From: Brent Davis (1996). Teaching Mathematics: Towards a sound approach. Pgs 2-15.
Section A Enactivism
[O]rganism and environment enfold into each other
and unfold from one another in the fundamental circularity that is life itself.
– Francisco Varela, Evan Thompson, and Eleanor Rosch (3)
I listen, because it reminds me again and again that the whole world runs by rhythms
I have not yet learned to recognize.
-Thomas Merton (4)
Much of the recent activity in the field of mathematics education has consisted in efforts to negotiate a series of impassable dichotomies – dichotomies that seem to be the direct and inevitable consequences of a collision between traditional objectivist perspectives and more recent subjectivist proposals. In this section, I examine the origins of the mode of bipolar thinking that has given rise to these tensions, seeking not to resolve them but to understand them more deeply. It is by endeavoring to develop such understandings, I suggest, that we open up possibilities for not merely healing the “gaps,” but for side-stepping the mode of thinking (and acting) out of which they arise.
In beginning the discussion in this manner, I do not wish to suggest that the dichotomies that we construct and through which we make sense of the world are unhelpful, but that they should not be regarded as absolute. Thought and language, it can easily be argued, are founded on our capacity to separate “this” from “that” – and so, to deny the possibility of such distinctions is to render human experience completely incomprehensible. It is thus that, rather than suggesting that mind/body, self/other, fact/faction, knower/ known and other binaries are false – a move which is, in itself, dichotomizing – I invoke the pragmatist measure of truth and argue that such constructions are valid insofar as they are useful.
The point I attempt to make in this section, then, is that, in terms of mathematics teaching, many dichotomies have outlived their utility, having locked us in a modern mind-set which posits us as essentially autonomous entities: Not only are we isolated from one another, but we are set apart from the universe. The foundation of this sort of dualistic thought is the topic I turn to presently. That discussion serves as a precursor to a brief introduction of an alternative orientation to issues of identity and cognition which, for the moment, I will describe as embracing complex and unpredictable evolutionary dynamics rather than imposing orderly and calculable mechanical processes.
Of all the parts of this book, this section is perhaps the most densely written. As such, I must ask for the indulgence and patience of those for whom some of the ideas are new. Once again, I have structured the writing so that each of these notions will be further developed in a context where they are applied to an issue of mathematics teaching. My main purpose here, in addition to situating some of the ideas philosophically, is to announce some of the central principles that will be used in the subsequent discussions.
Our Modern Heritage
The predominant epistemological perspective of the “modern” era was first announced by mathematician and philosopher René Descartes in the seventeenth century. Descartes, a contemporary of Galileo, Bacon, and Locke, and a predecessor of Newton, articulated two breaks from earlier perspectives on knowledge and modes of inquiry – perspectives which he rejected as inconsistent and unreliable mixtures of fact and fancy.
The first point of departure was on the issue of method, whereby Descartes denounced tradition, hearsay, mysticism, and religion as he called for the pre-eminence of the “natural light” of (mathematical) reason. Voicing a disdain for all other intellectual authorities, Descartes argued that all previous speculation should be rejected until indubitable principles, against which all other knowledge claims could be measured, were derived. In calling for this shift to a particular and narrow conception of reason, Descartes introduced many concepts and arguments which are foundational to modern science and analytic philosophy.
In this regard, perhaps his most noted contribution is his cogito - “I think” – which also marks his second break with tradition. Briefly, in his quest for a certain foundation for his epistemological system, Descartes arrived at the self-evident and self-verifying truth of the statement, “I think”, and this axiom became the solid ground on which he sought to verify or refute all other knowledge.
It is important to note that Descartes’ project was built upon a distrust of the evidence of the senses – a suspicion that was inherited from the ancient Greeks(5). Because one’s knowledge of the world was always and inevitably filtered through untrustworthy sensory organs, one could never know – in any absolute way – the “truth” of the (external) universe. At best, one could build better and better mental representations of the physical world, and the process of assembling those representations demanded a persistent attitude of questioning – an attitude that Descartes introduced as foundational to scientific inquiry. This “method of doubt” was offered as a screen to sift out truths from those knowledge claims that could not be validated. Rational thought was thus offered as a way of knowing that was both superior to and independent of a reliance on the senses. Descartes’ model of reason – and the one that was to become the model of rationality in the modern world – was found in geometry, a discipline that offered a process of verification that Descartes regarded as the only route to unimpeachable fact (6). For him, geometric reasoning offered a means of deducing the nature of the entire universe from foundational principles, with each deductive step bound to preceding steps in an irrefutable sequence of logical moves. It was thus that, according to Palmer, truth for Descartes was “more than merely the conformity between knower and known, it [was] the subject’s rational certainty of this conformity” (7). Rational reflection (“rationalism”) rather than empirical observation (“empiricism”) was the key to knowledge.
In establishing the cogito as the foundation and geometric reason as the means of construction – that is, in specifying both the axioms and the logic of this manner of determining truth – Descartes advanced a mode of dualistic thought that permeates modern perspectives of the universe (8). Positioning the radical subject (i.e. the modern ideal of “self” or identity as solitary, coherent, and independent of context) as the reference point for all that is known, for example, compelled him to propose the existence of at least one object – an Other – that was independent of himself and relative to whom he could situate himself as part of an objective world. Stated in different terms, in distinguishing the figure of the “self”, Descartes also distinguished the background of the “not-self” which was collected under such names as “other” and “world”. Thus arose the fundamental subject-object dichotomy. Paradoxically, this dichotomy also provided the impetus for the empiricist tradition which, contra Descartes’ rationalist proposal, relies on observation and experiment as the basis of knowledge production.
Another split initiated by Descartes’ thought was the dichotomizing of mind and body. In arguing that thinking is the basis for all truth, and hence of existence, Descartes was suggesting that a person is essentially a thinking thing – one that is capable of conceiving of itself as existing without a body. Put differently, it is not essential in Descartes’ formulation that we have a corporeal existence. Of course, this mind/body separation finds its roots in earlier philosophical and religious thought. However, in giving it a rational “scientific” basis, Descartes set the stage for a series of tensions that now, collectively, serve as a pervasive and resilient backdrop to much, and perhaps most, of Western academic thought.
To elaborate, in constructing the world on the foundation of the cogito, Descartes articulated more than the separations of mind from body, self from other, and representation from reality – all of which might be described as manifestations of a mind/body dualism. In addition to the essential distinction between mental and physical objects (with the consequent priority being assigned to the former), Descartes also contributed to the foundation of a host of other dichotomies, including knower versus known, organism versus environment, human versus nature. Further, the Cartesian orientation contributed to a view of the Self as a unified coherent subject: an autonomous entity that is isolated from others, independently constituted, essentially static, and able to maintain its integrity through diverse experience.
Other consequences of this perspective, which I shall heretofore refer to as “modernist”, included an empirical emphasis on the trustworthiness of methods used to develop knowledge (i.e. more accurate representations of reality). As such, method came to be seen in increasingly mechanical and technological terms; the universe, correspondingly, was reduced to a similarly technical form. Today, machine metaphors frame and reveal Western perspectives on the universe, the earth, nature, our bodies, and – ultimately, with the development of the computer – our minds. With this technical mind-set, the aim of inquiry has grown beyond the desire to better our understandings. The primary goal is now to control the objects of our inquiry. As Palmer elaborates, with the widespread acceptance of Descartes’ conclusion that “the world has meaning only with respect to man”, our relationship to the world is no longer cast in terms of open responsiveness, but in “restless efforts to master it” (9).
And, perhaps most significantly, with thought being afforded priority over being in Descartes’ cogito, epistemic issues began to overshadow ontological concerns – a reversal that has had profound implications for our modern conceptions of both knowledge and education. I will return to a further exploration of the consequences of modernist philosophies in later chapters that deal more specifically with these topics.
Before moving on, though, an important point of clarification should be made. In this analysis of Descartes’ work and the manner in which it figures into current thinking, my intention is not to denigrate his conclusions. Rather, I believe that he was a luminary, a critical figure in our cultural heritage. His contributions to our philosophical, mathematical, and scientific thought – not to mention our commonsense – have greatly enabled the technologies that permit our current standards of living. However, as I develop in this text, the truths of Descartes, while well-suited to the circumstances of the seventeenth century, do not fit well with the situations in which we find ourselves today. It is thus time to interrogate his legacy, asking ourselves about what we have been taking for granted and how our assumptions have shaped our perceptions and actions. Prompted by a host of crises, ranging from the personal to the planetary, we are coming to the slow realization that we must behave differently
Foundations of an Alternative
It is interesting to note the prevalence in research reports of the claim that particular methodologies or perspectives are “anti-dualist”. Reacting to a philosophical backlash against Cartesian (modernist or analytic) bipolar thinking, researchers and theorists are quick to point out that they have not succumbed to a “this-or-that” way of thinking. Nevertheless, within mathematics education at least, there seems to be an irresistible tendency to grant priority to one or the other of the “real” known (material or abstract) or the ideal knower – tendencies which, like the favoring of either empirical or rational modes of inquiry, find their origins in the same system of beliefs.
And so, in spite of their apparent diametric opposition, these modernist perspectives can quite easily be shown to be on the same rational loop – a loop which begins with the epistemic primacy of “I think”. The consequences of dualistic ways of thinking, along with extensive critique of such thought, are offered by a group of theorists who tend to be gathered under the title of “postmodernism” (although not always by their own choosing). Unfortunately, while postmodern discourses have offered valuable critiques of Descartes’ legacy, it seems that one of the precepts of postmodernism – that is, that the quest for new groundings is doomed to failure – has been profoundly misinterpreted as suggesting that we can say very little about anything. Not surprisingly, this conclusion has prompted numerous and zealous attempts to destroy the foundations of existing structures – thus demonstrating the temporal and contextual nature of all knowledge – while offering in their place the unsteady (and unsatisfactory) ground of fallibilist, relativist, and individualist accounts of knowing.
The new challenge thus seems to be the development of alternatives which abandon Cartesian assumptions but which do not give in to the temptation of establishing a new and irrefutable foundation. In this writing, I would like to explore one possibility that seems to be emerging from some convergent streams of thought that flow from such diverse disciplines and discourse fields as continental and pragmatist philosophy (10), cognitive psychology (11), ecological thought (12), biology (13) and mathematics. Growing numbers of theorists in these areas are starting from the evolutionary metaphors of Darwin rather than the analytic and reductionist model of Descartes. Their focus is thus on the dynamic interdependence of individual and environment, of knowledge and identity, and of self and other, rather than on their autonomous constitution. Variously referred to as “pragmatism”, “enactivism”, and “(deep or social) ecology”, these strands of thought join with postmodern discourses to offer a critique of modern dichotomous thinking, calling for a remembering of our past and an embracing of the complexities of existence. Enactivist (14) theorists thus offer descriptions of knowledge and communication and models of cognition and learning which are historical, situational, dynamic, intersubjective, and consensual. More importantly, perhaps, and in sharp contrast to the modernist foci, they acknowledge the centrality of the phenomenal and experiential rather than fixating narrowly on the formulated.
In brief, then, my project is to explore an alternative way of interpreting ourselves in the universe – in the process seeking to displace both the I (as the starting point for any systematic knowledge of the world) and the eye (as the primary sensory modality in assembling such knowledge). In problematizing the disembodied-I of Descartes and the disembodied-eye of modern science, I argue for a movement toward a conception of our participation in the universe that is founded on the shifting boundaries of us within the negotiated and evolving space of listening.
The remainder of this section consists of an introduction to a few of the important elements of enactivist theories. Because these points will be elaborated upon in subsequent sections and chapters as they are applied to various issues related to the teaching of mathematics, this introduction is deliberately brief.
Searching for a Middle Way
Maurice Merleau-Ponty, a preeminent postwar French philosopher, has provided us with a radical reinterpretation of Descartes’ cogito. Merleau-Ponty’s goal in this project was to find an alternative to the bipolar divisive way of thinking that dominates Western scholarly thought.
In Phenomenology of Perception, Merleau-Ponty rejects both rationalist and empiricist accounts of perception – the former because it focuses too narrowly on the cognizing agent (thus failing to provide an adequate account of the “world”); the latter because it demands too great a correspondence between a real world object and the resulting perception. Seeking a middle ground between the mental and the physical (the inner and the outer), Merleau-Ponty suggests that the body is that which renders the mind and the world inseparable. Far from representing a discrete demarcation between subject and object, one’s body is simultaneously of oneself and of the world. For Merleau-Ponty, then, the body is our means of belonging to our world – a world that shapes us and a world that we participate in shaping.
Taking up this notion, Varela, Thompson, and Rosch (who bring together biology, continental philosophy, and Buddhist thought) have endeavored to elaborate upon Merleau-Ponty’s “fundamental intuition of double embodiment” (15). In this conception, the body is understood both as an outer (physical-biological) and as an inner (lived experiential-phenomenological) structure. These structures are not opposed; rather we “continuously circulate back and forth between them” (16).
An understanding of their use of the term “structure” is critical here. Briefly, one’s structure comes about from the combined influences of biological constitution and one’s history of interaction in the world – a notion that recalls Vygotsky’s contention that human identity is subject to the dialectical play between biology and history (17). For the current purposes, a person’s structure may be thought of as being loosely synonymous with his or her personality or self. For the time being, however, I will be avoiding these terms because of the inflexibility and the permanence they connote when used in a modern context. One’s structure, in contrast, is thought to be fluid, temporal, and necessarily undergoing change. As Maturana and Varela put it, “Ongoing structural change of living beings . . . is occurring at every moment, continuously, in many ways at the same time. It is the throbbing of all life” (18). Unlike modern conceptions of identity whereby one’s self is regarded as a product, then, one’s structure is product, producer, and process. Structure is thus a fluid notion that is analogous to our experience of the object/event of sound – an idea that might be reflected in the presence of sona in the word “person” (19).
To foreshadow one of the principal implications of this text, a person’s range of possible action is determined by his or her structure. Hence, in an interaction with another person, how he or she acts is not primarily a function of the other person’s actions (as is presumed in transmission models of communication and teaching), but a consequence of his or her own structural dynamic. As Maturana and Varela explain:
[The] perturbatiions of the environment do not determine what happens to the living being; rather it is the structure of the living being that determines what change occurs in it. This interaction is not instructive, for it does not determine what its effects are going to be. . . . [The] changes that result from the interaction between the living being and its environment are brought about by the disturbing agent but determined by the structure of the disturbed system (20).
To the observer, however, it may appear that one person is functioning according to the directions given to him or her by another person. Nevertheless, it is more appropriate to think of the interaction as a choreography in which one influences and is complicit in, but cannot determine, the other’s actions. Put differently, one does not “pick up information” from the environment; rather, one’s structure specifies which environmental patterns will trigger action. Furthermore, these environmental patterns (or, in Maturana and Varela’s terms, “perturbations”) do not cause the person’s actions. Rather, they present an occasion for the person to act according to his or her structure.
In such interactions, one’s structure is necessarily affected, although not always visibly, and he or she thus emerges a “different person”. The other person involved in the interaction is similarly affected, and so the two “co-emerge”. Varela, Thompson, and Rosch use the term co-emergence to call attention to the manner in which organism and environment, self and other are “bound together in reciprocal specification and selection” (21). That is, the world’s relationship to the organism is not merely uni-directional and constraining; the organism also initiates or contributes to the enactment of its environment. They specifY one another. In this mutual specification, they co-emerge. The “subject”, in this conception, is not and cannot be considered as disembodied or as objectively separated from the world. Both are entwined in the “fundamental circularity” of existence.
The full import of this notion is found in Varela, Thompson, and Rosch’s interpretation of the word “embodied”, which is used to highlight two points. First, as is more commonly acknowledged, “cognition depends on the kinds of experience that comes from having a body with various sensorimotor capacities” (22). On this point, Mark Johnson goes so far as to suggest that words and concepts are metaphorical extensions of originary bodily experiences (23). The second and, perhaps, more critical point is that our sensorimotor capacities are embedded in and continuously shaped by broad biological, social, and historical contexts. Our knowledge and our identities – our structures or our embodiments – are thus dependent on “being in a world that is inseparable from our bodies, our language, and our social history” (24).
A more naive form of these ideas is found in current debates on the relative influence of nature and nurture on personality, intelligence, and other (ostensibly fixable) traits. Viewed through the lens of enactivist theory, these debates miss two essential points: First, they tend to separate biological from social or historical factors, thus implying that the contribution of “nature” is fixed from birth. The separation of the terms “nature” and “nurture” suggests that they can somehow be held distinct – that nurturance is not natural and that what is natural is singularly and automatically constituted. Second, the debates presume a passive cognizing agent who is shaped by various forces, but who plays no role in selecting or affecting those forces. Enactivist theory denies the possibility for these easy separations, arguing that such distinctions are both impossible and unnecessary.
With regard to the nature of the individual knower, Varela, Thompson, and Rosch suggest that the basis of cognition is not to be found in the Rationalist “I think” nor in the Empiricist “I observe” – both of which are founded on the premise of the detached knower (or disembodied I/eye) – but in the enactivist “I act”. Acting encompasses both thought and observation; acting presumes both actor (subject) and acted upon (object). In brief, acting demands reunions of mind and body and subject and object. It is this notion of embodied action that allows us to bypass the extreme positions of cognition as either recovering what is outer or projecting what is inner without seeking recourse in the supernatural or in metaphysics. The upshot is that cognition cannot be a matter of internalizing or forming accurate representations of things of the world. Rather, cognition is inseparable from and fundamental to perception and action. Perceptions guide actions; actions enable perceptions. This inseparability is expressed in Maturana and Varela’s aphorism, All doing is knowing, and all knowing is doing (25), and the term “enaction” is intended to remind us of the primacy of action in shaping our experiences, our perceptions, and our world.
The individual’s cognition, in this conception, is analogous to the evolution of a species, whereby an idea or an action comes about not because it is “correct” or optimal, but because it is possible in the given context. In making this provocative association between cognition and evolution (and their use of this comparison does appear to be more than metaphorical), Varela, Thompson, and Rosch are also indicating that enactivist notions can be applied at levels other than the organismic. Stated otherwise, the “cognizing organism” of their discussion need not be a single self-contained and visibly bounded unit. Although the time scale varies, the developmental processes of the species and the individual are, in essence, the same: the cognizing agent (i.e. species or person) is engaged in a perpetual process of adapting itself to a similarly dynamic and responsive environment.
The re-unification of the mind and body in contemporary discourses carries with it another important consequence: just as mind and body are inseparable, the distinctions we draw between ourselves are problematic. Our bodies serve to separate our individual selves from one another at the same time that they come together in complex action – in effect establishing a collective corpus which can be thought of as a body with its own integrity. This notion is often difficult for those in modern and Western societies where the notion of an autonomous and independent self is idolized (no doubt partly because of Descartes’ philosophy). But it is an idea that is implicit in many parts of our language and, in particular, in our (metaphorical or literal?) uses of the same sorts of terms to describe both individual and collective action.
Natural Selection versus Natural Drift
Enactivism draws upon various principles of a theory of neo-Darwinian evolution that Varela, Thompson, and Rosch refer to as “Natural Drift” – an interpretation of evolution that might be contrasted with the more prominent and popularly accepted “Natural Selection”.
The commonsensical Natural Selection is founded on a conception of the environment as a monological authority. Species are thought to adapt to their essentially fixed contexts and, in the process, those organisms that are the best suited to the circumstances – the fittest – will inevitably prevail over their competitors. Evolution is thus thought to be a more or less linear progression toward increasingly sophisticated and complex life forms.
Natural Drift begins by rejecting the simple separation of organism and surroundings that is implicit in the notion of a fixed environment. Noting that species not only adapt to but participate in the altering of their contexts, those taking a Natural Drift perspective on evolution argue for an ecological – that is, a collective and mutually specifying – process of determination, rather than a monological authority. So framed, evolution is more a complex choreography than a linear progression, with the criterion for survival being an adequate fit with a dynamic setting rather than an optimal fitness to a fixed setting. In other words, it is not competition with fitter species that leads to extinction; it is the inability to keep pace (i.e. maintain fitness) with a changing environment – a logic of “good enough” that is in accord with current (constructivist and social constructivist) orientations toward the emergence of personal conceptions and collective knowledge. Natural Drift thus emphasizes cooperation over competition, ecologies over monologues, adequacy over optimality, movement over progress, and complexity over linearity.
An important upshot of this move from “the world as given” to “the world as unfolding through a choreography of action” is that the modern Cartesian desire to know the world as it is is thoroughly frustrated. The universe is constantly evolving, forever eluding any attempt to fix and to know it. It is thus that, for the enactivist, the world is not preformed, but performed. We are constantly enacting our sense of the world – in the process, because we are part of it, altering it. Varela, Thompson, and Rosch refer to this phenomenon as the “fundamental circularity”, whereby they suggest that the universe changes when something as minuscule as a thought changes – because that thought is not merely in the universe; it is part of the universe. Hence, our efforts to understand the world are perhaps better thought of in terms of interpreting our own perceptions and patterns of acting within a dynamic context than in terms of coming to know that context as somehow independent of our participation.
The Contribution of Mathematics
Just as the mathematics of the seventeenth century was integral to the formulation of Descartes’ philosophy, so the mathematics of today is making a vital contribution to the reformulation of our understanding of the universe. Three branches of inquiry, in particular, are worthy of note: fuzzy logic, non-linear dynamics, and complexity theory.
Fuzzy logic – which is not a system of logic that is fuzzy but one which embraces that fuzziness of our experiences and perceptions – challenges the 1-0, true-false, all-or-nothing manner of thinking that is implicit in modern world views. Fuzzy logicians have alerted us to the rough edges of our crisp categories and to the gaps in our precise definitions, in the process reminding us that the clear-cut boundaries we perceive tell us more about the observer than the observed. Even the primary category – the I – is regarded as hazily bounded, as we are called to recognize the dynamic and fluid natures of our individual identities. In challenging the most fundamental of principles underlying our mathematics, the possibility of crisp categories, fuzzy logic is in harmony with the cross-disciplinary movement toward problematizing simplistic distinctions.
It is thus that the imagery offered by non-linear dynamics – or chaos theory – is displacing that of the linear equation to serve as a visual metaphor for our understanding of phenomena and relationships. As one closes in on the “edges” of a fractal figure, one does not find simple – or even simpler – boundaries. Rather, in our analysis of such boundaries, it becomes clear that their complicatedness is more a matter of how we choose to regard them than of their particular qualities, thus placing “observation” into the dialogical space of observer-observed rather than in the monological space of either.
Fractal geometry can also be used as a model to better understand enactivist theory. There is a certain fractal quality to this framework, as suggested by this self-similarity of processes across such conceptual levels as organism, collective, and species – a self-similarity that some have extended in both directions, from the subcellular to the biosphere. Of note are James Lovelock’s (26) “Gaia hypothesis” in which the ideas are applied on a planetary scale and Varela’s seminal work with Maturana (27), which focuses on the interdependent and subordinate elements comprising our own biological structures.
It is this sort of layered self-similarity among phenomena that has become the focus of those investigating complexity theory. Concerned with those systems that exist on the edge of chaos -“where the components of a system never quite lock into place, and yet never quite dissolve into turbulence” (28) – complexivists have begun to forge a rigorous alternative to the linear and reductionist thinking that has dominated academic thought since the time of Descartes. Significantly, complexity theorists distinguish between systems that are complex (i.e. which are self-organizing, adaptive, and spontaneously evolving) and those that are merely complicated (e.g. mechanical devices), in the process dislodging the “clockwork universe” metaphor that was initiated by Descartes and his contemporaries and replacing it with a more holistic, organic model of the cosmos that hearkens back to premodern conceptions.
These sorts of ideas, all of which will be further developed in subsequent sections and chapters, point to a certain synchronicity across academic disciplines – and, indeed, throughout Western culture. (This sort of “grand coincidence”, it might be noted, is one of the phenomena under study by complexity theorists). Framed by some of the central principles of enactivism, I will be applying several of these ideas and exploring some of their implications for the teaching and learning of mathematics. (To this end, it is worth noting at this point that, with these sorts of developments, a new status to the discipline has been established. No longer believed to be the foundations to or the building blocks of reality, mathematical objects are now offered as possible models of or metaphors for varied phenomena (29).
Section 1A: Enactivism
- Maurice Merleau-Ponty, Phenomenology of perception (London: Routledge, 1962), 347.
- Gretel Ehrlich, cited in Hannah Merker, Listening: Ways of hearing in a silent world (New York: HarperCollins, 1992), 179.
- Francisco Varela, Evan Thompson, and Eleanor Rosch, The embodied mind: Cognitive science and human experience (Cambridge, MA: The MIT Press, 1991). 218.
- Thomas Merton, Raids on the unspeakable (New York: New Directions, 1964).
- Plato had already argued that the world is revealed to reason alone and not to ordinary sense perception. See Roger Scruton, A short history of modern philosophy: From Descartes to Wittgenstein (London: Routledge, 1981).
- It is interesting to note that. at the time. mathematics was not highly regarded. In fact, it was neither seen as an independent discipline nor as a means of deducing unquestionable truths, and was as closely associated with mystical endeavors as with scientific. I return to this issue in the first section of the next chapter.
- Richard E. Palmer, Hermeneutics: Interpretation theory in Schleiermacher, Dilthey. Heidegger and Gadamer (Evanston, IL: Northwestern University Press, 1969). 144 (original emphasis).
- Explicit formulations of this mode of dualistic thinking extend back at least to the Pythagoreans of Ancient Greece. They identified ten fundamental opposites: odd/even, male/female, good/evil. wet/dry. right/left, rest/motion. hot/cold, light/ dark, straight/curved, limited/unlimited. See John McLeish, The story of numbers: How mathematics has shaped civilization (New York: Fawcett Columbine, 1991) for a more detailed account of these notions.
- Palmer, Hermeneutics, 144, 146.
- For example: Merleau-Ponty, Phenomenology of perception; Rorty, Objectivity. relativism, and truth, volume 1; Hans-Georg Gadamer, Truth and method (New York: Continuum. 1990).
- For example: Jean Piaget and Bärbel lnhelder, The psychology of the child (New York: Basic Books, 1969); L.S. Vygotsky. Mind in society: The development of higher psychological processes (Cambridge, MA: Harvard University Press, 1978); and Jerome Bruner, Actual minds. possible worlds (Cambridge, MA: Harvard University Press, 1986). Jerome Bruner, Acts of meaning (Cambridge, MA: Harvard University Press, 1990).
- For example: Berry, The unsettling of America; Gregory Bateson, Mind and nature: A necessary unity (New York: E. P. Dutton, 1979); James Lovelock, Gaia, a new look at life on Earth (New York: Oxford University Press, 1979).
- For example: Humberto Maturana and Francisco Varela, The tree of knowledge: The biological roots of human understanding (Boston: Shambhala, 1987); Varela et al., The embodied mind.
- Following Varela et al., in The embodied mind, I will be using” enactivism” as a gathering term to refer to the related schools of thought.
- Varela et al., The embodied mind, xvii.
- Ibid., xv.
- Lev S. Vygotsky, Thought and language (Cambridge, MA: The MIT Press. 1962).
- Maturana and Varela, The tree of knowledge. 100.
- The etymology of “person” is uncertain. Most etymological dictionaries do suggest a relationship to the sonorous realm.
- Ibid., 96 (original emphasis).
- Varela et al., The embodied mind, 174.
- Ibid., 172.
- Mark Johnson, The body in the mind: The bodily basis of meaning. imagination, and reason (Chicago: University of Chicago Press, 1987).
- Varela et al., The embodied mind, 149.
- Maturana and Varela, The tree of knowledge, 26.
- Lovelock, Gaia, a new look at life on earth.
- Maturana and Varela. The tree of knowledge.
- 28. Mitchell Waldrop. Complexity: The emerging science at the edge of order and chaos (New York: Simon and Schuster, 1992). 12.
- See Jack Cohen and lan Stewart, The collapse of chaos: Discovering simplicity in a complex world (New York: Penguin Books, 1994).