Footnotes

1

No. 150 in the Red Feather Institute Series. Return

2

Kuhn's more recent comments on the nature of scientific paradigms (May, 1991:40-49) suggests that his present position is much more akin to that presented here. However, the implications of his present remarks do not inform that exceedingly large body of existing literature. Wherever Kuhn goes with paradigm theory in the future, his point about the intellectual utility of paradigms remains; they do help focus the knowledge process and they do give a key to how one may sort out a vast array of data. Return

3

Postmodernism comes in two parts: one is a critique of the knowledge process (Manning, 1989) and one is a catalogue of the negativities of postmodern experience (Kroker and Cook, 1988). What is offered here is a postmodern critique. The critique grounds an understanding of postmodern conditions which have both affirmative and nihilistic moments. See Pauline Rosenau (1992) for an excellent survey of both dimensions. Return

4

Today, ramsey theorists work to determine how many members of a set are required to generate a given desired substructure. Their results are important for communications theory, computer networking and game theory. Return

5

Graphics in Figures 1 and 2 are courtesy of The Scientific American. The graphic used in Figure 3 is courtesy of James Gleick. Return

6

Gestalt psychology itself is, in turn, under critique for its insistence that these 'rules of perception' are stable and universal. Human beings may (or may not) be designed in such a way that such rules of observation hold but that does not mean that other beings could not exist which uses different rules of perception to order the same sense data. Animals may well use differing 'universal' rules of perception and plants have yet to be heard from. Return

7

There is a story in biology about two rats talking about their life; one rat is discontent and the other is quite happy. When asked about the source of her contentment, the happy rat answers that she has trained a human being to feed her every time she pushes a button. While such a story is supposed to be humorous, I wonder why it is not proper to view that story as a straight narration of 'what actually happens' from the point of view of the rat. I discount actually planning and abstract understanding from this example of 'interaction.' Return

8

See Kline, Morris. 1972 Mathematical Thought: From Ancient to Modern Times. In three volumes. New York: Oxford University Press. Return

9

It is a matter of some irony that Poincaré, Mandelbrot, Lorenz and others in Chaos theory looked for ways to solve problems of order, predictability and control. Nature failed to cooperate and the science of Chaos was born. Return

10

For an easy read, see James Gleick, 1987 Chaos: Making a New Science N.Y.: Penguin Books. For a more technical reading, see Peter B. Kahn, 1990, Mathematical Methods for Scientists & Engineers. Return

11

Stephen Hawking (1988: 169) asserts that the goal of science is to discover one grand unified theory which explains everything. John Casti denies the possibility since most systems entail incomputable parameters (1990:406). See the work of Isaiah Berlin (1990) for an overview of the romantic critique of modern science. He uses Vico and von Herder against those who affirm the Enlightenment Ideal of timeless truth and a single coherent universal paradigm yielding a final theory which can predict on terms demanded by such as Penrose. Return

12

See Roger Penrose, 1989:286, for a discussion about the relationship between the objectivity and measurement. The hard position, attributed to Einstein, is that there are real objects out there that can be measured precisely and the dynamics can be modelled precisely; all of history is contained in these measurements. The soft position, represented by Nils Bohr, seems to be that the quantum state of an object has no physical reality but is just the sum of the measurements one makes. What is a matter of speculation for physical science is a matter of fact for social science. All social reality is a calling forth of itself by a four-fold process which involves human beings defining the nature of a situation, treating it as if it were real, organizing their behavior in ways (loosely) compatible with that definition and then seizing upon that reality as if it had always existed as a fact. Marriages, funerals, parties, and multinational firms are, equally, social facts and human artifacts. See also, John Barrow, 1992:94 for a more current account of the intersubjectivity of science. Return

13

I retain the convention in the philosophy of science that it is the statements about reality which are true (or false); not reality itself. Reality can have only differing attributes; there is no such thing as a false tree or a true rose, Gertrude Stein notwithstanding. Return

14

Falsification remains possible for point attractors; for torus shaped attractors, falsification becomes less useful. For systems with 2,4, and 8 attractors, it becomes a matter of choice which end-state one prefers to call normal and which one prefers to call error or false. After a torus explores, i.e., after 3.6 bifurcations, the number of attractors is so great that the idea of falsification becomes nonsense. Return

15

A lyapunov number is a measure of how fast points in any dynamical system vary from each other. It will, thus, give a measure of how quickly correlations between two or more points fade (or increase). See Briggs and Peat (1989:87). Return

16

Some will take this to mean that positive knowledge is impossible. I tend to be more optimistic. See the fine essay by Ben Agger on the forms of positivism in the Berkeley Journal of Sociology, "Is Wright Wrong," V.XXXIV, (1989:190). Agger identifies Positivism I as an assumption that the knowledge process can and should mirror reality. Positivism II, logical positivism, claims that scientific statements must be if-then statements. Positivism III holds that empirical research is possible and has great utility even if the ontology at hand is constructed in the first instance and changed by research in the second instance. Agger says that all progressive scientists should be Positivists III. I agree since Chaos findings tend to diminish the writ of the first two positivisms, however I extend Agger's recommendation to postmodern science generally whether an explicit political agenda is undertaken or not. Return

17

Somewhere in the back of my mind is the nagging idea that the seven catastrophes of René Thom serve as a ramsey set for the conceptualization of system dynamics when a system has six variables. More than that, I think that the structureness of structure is a function of the number of dimensions we chose to perceive. If we count four or fewer dimensions (time counts as a fourth), then we do not 'see' structure at a more complex scale of observation. If we count more than four, then we readily perceive structure where before we could only see process. If we consider the physiological processes of the frog or ape one at a time, we see blood cells in motion, oxygen transformations and osmotic passages. If we change scale and change dimensional perspective, voila, there is structure. I include these points in a footnote since, although I believe them to be most important, I do not know why I think so. I cannot explain how such could be the case in any reasoned manner. Actually, I awoke at 4:am and found myself explaining the concept of structure in terms of the number of dimensions we were willing to take into account. Since we ordinarily live in three dimensional space, we fail to grasp the fact that we now apprehend 'reality' at more complex dimension. All this presumes some validity to string theory since at low energy, tension is high enough to accommodate structure-like features while at high energy levels, tension is low and 'stringiness' is evident (Barrow, 1991:78). Return

18

Landsat satellites scan the earth with seven remote sensing bands. The bands are chosen to reveal things that ordinary photos do not reveal. They can image underground water sources, population densities, geological faults, and other physical characteristics that are not visible with the sensing bands ordinary persons and cameras use. The number and capacity of such scanning bands are themselves, in part, a human artifact. Return