A 4-Dimensional Bifurcation Map

A postmodern paradigm of paradigms would suggest that it is not, as Kuhn suggested (1962:22), that 'more mature' paradigms replace 'immature' paradigms which then subsume and explain ever more of the data of empiric observation. Rather there are, in the postmodern view, any number of co-existing paradigms none of which can encompass all of the data for several reasons. The first and most compelling is that nature and society are so complex and so richly interconnected that the choice of where the boundaries of theory are to be set is a matter of research interest and research capacity more than the underlying ontology at hand. Then too, there is the most inconvenient fact that the very data used to build theory are, in part, generated by the use of differing paradigms.

John Horgan, Senior Writer for Scientific American, has noted that new experiments challenge our notions of reality; Photons, neurons, and objects large enough to be 'seen' appear to lack form until they are observed. Stranger still, observation can alter the outcome of behavior; what happens depends in part on whether we watch it. The most bizarre phenomena reported is that the measurement of one entity can affect the form of another remote entity apparently unconnected to the first (July, 1992:94).

The relationship between object of study and the knowing subject is further confounded in the case of the many forms of social reality by the fact that all social reality is a product of a very complex and subtle symbolic interactional process in which social facts defined as true become true in the consequence of one's belief in them and in the activities one pursues in consequence of that belief. Though Mead, Cooley, Blumer and other have given us the basic conceptual tools with which to sort out that reality creating process still there remains much to do in order to know the points and times at which human agency may shape the world in which they live and, when and why human agency is limited in the construction of social reality.

The theoretical point advanced here is that the same point made be made about paradigms; not only are data, in part created by the research act but paradigms themselves are, in part, created by the research design. The knowledge process, in its entirely, is a human product but a human product that is tied to an ontology which has a life of its own. Chaos theory offers a point of departure for reconsidering the way in which paradigms develop including the basic question of why they change. I want to begin a dialogue in the philosophy of science with which to sort out such questions. The first place to start, it seems to me, is with an understanding of how many paradigms are possible. Ramsey theory gives us insight into that question.

II. Paradigms and Ramsey Theory.

In 1928, Frank Plumpton Ramsey proved that the number of astrological patterns one can generate depends upon the number of stars one can see. Ramsey theory (Graham and Spencer: 1991) explains how ancient Sumerians could look at the stars and 'see' lions, bulls, bears or scorpions. Traditional interpretations were that some mysterious force or Being generated an ontologically existent constellation and that that constellation influenced the course of real events among humans. Ramsey theory suggests that given a finite number of stars, there is a smaller finite number of constellations possible. Human imagination and human interest focus upon one such set to the exclusion of other such sets. 4

While Ramsey theory tells us that the number of complete paradigms possible depend upon the number of events in a set, Gestalt theory teaches us that conception of the whole does not depend upon the nature or number of the parts. When we look at the skies and see the big dipper, the notion of a dipper is qualitatively different from and cannot be inferred from a complete knowledge of stars as stars. Just so, there is no ontological basis in the nature of points for the concept of a square or in the character of sound waves of a melody; these conceptual patterns are, in part, a result of human activity.

Gestalt psychology arose out of the work of Max Werthheimer and others who tried to explain why, when we observe a series of still photographs, we 'see' motion. This work lead to the idea that perception and conception were a function of the observing human as well as the observed object; objectivity and subjectivity join to produce knowledge in a delicate and ever changing intersubjective format. Neutral observation of a stable underlying ontology, so central to modern science and to the method of successive approximations is displaced in both natural and social systems. It takes human labor to 'see' the objects in nature and society. 6

A. For purposes of a postmodern paradigm theory, Gestalt psychology abstracts a few simple rules (out of all possible rules) of perception, that human beings have come to use in pattern recognition. Rock and Palmer (1990:89) list the Gestalt 'laws' of grouping. They include (Fig. 2, below) proximity, similarity, closure, and continuation. Two new such laws are put forward in their article: region and connectedness.

B. There is another rule essential to the discussion of chaos theory which follows. It says that pattern recognition

Figure 2 Paradigms and Patterns

depends upon scale of observation. If we observe the first row in Fig. 2, we can see either eight separate dots or we can see a line of eight dots depending upon the scale we choose to take. In the third box, we can either see one set of two black dots, another set of two white dots then a second set of two black dots and finally a second set of white dots as, clearly Rock and Palmer wish us to. A less tractable observer might well change scale and 'see' sets of black and white dots alternately in a continuing line.

C. While the rules above account for structure (most important to the discussion below), none of the rules mentioned by Rock and Palmer account for conception of motion. If we were to generate a set of rules of dynamic conception, we could 'see' an ever moving line of white dots coming into view and moving across a screen to disappear at the end. The same rule of scale connects dynamics to structure. If we slow the scale of action down far enough, we get structure; if we speed it up, structure converts into process; if we speed it up more, we again get enough structure to fuel the conceptualization process.

IV. The Sociology of Knowledge

When we join Ramsey theory with Gestalt psychology, we have a beginning of a theory of paradigms. Together they tell us how many paradigms we could perceive and how we go about selecting one subset out of the total set of paradigms we could conceive. In order to understand the process of subset selection, we have to turn to the sociology of knowledge; more precisely, to the concepts of culture, socio-linguistics, symbolic interaction and construction theory, interest theory and stratification theory.

As we deconstruct the process by which paradigm selection is made, we encounter an ever increasing intersubjectivity between observer and observed. It is unlikely that physical objects and animate objects in more simple forms of life 'interact' with their environment to create it but, for social life worlds, intersubjectivity begins with Gestalt theory and is fleshed out with anthropological insights about language and cultural themes. 7

A. Culture Complementary to formal and psychological sources of paradigm identification and selection are cultural imperatives which point a people toward one paradigm out of an array of paradigms which nature and society present them. The concept of culture, understood anthropologically, considers a way of life of a given people. It is the themes of a culture which help pre-decide which aspects of any given complex of events we will find of interest to us.

If we are an agricultural society, we will be interested in those features of soil, climate, and crops which support our way of life. Such interests, in turn, shape the kind of mathematics, genetics, biology and climatology and the knowledge process generally. Morris Kline (1972), in his magnificent survey of mathematical thought, helps us understand how numbering systems are formed by human interests and how different kinds of mathematics arise out of the practical interests of a given socio-cultural way of life. 8

Kline (p.10 et passim) informs us that Babylonians developed numbers, geometry, quadratic equations and cube roots to solve problems of land tenure, share in grain storage, interest rates, requirements for workers on irrigation projects and other agricultural questions. The work of Kline illuminates the renewed interest in astronomy as colonial powers sent ships around the world to search for new lands in which to grow sugar, timber and grain after lands in the near East were denuded and eroded beyond repair (p. 216 et passim). Columbus and his successors used this new technology to go West or South rather than East looking for spices and cane since the Mediterrean basin was dominated by Muslim society.

Cultural themes also provide the metaphors with which ramsey sets are named. Hunting societies name constellations for the animals important to them. Warlike societies would use weapons with which to name patterns. Religious societies might well give the stars and constellations the names of saints. Darwin, living in the same predatory economic system as Dickens, looked at nature and saw competition and a bloody struggle for existence as the best way to explain change; Kropotkin living in a much more cooperative social order, looked at nature and saw mutual aid as a way to explain survival among the species. Nature and society are variable enough and complex enough to support both paradigms. Had Darwin looked, he could have found a lot of cooperation in the animal kingdom; had Kropotkin tried, he could have found enough conflict within and among species to sustain other paradigms of explanation. Nature is always more complex and contradictory than paradigms permit.

Given the cultural components to such naming of ramsey sets, one can see that it is a human poetics rather than ontology which is used in the labelling of paradigms. The naming of the current epoch in the knowledge process, that of 'modern' science is itself both a politics and a poetics in which linear models of thinking and organizing are privileged. If we deconstruct the rise and centering of modern, linear science, we can see that it is admirably suited to serve the interests of those who prefer to maximize technical rationality in a system. The essential thing about prediction, from the point of view of human beings, is that it lends itself to control.

Profit driven systems work best when workers respond mechanically to engineering specifications; when customers respond linearly to advertisements using depth psychology and when competitors respond automatically to market demand. Bureaucratic systems work best when a staff follows policy lain down by management and process clients through the routines of the system impersonally and objectively. If nonlinearity is to be found in the market system or the bureau, it is minimized and dismissed as observer error, faulty research design, poor instrumentation, chance or just bad theory. Chaos theory offers another understanding of nonlinearity and informs quite a different politics of management.

B. Socio-linguistics instruct us on the way language helps us view the world in which we live. Each language system dissects the physical world a bit differently from other language systems. If we are Eskimo and live within the linguistic framework of that society, we may invent 30 concepts which guide our conception (and study) of snow; if we are Arabic and live within the linguistic framework of that society, we may have to import words with which to refer to snow while we may have 15 concepts about female camels which help us grasp those aspects of the life cycle and activities in which we are interested and which are needed to reproduce our camel culture.

It is mode of production, climate, geography, and culture in ever changing variation which informs our interest in conceptualization linguistic devices. These work within the larger frameworks of Ramsey theory and Gestalt theory to shape the paradigms of knowledge with which we dissect nature and society.

In general, the more complex the society, the more concepts we must create to analyze it; the more variables we can find and the more paradigms we have available to name. Ramsey theory sets the lower limits for conceptualizing; human interests pick and choose among the patterns recognized.

Given a society with a complex division of labor, say of advanced industrial Germany, then it needs a complex speech capacity to function. German has 350,000 or so concepts; together with a few rules of grammar and syntax, one could build an Ackerman tower of linguistic statements which exceed the analytic capacity of a computational system larger than the solar system. Given the variety of languages in human history the set of concepts is available with which to construct social life worlds becomes infinite while the subset of possible social life worlds is variable beyond imagination.

Any effort in social science to study such a set of concepts tied to existent social life worlds and to specify its 'universal laws,' is a most political act. This politics is made still more intersubjective in that, as the language system expands, it increasingly enters back into the social life world from which it was taken to shape both physical, natural and social systems. There can be no theory of lasers until human beings produce lasers. There can be no theory of bureaucracies until human beings construct and reflect upon what they have constructed.

C. Construction theory Berger and Luckmann, (1966) instruct us that most if not all social forms are created by human beings. For social reality at least, there is no objective social ontology which antedates believing, thinking, wanting, acting human beings. Symbolic Interaction theory (Mead, [1934] 1972) tells us that all such creation is a collective process; the self system, the social role, the social formation require at least two persons. The mother without the child, the teacher without the student, the doctor without the patient are absurdities. Each child, however young, creates itself as a son or a daughter and an adult as a parent in response to the presentments of that adult as parent; the process gains in symmetry as the child grows older but its behavior is always an essential component in order that this social reality (son/mother) emerge.

In symbolic interactional theory, it is the meaning assigned to an infant in a social process which 'makes' it a son or daughter rather than its physiology, biology or genealogy. We construct our children as our kin in a rite of passage in which we name them one of us and in which we accord them status as an 'us.' We give facticity to the social object and we terminate its social facticity in another social routine. There is a physical ontology which answers to the name of the child but that physical reality may or may not be conceptualized to be kin. If not, then there is no social fact which answers to the name of a son or daughter apart from the fate of that physical entity. The same is true of all social facts; marriages are constructed and terminated apart from the physiological base from which they are made. As Durkheim said, they are reality sui generis.

Private agreements, property law and the authority of the state are used to construct complex market systems. Before such activity, capitalism as a system did not exist. Land, buildings, and labor did exist but were used and understood within other economic formations: low tech communalism, slavery, feudalism and colonialism. There can be no 'laws' of capitalist formation or market dynamics with first human beings creating capitalism. The same may be said of patriarchy or socialism. What can also be said is that, having constructed such systems, they could have been constructed differently with different organizing principles. In neither class, race or gender is biology or psychology destiny. It is from its culture and its sociology that any stable and 'objective' laws of a society derive.

In politics, an embodied political form will tend to produce the kind of behavior which, empirically, validates the theory attached to it. If authoritarian relations are prophesied and embodied, leaders act decisively and arbitrarily while subjects tend to be dependent and uncertain. The observations of Harold Lasswell in the Encyclopedia of Social Sciences (that one should not 'submit' to democratic paradigms in which the public is said to be the best judge of its own interests) are sensible and factual in an elitist political formation. It is ill advised to allow people to think for themselves if they believe that they are incapable of thinking; if in fact they are ignorant and if in fact they are excluded from the means to implement policy. That the masses are incompetent is an empirical truth made true, in part, by the actions of elites. Lasswell, in his objective reporting of a positive fact, overlooks the prior question about the possibility of constructing democratic social life worlds in which people are perfectly competent to judge its own best interests; are well informed about the macro-structural effects and have effective means to implement policy. In his conflation between that which is and that which could be, Lasswell moves out of his role as a scientist and becomes an advocate for that which is.

D. Interest group theory

adds an important twist to Ramsey theory, Gestalt theory and Construction theory. It tells us that each distinct group occupying a distinct niche in a larger social formation will have its own special interest in observing and conceptualizing a body of factual knowledge about given topics.

How each interest group will conceptualize crime will depend upon its own organic intellectuals. In any given group, all persons having direct participation will have sufficient understanding to mount an effective truth claim. Each group will have a view of truth and fact which is fractally true. On a wide range of subjects, there will arise as many viewpoints as are there groups with distinct interests.

In a society oriented, for example, to private ownership, theft will be understood by owners as a moral delict while for workers, theft might be understood as a form of redistributive justice. In a communal society, absent class divisions, theft as a concept may be difficult to grasp; as the Chief Seattle put it, "Who can own the clouds, the forests, the rivers or the deer? If there are no 'owners,' there can be no theft. If everybody owns something, then everybody is entitled to its use and its fruits. Today, we dispute ownership claims to fetuses, genomes and Antarctica where yesterday they were, unambiguously, outside the scientific process hence outside the legal process.

E. Power

Figure 3 An Outcome field with expanding numbers of basins

However if one examines other attractors in a field of outcomes, predictivity is, progressively, lost. These other attractors are called 'strange attractors.' It is said to be an 'attractor' since a system or a set of similar systems tend to take the same path given similar initial conditions. A strange attractor is strange since a system in a given dynamical regime may take a similar pathway as do systems with similar initial conditions but it never takes exactly the same pathway. Causality fades and fails as simple dynamical regimes transform into more complex.

In portions of a causal basin in which there are two and only two outcomes possible, causality is fairly tight (Box 2, Figure 3). The geometrical form which describes loose but still limited dynamics is a strange attractor called a torus. In the case of a fish, we could see its dynamics take the shape of a torus in its yearly growth cycles. It would tend to grow at about the same rate in any given day, week, month or year given a steady supply of food. We can see a torus in a stream of water but we call it a whirlpool. Each such cycle in the life of a fish or a water molecule is called an iteration. If there are 8n outcome basins, i.e., a field with eight attractors, causality is a bit looser; a fish or water molecules in a stream could go to eight different semi-stable regions in a stream.

Figure 4 Cross Section of a Torus Attractor.

Figure 4 depicts a cross section of a torus. There is a central but limited pathway which any iteration of any given system might take. In a section such an attractor, we can see that there is no one and only one point which is natural as an endstate.

One can rotate the cross section of the torus mentally and obtain the familiar 'normal curve.' In this case, the curve would beskewed to the left. The tongue to which the arrow points constitutes a bifurcation point. It would produce a second geometrical figure called the Butterfly Attractor. If one were to make a section of it and rotate that section in one's mind, one would get a bimodal distribution. Each of the dynamic states in Chaotic systems yields a familiar display but, in such displays, the full range of outcome states is lost and the dynamical character of all social processes is lost. In such slices and with such techniques, process is frozen in time and space. Phase-space analysis restores those dynamics and, in so doing, generates an entirely new way of doing social research.

In turn, that new way of doing research reopens paradigm theory and informs us that the ontological field in which natural systems is found is so variable that n number of paradigmatic fields are possible...when observes the entirety of phase-space including a fully chaotic field, the number of scientific paradigms available vary with region and with scale of observation.

Modern science gives preference to stable causal connections and chooses to privilege the first two states in Figure 4; Chaos theory tells us that stability is lost as bifurcations unfold. In metallurgy, hydrology, and astronomy, causality opens and closes much like the smile on a Cheshire cat as bifurcations unfold in phase-space. In some regions of an outcome basin, prediction is stable (white region in Fig. 4); in other regions, not so (grey regions).

However, linear systems are inherently unstable since linear feedback explodes into chaos after the third bifurcation. As positive linear feedback amplifies deviation, there comes a time when new dynamics emerge (Thom, 1974; Feigenbaum, 1978). After four bifurcations (far right of the outcome basin) orderly causality and 'superb' predictability occupy but a small portion of phase space.

Nonlinear systems can be very stable even if not tractable to linear quantification. Absent linearity, no amount of research can 'raise' some theories to the superb category yet such findings are valid and deserve a place in the knowledge process. The preference for superb theories over 'weaker' theories is a preference for order (and thus control) rather than a pursuit of truth.

Modern science presumes that, through the method of successive approximations, the various disciplines converge upon one and only one superb theory which is valid for all time to come. 11 Chaos theory instructs us that there are at least four dynamical states (Fig. 4) into which any system can fall; each state has a differing set of dynamics. If one happens upon the system in a stable region of an outcome basin, modern science would tend to terminate the knowledge process as completed. All other system dynamics would register as error, faulty theory, observer bias, incomplete measurement or inadequate computational technology.

Modern science tells us to prefer a parsimonious set of parameters over a more complex set. One uses Ockham's razor to shave parameters down to a manageable set. Chaos theory tells us that linearity and precision are achieved by the arbitrary exclusion of parameters which drive a system (Ekeland, 1988:16). Indeed, the bias in analysis is that it deliberately excludes the full complement of factors which drive really existent systems.

Modern science tells us that the laws of nature and society fit together coherently. Given such coherence, one can use Aristotlean logic with which to derive propositions and draw inferences. Chaos research discredits aristotlean logic for 8n causal basins. Aristotlean logic excludes middle values in making truth claims. A thing must be true or it must be false. In 'superb theories,' truth values can be precise to the 10¹⁴ place. One can use Aristotelian logic to make inferences. In nonlinear dynamical states, there are middle values which makes a truth statement fractally true. Chaos theory tells us that the exclusion of observed facts which fall between regions in an outcome basin is an arbitrary act. Binary truth values are thus gained at the expense of fractal truth values.

Modern science sees chaos as a nuisance, an error in research design, an error in measurement or observer error. Postmodern science sees chaos as an integral feature of real systems out of which qualitative change emerges.

Chaos Theory and Ramsey theory together suggests and chaos research tends to confirm that several realities can occupy the same time-space continua. The concept of the soliton (Briggs and Peat, 1989). In the USA, market dynamics exist side by side with other economic systems. The separation of market dynamics from other exchange acts within a social formation is an arbitrary isolation of a basin of very differing economic systems. Interest group theory helps us understand why societies are called capitalist societies when most of its economic activity is outside the logics of for-profit market activity.

Parallel economic systems such as crime, state welfare programs, charity, mutual aid within kinship groupings as well as gray market exchanges are outside the formal logic of a free market economy but which supplement (and simplify) free market systems. The boundaries between market driven activity, state welfare, private charity, crime or kinship aid are connected in the ways suggested by the outcome basin depicted in Fig. 5,

Figure 5 Paradigms and Outcome basins

Postmodern philosophy of science concedes only a small special region of natural and social dynamics to logical positivism and the canons of science which guide it. By itself, Chaos theory calls such canons into question. Add Ramsey theory, Gestalt theory as well as insights from the sociology of knowledge and the knowledge process is opened up to accommodate change, variability and creativity in ways not possible while imprisoned in the modern paradigm. The next few years will see the knowledge process shaken open, re-examined and responsibility for the social life worlds in which they live, returned to human beings as the intersubjective nature of the knowledge process feeds back into all social sciences including politics, economics and religion.

The utility of quantification, of binary truth statements, of falsification and replication as well as the venerated scientific method--that of successive approximations to a final stable theory; all are called into question. All are seen as special to a set of human interests which lead the knowledge process toward those regions in phase-space in which order and predictability are found while dismissing those regions in phase-space where looser, fuzzier, more fractal dynamics are found.

A. Quantification Chaos research reduces the utility of rational (linear) numbering systems to the knowledge process. Rather it turns to fractals (Mandelbrot, 1977) and 'rubber math' as the measurement tools of choice (Briggs and Peat, 1989:83). This decenters physical sciences as the 'hard' sciences. Compared to the study of physical systems, good research in the social sciences poses by far the greater problems for the knowledge process.

Qualitative research employing the richness of human language as well as qualitative mathematics can restore to social research the variability, openness, discontinuity and reversibility of social systems that interval and rational numbering systems and the preferences for Iron Laws and Formal Theory strip from human interaction.

B. Objectivity Penrose notes that quantum theory questions the facticity of the object under observation. 12 Some suggest, following Nils Bohr, that there is no object out there until one undertakes the measurement of it. Prigogine makes a similar but more comprehensive point when, in his New Delhi Lecture (1982), he quotes Kothari with approval:

The simple fact is that no measurement, no experiment or observation is possible without a [paradigm].

1. It is the paradigm which calls forth and limits the data found; data do not validate paradigms so much as help legitimate them. The objects of study and thus the data itself within a paradigm are, in part, selected by the paradigm. Perhaps the most revealing experiment is the well known conundrum of wave versus particle; when a beam of light (or a beam of electrons) passes through one slit, it behaves as a particle; when it passes through two slits, it exhibits wave characteristics. The paradigm one uses to understand the objects of study is a human choice; both observations yield true statements. 13

2. Construction theory informs us that the objects of social research are called forth by a self-fulfilling process. Ramsey theory suggests that the universe of possible social life worlds to be constructed approaches infinity. The universe of paradigms with which to study social life can be modelled by a mandelbrot fractal with infinite scale, infinite detail, infinite length and infinite centers.

3. Social science plays a role in the construction of various social forms by limiting itself to the study of what is at a given time and by preference to those regions of order within the universe of that which is. Then, reentering the social life world from which it came, theory becomes part of a self fulfilling prophecy in which power plays a varying role.

C. Facticity and Validity Chaos research and fractal geometry join with Gestalt theory and the sociology of knowledge to instruct us that the facticity of any object as well as the validity any statement which purports to describe its behavior depends upon the region in phase-space one chooses to study. The geometry of phase space depends, in turn, upon the number of parameters one selects to guide the search for facts.

Truth is a human artifact; not an impersonal mirroring of nature. More succinctly, in the postmodern sensibility, epistemology shapes ontology via human practice; some of which answers to the sociology of fraud, some of which embodies the prophecies contained in its religious and/or scientific paradigms.

D. Falsification The important thing to note, in terms of the method of successive approximations, is that with fractal truth values and fourth-order bifurcations, falsification is lost as an aide to the knowledge process. 14

1. Karl Popper (1965) has made much of falsification, in the logic of scientific discovery, as a one sided way to zero in on truth statements. Chaos findings teach us differently. When one obtains contradictory results in two or more observations, it may be the case that one is sampling differing regions of a fractal basin of outcomes. It is not the case, in Chaos theory, that if a proposition about the dynamics of a system is not validated by observation in each and every observation of a system, that the truth-value of the proposition is null. Two different findings, each incompatible with the other, may both be valid with a truth-value of close to 1.0 depending upon which region of phase-space one samples.

2. Paradigms then, do not become mature by means of falsification. Falsification can be a political act arising from a given practice (informed by human desire and interest and often power) in which one arbitrarily chooses which region of a basin of outcomes to reject as abnormal and in which another region is selected as the normal, natural, or god-given set of outcomes.

3. Given the human role in construction of paradigms and their embodiment in nature and society, postmodern science ceases to be oriented to falsification and becomes instead, oriented to a critique of the degree to which human intersubjectivity is successful or unsuccessful in creating given paradigms.

E. Replicability If one finds a pattern of outcomes in one region of a basin of outcomes, standard scientific paradigms would insist that the same results be found in every sector of the basin of outcomes for systems with similar initial conditions.

1. It is an attribute of such nonlinear systems that small differences can produce very different outcomes in end-states (at any given sector of phase-space). It is an important consequence of chaotic dynamics that one is unable to predict which small changes will be absorbed and which will produce large changes; thus, the basin of outcomes of any such sets of nonlinear social systems will consist of some portion of outcomes that are expected and some unpredictable region of outcomes that are new and very different (Fig. 4).

2. Ordinarily one explains contradictory findings in terms of deviancy, observer bias, faulty research design, poor theory, inadequate technology or inept operationalization. These continue to plague the knowledge process however, Chaos findings would suggest that the ontology of everyday life is varied enough to produce contradictory findings; the findings are valid depending upon which sector of a basin of outcomes one happens to sample.

3. The idea that a paradigm becomes mature through a series of replications in which a discipline hits upon a region of stable findings which are true for all time and space is decentered as the mission of the knowledge process. Instead, in postmodern science, a critique is mounted concerning which, among alternative iterations, attractors and paradigms is/are most congenial to the human project. A grounding for such transcendent critique is again, a human project.

F. Successive Approximations The method of successive approximations assumes that it is possible to zero in on a final set of interconnected, stable and logically coherent propositions. It assumes that there is, in fact, one set of universal and eternal propositions which govern all cases of a kind for all time. In modern science, this method assumes that all systems in nature embody rational, lawlike social processes.

1. Chaos research teaches us that the dynamics of real systems are such that relationships and endstates vary over a basin of outcomes in phase-space; there is no one stable end-state to which even the best research methods guide the knowledge process.

2. The method of successive approximations remains useful if one specifies the attractor and the region of outcome in which one is interested. One can use Lyapunov number to determine the boundaries of an attractor and thus, have a tool with which to set a boundary on all the statements made about that region. 15

3. Postmodern science, as practice, may be used to help to move a society or a natural system toward any given region in phase space but emancipatory knowledge processes require this choice be a matter of democratic politics.

G. Rationality Rationality is not the essence of science. Chaos research teaches us that some scientists may choose one region in a fractal field of outcomes in which to work while another chooses a different region--when their findings are different from or contradictory to each other, the choice is not as between either paradigm but rather how useful each finding might be were it to address human concerns.

1. In his New Delhi Lecture, Prigogine noted that both Boltzmann (and his views on reversibility in the laws of thermodynamics) on the one side and Darwin (and the irreversibility in his laws of evolution) were both true and contradictory. Prigogine was not the first to ask how it could be true that all systems tended toward probability and, at the same time, some systems tended toward irreversibility. Lars Onsager answered the questioned by postulating temporary order drawn from the partial order in a larger system of disorder. Prigogine answered it by saying that both processes of disorder and of order were found in nature. Both won Nobel prizes for their work: Onsager in 1928 and Prigogine in 1977.

2. In social life, the same contrariety is found; if one study reveals the correlates of crime to be located in personality while another study reveals the correlates of crime to be found in social variables, it is entirely possible in postmodern science for both to be (fractally) true. The impact of personality variables may differ in a different political economies. Poverty can bring out the worse in a person or the best depending upon how poverty is generated and how it is met. Thus in a competitive, solipsic, and possessive society, personality factors may have small and variable effect on the forms of crime. In an egalitarian, sharing, caring society, however poor it might be, psychological variables may stand out as causal to theft, rape, vandalism or murder.

3. In economics, it is entirely possible for modernization theory, marxist theory and dependency theory to be 'valid' depending upon which parameters are selected and which regions of phase space are explored. In criminology, it is possible for a variety of contradictory theories to be fractally true. Order theories, Racist theories, Conflict theories, and Control theories each select some band on the spectrum of human interest upon which to focus research, within which to observe its facts, and from which to form and test its hypotheses. In postmodern science, it is entirely possible that each such paradigm has a fractal truth value, the validity of which changes as one moves from region to region in a basin of facts and as one changes scale of observation and as the dynamics of the system under study change.

4. There is a meta-rationality possible in nonlinear dynamics. In another place, I have discussed two levels on which one can cope with nonlinearity in an effective mode. In brief, it is possible to determine the dynamical key which marks a given nonlinear regime and use it to stabilize a field. The larger strategy requires that one determine the key parameters which drive a system (or set of systems) to far from stable dynamics and institute macro-policy by which to forestall such bifurcations. In none of this is the kind of rationality informed by consistency, precision and universality to be found. Yet Chaos can cope with chaos in a loose and variable way.

H. Truth-Values In modern science, the standards of truth center around binary truth values. In Newton's paradigm, the future was knowable with a truth value of 1.0; if findings did not confirm previous results, the truth value of a new hypothesis was 0.0. In a chaotic universe, the truth value of a prediction varies from 0.0 to .999 but is seldom 1.0 (except in the special, temporary and unlikely case of a perfectly stable state point attractor). Truth-statements in social science are thus fractal.

1. Chaos theory instructs us that there is a fractal basin of outcomes for any given kind of human behavior (Fig. 3). The truth value of any statement about that behavior depends upon which region of phase-space one chooses to study. If we find a region in which stable relations are present, we say we have a valid finding. If we happen upon a region in which causality is loose, we say that we must reject a hypothesis as false. A better view is that causality loosens as bifurcations propel a system toward far from stable dynamics (Feigenbaum: 1978).

For example, if we were to study the outcomes of poverty on Black children in Chicago, we might be able to come up with truth-values close to 1.0 in that part of a basin of outcomes in which a limit cycle or torus attractor is found; close enough to lead us to think that variation is due to observer error or to another, unknown variable. However, if we expand our research horizon, we might find that the connection between poverty and race change as economic cycles come and go.

The same is true for the connection between crime and age. If we study only that portion of phase space in which burglary, theft or robbery occurs, we find a tight connection between the two variables. If we open up phase space to include the S & L scandal, stock market fraud, or tax evasion, we find crime crosses all age categories. We are then left to consider whether crime varies more with social justice than with age, race, or gender.

2. In the boundaries between torus and strange attractors in a fractal basin of outcomes, truth values of propositions are very hard to determine. Looking at that region from the modernist paradigm, we would tend to think that nothing of interest to the knowledge process was happening. If we look at the geometry of but one stable attractor, we tend to see order and predictability of the sort we prefer; taken as a whole, the entire basin of outcomes reveals fractal nature of truth-values since the outcome basin itself is fractal.

I. Parsimony Ekeland (1988:86-87) notes that Mariotte's paradigm with three parameters is used in most analyses to define the behavior of a gas (temperature, pressure and volume). He says these variables are well defined only at equilibrium. If we want accuracy of the sort that Penrose demands for dynamic systems, one needs to use another paradigm, Boltzmann's, in which 6ⁿ variables are used (six variables per molecule; three of position and three of velocity times the number of molecules). If we are dealing with 10²³ molecules (a reasonable quantity of a gas), we need a model with 6 x 10²³ variables. Which of two paradigms one uses is a human choice.

In sports, in accounting, in criminology as in physical science, there is no natural end to the number of parameters one may specify, measure and correlate. Each day, sports analysts come up with ever new parameters about the game they study: in basketball, one can program in an infinite number of variables, correlate each in every permutation and fill in the moments between commercials, action and the next program scheduled. The number of games won in the last three seconds by three point shots made by point guards is of keen interest to those who follow Isaiah Thomas or Magic Johnson. In baseball, it might be the number of games won by left-handed pitchers under lights in open stadia; in football, it might be the number of times a quarterback has won a game in the last two minutes using passes to a tight end.

If there is no limit to the variables one can specify, then perfect prediction and perfect knowledge based upon the complete measurement of all variables is an impossible goal. Emancipatory postmodern science opens up the research process to include those parameters of interest to oppressed peoples. Cathleen Armstead and Francesca Cancian (1990) have assembled a bibliography of such participatory research which serves as a guide to the specification of such parameters for intersubjective research.

J. Cumulativity The assumption of a neat and ordered connectivity leads modern science to assume that each experiment at each level of systems organization and functioning fits into a larger whole. The task of the knowledge process, given such an assumption is to generate a portrait of the whole in theoretical terms. Each new finding is presumed to fit into that larger whole and to add a new detail to the larger theory.

Such is not the case in Chaos research. What one finds in one region of phase-space will be very different from what one finds in another region. What one finds at one magnitude will have some more or less loose similarity to what one finds at another scale but will have intractable, unpredictable, inexplicable differences not compatible with the dynamics in other chaotic regimes.

Thus the idea that one day the knowledge process will end as we fit the last piece into a grand unified theory encompassing all realities and coherently translatable across domains by a propositional logical positivism is an authoritarian dream. It is just not on. Nature is simply too messy and too complex to fit into the neat and tidy boxes of logical positivism.

K. Intersubjective Methodology The methods of postmodern science are very different from the methods of modern science. Epistemology and ontology; knowing and being, come to have, in the postmodern version of the knowledge process, an interactivity that is not found in modernist worldviews and knowledge processes. Interactivity means that all truth statements are, in part, constructed by the people who are the subject of the truth statement. 16

Such an insight means that the mission of the knowledge process in postmodern science is very different from the mission of the knowledge process in modern science. In modern science, the mission of the knowledge process was to built a grand unified theory from which human beings could accommodate themselves when necessary and with which human beings could fashion a 'Great Instauration' as Francis Bacon put it. Bacon's Utopia, called the New Atlantis, featured a collegium of scientists, called Solomon's House who would advise princes and Popes on social policy.

Postmodern philosophy of science offers a much wider and more uncertain set of missions for the knowledge process. Absolute truth is out; firm and final control of nature and society is foregone; sure and certain knowledge of everything is abandoned but. But in exchange, the human hand and the human soul become visible as architect of the very knowledge process which describes existence. Given the human element in creating the realities of which we have knowledge, the future is open and it is increasingly amenable to human agency. Whether that is a cause for optimism or deep despair is also a human choice. We have it in our power to fashion a society and a knowledge process congenial to the human project or one in which we are all strangers and enemies each to the other.

L. Deep Structures The ancient polemic between structure and process is, in part, resolved by an examination of a fractal basin of outcomes. Only process from one iteration to another is found in real time while fractal structures emerge in phase-space. Structure emerges ever more clearly and sharply as iterations pile up in phase-space but edges and openings continue to be fractal at some scales of observation. Another way to put it is that process trumps structure in real time while, in phase-space, structure consolidates or loosens depending upon the regions explored in phase-space.

However, the fact that structure is found only in phase-space does not mean that process is not affected by those same structures. Human conception and human memory store such structures in speech and in social rules. Thus structure returns from phase space to shape and reshape human processes to the extent that human insight and human genius extracts them from the totality of events that suffuse human life. 17


VII. Conclusion The number of research paradigms is a function of the complexity of the research field we choose to study. Perception of a paradigm is a function of the rules of observation we use. The conceptualization of research paradigms is a function of the cultural field in which the scientist works; language and practical interests preshape the knowledge process and fit it to a given paradigm. The choice of paradigms is a function of power and social control activities as well.

The complexity and variation of natural and social systems mean that the ontological firmament which is the topic of scientific research offers an infinite number of perspectives from which to make truth claims. The facts upon which such truth claims are founded are themselves human artifacts in that one may choose, for example, which part of the electro-magnetic spectrum to use to study a physical system and thus select the nature of the facts which emerge. 18 More than that, there are those who think that a given feature of natural systems is called forth by the measuring tactics used; such facts, while part of the ontology 'out there,' have no prior ontological existence independent of the research act.

Since the dynamics of nature and society are inhospitable to formal axiomatic theory and logical positivism of modern science, a central implication for the philosophy of science is that all theories of social behavior have to be nonlinear Change Theories. The quest for Grand Unified Theory true for all time and space is seen to be, given the structure of postmodern science, a partisan and fruitless quest for an end to history.

Each field of study in science is a separate paradigm not shaped by the number of naturally occurring phenomena or by the natural boundaries between fields of study but rather a function of the changing foci of human interest. Astronomy, Law, religion, philosophy, and medicine were the earliest 'sciences'. Geology, biology, psychology, anthropology and sociology were added later. They were developed and began probing for facts and creating theories as required by practical interests in mining, animal husbandry, warfare, colonialism and industrial production.

As new social forms and social interests emerge, there will be new sciences to describe and explain them and in that knowledge process, help such facts to consolidate, change and expand still newer sciences. There can be no theory of formal organizations without the human creation of formal organizations. There can be no sociology of disaster until human technology develops such that humans can generate the technology with which to despoil the earth, themselves and other societies...and then gauge that damage. By the year, 2100 a.c.e., there will be 10ⁿ sciences for every one now absorbing the lives and genius of its practitioners. Living systems are so complex, so interconnected with each other and their environment that the number of scientific disciplines possible is limited only by human imagination and human effort.

Part of the postmodern project in physical as in social science is to emphasize and explore the human authorship of both reality and theories of reality. It is that very interactivity between object of knowledge and subject of knowing which means that science, philosophy, theology and sociology are but different names for different aspects, different regions of the same knowledge process. The boundaries between social science, social philosophy and social policy are fractal and everchanging.

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