I come to you from the
exhilarating experience of participating at one of the frontiers of
human realization. Work on nonequilibrium thermodynamics and
nonequilibrium systems theory spearheaded by Dr. Ilya Prigogine of
Texas University, Austin and Université
Libre, Brussels, for which he just received the Nobel Prize, has opened
a new perspective on the physics of evolving systems and the
mathematical modeling of evolution. My coworkers and I in the CSR World
Model project are applying this nonequilibrium theory to the vast
planet-spanning system which human society has lately become.

What has this to do with "Small Is Beautiful" and "Limited Growth" economics? A great deal, it turns out, though the connection was far from apparent to us in the beginning.

The connection centers on the notion of "equilibrium". Much of the claim to ecological authority of "Small Is Beautiful" and "Zero Growth" advocates rests on the assumption that the rest of Nature is in a static, or at most a cyclically changing, equilibrium; that only human Nature is restless, bustling, dynamic, progressive, and growing. And their prescriptions, in the main, boil down to having human society "rejoin" the rest of Nature in this supposed state of permanent "global equilibrium".

But suppose this supposition is not the case? What if Nature all along, both before and during the time of humanity's emergence, is__inherently__
evolutionary? Suppose there have been "energy crises" and "ecology
crises" before, in which the human race played no part because it did
not yet exist?

Our studies, drawing on well-known findings of modern science, indicate that Nature is, and inescapably so, not a static but a__growing__ entity, and that such "crises" are an intrinsic feature of its growth.

In that case, humanity's growth 'is no longer__ipso facto__ incompatible with Nature. The issue is no longer social growth versus no-growth equilibrium, but social growth which __fits in with__ and __contributes to__ the overall ongoing creation of Nature, versus that which does not.

Further on I will recount some of the history of Nature prior to humanity's emergence, which graphically illustrates that Nature in the large is far from being in "Zero-Growth" equilibrium. But first I want to present an explanation as to why the nonequilibrium character of the Universe has been underemphasized, in human thought, for so long.

Nonequilibrium systems generally correspond to nonlinear mathematical models, equilibrium systems to__linear__
models. Recalling high school algebra, where nonlinear simply meant
"curved" - a curved graph - "linear" a straight-line one, gives a
pretty good idea of what the terms mean in what I'll call "systems
algebra" -- the language of integrodifferential equations -- once, that
is, we specify a way to "graph" the evolution of a system.

We do this by imagining a "state space" for a system, a space in which every <>point stands for a definite possible state of the system in question. The coordinates of such a state-point are just the important__measurements__
about the system. If the system in question were the world economy,
these might include inflation rates or investment rates for example. If
it was your body's physiological system - blood sugar level, blood
pressure, etc. The evolution of a system, defined as a movement from
state to state, can be mapped then as a movement from point to point in
the system's state-space. This movement sweeps out a "track" or
"trajectory", having a definite shape or "signature" __characterizing__ the inherent dynamic or "law of motion" of the particular system in question.

Now a linear system's trajectory typically converges to a single final point as time increases, an end point where it stops moving. This "fixed point" or point of no further change is the state-space representation of an "equilibrium". Add a time axis to the state-space, and the fixed__point__ becomes just a straight __line__
rising up out in the direction of the time axis, "flat" like the lines
on the "vital-signs" oscilloscope next to the operating table when the
patient dies.

Equilibrium is like death: no change, no life. The actual trajectory of the linear system is usually distinct from the fixed point "until ", that is, the trajectory converges asymptotically on that fixed point or straight-line as t increases.

A nonlinear system's trajectory, on the contrary, typically converges on, not a fixed__point__, but a fixed __trajectory__,
a curved, and closed (self-reentering) path called a "limit-cycle". A
limit-cycle is "state-space-ese" for a sustained undulation in the
system-measurements that does not die away. That is, it represents a
rhythmic "beating" in the values of the various state-variables through
time. Adding an explicit time-axis to the state-space transforms the
closed trajectory of the limit-cycle into a helical spiral rising up
out along the time dimension. The actual trajectory of the system, as
distinct from the limit-cycle itself, is a __vortical__ helix, ever
winding in upon this final fixed spiral of the limit-cycle proper, but
never reaching it in all finite time, that is, reaching it and merging
in it only "in the limit" as (hence the name "__limit__" cycle).

This means that such nonlinear systems are "cyclical" in their pattern of change only "in the limit". For all finite time they are__noncyclical__.
The frequency and amplitude of their changes themselves keep changing
with each "beat". Such nonlinear systems are "aperiodic". An example of
a helical trajectory is the seasonal procession of states. We often
speak of this as a "cycle", and yet, though Spring, Summer, Fall, and
Winter come around again and again, no two Springs, no two Summers,
etc. are ever quite the same.

The "nonperiodic" character of limit-cycle systems has recently been dubbed "chaos". Yet the limit-cycle pattern of change is quite deterministic, and the equations which describe it are often quite simple and brief, though nonlinear. Many phenomena previously classed as merely random or probabilistic under the reign of linear theory (including possibly those of "quantum mechanics") may soon find a nonstatistical, causal, but nonlinear, explanation.

Nonlinear systems thus exhibit an everchanging, ever new quality, never exactly repeating states. Recalling the sayings of the ancient pre-Socratic philosopher Heraclitus, such as "you can't put your foot twice in the same river", we might call this the "Heraclitean" property of nonlinear, nonequilibrium systems, but with one*proviso*.
The "river" image may connote mere turbulence, random or chaotic
change. But the irreversible, nonrepetitive continuum of change we wish
to describe is highly ordered. Its characteristic gradient is
development or evolution. So when we talk of the "Heraclitean
continuity" of nonlinear systems, we mean a pattern of growth of
systems into ever-higher levels of organization and internal
interconnection.

Now, while it has become a truism that Nature is nonlinear, in general we do not yet know how to solve nonlinear equations. We must resort to linearized models which we can solve, though essential qualitative information is "lost in translation".

My work in the Modeling Group involves studying certain recently discovered operators which, in continuous self-operation, that is, operating on themselves recurrently, generate trajectories which look suspiciously like limit-cycles. The new kind of function so constituted may provide the long-sought solutions to nonlinear equations of limit-cycle type, promising enormous savings in computer time^{1}
and memory "space" needed to run realistic mathematical models. Such
solutions, however, would force us to relinquish the idea of R^{n},
the traditional "real" number space of n dimensions, as the exclusive
habitat of state-spaces, and to at last admit the full "reality" -- or
conceptual necessity - of the misnamed "imaginary" numbers, i ( ) and beyond.

Mathematics and science in general have been reproached for inability to describe organic, holistic, living, and self-conscious or "self-reflecting" phenomena. This reproach grounds a romantic reaction against science, and I regard the "Zero Growth", "pro-Smallness", and "Back To Nature" movements as among the ultimate fallout of that Romantic reaction. This Romanticism, on the one side, conceded to mechanistic science, without contest, the whole realm of supposedly "inorganic" and "nonliving" Nature while, on the other, taking as its own the realms of "organic life" which it proclaimed science could never enter -- both wrongly, as it is turning out. For, as Prigogine has shown, nonlinear mathematical models can grasp that "organic" realm. They can map the "whole more than the sum of its parts" property. They can model__qualitative__
evolution. That is, they can depict spontaneous movement through a
succession of phases of gradual, quantitative growth, each bounded by
the sudden onset of relatively short periods of __leap__ into a higher and __qualitatively__
different level of organization and of renewed quantitative growth.
Nonlinear models can describe in this way a development like that of
biological life, __away from__ thermodynamic equilibrium and
maximal disorder or "entropy", in contradistinction to the entropic,
devolutionary trajectories described by linear, fixed-point,
equilibrium systems.

Thus the answer to the Romantic reproach has been waiting__inside__ modern science all along. We are moved to ask: why has it remained undeveloped for so long?

We have, then, an explanation for the neglect of the nonequilibrium aspect of Nature in the inability to solve nonlinear integrodifferential equations. But I did not come here today to discuss mathematics (however much it may seem the contrary). Moreover, to leave the explanation at that would be superficial. Mathematical solutions and symbols are not the crux of the difficulty with non-equilibrium processes. This difficulty resides at the__conceptual__
level which alone provides these symbols with meaning in the first
place. And our lack of the requisite symbols and solutions in the
nonlinear domain signifies a lack of or a difficulty with mastering the
requisite __concepts__. So let us look at the __symbolization__ of nonlinearity in mathematical notation in order to get at the __conceptual meaning__ of non-linearity, a meaning which is not confined to mathematics.

In ordinary algebraic equations, when the unknown, x, occurs not raised to any power or, as we also say, raised to the__first__ power, the equation is __linear__.
But when x operates on itself one or more times, i.e. is raised to a
higher power, nonlinearity sets in. In the context of
integrodifferential equations describing a system's evolution, the
unknown, x(t), is a __function of time__. The integrodifferential equation tells how some key measurement(s) of the system change in time.

Nonlinearity here means that the function x(t) occurs raised to a power in the equation, i.e. that the values of the function operate on themselves in determining the change-rates of the system.

In nonmathematical terms, nonlinearity refers to the__self-interaction__ of the states of a system. The growth of a nonlinear system is generated and controlled by the __self-interaction__ of its states.

Now, our difficulties with the concept of self-interaction are hardly confined to mathematics. In logic, for example, we find self-interaction at the heart of the paradoxes which have beset that subject throughout recorded history, but with special intensity in the Twentieth Century. The paradoxes typically involve propositions which deny themselves, adjectives which apply to themselves, sets which include themselves, and the like. Interestingly enough, in sufficiently mathematics-resembling versions of symbolic logic, they take the form of "higher degree" or nonlinear logical expressions, which are, typically, unsolvable in terms of classical truth-values, that is, in terms of ordinary "logical numbers".

Well, the "artificial languages" of mathematics and logic grow out of the "natural language" of everyday speech. And the grammar of our native language commits us, largely unconsciously, to habits of thought which make the nonequilibrium quality of Nature, and its source in self-interaction, difficult to grasp. Indo-European grammars of present vintage divide reality into nouns and verbs. Consequently, we tend to think of "things" -- represented by nouns -- as solid, stable, inert substances which "go through" change only superficially; which participate in "events", described via verbs, only__externally__, "from the skin out." "The bat hits the ball." -- but bat and ball supposedly remain unchanged in their __substance__.

There have been languages, however, which "saw" things differently -- which eschewed this sharp division into noun versus verb, "thing" versus action or event -- languages which, in fact, "saw" things__as__
actions. Benjamin Lee Whorf describes, for example, the Nootka language
in which, for instance, you don't say "the house" but "a house occurs"
or "it houses" -- in which all objects are conceived as
process-objects.

In fact, most languages, in their primitive stages, lack the sharp division between noun and verb categories prevalent today, including ancient Chinese, ancient Aryan, and the Greek of Heraclitus' time. And, in the helical trajectory of historical development, it is coming time for those primitive seeds of a process-grammar and of a process-conception of the universe to return on a higher level, and germinate.

Modern physics, with its notions of "matter-waves" and of matter as "frozen energy" or "dense field", already encourages us to think of objects as events, processes; of mere "being" itself as__activity__; of matter as __vibratory__,
of apparently solid substance as "made of motion". If we carry this
thought through, self-interaction emerges as the cause of
nonequilibrium, of evolution. For, if we think of substance as inert,
then it may exist without interacting with or affecting anything. In
that case, there can be changelessness, equilibrium. This case
expresses an "all-noun" view of reality. But if we conceive actual
things as unities of noun and verb qualities, as process-objects, then
any being is already a "doing", an activity, a __disturbance__ of
equilibrium. Its very existence, being an activity, will have
consequences for its surroundings and, therefore, via reactions, for
itself. It will continually send out ripples of effects and receive
their echoes, whose impact will modify it, that is, its activity,
sending forth modified ripples whose echoes, in turn, will modify its
rippling yet more, and so on. This is self-interaction. Being in a
state is what forces a system out of that state. Existence implies
change. Existence in a state is the cause of the movement beyond that
state.

This discussion of nonequilibrium phenomena may have seemed abstruse. But, applied to the history of Nature, it yields some startling new insights, revealing both the fallacy of the "Small Is Beautiful" argument, and alternative, growth-preserving solution to the ecological and energy crises of today.

If "things" are really__processes__, then the consequences of their existence must include __depletion__ of something on the one hand, and accumulation of something on the other. For a process is, by definition, a __transformation__ of something into something else. With this simple corollary, our application can begin.

What has this to do with "Small Is Beautiful" and "Limited Growth" economics? A great deal, it turns out, though the connection was far from apparent to us in the beginning.

The connection centers on the notion of "equilibrium". Much of the claim to ecological authority of "Small Is Beautiful" and "Zero Growth" advocates rests on the assumption that the rest of Nature is in a static, or at most a cyclically changing, equilibrium; that only human Nature is restless, bustling, dynamic, progressive, and growing. And their prescriptions, in the main, boil down to having human society "rejoin" the rest of Nature in this supposed state of permanent "global equilibrium".

But suppose this supposition is not the case? What if Nature all along, both before and during the time of humanity's emergence, is

Our studies, drawing on well-known findings of modern science, indicate that Nature is, and inescapably so, not a static but a

In that case, humanity's growth 'is no longer

Further on I will recount some of the history of Nature prior to humanity's emergence, which graphically illustrates that Nature in the large is far from being in "Zero-Growth" equilibrium. But first I want to present an explanation as to why the nonequilibrium character of the Universe has been underemphasized, in human thought, for so long.

Nonequilibrium systems generally correspond to nonlinear mathematical models, equilibrium systems to

We do this by imagining a "state space" for a system, a space in which every <>point stands for a definite possible state of the system in question. The coordinates of such a state-point are just the important

Now a linear system's trajectory typically converges to a single final point as time increases, an end point where it stops moving. This "fixed point" or point of no further change is the state-space representation of an "equilibrium". Add a time axis to the state-space, and the fixed

Equilibrium is like death: no change, no life. The actual trajectory of the linear system is usually distinct from the fixed point "until ", that is, the trajectory converges asymptotically on that fixed point or straight-line as t increases.

A nonlinear system's trajectory, on the contrary, typically converges on, not a fixed

This means that such nonlinear systems are "cyclical" in their pattern of change only "in the limit". For all finite time they are

The "nonperiodic" character of limit-cycle systems has recently been dubbed "chaos". Yet the limit-cycle pattern of change is quite deterministic, and the equations which describe it are often quite simple and brief, though nonlinear. Many phenomena previously classed as merely random or probabilistic under the reign of linear theory (including possibly those of "quantum mechanics") may soon find a nonstatistical, causal, but nonlinear, explanation.

Nonlinear systems thus exhibit an everchanging, ever new quality, never exactly repeating states. Recalling the sayings of the ancient pre-Socratic philosopher Heraclitus, such as "you can't put your foot twice in the same river", we might call this the "Heraclitean" property of nonlinear, nonequilibrium systems, but with one

Now, while it has become a truism that Nature is nonlinear, in general we do not yet know how to solve nonlinear equations. We must resort to linearized models which we can solve, though essential qualitative information is "lost in translation".

My work in the Modeling Group involves studying certain recently discovered operators which, in continuous self-operation, that is, operating on themselves recurrently, generate trajectories which look suspiciously like limit-cycles. The new kind of function so constituted may provide the long-sought solutions to nonlinear equations of limit-cycle type, promising enormous savings in computer time

Mathematics and science in general have been reproached for inability to describe organic, holistic, living, and self-conscious or "self-reflecting" phenomena. This reproach grounds a romantic reaction against science, and I regard the "Zero Growth", "pro-Smallness", and "Back To Nature" movements as among the ultimate fallout of that Romantic reaction. This Romanticism, on the one side, conceded to mechanistic science, without contest, the whole realm of supposedly "inorganic" and "nonliving" Nature while, on the other, taking as its own the realms of "organic life" which it proclaimed science could never enter -- both wrongly, as it is turning out. For, as Prigogine has shown, nonlinear mathematical models can grasp that "organic" realm. They can map the "whole more than the sum of its parts" property. They can model

Thus the answer to the Romantic reproach has been waiting

We have, then, an explanation for the neglect of the nonequilibrium aspect of Nature in the inability to solve nonlinear integrodifferential equations. But I did not come here today to discuss mathematics (however much it may seem the contrary). Moreover, to leave the explanation at that would be superficial. Mathematical solutions and symbols are not the crux of the difficulty with non-equilibrium processes. This difficulty resides at the

In ordinary algebraic equations, when the unknown, x, occurs not raised to any power or, as we also say, raised to the

Nonlinearity here means that the function x(t) occurs raised to a power in the equation, i.e. that the values of the function operate on themselves in determining the change-rates of the system.

In nonmathematical terms, nonlinearity refers to the

Now, our difficulties with the concept of self-interaction are hardly confined to mathematics. In logic, for example, we find self-interaction at the heart of the paradoxes which have beset that subject throughout recorded history, but with special intensity in the Twentieth Century. The paradoxes typically involve propositions which deny themselves, adjectives which apply to themselves, sets which include themselves, and the like. Interestingly enough, in sufficiently mathematics-resembling versions of symbolic logic, they take the form of "higher degree" or nonlinear logical expressions, which are, typically, unsolvable in terms of classical truth-values, that is, in terms of ordinary "logical numbers".

Well, the "artificial languages" of mathematics and logic grow out of the "natural language" of everyday speech. And the grammar of our native language commits us, largely unconsciously, to habits of thought which make the nonequilibrium quality of Nature, and its source in self-interaction, difficult to grasp. Indo-European grammars of present vintage divide reality into nouns and verbs. Consequently, we tend to think of "things" -- represented by nouns -- as solid, stable, inert substances which "go through" change only superficially; which participate in "events", described via verbs, only

There have been languages, however, which "saw" things differently -- which eschewed this sharp division into noun versus verb, "thing" versus action or event -- languages which, in fact, "saw" things

In fact, most languages, in their primitive stages, lack the sharp division between noun and verb categories prevalent today, including ancient Chinese, ancient Aryan, and the Greek of Heraclitus' time. And, in the helical trajectory of historical development, it is coming time for those primitive seeds of a process-grammar and of a process-conception of the universe to return on a higher level, and germinate.

Modern physics, with its notions of "matter-waves" and of matter as "frozen energy" or "dense field", already encourages us to think of objects as events, processes; of mere "being" itself as

This discussion of nonequilibrium phenomena may have seemed abstruse. But, applied to the history of Nature, it yields some startling new insights, revealing both the fallacy of the "Small Is Beautiful" argument, and alternative, growth-preserving solution to the ecological and energy crises of today.

If "things" are really

___________________________________________

**Notes**

The intervening portion of the
exposition presents substantially the same material as the "Crises of
Nature Prior to Man" section of __Humanity's Role in the Biosphere: Intrusion or Completion?__ and so is not reproduced here.

________________________________________

**Annotations
**

** ^{1}** See
"Power-Orbits and Limit-Cycles: Hypernumber-Valued Reflexive Functions
as Solutions to Nonlinear Integrodifferential Equations"

___________________________________________