­ing  these  issues  one  quickly  encounters  the  mind/brain  relation and  its  forebears  and  cognates – mind/body, psyche/soma, mental/material, idea/thing, heaven/earth,  spirit/flesh, signified/signifier, and so forth. The relation is of course one of the founding binaries of Western culture; no surprise, then, that it impinges on and problematizes what we mean by “doing mathematics” and by “counting,” and that this occurs as soon as we write down the fa­ miliar progression of ideograms 1, 2, 3, … symbolizing the so­ called natural numbers and start to ask semiotic questions about it.

The deepest  difficulty concerns not so much the character and status of the individual numbers themselves, as the interpretation of the continuation symbol “…” operative on all of  them: they form a progression. How are we to give a linguistic account of un­ limited progress of numbers? What would be the appropriate semi­ otic response to infinity? How can we witness it in terms of written signs? And, perhaps more opaque at this stage, what has the mind/body relation to do with the problem? I shall return to these questions later.

A second difficulty can be located in the inscription of the mind/body binary within classical Greek formulations of number. One has the mind giving rise to what was called arithmetica (numbers in heaven, ideal objects of philosophical contemplation); and one has the body giving rise to logistica (numbers on earth, lowly things to calculate with). Derivative and supportive of this is the whole Platonic philosophy of mathematics, which contends that mathematical objects – points, lines, numbers, spaces, functions, surfaces – exist in  some ideal realm, independently of any human presence, activity, or knowledge; prevalent among mathematicians, scientists, and the general populace, it is a belief strongly and confi­ dently upheld. Witness, for example, the influential writer on mathematics Martin Gardner: “How mathematicians who pretend that mathematical structure is not ‘out there’ independent of human minds, can view successive enlargements of the M[andelbroit]-set and preserve their cultural solipsism is hard to comprehend. 2

  1. Martin Gardner, “Beauty in Numbers” New York Review of Books, March 16, 1989, pp. 26-28; p. 26. Also Gardner’s statement, “Penrose finds it incomprehensible (as do I) that anyone could suppose that this exotic structure is not as much ‘out there’ as Mount Everest is, subject to exploration in the way a jungle is explored” (p.

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For Platonists, then, mathematics discovers truths about this pre-given external realm in just the same way that (realist con­ ceived) science studies the world of so-called external reality. And for both, language is no more than a transparent, inert medium whose features, interesting though they might be, have only a sur­ face, marginal, and in principle always eliminable bearing on what is “out there.” In short, for Platonic realism, language is ultimately nothing other than naming.

According to this, numbers are objects, signifieds, which preexist the names – numerals, signifiers – that we  assign  to  them.  What would it mean to challenge the hierarchy and the separation  of terms of this hallowed number/numeral coupling? Is it not the case that even anti-Platonists (such as mathematical constructivists) are not prepared to give up on an autonomously generated  succession of integers outside and before language? How could the picture of a preexisting world of numbers and a posterior assignment of names be overturned? Could numbers somehow owe their  existence  to that which merely notates them? Is this not absurd: do not the sig­ nifiers “5,” “V,” “five,” “funf,” “cinque” name  the same  number­ one that, however it is referred to, exists prior to  and  indepen­ dently of the attachment of such signifiers to it? Does not any alternative  ask  us to  believe  in  not  one,  pre-given  progression  of natural numbers, but many artificial progressions,  each inseparable ·from the particular system of signifiers used to bring it into existence? For almost all mathematicians this and, by  implication, any attempt to dissolve the separateness and inviolable priority of the signified over the signifier-would go beyond sense.

But why should the situation of mathematics be so special and singular, so intransigently different from  language in relation to this possibility? Is there nothing to be gained from two decades of poststructuralist, deconstructive theory devoted to precisely such a dissolution? Thus, consider a familiar linguistic rather than mathe­ matical example: the opposition of literal and figurative. Here again one has a privileged  term, a presumed priority; the  literal serving as the ground  from which  figurative language,  such as tmetaphor, is traditionally derived. But, as Jonathan Culler observes, such an “asymmetry turns out to be unstable, and as one explores the logic of the situation further, one discovers that the term treat-

vi), in Roger Penrose, The Emperor’s New Mind (New York: Oxford University Press, 1989), which is an extended apology and defence of Platonism by a distinguished mathematician. l prefer to quote Gardner (rather than Penrose himself or some other professional mathematician) precisely because he writes for such a general audience.

ed as secondary and derivative can be seen as basic.” 3 The result being that the very ability to proclaim an absolute separation be­ comes untenable, and one is left with a reconstituted reading of the literal/figurative opposition as a codependent or codetermined cou­ pling that denies the possibility of an atemporal structure of oppo­ sition. The move is quite general, and Derrida, who calls this decen­tering and relocation of a supposed supplementary term the “logic of the supplement,” has famously enacted this logic on the speech/writing opposition and, more to the present point, on the signifier/ signified binary that constitutes Saussure’s understanding of the sign.

Trying to use Derridean deconstruction, however, with its spe­cialized lexicon of “logocentrism,” “supplement,” “trace,” “dif­ ferance,  “presence,” and so on, directly in order to critique the mathematical binary of number/numeral is not  a  good  idea. One has to go indirectly and from behind, as it were, deconstructing (though that is not the word I’d use) mathematics within its own already highly specialized terms.

Let me leave that hanging and put mathematics  in abeyance for the present, and go instead on a curve through certain projects in robotics, evolutionary and developmental biology, artifical intelli­ gence, cognitive science and social anthropology. Thirty years ago Warren McCulloch wrote his celebrated Embodiments of Mind. Since then, especially in the last decade, a slew of books echoing, advanc­ing,  and  reconfiguring  his  assault  on the  mind/body  binary  have appeared, their titles and subtitles – The Embodied Mind; On the Mat­ ter of the Mind; The Body in the Mind; The Biology of Mind; The Mindful Brain – looking to exhaust the list of possible anti-Cartesian slogans.4  The projects  that I am going to talk about are part  of these

3. Jonathan Culler, Tile Pursuit of Signs: Semiotics, Literature, Deco11struction (Ithaca, NY: Cornell  University  Press,  1981), p.  206.

  1. W. S. McCulloch,  Embodiments  of Mine/  (Cambridge,  Mass.: MIT  Press,  1965); Gerald

M. Edelman and V. B. Mountcastle, Tile Mindful Brain (Cambridge, Mass.: MIT Press, 1978); Francisco J. Varela, Evan Thompson, and Eleanor Rosch, Tile Embodied  Mind: Cognitive Science and Human Experience (Cambridge, Mass.: MIT Press, 1991); J .-P. Changeux,  Neuronal  Man: Tile Biology  of Mind  (Oxford:  Oxford  University  Press, 1986); Mark Johnson, Tile Body in tile Mind: Tile Bodily Basis of Meaning, Imagination, and Reason (Chicago: University of Chicago Press, 1987); Gerald M. Edelman, Bright Air, Brilliant Fire: On tile Matter of tile Mind (New York : Basic Books, 1992). Of course, the  agendas,  themes,  methodologies  and  intellectual   concerns  vary  enormously  be­ tween  these  works.  Thus,  to  take  a  single  example,  Varela,  Thompson,  and  Rosch’s Embodied  Mind invokes Merleau-Ponti’s phenomenology  of perception  as well as em­ bracing   a  particular   Buddhist-inspired   idea  of  the  (non)self,  whereas  Edelman   in Bright Air, Brilliant  Fire (and  elsewhere)  makes a  point  of  distancing  himself  from

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efforts and are linked to each other by their allegiance to a set of key terms, principally “situated,” “enacted,” “embodied,” and “codetermined,” and their characterization in terms of a certain move-at once a metaphor, an algorithm, a principle, described as bottom-up (body) as against top-down (mind).

Before I start, a note of caution on the “body.” Recently, Kather­ine Hayles, in an essay on materiality, suggested that “one belief from the present likely to stupefy future generations is the post­ modern orthodoxy that the body is primarily, if not entirely, a lin­ guistic and discursive construction.” I suspect  future  historians will be spoilt  for choice when  it comes to being stupefied by our present. Nevertheless, she is surely right to emphasize the servitude to theory and high-mindedness (for which Foucault holds not a lit­tle responsibility) that excludes the biologically material, the expe­ riential, the corporeal, and the enactive from what we mean by em­ bodiment. No such exclusions are intended here.

Let us start from a certain parallel between forms of immersion and separation: minds inside and distinct from bodies (organisms), and bodies immersed in and distinct from the world (environ­ ment). Pursuing  this, one approach to the binary of mind/body would be through the biological opposition of organism/environ­ ment. The prevailing evolutionary picture is the familiar neo-Dar­ winian synthesis, which posits a genetically determined organism immersed in and selected for by a preexisting and effectively inde­ pendent environment. Recently, several proposals for rethinking this relationship have been suggested. Perhaps the least con­ tentious is that which foregrounds the phenomenon of coevolution. Here, species are quite literally parts of each other’s environments, hence the ecological reciprocity of prey and predator couplings, or the coevolution of insects and flower-bearing plants, and so on. And this applies not only to species: DNA and RNA themselves are now thought to have come into being and evolved reciprocally in relation to each other.

Others urge a more theoretically radical, full-blown codetermi­ nation between life forms and the environment in, through, by, on, which (the prepositions here are part of the point) evolution takes place. As Richard Lewontin puts it, every philosophical position he can think of and would, one imagines, be very un­ comfortable in the same bed with Buddha and Merleau-Ponti. And so on. Nonethe­ less, for the purposes of the broad-brushed picture here, and in relation to the cogni­ tive focus they all have, the juxtaposition of these works is more than justified.

5. N. Katherine  Hayles, “The Materiality of Informatics,” 1 (1993): 147.

The organism and the environment are not actually separated. The environ­ ment is not a structure imposed on living beings from the outside but is in fact a creation of those beings. The environment is not an autonomous process but a reflection of the biology of the species. just as there is no or­ ganism without an environment, so there is no environment without an or­ ganism.6

By this is meant that in a sense our bodies enfold past environ­ ments and vice versa. And this not only within species at the level of populations, but in the development of individual organisms; so that, far from being separate and independent, the organism, in its passage from genes to adult, and the environment that mediates this passage, are codetermined and mutually enfolded. In Susan Oyama’s words, “genetic and environmental influences [are made to be] interdependent as genes and gene products are environments to each other, as extraorganismal environment is made internal by  psychological  or biochemical  assimilation, as internal state is externalized through products  and  behavior  that  select and organize the surrounding  world”7

This refusal of a preformed and independent world is given a cog­ nitive dimension by Francisco Varela, Evan Thompson, and Eleanor Rosch in The Embodied Mind. For them the crux is that “cognition is not representation but embodied action and the world we cognize is not pre-given but enacted through our history of structural coupling.”8 By “representation” they mean mirror-realism, and rejecting it forms part of repudiating the core claims of cognitivism; the prevailing mind-as-a-computer paradigm in artificial intelligence. Against the model of thinking as rule-based information-processing and manipulation of symbols mirroring an autonomous, preexist­ ing reality, thinking becomes more like perception-in-action, where what we do guides thought rather than the reverse. And instead of a functionalism that has the thinking device (hardware) as irrelevant to what it thinks (software)-leading to such eye-widening wonders as downloaded minds and backup copies of the self-one has the organic and messy wetware of neuronal pathways. In short, action and intelligence are not to be seen as the result of rational thought-such as representation, beliefs, symbols  and  planning­ but the cause and source of rationality itself . This means that cognition,  in  its  moment-to-moment  operations  as well  as its overall

6. Quoted in Varela, Thompson, and Rosch, Embodied Mind (above, n. 4), p. 198. 7. Quoted in ibid., p. 199.

8. Ibid., p. 200.

structure, is not controlled by a plan from the mind above, but bubbles up as it were hesitantly and blindly from the body below.

For Gerald Edelman in Neural Darwinism and Jean-Paul Changeux in Neuronal Man such an upward vector, working via se­ lection, operates throughout the evolutionary morphology of the brain.9  Edelman  cites the immune system as the motivating exam­ple. In order to make antibodies that match foreign proteins, the

immune system can operate in two opposed ways: either it can, in advance and perhaps randomly, produce millions of different anti­ bodies and then increase production of the one that matches the alien protein, or it can wait and see, checking out the structure of a foreign protein as it arrives and using the information it extracts to manufacture the appropriate antibody. It turns out that the second strategy, rational and surely more intelligent, is not the one used; the immune system, we learn, uses the principle of selection ap­ plied to the mindless proliferation of possibilities. The thesis Edel­ man promotes is that a parallel phenomenon (only vastly more complex) involving not antibodies but groups of neurons takes place within the organism at the level of perception, memory, cog­ nition, and eventually consciousness. In other words, selection from below, rather than rational guidance and information­ processing from above, is the mechanism by which the mind/brain system-as well as the immune system itself-came  into  existence and evolved.

The reliance on selection here is a crucial move. Selection, as Edelman points out, marks the methodological gulf  between physics, whose particles are identical, and biology, where individual variation within populations, far from being an unwelcome fuzziness to be idealized away, is the sine qua non of all evolution­ ary change. Thus, not only is computer-inspired functionalism as a matter of fact explanatorily inadequate, but the computer, operat­ ing like physics on identities as it does, is in principle misguided as a model of mind. “Selection,” Edelman says, “contrasts starkly with platonic essentialism, which requires a typology created from the top down; instead, population thinking states that evolution pro­ duces classes of living forms from the bottom up.” 10

It is precisely this bottom-up versus top-down figure, conceived as an overriding methodological principle, that MIT robotics engi­ neer Rodney Brooks nails explicity to his mast. But before I intro-

  1. Gerald M. Edelman, Neural Darwinism: Tire Theory of Neuronal Group Selection (New York: Basic Books,  1987); Changeux, Neuronal Man (above, n. 4).

10. Edelman, Bright Air, Brilliant Fire (above, n. 4), p. 73 (emphasis added).

duce his work, let me enlarge on the figure itself. The range of its surface occurrences, from the Pentagon’s recent expenditure  re­ view, to attacks on the autonomy of school boards, to corporate downsizing, to current work in artificial intelligence and neurobiol­ ogy, suggests a deep-lying and apparently much-needed  image. What is the preference or prejudice at work here? What would top­ down over bottom-up (to invert the order)-as intellectual method, organizing metaphor, or cognitive style mean? One might gloss it as the ranking of the global, panoptic, abstractly analytic over the concrete, limited, and locally synthetic; of posterior description, morphology, and structure over history, evolution, and genesis; of plans over objectives and goals; of general laws over incidents and cases; of context-free reason over situated knowledge; of  realist truth over constructivist emergence. But the figure is also reflexive and applies at once to descriptions of itself: the summary I have just given (which is perhaps less than helpful as an explication) is very much a top-down take on the bottom-up/top-down difference. So let us try it the other way through  a bodily situated example.

Suppose you drive in both France (or the United States) and Britain (or Japan) and wish, for whatever reason, to avoid getting killed, particularly through a head-on collision. Then you need to know a simple top-down  rule: drive on the right in France and on the left in Britain.  But what if you cannot remember which way round the rule goes, or (common in academia) you get your left and right mixed up, or, jet-lagged in the middle of a dark nowhere, you get confused  as to whether you are in Britain or not; how are you  going to drive and stay alive? One possible  response: get off the road; if you are unable to remember a one-line rule, do not know your right hand, and forget which country you are in, you should not be driving. Well, don’t despair: here is a bottom-up procedure you can follow. When entering a road you look through your own-the driver’s-window to see oncoming traffic, and you join it; on the road, you make sure you are always sitting in the middle (equivalently: the roadside isn’t rushing past next to your window). That is all you will ever need. Observe: the suggested strategy is about as mindless as you can get-you do not have to re­ member an abstract rule, only a procedure tied to your physical sit­ uation; a knowledge of right and left is not needed; and the strate­ gy works for whatever country you are in even if you don’t know which. Of course, there is a joker: you need to be in a French car in France and a British one in Britain; the position of the steering wheel is obviously crucial.

Now that I have helped  you drive abroad, let me return to our theme. How is it, Rodney Brooks asks, that after thirty years of prodigious effort, orthodox or classical AI, the  current  paradigm, has produced robots that can do no more than make their way clumsily across a carefully prepared, never varying toy world, even though they incorporate any amount of fancy hardware and have access to vast computing power running at  state-of-the-art mega­ hertz speeds-whereas a bee, for example, employing truly ancient technology, with a tiny amount of on-board computational power running at a few kilohertz, can navigate a previously unseen, noisy, and possibly hostile environment as well as perform a host of tasks, from  foraging  and  communicative   dancing  to  returning   to  the hive?

Has evolution, then, a better purchase on robotics than AI ortho­doxy? Over the past half-dozen years Brooks has been demonstrat­ ing how and why the answer might be a very sure yes. In a paper punningly entitled “Intelligence Without Reason” he dismisses tra­ ditional AI for “trying,” as he says, “to tackle the problem of build­ ing artificially intelligent systems from the top down” by relying on the introspective notions of thought and reason and their comput­ erized interpretations via information processing, conscious plan­ ning, and problem solving; instead, he advocates an approach based partly on biology, on an older cybernetic tradition and on engineering from first principles-one that starts “from the bottom up concentrating on physical systems … situated in the world, au­tonomously carrying out tasks of various sorts.”11 This has funda­ mental design implications: instead of a centrally controlled hierar­ chy of connected systems-a robot with a “mind”-Brooks  uses what he calls “subsumption” architecture: robots are built in layers consisting of simple machines that are minimally interconnected, each layer independent of any above it and each, by autonomously controlling its own pattern of sensors and actuators, engaged di­ rectly with the world.

The approach has paid off: using a half-dozen or so layers Brooks has constructed a variety of robots, from six-legged proto-insectoids that scuttle successfully across littered desktops to machines that roam his laboratory looking for and collecting empty soda cans. (Aside: One should be wary of locutions like “looking for”: each in­ternal state of Brooks’s robots lasts no longer than a fraction of a second and any intentionality is therefore, as he is quick to point out, in the mind of the beholder.) His immediate goal is to reach

  1. Rodney A. Brooks, “Intelligence Without Reason,” MIT Artificial Intelligence Labo­ ratory, Memo 1293 (1991): 1.

the level of full insect intelligence, which he estimates would re­ quire some fourteen layers. Given the coevolution with trees, flow­ ers, and insects of our evolutionary ancestors, building a bee (or bee-aid, one should say) would surely be an achievement, with im­ plications well outside pure robotics.

Brooks enunciates “embodiment” and “situatedness” as the or­ ganizing principles of his bottom-up, behavior/action-based ap­ proach to building intelligence. In his usage, the terms are separate. Embodiment means that “the world grounds regress,” 12 which en­ tails that what his robots do is inseparable from the details and constraints of their construction, thus ensuring that their own physicality, rather than prior desiderata, will govern whatever sym­ bolic meaning is to be attributed to their activity. One can see em­ bodiment as a thinned-out version of individual human agency: a contained, palpable projection of body behavior onto the space of an “it” with a small number of degrees of freedom; for example, a paint-spraying robot on a production line13 Situatedness, on the other hand, embraces for Brooks an ongoing relation to a changing world that has surprises and a history: instead of a pregiven plan relating to a representation of an oversimplified environment, a sit­ uated machine or agent must use its immersion and changing per­ ceptions in the here and now as the basis of its response; “The world is its own best model” is how Brooks expresses it.14 Recall our driving example where incorporating the world-oncoming traffic, steering wheel-was crucial. More generally, Brooks gives an airline booking system as an exemplar, making situatedness a thinned-out version of social agency-a dispersed, invisible, ticket-processing “they.”

Brooks’s attachment to the bottom-up procedure is also perfor­ mative, ruling the description as well as the content of his ap­ proach. Thus, not only is mind-problem  solving, central control, representation-subordinated within his model of intelligence, but its socio-cultural correlates-philosophy, abstract thought, theory­ are likewise invoked by him on a minimal, need-to-know basis. Such double mindlessness is a powerful head clearer, enabling the binaries of real/artificial or natural/machine intelligence to be pro­ductively opened up.

12. Ibid., p. 16.

  1. For the citation of this example as paradigmatic of what he means by embodi­ ment, and likewise the following one of situatedness, see Rodney A. Brooks, “New Approaches to Robotics,” Science, 253 (1991): 1227-1232.

14. Brooks, “Intelligence Without Reason” (above, n.  11), p. 15.

One aspect of these binaries, “the problem of human/machine communication,” as she puts it, provides the subtitle of social anthropologist Lucy Suchman’s essay Plans and Situated Action. In order to think our connection to machines, Suchman argues the need to set up a theoretical framework for understanding how action always is what she calls “situated” and, correlatively, how we understand and use plans. It should be evident by now that the theories of cognition and robotics just discussed reject plans, detailed re­presentations, predetermined courses, or blueprints of action, as the principal or even necessary component of what we call intelligent behavior. Suchman’s contribution – and her concern is neither robotics nor cognition, but socially constructed meanings – is not that plans are absent from purposeful social activity, but rather that our engagement with them vis-a-vis communication and shared understanding is other and more complicated than common sense (and hence cognitivism and orthodox Al) imagines it to be. In her words:

The confusion in the planning literature over the status of plans mirrors the fact that in our everyday action descriptions we do not normally distinguish between accounts of action provided before and after the fact, and action’s actual course. As common-sense constructs, plans are a constituent of practi­ cal action, but they are constituent as an artifact of our reasoning about ac­ tion, not as the generative mechanism of action. Our imagined projections and our retrospective reconstructions are the principal means by which we catch hold of situated action and reason about it, while situated action itself, in contrast, is essentially transparent to us as actors.15

Such a reversal and displacement – plans as action-driven rather than actions as plan-determined – requires a radical repudiation of classical (Durkheimian) and therefore much current sociological or­ thodoxy; a repudiation that,  as Suchman  observes, was prefigured by G. H. Mead sixty years ago and extended some thirty years later into the basis of what is now termed “ethnomethodology.” Accord­ ing to classical sociology, there is “an objective reality of social facts” that individual actors obey or conform to,  received  norms that determine our actions and attitudes. Sociology is the would-be science of this objective realm. To reverse this is to hold that what is called objective social reality is the problem of sociology – how did and do we manufacture objectivity? – not  a given  from  which the subject starts. Suchman sees her notion of situated action as de-

1S. Lucy A. Suchman, Plans and Sihiated Actions: The Problem of Human-Machi11e  Com­ m1micatio11  (New York: Cambridge  University  Press,  1987), pp.  38-39.

constructing a classically sanctioned binary: producing a third way between a behaviorism that reduces the significance of actions to uninterpreted body movements and a mentalism that ranks actions within the physico-social world as secondary and epiphenomenal.

Suchman develops this insight in different directions: from a Heideggerean recognition that action becomes visible-and hence subject to rational scrutiny and after-the-fact narrativization, only when dysfunctional or otherwise interrupted, to  an  insistence  on the indexicality of signs, the fact that “the sign is actually a con­ stituent of the referent” in any  account of mutual intelligibility; particularly at the interface between persons and  machines. 16  It is the latter that enters into the practical application of Suchman’s understanding of the idea and mechanics of  situated  action.  In fact, Suchman is an employee of the Xerox corporation at Xerox PARC, so let me elaborate the application by moving to John Seely Brown, director of PARC, and his prize-winning piece in the Har­ vard Business Review that puts Xerox’s employment of anthropolo­ gists like Suchman in a certain corporate context.

Brown’s paper opens with a startling declaration: “The most im­ portant invention that will come out of  the corporate research  lab in the future will be the corporation itself.” 17 By this he means us to understand that it is new organizational and technological archi­ tectures, corporate reconfigurations conducive to product innova­ tion, rather than the products themselves, that need  to be  invent­ ed. “Corporate research must,” he insists, “reinvent innovation.” Why is this necessary? The short answer is that the innovation in question is being occluded by the structure of the corporation it­ self. Unquestionably, like the military, the Catholic church, the modern state, prisons, universities, schools, and hospitals, corpora­ tions are top-down outfits with  power  and  control  flowing  from the CEO (pope, ruler, governor, principal, provost, and so on). Changing this would surely be revolutionary:  recalling the Trotsky­ ist slogan of the early seventies – “Invert the vector: power to the people” – one might well  take Brown  for a corporate anarchist. But the business community need not fear: it is the vector of ideas, novelty, information, and invention, not that of power, that Brown wants to invert.

How did the need for this inversion  become evident? “Some of

  1. Ibid., p.62.
  2. John Seely Brown, “Research That Reinvents the Corporation,” Harvard Business Review 69 (1991): 102. For a useful and interesting set of responses to some of Brown’s ideas, see the debate “Can Research Reinvent the Corporation?” Harvard Business Re­ view 69 (1991): 164-175.

our most important research,” Brown writes, “has been done by an­ thropologists.”18 Enter Suchman. Thus, consider the question never far, one supposes, from corporate lips: “What do workers do?” Ask the CEO, the management, or the workers themselves,  and the an­ swer  is  more  or less  in  line with  the job  description,  the  formal, generic characterization  of procedures  that are to be followed; and indeed, this is what Suchman got when she put the question to of­ fice clerks at Xerox. But when she observed them, a very different answer emerged: the clerks were  not  following  the procedures  in the manual but instead were using a variety of informal methods to get the work done. They were, in Brown’s description,  “constantly improvising, inventing new methods to deal with unexpected  diffi­ culties and to solve immediate  problems” 19; being, in other words, innovative in ways nobody, including themselves, had been aware of (or, at least, able to articulate). Like plans in Suchman’s analysis, the job-manual description was either after or before the fact, use­ ful for comparison, levels of remuneration, insurance, and the needs of management, but neither a determiner nor an accurate de­ scription of working reality.

A  similar finding emerged  about  tech-reps,  the people  who go out and service Xerox machines. Since the rate at which they can be trained puts an upper bound on the speed of new product intro­ duction, their education became the focus of Suchman’s ethno­ methodological eye. Here again a crucial disparity between formal and informal, situated and unsituated action, came to light. It was not, apparently, through Xerox’s well-funded, state-of-the-art teaching methods, but through casual, unofficial, locally produced stories – coffee-break gossip, and situated on-the-job narratives of breakdowns and unforeseen glitches – that they learned what they needed to know. Circulated informally among themselves, the self­education of the reps, like the innovations of the clerks, was invisible to management.

It was invisible, that is, to management governed by the top­ down vector of the traditional  corporation and,  more  importantly for Brown, even when made visible, inaccessible because of the lo­ cal, unplanned, bottom-up generation of such  information.  And this is the burden of Brown’s song: the corporation’s necessary self­ reinvention in order to harvest its own, internally produced inno­ vation, a reinvention obliging the corporation to dissolve aspects of the management/worker  binary  and involve  itself  in a process  of

18. Brown,  “Research,” p. 108.

19. Ibid .

mutual determination that he calls coproduction-first coproducing itself with its own workforce, and ultimately coproducing with its customers the very technology and potential applications that it sells to them.

Let me summarize. We have gone from the organism/environ­ment binary reconfigured as a codetermined coupling within post­ Darwinism, through the embodied minds, cognition-in-action, and neuronal selection of postcognitivism, to the mind-less robotics of a situated and embodied postclassical AI, to, finally, the situated ac­ tion of a post-Durkheimian social anthropology. In each case the move was explicitly formulated by its adherents as one that refused the flow of ideas, choices, information, decisions, or control from above, refused a theory or perspective laid out in advance of what it would apply to, in favor of what can only be called an emergence of action, intelligence, and conscious awareness from below.20

But there is an important source of bottom-upness that I have not yet mentioned; one more concerned with cultural-political re­ writing than with furthering scientific or corporate instrumentality, and one that has become so omnipresent as to be no longer identifiable within the top/bottom matrix of its original emergence. I mean of course feminism, which in the above list would be glossed as postpatriarchy, would appear as the attempt to completely re­ configure Western culture from the bottom-female-up, as well as being literally and figuratively the mother of all manner of embodiments. But it is quite impossible to do more than mention feminism here; I will make do with a single image taken from one of Donna Haraway’s feminist manifestos, relevantly titled “Situated Knowledges.” 21  For Haraway, being situated involves privileging

  1. Readers interested in a more detailed account of the issues behind the anticogni­ tivism sketched here should consult the replies by Lucy  Suchman,  Philip  Agre,  and others in the same volume of the journal Cognitive Science to a stand-up defense of it by A. H . Vera and H . A. Simon, “Situated Action: A Symbolic Interpretation,” Cogni­ tive Science 17 (1993): 7-48. Differently, and on a broader cultural terrain than that under observation here, the move being  summarized  maps  onto  the  refusal  of  the idea of a controlling, overarching “story” and the embrace of locally determined con­ tingency as an irreducible effect-thus,  among others,  the  loss of  what  he calls  “mas­ ter narratives” in jean-Frarn;ois Lyotard, Tile Postmodern Co11ditio11 (Minneapolis : Uni­ versity of Minnesota Press, 1984); the  insistence  on  contingent  evolution  in  Steven Jay Gould, Wonderful Life (New York: Norton, 1989); and the argument for the essen­ tial contingency of human consciousness in David Layzer, Cosmoge11esis: Tile Growt/l of Order in tile Universe (New York: Oxford University Press, 1990).
    1. Donna J. Haraway, “Situated Knowledges : The Science Question in  Feminism and the Privilege of Partial Perspective” in idem, Simians, Cyborgs, and Women: Tile Rein­ vention of Nature (New York: Routledge, 1991), pp. 183-201.

what is always and necessarily a “partial perspective,” and  her name for the rhetorical sway of the unsituated, the totalized, the disembodied – at least as it appears in science’s claims to unlocated, context-free, objective knowledge, spoken in what the philospher Thomas Nagel calls a “voice from nowhere” – is the “god trick”; which, given the direction of heaven – is  surely  the most  sanctioned,  sacred,  and  deeply  defended  top-down  move  in Western   culture .

I want to use Haraway’s image to link my curve through bottom­ uppery back to mathematics. Where in  mathematics,  one  might ask, is the god trick turned? The short answer: at and through in­finity. A longer answer would reveal two god tricks-Plato’s and Aristotle’s-corresponding to two ideas of number and hence of the infinite: Plato’s corresponding to the numbers’ being and always having been “out there” (actual infinite), and Aristotle’s corre­ sponding to an endless coming-into-being of the numbers  (poten­ tial infinity). The latter is logically weaker and taken as metaphysi­ cally unobjectionable; there is more mileage, then, in exposing the god trick it rides on. To do so one needs to say how god is smug­ gled into mathematics incognito inside the innocent-looking con­ tinuation  symbol “…” that we write after 1, 2, 3.

Let us ignore the usual job description given of mathematics (ex­ ercise of pure reason, pursuit of objective truth, free play of the mind, and the like) and operate ethnomethodologically. We ob­ serve that mathematicians spend their time scribbling and think­ ing: writing or manipulating or disseminating (to themselves and others) a prodigious range of symbols, as well as thinking about all manner of imagined worlds and the objects/processes within them. The two activities are obviously related, though how and why seems opaque if not mysterious. To go further one has to look more closely at the kind of writing that takes place, at its rules, protocols, and symbolic conventions and the way these semiotic features hook onto, feed and are fed by, the mathematical imaginary. A closer look reveals two purely grammatical features of rigorous mathematical texts: they are dominated by the imperative mode, amounting at times to little more than networks of injunctions;and they exclude all indexical signs such as “I,” “here,” “now,” and “this.” Two questions, then: First, who or what utters these injunc­tions, and to whom are they uttered? Second, what does it mean to lack the kind of subjectivity and self-location that indexical terms make available? These questions lead inevitably to the interiority of mathematical practice, to the demand for a quasi-phenomenology of symbol manipulation that operates from the mathematician up.

The basis for such a phenomenology can be found in the semi­ otic writings of Charles Saunders Peirce. Elaborating a suggestion of his, one can describe mathematical reasoning as a species of wak­ ing dreams or thought experiments.22 Such “reflective observations,” as Peirce called them, are like those we use every day when, in imagination, we manipulate a proxy or surrogate of ourselves in or­ der to figure out what would happen if we – our material  selves ­ did the thing in question. For mathematics, the imagined scenario is a world conjured into being through written signs, and the thing being mentally tested is a mathematical statement, which, on this view, becomes a prediction about the mathematician’s future en­ counters with signs. In other words, mathematics is a rigorous inscriptional fantasy: the insistence on writing determining what can be legitimately imagined, and the ongoing process of imagining controlling what mathematicians can meaningfully and usefully write down.

How, according to this, are we to understand counting? In par­ ticular, how do we give meaning to the repeated and always repeat­ able adjunction of an idealized mark coded into the interpretation of  the  ideogram  11  of  endless  Aristotelian  counting?  Who  is, could be, or will be making these marks? Evidently not us-easily bored, error prone, confused, and mortal-but the surrogate self of a thought experiment. Such a self is idealized to avoid the vicissi­ tudes of actual counting: we require it to perform thought experi­ ments, not real ones. Now, if it is to be persuasively predictive for us, then it must be a human simulacrum, an idealized self that re­ sembles us-we who send it on its journeys-in  some essential way; otherwise, why credit its activities as meaningfully related to, im­ pinging on and affecting, us? But idealized in what way, to what extent? What about us, as counting, signifying agents, is essential? Should we-and the answer determines whether the god trick, Aris­totle-style, is turned or not-insist that our embodiment, our incar­ nation within the physical and material universe, be a feature of any human simulacrum deserving that name?

If we answer no, then we conjure into being a totally disembod­ ied agent, an imagined surrogate of ourselves that operates outside the regimes of time, space and any kind of material presence -a  be-

  1. See Brian Rotman, “Toward a Semiotics of Mathematics,” Semiotica  72 (1987): 1-35, for the original construction of a model of mathematical reasoning as thought­ experimental along Peircean lines; and idem, Ad /11fi11itu111 t11e Gllost i11 Turing’s Mac/line: Taking God out of Matllematics and Putting tile Body Back in (Stanford: Stan­ ford University Press, 1993), for the particular application of  the  model  to  the ques­ tion  of  infinity  in  mathematics.

ing unsituated and free of any context we can name, godlike in its ghostly and untouchable otherness. This negative response is the only one that contemporary infinitistic mathematics can give if it wants, as it does, the means to count endlessly. This is because if we answer yes, if we insist that any surrogate of ourselves be – in no matter how idealized a way – embodied, then it will be part of the regularities and limitations of the universe we all inhabit. It will operate in accordance with time, space, and materiality in the presence of noise and error and will, therefore, be unable to count endlessly.

One choice, then, confirms contemporary Aristotelian infinitism with its endless progression of numbers. The other choice leads to a nonclassical, non-Aristotelian understanding  of  iteration, a  mode of repetition that I have elsewhere called “counting on non-Euclid­ ean fingers.”23 I shall not try to spell out what such  a  counting from the bottom up produces in the way of a new configuration of “number.” This is, after all, not the place or occasion to go into the consequences that flow from the repudiation of infinitistic mathe­ matics, and the details of the non-Euclidean arithmetic through which one might think an alternative idea of number. Suffice it to say that they exhibit an organization and a set of mathematical questions radically different from those presented by the conventional picture of number. What can instead be emphasized here is the overall strategy involved, the idea built in from the very start, of the embodiment, the materiality of  the counter.

We must think of this materiality in two senses. The term “counter,” like “signifier,” denotes both the agency-the one-who­ counts, the one-who-signifies-and  the object used-tally mark, sign vehicle, and the like. To insist, then, on the materiality of the signifier cuts the ground from under both the god tricks in mathe­ matics. No longer are the whole numbers given, natural, and before us, the work of a god-mind whose plan for them, in place from the beginning, preempts all understanding of what it means to count. Numbers no longer simply are, either in actuality or in some ideal­ ized potentiality: they are materio-symbolic or techno-semiotic en­ tities that have to be made by materio-symbolic creatures. They and their arithmetic  are always part of the larger and open-ended hu­ man initiative of constant becoming; an enterprise never free from choice, contingency, the limits of our (always material) resources, and the arbitrariness of history. To understand  numbers in this way

  1. Brian Rotman, “The Rise and Fall of Infinity,” lectures delivered Spring 1993 at MIT.

is to move toward an ecologically more sensitive, greener, more biological mathematics; one in which neither Aristotle’s prime mover nor Plato’s divine mind makes its infinite appearence, since in it numbers have to be grasped bottom-up from the living body of the counting subject.

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