REFERENCES
- The autopoietic organization is defined by Varela el al3., as “a network of productions of components which (i) participate recursively in the same network of productions of components which produced these components, and (ii) realize the network of productions as a unity in the space in which the components exist.” The definitions vary somewhat from paper to paper (the reader should consult references below) .
- H. Maturana and F. Varela , “ Autopoiesis The organization of the living ” [ 1973 ]. In Autopoiesis and Cognition—The Realialion of the Living , edited by H. Maturana and F. Varela, D. Reidel , Dordrecht , Holland , 1980 , pp. 73 – 141 .
- F. Varela , H. Maturana , and R. Uribe , “ Autopoiesis The organization of living systems, its char-acterizalion and a model. ” BioSystems , 5 , 1974 , pp. 187 – 196 .
- H. Maturana , “ The organization of the living A theory of living organization. ” Int. J. Man-Machine Studies , 7 , 1975 , pp. 313 – 332 .
- E. Jantsch , The Self-Organizing Universe . Pergamon Press , New York , 1980 .
- E. Jantsch , “ The unifying paradigm behind autopoiesis, dissipative structures, hyper- and ultra-cycles. ” In Autopoiesis, Dissipative Structures, and Spontaneous Social Orders , edited by M. Zeleny , Westview Press, Inc. , Boulder , CO , 1980 , pp. 81 – 87 .
- M. Zeleny , “ Autopoiesis A paradigm lost? ” In Autopoiesis. Dissipative Structures, and Spontaneous Social Orders , edited by M. Zeleny , Westview Press, Inc. , Boulder , CO , 1980 , pp, 3 – 43 .
- M. Zeleny , “ Spontaneous social orders. ” Int. J. General Systems , 11 , 1985 , pp. 117 – 131 .
- N. Luhmann , “ The autopoiesis of social systems. ” In Sociocybernetic Paradoxes , edited by F. Geyer and J. van Zouwen , Sage Publications Ltd. , London , 1986 , pp. 172 – 192 .
- Luhmann (above) even advocates developing “a truly general theory of autopoiesis” as self-organizing systems that “carefully avoids references which hold only for living systems” (p. 172 ).
- See V. Kenny and G. Gardner , “ The constructions of self-organizing systems. ” The Irish Journal of Psychology , 9 , 1 , 1988 , pp. 1 – 24 .
- G. R. Fleiscnaker ,P “ Autopoiesis The status of its system logic. ” BioSystems , 22 , 1988 , pp. 37 – 49 .
- M. Zeleny and K. Hufford , “ The application of autopoiesis in systems analysis Are autopoietic systems also social systems? ” International J. General Systems , 21 , 1991 , pp. 145 – 160 .
- Consistent with most of the literature (e.g., Weiss47, who uses emergent collective order and self-organization synonymously), spontaneous order production is taken to be emergent collective order and both are used synonymously here with self-organization .
- J. Mingers , “ An introduction to autopoiesis—Implications and applications. ” Systems Practice , 2 , 2 , 1989 , pp. 159 – 180 .
- H. Maturana , “ Everything is said by an observer. ” In Gaia, A Way of Knowing , edited by W. Thompson , Lindisfarne Press , Great Barrington , MA , 1987 , pp. 65 – 82 .
- Maturana , 1987 , p. 71 .
- H. Maturana , “ Autopoiesis Reproduction, heredity and evolution. ” In Autopoiesis, Dissipative Structures, and Spontaneous Social Orders , edited by M. Zeleny , Westview Press, Inc. , Boulder , CO , 1980 , pp. 45 – 79 .
- G. Klir , [ Architecture of Systems Problem Solving . Plenum Press , New York , 1985 ] as cited in Zeleny and Hufford, 1991, p. 146 .
- Maturana , 1975 ; Maturana and Varela , 1980 .
- To say that a system is homeostatic and not goal-directed (end-directed or purposive) simply flies in the face of what these two words are conventionally taken to mean (e.g., see E. Nagel, “Te-leological explanation” or A. Rosenblueth N. Weiner, and J. Bigelow, “Behavior, purpose, and teleology.” In Purpose In Nature, edited by John Canfield, Prentice-Hall, Inc. Englewood Cliffs, NJ, 1966, pp. 67–88 and pp. 9–26 respectively). It is important to note that goal-directed behavior does not assert intentional or “end-in-view” behavior, although it can be intentional. To say, as Maturana does, that all the dynamical behavior of an autopoietic system is subordinated to the homeostatic maintenance of its autopoiesis is precisely equivalent to saying, as Jantsch6 does, that the function of an autopoietic system is its autopoiesis. To invoke end-directed behavior on the one hand and deny it on other is paradigmatic nonsense. Although it may be simpler to deny it than explain it, autopoiesis can hardly be said to explain the living without accounting for this end-directed behavior .
- General system theory has long recognized homeostasis (time-independent dynamical steady states) and equilinality (the convergence from different starting points to the same dynamic end state as the product of time-dependent behavior) as a spontaneous property of open-systems. In the more modern language of dynamical systems theory and nonequilibrium thermodynamics, such end states are the “atlractors” of dissipative systems. The range of perturbations from which a system returns to its original state, or the range of starting points from which the system will converge on the same limit set, defines the “basin of the attractor.” .
- Maturana and Varela , 1980 , p. 80 .
- Luhmann , 1986 , p. 174 .
- Kenny and Gardner , 1988 .
- H. Maturana and F. Varela , The Tree of Knowledge . New Science Library , Shambhala , Boston , 1988 , p. 242 .
- Maturana and Varela . 1988 . p. 49 .
- For the impossibility of a circular process that is not entropy producing, see R. Clausius , “ On the Application of the Theorem of the Equivalence of Transformations to the Internal Work of a Mass of Matter. ” Philos. Mag., ser. , 4 , 24 , 1862 , pp. p81–97, 201–213 p.
- Maturana and Varela , 1988 . p. 48 . The more general definition of autonomy in systems theory is taken to be the range of perturbations over which a system will return to the same state (see basin of attraction“). This merely underscores not only the existence of a field within which the attractor is embedded, but, given that a change in field conditions can result in the production of a new “autonomous” entity, (he creutive role of the field in the production of the new entity (see further discussion in text) .
- Zolo , as cited in Kenny and Gardner , 1988 , p. 8 .
- Kenny and Gardner . 1988 , p. 7 .
- M. Zeleny , “ Cybernetyka. ” Int. J. General Systems , 13 , 1987 , pp. 289 – 294 .
- M. Zeleny , “ Tectology. ” Int. J. General Systems . 14 , 1988 , pp. 331 – 343 .
- J. Smuts , Holism and Evolution . Macmillan , New York , 1926 .
- Smuts , 1926 , p. 106 .
- C. Lloyd Morgan , Emergent Evolution . Williams and Norgate , London , 1927 .
- W. M. Wheeler , Emergent Evolution and the Development of Societies . W. W. Norton , New York , 1928 .
- R. Swenson , “ Emergent attractors and the law of maximum entropy production Foundations to a theory of general evolution. ” Systems Research , 6 , 3 , 1989 , pp. 187 – 197 .
- H. Spencer , First Principles . Williams and Norgate . London . 1862 .
- Spencer , 1862 , p. p148/149 p.
- H. Spencer , “ The social organism ” [ 1860 ]. Reprinted in Essays—Scientific, Political, and Speculative . D. Appleton and Co. , New York . 1892 .
- Spencer , 1860 , p. 273 .
- H. Spencer , The Principles of Sociology [ 1876 ]. Reprinted in Herbert Spencer Structure, Function and Evolution . Charles Scribner's Sons , New York , 1971 .
- Spencer , 1876 , p. 114 . Spencer stressed the conservation of the ongoing relations, not the individual components which were continuously reproduced and replaced (E.g., new cells in organisms, and new humans and commodities in sociocultural systems). It is “ ... the constant relations among its parts (that) make it an entity... the general persistence of the arrangements among them throughout the area occupied ” (p. 102), said Spencer, with regard to both organisms and sociocultural systems .
- Zeleny and Hufford . 1991 . p. 147 . p. 156 .
- Spencer , 1876 , p. 109 .
- P. Weiss , The Science of Life . Futura Publishing Company , New York , 1973 .
- P. Weiss , “ 1 + I = 2 (When one plus one does not equal two). ” In The Neuroscicnccs A Study Program, G. Quarton . edited by T. Melnechuk and F. Schmitt , Rockefeller University Press , New York , 1967 , pp. 801 – 821 .
- L. von Bertalanffy , General System Theory . George Brazilter , New York , 1968 .
- L. von Bertalanffy , Problems of Life . Watts & Co. , London , 1952 .
- von Bertalanffy , 1952 , p. 124 .
- Weiss , 1973 .
- Weiss originally used “micromdeterminacy,” but I have substituted “micronortdeterminacy” so as to render his meaning more precise. In the sense of their use today, indeterminacy can refer to an observer's ignorance even where a system is micromechanically deterministic where Weiss's sense is observer-independent. For Weiss, macrodeterminacy “can demonstrably exist on a higher level without being based on any correlated microdeterminacy on the next lower level.” This fundamental point, essential to Weiss's argument against mechanistic explanations, also holds the key whereby stochasticities provide seeds at critical thresholds for symmetry-breaking events signaling the discontinuous emergence of new levels of order (new higher-ordered attractors) .
- Weiss , 1973 . p. 52 .
- Inherent in the idea of self-organization, and recognized by Spencer and also by Bertalanffy, “progressive mechanization” refers to the progressive loss or “freezing out” of degrees of freedom that occurs during spontaneous ordering–the bridge between micronondeterminacy and macrodeterminacy .
- Emergence and self-organization entail the progressive addition of constraints on previously disordered or less-ordered components to produce more-highly-ordered states. At the final state (at-tractor), constraints are maximal and the internal degrees of freedom are minimal. “The direc-liveness,” said Bertalanffy/9 “which is so characteristic of life-processes that it was considered the very essence of life, explicable only in vitalistic terms, is a necessary result of the peculiar system-state of living organisms, namely, that they are open systems.” .
- With the recognition of equifinality, by which open systems converge on the same end states regardless of the initial conditions within some range of initial conditions, viz,, the “basin of attraction,” homeostasis, and progressive mechanization (which 1 will call “progressive determinism”), end-directed behavior as the spontaneous result of dissipalive flows was acknowledged as playing a fundamental role in the production of living order .
- Waddington later introduced the term homeorrhesis (rrhesis from the Greek “rhein” to flow) which characterizes the same dissipative attractor dynamics as equifinality [ C. Waddington , “ The theory of evolution today .” In Beyond Reductionism New Perspectives in the Life Sciences , edited by A. Koestler and J. Smythies , The Macmillan Company , New York , 1969 , pp. 357 – 410 ],
- Maturana , 1975 . p. 317 .
- Maturana and Varela , 1973 , p. 78 .
- H. Maturana , “ Biology of language; The epislemology of reality. ” In Psychology and Biology of Language and Thought , edited by G. Miller and E. Lenneberg , Academic Press , New York . 1978 , p. 61 .
- B. Russell , History of Western Philosophy . George Allen & Unwin Ltd. , London , 1961 .
- See, e.g., Bacon's disdainful remark that “ inquiry into final causes is sterile, and, like a virgin consecrated to God, produces nothing ” [ A. Woodfield . Teleology . Cambridge University Press , London , 1976 ].
- The constraint of summativity excludes collective behavior, the constraint of analytically continuous functions excludes discontinuous (thus creative) behavior, and reversibility precludes goal-directed or purposive behavior .
- R. Swenson , “ End-directed physics and evolutionary ordering Obviating the problem of the population of one. ” In The Cybernetics of Complex Systems Self-Organization, Evolution, and Social Change , edited by F. Geyer . Intersystems Publications . Salinas , CA , 1991 , pp. 41 – 59 .
- As Boyle pointed out, such a mechanical world, which he compared to the “ ingenious clock of Strasburg-Cathedral ” [ F. Lange , The History of Materialism . The Humanist Press , NY , 1950 (trans. 2nd edit orig. publ. 1877, p. 303], like the Strasburg-Cathedral clock, must have an intelligent creator or, in Newton's own question, given “these things being properly ordered, do not (these) phenomena show us that there is an incorporeal Being, living, intelligent, omni-present, who in infinite space as His sensorium sees things themselves intimately, knows them completely, and thinks?” [ E. Gilson, From Aristotle to Darwin and Back Again. J. Lyon (trans). University of Notre Dame Press, Notre Dame, IN, 1984, p. 159] .
- R. Swenson , “ A robust ecological physics needs an ongoing crackdown on makers conjured out of thin air. ” PAW Review , 5 , 2 , 1990 , pp. 60 – 65 .
- R. Swenson , “ Order, evolution, and natural law Fundamental relations in complex systems theory. ” In Handbook of Systems and Cybernetics , edited by C. Negoita , Marcel Dekker, Inc. , New York , 1991 , pp. 125 – 148 .
- R. Swenson , “ Engineering initial conditions in a self-producing environment. ” Proceedings of the IEEE and SS/T Conference “A Delicate Balance Technics, Culture and Consequences , 20 – 21 Oct. 1989 . Los Angeles , edited by J. Biddle , IEEE SSIT , Los Angeles , 1990 , pp. 68 – 73 .
- B. Weber , “ Implications of the application of complex systems theory to ecosystems. ” In Proceedings of the 8th International Congress of Cybernetics and Systems , 11–15 June 1990 , New York . F. Geyer (ed) , Intersystems Publications , Salinas , CA ( in the press ).
- B , Weber , and D. Depew , “ Evolutiongeneral systems theory Towards a robust synthesis. ” Proceedings of the 33rd Annual Meeting of the ISSS , 33 , 3 , 1989 , pp. 38 – 45 .
- E.g., Kant's nebular hypothesis, advanced nearly a hundred years earlier, that the solar system had come into being from an incoherent gas within the context of a larger universe (thus meaning thai the Eanh and the life upon ii had come into being, too) was now accepted although not necessarily in its details .
- C. Darwin , On The Origin of Species by Means of Natural Selection, or the Preservation of Favored Races in the Struggle for Life . John Murray . London , 1859 .
- R. Clausius , “ Ueber verschiedene fiir die anwendung bequeme Former der Hauptgleichungen der mechanischen Wiirmetheorie, ” Annalen der Physik und Chemie , 7 , 1865 . pp. 389 – 400 .
- W. Thomson , “ On a universal tendency in nature to the dissipation of mechanical energy. ” Philosophical Magazine and Journal of Science , 4 , 4th series, July-December , 1852 , pp. 304 – 306 .
- In fact the word ‘evolulion’ is not found in the twenty-one chapters of The Descent of Man nor in any of the summaries of these chapters nor in the summaries of the fifteen chapters of The Origin [Cilson, 1984] .
- As Carniero has noted, the term ‘evolution’ was first used by the Swiss preformationist Bonnet with reference to embryology (to mean the “unfolding” of the embryo). Lamark did not use the word at all in his theory on the transformation of species first put forward in 1809. It was Spencer who introduced the first comprehensive theory of evolution and who popularized the word in the 19th century [R. Carneiro, “The devolution of evolution.” Social Biology, 19, 1972, pp. 248–258; Gilson, 1984]. Spencer gave a precise (non-preformationist) definition to evolution as “a change from an indefinite, incoherent homogeneity, to a definite, coherent heterogeneity; through continuous differentiations and integrations” [11862. p. 216]. This is precisely what we mean today as “spontaneous ordering.” As the title of Darwin's 1859 work makes clear, his intention was only to show i) that species had changed over time, and ii) that they were modified by a process he called “natural selection.” It was later that the word evolution became identified with his work and evolutionary theory became reduced to natural selection .
- In fact Darwin borrowed the phrase “struggle for existence” from Malthus who had already claimed the “|atstruggle” as a general property of the “animal and vegetable kingdoms” Cilson. 1984. p. 76) (Darwin also borrowed the “survival of the fittest” from Spencer) .
- L. Margulis and J. Lovelock , “ Biological modulation of the Earth's atmosphere. ” Icarus . 21 . 1974 . pp. 471 – 489 .
- J. Lovelock and L. Margulus , “ Atmospheric homeostasis by and for the biosphere The Gaia Hypothesis. ” Tellus . 26 , 1974 , pp. 1 – 10 .
- L. Margulis and J. Lovelock , “ The biota as ancient and modern modulator of the Earth's atmosphere, ” Pure and Applied Geophysics , 116 . 1978 . pp. 239 – 243 .
- R. Swenson , “ Emergent evolution and the global attractor The evolutionary epistemology of entropy production maximization. ” Proceedings of the 33rd Annual Meeting of the ISSS , 33 , 3 , 1989 , pp. 46 – 53 .
- R. Swenson , “ The Earth as an incommcnsurale field at the geo-cosmic interface. ” In Geo-Cosmic Relations The Earth and its Macroenvironntent . edited by G. Tomassen, W. de Graaf, A. A. Knoop, and R. Hengeveld , PUDOC Science Publishers . Wagengingen , The Netherlands . 1989 , pp. 46 – 53 .
- This has been called the “Vemadsky Paradox” by C. Barlow and T. Volk , “ Open systems living in a closed biosphere A new paradox for the Gaia debate.” BioSystems . 23 . 1990. pp. 371 – 384 .
- R. Dawkins , The Extended Pheiwtype . W. H. Freeman . San Francisco , CA . 1982 .
- R. Swenson , “ Gauss-in-a-box Nailing down the first principles of action. ” PAW Review . 4 , 2 , 1989 , pp. 60 – 63 .
- Micro and macro in the level-independent theory presented here are defined relative to each other and not in any absolute sense .
- L. Boltzmann , “The second law of thermodynamics.” Populare Shriften. Essay 3, address lo a formal meeting of the Imperial Academy of Science, 29 May 1886. Reprinted in Ludwig Boltzmann, Theoretical Physics, and Philosophical Problems, S. Brush (transl.), D. Reidel Publishing, Co., Boston, 1974, pp. 13–32. Consequent to his claim lo have reduced the second law to a law of probability, Boltzmann said that molecules or “bodies” moving at the same speed and in the same direction is the most improbable case conceivable an infinitely improbable configuration of energy [pp. 22] .
- A. Koestler , discussion following the lecture by P, Weiss , “ The living system Determinism stratified, ” in Beyond Reductionism New Perspectives in the Life Sciences , edited by A. Koestler and J, Smythies , The Macmillan Company , 1969 , p. 30.
- Bertalanffy , 1952 , p. 145 .
- A living system, said Schrodinger, feeds on “negative entropy” thereby “freeing itself from all the entropy it cannot help producing while it is alive” [ E. Schrödinger , What Is Life? The Macmillan Company, New York, 1945, p. 72]. Later, Prigogine used the term “dissipative structure” for the general class of phenomena which Bertalanffy had addressed as “open systems” (1. Prigogine, Thermodynamics of Irreversible Processes, lnlerscience Publishers, New York, 1955]. Because a theorem of minimum entropy production proposed by Prigogine was erroneously thought to apply to order-producing systems, some confusion still remainsneeds some comment. Very briefly (for a fuller account see Swensonw) the theorem states that for a system extremely close to equilibrium with more than one force (potential) driving flows (and thus producing entropy), if one force is maintained constant but the others are allowed to dissipate, the entropy production will decrease until it reaches a minimum (relative to its earlier states) in the steady state. This statement is completely unsurprising since close to equilibrium the flows are a linear function of the forces, as the forces dissipate the flows will necessarily decrease until only the one held constant remains. This tell us only that the flows are linearly dependent on the forces in the near-equilibrium regime (a fact well-known since Onsager) and that potentials are spontaneously minimized—the second law. It does not tell us which flows or paths to equilibrium are selected, given these facts. The answer to that question (shown below in the text) is the pathways or flows, given the constraints, that get it to equilibrium or minimize the potentials the fastest. This is the principle that accounts for ordering .
- K. von Baer , “ Controversy over Darwinism. ” In Darwin and His Critics The Reception of Darwin's Theory of Evolution by the Scientific Community , edited by D. Hull , Harvard University Press , Cambridge , MA , 1973 , pp. 416 – 427 .
- von Baer , 1973 , p. 421 .
- M. Bunge , Causality in Modern Science . Dover Publications , New York , 1979 .
- M. Planck. Scientific Autobiography and Other Papers , Philosophical Library . NY , 1949 .
- H. Callen , Thermodynamics and an Introduction to Thermostatistics . John Wiley & Sons , New York . 1981 .
- W. Malkus , “ Discrete transitions in turbulent convection. ” Proceedings of the Royal Society of London , 225 , 1954 , pp. 185 – 195 .
- I have recently re-introduced autocatakinetics into the literature65 in its updated and more precisely defined form. The term, first used by Oslwald, was noted later by Lotka [ A. Lotka , Elements of Mathematical Biology , Dover Publications , New York , 1956 ].
- In terms of global evolution, this fact is easily seen. There had to be prokaryotic substrate before eukaryotes, a eukaryotic substrate before higher-ordered forms, agriculture before the state, etc .
- The need for an ecological physics was a central insight of the ecological psychology of J. J. Gibson with his recognition of the operation of lawful relations scaled to the ecological level of perceivers [ J. J. Gibson , The Ecological Approach to Visual Perception . Houghton Mifflin , Boston , 1979 ].
- M. Turvey and C. Carello , “ Cognition The view from ecological realism. ” Cognition , 10 , 1981 , pp. 313 – 321 .
- Such considerations immediately lead to a reconsideration of physics itself. They imply that die basis for physical interactions between systems is far wider than has been considered heretofore a wider class of observable quantities Physics has been dominated by the idea that there is ultimately a single mode of system analysis, involving the isolation of “elementary particles'; the laws of interaction between these particles ultimately determine every property of any natural system.” [R. Rosen , Fundamentals of Measurement and Representation of Natural Systems , North-Holland , New York , 1978 , p. pxii–xiii p].
- P. Kugler , M. Turvey , C. Carello and R. Shaw , “ The physics of controlled collisions A reverie about locomotion. ” In Persistence and Change , Erlbaum , Hillsdale , NJ , 1985 , pp. 195 – 229 .
- Thus the global Earth system as a whole is a living system,so is a mule even though it does not reproduce (it does not produce more mules). Whereas fission or the splitting of entities into two is a property of autocatakinetics (physics) under the appropriate field conditions (e.g., B6nard cells (see Figure 8)villages), it is not diagnostic of the living. The generality of the replicative autocatakinetic definition of the living should also be noted; it does not require specifying any particular scale or components, or a boundary. On this view cultural systems (human social systems) are living systems, too. although of a particular kind their autocatakinetics is specified by second-order kinematic fields (see text) .
- The adjective “ replicative” as distinct from “replicating ” is borrowed by Csanyi [ V. Csdnyi , Evolutionary Systems and Society . Duke University Press , Durham , NC , 1989 ].
- R. Swenson , “ Evolutionary systems and society. ” World Futures , 30 , No. 3 , 1991 , pp. 199 – 204 .
- The primitive nature of this particular kind of constraint was introduced by Polanyi [ M. Polanyi , “ Life's irreducible structure. ” Science , 160 , 1968 , pp. 1308 – 1312 ] and has been refinedargued with great force over the years by Pattee [H. Pattee, “Dynamiclinguistic modes of complex systems.” International Journal of General Systems, 3, 1977, pp. 187–197;“Cell psychology An evolutionary approach to the symbol-matter problem.” Cognition and Brain Theory. 5. 4, 1983, pp. 325–341] .
- E.g., the amount of ATP required to replicate DNA is the same regardless of the particular sequence; the difference in the amount of entropy production involved in writing or printing two alternate phrases on this page (even if they have completely contradictory meanings) is inconsequential with regard to which one gets written; in geological time this page and the ink (the chemistry) that instantiates the words will dissipate extremely rapidly regardless of the sequence of the words, but they are absolutely stable or inert (rate-independent) relative to the rate at which they are written (printed) or read, regardless of the sequences .
- It is precisely this rate-independence of DNA strings relative to the rest of the cellular dynamics that renders illegitimate, a priori, the attribution of any kind of active, purposive behavior to genes, e.g., “striving,” “selfish” replicators it is denied absolutely by their function. They are inert relative to the dynamics of the cell. The autocatakinetic cycle is the minimal selfish unit; genes are harnessed by autocatakinetics towards their own dissipative ends.65,67 .
- F. F. Robb , “ On the application of the theory of emergence and of the law of maximum entropy production to social processes. ” Systems Practice , 3 , 4 , 1990 , pp. 389 – 399 .
- F. F. Robb , “ Are institutions entities of a natural kind? ” In Handbook of Systems and Cybernetics , edited by C. Negoita , Marcel Dekker, Inc. , New York , 1991 , pp. 149 – 162 .
- R. Swenson , “|atComments on Robb's research note, ‘On the application of the theory of emergence and of the law of maximum entropy production to social processes’. ” |btSystems Practice , 3 , 4 , 1990 , pp. 401 – 402 .
- I have some sympathy for the authors' position here, having used the word “sociophysics” for such purposes myself.69 .
- A double-dual formalism (not a dualism!.) has been proposed by ShawAlley to capture the ecological relationship between perceptionaction where the values of X are environmental properties,the corresponding values of Y are properties of a living system. As a mathematical duality, D is an operation establishing an isomorphic correspondence between XY such that for any function/that establishes a value in X there is another function g that establishes a corresponding or dual value in Y. The duality operation, ƒ X → Y, is perception and its values, affordances; the inverse duality operation, g Y → X, is action and its values, ejfectivities. The operations/and g, D X → Y, designate a system of constraints comprising the ecological relation between perception and action [R. E. Shaw, and T. Alley, “How to draw learning curves Their use and justification.” In Issues in the Ecological Study of Learning, edited by T. Johnston and A. Pietrewicz, Erlbaum, Hillsdale, NJ, 1985, pp. 275–304].