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| The End of Certainty by Ilya Prigogine |
The late Ilya Prigogine (1917-2003) was a Nobel laureate physicist and chemist whose major scientific contribution was his theory of "dissipative systems." All living systems — you, me, a rhinoceros, an earthworm, a bacterium; an ant colony; a tree or a forest; an alga in a backwater; a brain or bloodstream; an ecosystem — are dissipative systems. So are many non-living physical and chemical systems. They all have in common that flows of matter and energy course through them, giving them the power to bootstrap themselves into stable, orderly states far from thermodynamic equilibrium.
However, as Prigogine shows, dissipative systems also pass through states that are unstable and disorderly, en route to their orderly, stable states. Inherent in systems that take themselves away from thermodynamic equilibrium is both the potential for self-organized order and the potential for chaotic instability.
When the system is (self-)organized into patterns of high physical order, it is highly predictable. But at times during the evolution of a dissipative system, instability rules. At times of system instability, the system may "choose" between two (or possibly more) alternative stable states, if it is not to remain unstable. Stability will come again only when the system opts for one or another of the available new stable states. Which option is chosen by the system is a matter of probabilistic uncertainty.
In the simplest case of a dissipative system which has reached such a point of instability, there are two equally probable options between which the system may choose. Which of the two options is actually chosen by the system will depend entirely on a tiny "fluctuation" or "perturbation" which comes along at just the crucial moment.
These fluctuations or perturbations which can turn a dissipative system manifesting instability into one with a new order of stability are intrinsically unpredictable; their origin is unknowable. Accordingly, which of several possible trajectories the system will take as it moves farther and farther from thermodynamic equilibrium cannot be foretold.
One of the key words in the above description of far-from-equilibrium dissipative systems, such as you or me as biological organisms, is "trajectory." A presupposition of the description just given is that the system has, in fact, a unique trajectory of successive states forming a discrete track within its overall "phase space." An abstract phase space is the graph of all states that the system might conceivably take on over the course of time.
Another conceptual description of such a system is not in terms of single actual paths, such as hard-and-fast trajectories, but of bundles of available paths called "ensembles." Each available path of successive states in the system's phase space has a certain probability associated with it. The available paths whose probabilities are not negligible (in Prigogine's phrase, "not vanishing") form an ensemble rather than a unique trajectory.
At any particular time, in the ensemble model, the state of the system is not a single point in graphical phase space, but a "cloud" of points with various associated probabilities. Hence, what the system does as it moves from state to state is no longer a matter of certainty, but one of statistical probability.
There are valid correspondences between these two ways of conceptualizing a dynamical system. For example, the mathematics of the first way can be transformed into the mathematics of the second way, as Prigogine shows. But when that is done, there is an interesting byproduct: a "breaking of equivalence." Specifically, when "the breaking of the equivalence between the individual description (in terms of trajectories) and the statistical description (in terms of ensembles)" happens, an "arrow of time" shows up where none was evident in the alternate model (see p. 89).
Prigogine gives as an example of this breaking of equivalence a system modeled as a "Bernoulli map." A Bernoulli map is a well-known way to generate "deterministic chaos" in a computer simulation, simply by performing a series of repeated simple mathematical computations.
In deterministic chaos, a trajectory-based model may be used. If so, the future trajectory of a system's states, based on the initial value(s) of the system's key variable(s), ceaselessly diverges from the trajectory based on a different initial value or values, no matter how tiny the difference between the two (sets of) initial values may be. Moreover, the distance between the two trajectories diverges at an exponentially increasing rate. The system's future states, when plotted, accordingly jump around helter-skelter and never settle into a detectable pattern.
Because of the helter-skelter succession of system states in deterministic chaos, it would at first glance appear that such systems have no arrow of time, no way to tell by looking at the system's historical or computed trajectory which way is "forward" in time and which way is "back." But if the mathematical model is shifted to one involving ensembles, not trajectories, as Prigogine shows (pp. 83-91), for systems in deterministic chaos the associated "probability distribution function" converges nicely to a single value over time.
That is, when the trajectory-based description of a chaotic system is converted mathematically to a statistical description, suddenly an arrow of time materializes out of the seeming chaos!
Prigogine favors the statistical description over the trajectory-based description in The End of Certainty, whereas in earlier books such as From Being to Becoming he favored what he now calls "a much less extreme approach to the conceptual problem associated with irreversibility" (p. 74). That is, in the past he took time's "irreversibility" — the fact that in our mental experience of the world there appears to be a directional "arrow of time" — as merely "a matter of convenience."
After all, the basic equations of Newtonian classical dynamics, starting in the 17th century, and those of 20th-century quantum mechanics betray no preference for "forward" and "backward" directions as realities of time. Neither does Einstein's mathematics in his theory of relativity.
All these theories have in common the assumption that what is being sought is, basically, a trajectory for each moving particle of subatomic stuff. True, in the quantum mechanics that apply to such particles in isolation — according to Prigogine's From Being to Becoming; but see p. 74 of The End of Certainty — "there are observations whose numerical value cannot be determined simultaneously, i.e., [the] coordinates [of the particle's position in physical space] and [the particle's] momentum."
In other words, according to at least one view of quantum mechanics — the so-called "Copenhagen interpretation" of Niels Bohr — there is an inescapable complementarity between certain numerical values that Isaac Newton, were he alive in the 20th century, would have wanted to ascertain independently of each other. For instance, the physical position or the momentum of an electron, which together compose its trajectory, can be ascertained only by trading off knowledge of one quantity for the knowledge of the other.
In The End of Certainty, Prigogine is no longer willing to settle for a view based on quantum complementarity, because accepting complementarity means there is more than one single valid description of events in the natural world. "If there is more than a single description," he asks (p. 74), "who would choose the right one?"
In quantum investigations, it is the observer — the scientist who sets up an experimental apparatus in preparation for making an observation — who in effect "chooses the right" description. That is, the experimenter decides whether he wants to measure position or momentum, since it is impossible to pinpoint both. In so deciding, he breaks the quantum complementarity in favor of one parameter or the other.
The act of observation, then, introduces an arrow of irreversible time into the quantum system. However, the equations of standard quantum theory do not themselves indicate that there exists an arrow of time. As Prigogine notes in the introduction to The End of Certainty (p. 5), "The role of the observer was a necessary concept in the introduction of irreversibility, or the flow of time, into [standard] quantum theory."
There is accordingly, it seems to me, a kinship between, on the one hand, the act of observation of a quantum phenomenon which breaks the position-or-momentum quantum complementarity and, on the other hand, the decisive influence of a fluctuation or perturbation in a dissipative system at a point of instability, when that fluctuation breaks what would otherwise be the symmetry of the existing situation and determines which future stable path the system will take.
Accordingly, it seems to me that Prigogine's worldview as expressed in The End of Certainty could be recast in a way that attributes both types of phenomena to the effects of an outside observer.
Suppose the outside observer is God. Suppose God observes the world system and all subsidiary systems within it. It is by virtue of this outside agent's acts of observational "choice" that the so-called quantum paradox — the perplexing "collapse" of the probabilistic wave function of subatomic particles into one or another definite value; see p. 48 — turns an ensemble of merely possible subatomic positions-cum-momenta into a single trajectory, such that the certainties of the macroscopic world we know can rest squarely on the uncertainties of quantum phenomena.
In other words, suppose it is God who "chooses" which of the many alternate "descriptions" of a quantum event in fact applies.
Suppose, too, that the fluctuations which break the symmetry of otherwise unstable states in far-from-equilibrium living systems and thereby result in new evolutionary orders of stability which are located even farther from equilibrium are, looked at another way, "seeds of surprise." Such seeds would be visualized as being sown by God, by virtue of the way in which he chooses to cast his all-seeing "eye," as an observer from outside this world.
If something like that is true, then there may be a lesson to be learned from Prigogine's "breaking of the equivalence between the individual description (in terms of trajectories) and the statistical description (in terms of ensembles)." To wit, when you adopt the mathematics which honors ensembles over trajectories, thereby setting aside the normal equivalence between the two models, you in effect create time as we experience it, with its ceaseless directional flow.
For temporal creatures such as we, the arrow of time is an immediate reality ... while the "eye" of God which sows such "seeds of surprise" into the otherwise deterministic flow of worldly events is not.
But isn't one of the tenets of theistic, biblical religious belief that "in the beginning" of time, God "created" the world we know, with all its panoply of living creatures? If we take the idea of the "beginning of time" as a metaphorical reference to the introduction of the irreversible arrow of time, à la Prigogine, into the world as we ourselves know it, and if we understand God's judicious sowing of "seeds of surprise" into the evolutionary history of the world as the way in which he "created" living kinds, then it isn't hard to construct a defense of theism from Prigogine's book.
Yet to insist on a theistic interpretation admittedly goes against the spirit of Prigogine's discussion. Prigogine, in fact, wishes to move in the opposite direction: one in which science sees the world in wholly physicalist terms.
That means that there can be no recourse to outside observers (such as God) whose acts of observation turn worldly probabilities into certainties.
Accordingly, Prigogine proposes in The End of Certainty a mathematical theory which can unite classical and quantum dynamics, chaos theory, the theory of dissipative systems, and also the statistical mechanics of thermodynamics. In deference to the general reader, his mathematics are merely described qualitatively, not given in detail.
The point of the mathematics is to render all the key equations in each of these disciplines in a probabilistic, ensemble-based form, rather than in a deterministic trajectory-based form. Only when that is done can the equations imply that the arrow of irreversible time is fundamentally real.
Accordingly, when an arrow of time is discovered in an overarching mathematical theory such as the one which Prigogine proposes, that theory cannot "see" God as (at least hypothetically) an outside observer who turns worldly probabilities into certainties by making observational "choices" from afar. Instead, how certainties emerge from statistical probabilities in the natural world, when and if they do, is a matter of chance alone.
To collapse that description of the situation even further, we can say that an arrow-of-time mathematics à la Prigogine is an intrinsically "godless" theory in which chance plays the role played by God in my earlier speculation about God-sown "seeds of surprise" arising in the evolving world. In Prigogine's arrow-of-time model, "seeds of surprise" are sown by chance, not by God.
It might seem that creationists ought to resent the "godless" arrow-of-time model. However, I do not think they would be well-advised to do so.
After all, religious believers are committed to the idea that we are "in" time and God is "outside" time. If by "in" time is meant that we experience time as irreversible, then nothing Prigogine says goes against that. If by "outside" time is meant that there is some level of "time" which pre-exists time's arrow of irreversibility, Prigogine would agree with that as well.
All Prigogine is trying to do is come up with a scientific theory that takes time's arrow of irreversibility with utmost seriousness. It should not be surprising to religious believers that such a theory may well be valid. Since science cannot peer beyond the natural world to "see" God anyway, a theory which is "godless" is consistent with that viewpoint.
On the other hand, an alternative mathematical model which assumes an outside observer as the source of (what we perceive as) the arrow of time may be equally valid, for the same reason as the "Copenhagen interpretation" of quantum theory may be valid. From that perspective, the quantum-mechanical equations which do not intrinsically manifest time as irreversible are supplemented by the effects of outside observation to produce an arrow of time.
Creationists and non-creationists might accordingly agree to disagree over these matters by viewing the two models as similar to the dichotomy in standard physics between light-as-waves and light-as-particles. In some situations, light behaves as if it were composed of overlapping waves, while in others it seems to be made of discrete photons. Both viewpoints are correct. Both are necessary. Neither can be discarded in favor of the other.
Likewise, the "godless" Prigogine model of dynamical systems exhibiting various forms of uncertainty and instability can, I hope and believe, coexist with a model which leaves open the possibility that an observer outside this world sows the "seeds" of evolutionary surprise which otherwise must be attributed to blind chance.
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