Chance ... or God?

The End of Certainty by Ilya Prigogine

In The World as Music I outlined why Ilya Prigogine's 1997 book The End of Certainty: Time, Chaos, and the New Laws of Nature leads me to the fanciful speculation that the world is basically made of music. Now I would like to show how such a view can be extended to include a God who calls the tune.

Briefly put, Prigogine asserts that many systems which are subject to change in the physical world naturally exhibit resonances of a mathematical variety originally discovered by Henri Poincaré. Poincaré resonances exist at points on a graph where equations describing dynamical systems — systems in the physical world made of objects in motion — intersect one another. At these points of intersection, crucial denominators become vanishingly small, such that algebraic divisions have undefined results. The customary approach to modeling dynamical systems, based on the idea that particles trace out hard-and-fast trajectories, breaks down.

Prigogine saves the day by substituting for the classical model a model based on probability distributions. His probability distributions substitute a "cloud" of system states for each single point ostensibly designating the state with certainty. They replace every simple, hard-and-fast trajectory with an expanding vapor trail of system pathways.

One of the desirable results of Prigogine's strategy is that the "instabilities" typically encountered by dynamical systems under particular sets of circumstances turn out to be related to and engendered by Poincaré resonances.

Basically, resonances are where waves — sound waves, light waves, mathematical waves — reinforce one another. When waveforms are coupled together, the amplitude (i.e., the height) of the combined waves can under the right circumstances be the (possibly very large) sum of the amplitudes of the individual waveforms. By envisioning certain mathematical constructs — "plane waves" and related "wave vectors" — as the basic stuff of which systems are built when they undergo motion and change, Prigogine is able to show that what we think of as particles occupying specific points in space and moving along specific trajectories in time are really made of these wavelike mathematical entities.

Such a wave-based mathematical model of the physical world achieves Prigogine's prime objective of introducing a directional "arrow" of time into the basic equations of physics, such that the clock of the universe cannot run backward after all.

Furthermore, the Prigogine model spreads out from the description of classical dynamical systems to include thermodynamic systems, chaotic systems, "dissipative" systems, and subatomic systems subject to the laws of quantum mechanics.


In traditional interpretations of quantum theory, subatomic particles such as electrons are never assigned trajectories per se. Instead, they have "wave functions." Different from the similarly named "plane waves" Prigogine alludes to, these attributes of the tiniest bits of matter are usually modeled using an equation named after its discoverer, Erwin Schrödinger.

Schrödinger's equation, notably time-reversible or time-symmetrical, does not allow for a directional flow of time. The irreversibility of time which we attribute to events (quantum, or at macroscopic scales larger than the subatomic) occurs, in traditional theoretical interpretations of quantum mechanics, as a result of external observation of those quantum events. Under observation, and only then, the supposed wave functions of quantum particles are said to "break down," creating a measure of scientific certainty about such things as the particles' velocities and positions where there had only been a fuzzy set of probabilities.

That reduction of the quantum wave function only under conditions of external observation is called the "quantum paradox" — for why should external observation radically affect the basic attributes of subatomic stuff?

Prigogine succeeds in sidestepping the quantum paradox by replacing specific wave functions à la Erwin Schrödinger with "ensembles" of possible states of a quantum entity, where an ensemble is represented mathematically by a probability distribution. His strategy simultaneously brings into the picture an irreversible arrow of time and removes from the quantum picture "the mysterious intervention" of an "observer" (p. 131).


When it comes to systems larger than the innards of an atom, Prigogine's selfsame model works equally well for any "nonintegrable" system — any system undergoing motion and change, whose overall behavior cannot be reduced to the sum of the behaviors of the individual pieces. And, for systems that are simple enough to be integrable, Prigogine's model can easily be transformed into the description given by classical Newtonian dynamics, whose equations turn out to be but a special case of Prigogine's more general ones.

Among the nonintegrable systems Prigogine discusses are systems in "deterministic chaos," a hot topic in physics over the last few decades. Also covered are thermodynamic systems, in which entropy and disorder increase over time. Meanwhile, one of the centerpieces of Prigogine's view concerns "dissipative" systems. Such systems hold themselves far from thermodynamic equilibrium by taking in and dispelling energy as well as matter over time. They can be merely physical or chemical systems, but all organisms are also dissipative systems.

Dissipative systems are wont arrive at crossroads of instability when they have been pushed far enough away from equilibrium by their characteristic energy flows. Prigogine calls these crossroads "bifurcation points" because the system will resolve each temporary instability by branching over onto one of two (or more) new pathways exhibiting increasingly complex order and stability. Which branch is "chosen" by the system at any particular bifurcation point is, according to Prigogine, a matter of irreducible probability — a coin flip, as it were.


Although Prigogine does not make this point explicitly, it looks to me as though he feels Poincaré resonances engender the "fluctuations" or "perturbations" which he says are associated with an evolving dissipative system's movement over onto one new branch or another at any given crossroads. These resonances seem to be intimately associated with the bifurcation points' fundamental character as instabilities. But Prigogine does not say how the resonances "decide" which of two possible paths the system will actually take.

Specifically, he does not say how a system undergoing characteristic junctures of instability makes its "choice" among alternate pathways in response to the minute "fluctuations" or "perturbations" that occur at a critical moment and that the reader must assume are associated with Poincaré resonances.

So, the reader is entitled to ask, if the resonances themselves do not "decide" which of two possible paths the system will actually take, what does? Prigogine seems to be content to allow that question to remain unanswered, except by reference to the influence of his "fluctuations" or "perturbations."

As for me, I consider "fluctuations" or "resonances," as an answer to the "what decides" question, to be just as mysterious as Prigogine calls the deciding influence of an outside observer, in discussing the paradoxes of quantum theory.


It looks to me as if there are two principled responses to this ambiguity. The first is the approach which Prigogine himself appears to take. He categorizes his probabilistic description of dynamical systems at various crossroads of mathematical instability as "primitive" and "irreducible."

You can't account for the primitive or irreducible elements that are necessarily found in every scientific theory. For example, the constant speed of light is a primitive, irreducible element of Einstein's theory of relativity. There is no way to explain why light alone should have an absolute velocity that is not relative to that of other moving entities. But when you predicate relativity theory on such an unexplained assumption, the theory then goes on to provide you with excellent explanations for, and reliable predictions about, many other phenomena.

Prigogine takes a like attitude toward the probabilistic-but-not-otherwise-predictable behavior of systems subject to Poincaré resonances. His main reason: only by taking the probabilistic uncertainty of the system as a given can you introduce the directional flow of time into a mathematical model of the physical world.

Put a different way, Prigogine settles for a theory that is stunningly powerful from a descriptive point of view, in that a single mathematical model adequately describes the behavior of systems from multiple disciplines: classical dynamics, thermodynamics, chaos theory, quantum mechanics, and the theory of dissipative, self-organizing systems.

Yet the model which describes the behavior of these systems does not explain it. It does not explain, in effect, why the probabilistic coin flip produces tails, not heads.

In fact, it undermines one of the assumptions classical physics traditionally made about coin flips and the like: if we could only ascertain the forces acting on the coin down to the nth degree of accuracy, we would be able to know in advance which coin flips would produce heads and which tails. The assumption was that coin flips and all other seemingly random events in the physical world were actually deterministic. Randomness was thought to be an illusion based on ignorance — as was the apparently unidirectional flow of time.

In undermining that assumption about the arrow of time being a bogus perception on our part, Prigogine likewise explodes the parallel assumption that coin flips are deterministic events. According to him, crucial events in the evolution of "nonintegrable" systems subject to Poincaré resonances are truly inexplicable except by reference to the laws of chance.


The other principled way to treat the "what decides" question is to decline to settle for the supposition that the choices made by systems at key crossroads of instability that stem from to Poincaré resonances have no further explanation outside the laws of chance.

What if there is a God behind these chance events?

According to Prigogine, if I read him aright, bifurcation points of dynamical instability match up with regions of "phase space" wherein chaos prevails (see p. 41). "Phase space" is an abstract realm made up of all the possible points on a hypothetical graph plotting the conceivable states of a system. These conceivable states in phase space are defined in terms of the instantaneous locations and movements of its constituent particles.

Under classical assumptions, at every individual moment the system as a whole occupies a single point in phase space. Since the system is "dynamical," it is subject to change over time. As it changes, in the classical view it traces out a "nice" (i.e., deterministic) trajectory, which amounts to a single, crisp line or curve through phase space.

In Prigogine's revision of the classical view, that single point in phase space has to be replaced by a "cloud" of possible points describing the system state probabilistically. The classical trajectory becomes an "ensemble" of possible trajectories through phase space. The "ensemble" can under some circumstances be viewed as a "random" trajectory, rather than a "nice," deterministic one. Under these circumstances, the system is chaotic. "Random" trajectories "wander erratically through regions of phase space" (p. 41). The future states of the system become exponentially less predictable as time marches on.

According to Prigogine, chaos, even when it is not "fully developed," can be associated with points in phase space that are "characterized by resonances" (p. 41). It is not clear to me exactly how closely the behavior of dissipative systems at resonance-engendered bifurcation points matches the "randomness" of chaos, but I assume the path the dissipative system "chooses" at such points of instability equates to a reemergence of order out of chaos.

Metaphorically, then, the choice could represent what naturally happens when God "stills the seas" of chaos.

The World as Music

The End of Certainty by Ilya Prigogine

Ilya Prigogine's 1997 book The End of Certainty: Time, Chaos, and the New Laws of Nature, which I last discussed in Seeds of Surprise, Part 2, leads me to the fanciful speculation that the world is basically made of music.

Consider this admittedly nearly impenetrable passage from pages 122-124:

We can now introduce the effect of Poincaré resonances into the statistical description of dynamics. These resonances couple dynamical processes exactly as they couple harmonics in music. In our description, they couple creation and destruction fragments ... which lead to new dynamical processes that start from a given state of correlations ... and eventually return to exactly the same state. ... [T]hese dynamical processes are depicted as bubbles ... .

These bubbles correspond to events that must be considered as a whole. They introduce non-Newtonian elements [into the theory of dynamical systems in physics] in that no analogue of such processes exists in trajectory theory. Such new processes have a dramatic effect on dynamics because they break time symmetry. Indeed, they lead to the type of diffusion that had always been postulated in phenomenological theories of irreversible processes ... .

I'll try to translate. The book talks about how Prigogine, the now-deceased Nobel laureate, had spent many years in search of ways to revise the basic theoretical understandings of physics inherited from Newton and, more recently, from the original investigators of the quantum theory of subatomic particles in the early 20th century. Their explanations found time's seeming directional flow, past to future, to be an illusion not borne out by the mathematical equations at the heart of their theories.

Prigogine, resisting that, revises Newton et al. by representing the motion of physical particles with "probability distributions" rather than hard-and-fast "trajectories." In the branch of physics known as thermodynamics — the study of energy as it relates to heat — one of the principal concerns is figuring out how heat and other forms of energy derive from the motion of molecules, particularly in a gas such as the earth's atmosphere in which the moving molecules are continually bumping into one another. The higher the temperature of the gas, the faster they speed about, and the more often they collide.

Colliding molecules in the real world, Prigogine says, do more than respond to heat sources. They establish persistent "correlations" with one another. This is a fact which scientific models based on individual trajectories lose sight of, and so it is generally ignored in the equations of standard theory.

I'll go over that again. Once two particles have collided, what happens to them next happens in part because they are now at least temporarily something of an "item," no longer totally independent of one another. If one them has yet another collision with a third particle, the motion of all three particles is thenceforth correlated. This is a fact models based on individual trajectories lose sight of, and so it is generally ignored in the equations of standard theory.

Correlations of particle motion can also be destroyed by ensuing events as well. Prigogine's "creation fragments" and "destruction fragments" are shorthand ways of referring to such beginnings and endings as they apply to correlations.

So are his "bubbles." They refer to the fact that a correlation's creation event (think of a bubble being blown) is thereafter "coupled" to its eventual destruction (when the bubble pops). The correlational bubble's very ability to exist, in turn, derives from a "Poincaré resonance."


In the mathematics of probability distributions, the location and motion of particles are expressed in terms of waves, and waves tend to have Poincaré resonances when they combine. Light (when it is not considered to be made of particles called photons) is made of waves. A vibrating guitar string produces audible waves in the air. The air molecules move in such a way that their collective density increases or decreases in proportion to the amplitude of the wave at any particular point in space and at any given instant of time.

Any physical thing that oscillates or vibrates, furthermore, will produce a wave whose wavelength, the distance between two adjacent crests of a wave, is inversely proportional to the frequency of vibration.

Prigogine abstracts from the mathematics of acoustical waves in applying the same sorts of equations to Newtonian dynamics and to thermodynamics as well. The equations used by Prigogine feature such constructs as "wave vectors" and "plane waves."

A wave vector is an algebraic term that shows up repeatedly in the equations; it is inversely proportional to wavelength. A plane wave, another mathematical construct, is a periodic function of the wave vector, and also of the spatial coordinate that specifies what position or location in space is being referred to. As the focus of attention moves to different positions in space, or as the wave moves outward from its source, the plane wave wiggles up and down between its maximum and minimum values. These maxima and minima depend on the wave amplitude or height ... which is also a function of the wave vector. (Yes, it gets complicated!)

Now, it's important at this point to realize that mathematical constructs such as wave vectors and plane waves are treated by Prigogine as if they were also physical things. In essence, he is suggesting that we ought to think of the pathways that moving physical particles trace — their trajectories, in old-fashioned physics — as no longer quite as real as the plane waves from which these pathways can be mathematically derived.

When plane waves of various wavelengths are superposed — stacked on top of one another — they wind up either increasing or decreasing the composite wave amplitude at any given coordinate in space, depending in whether their "interference" is constructive or destructive. If constructive interference is set up in just the right way, the otherwise cloudy, probabilistic locations of molecules in motion — making tracks as fuzzy as a vapor trail of a jet airplane — can resolve to distinct localized trajectories.

In a crucial special case, the superposed waves reinforce one another in such a way as to establish a "resonance," much as a violin's body resonates with a vibrating violin string and amplifies its sound. This is where things get really interesting. As physicist-mathematician Henri Poincaré (1854-1912) discovered, when systems resonate, key terms in the Newtonian equations that model them tend to develop "dangerous," verging-on-zero denominators, resulting in the inability to use these equations to calculate real-world trajectories.

Prigogine turns this minus into a plus by using Poincaré resonances as a springboard to a mathematics of dynamical systems which avoids dependence on trajectories and thereby succeeds in "breaking time symmetry." Prigogine's math gives scientists for the first time a principled way to see how dynamical systems can — or, indeed, must — evolve over the course of a time in a mathematically irreversible direction.


But what intrigues me personally is not so much the breaking of time symmetry, in the sense of establishing the flow or "arrow" of time as a valid physical reality. It is the admittedly rather fanciful notion that the world is basically made of music. That is, if Prigogine is right and "resonances couple dynamical processes exactly as they couple harmonics in music," then we are entitled to imagine the stuff of the universe as the same as the stuff of music.

The stuff of music boils down to composite audible waves succeeding one another in an orderly fashion over time. Each instrument or voice produces composite tones with harmonic overtones built atop a fundamental note frequency. When a piano sounds middle C, its struck string also sounds the C above that, and the G above that, and any number of higher harmonic frequencies which superimpose to give the piano its distinctive timbre or tone color. Resonances created by the sympathetic vibrations of the wood of the piano and the strings of other, unstruck notes add to the richness of the sound.

Then the pianist strikes the next note, and the stuff of the music instantly changes. Over time, a melody plays out which, thanks to the craft of the composer, usually ends by returning to the tonic note of whatever key the piece started out in — much as the "destruction fragment" of one of Prigogine's dynamical "bubbles" is coupled to its associated "creation fragment."

In music, resonances are what create timbre or tone color. They can also break glass. In Prigogine's physics, resonances break time symmetry. If there were no Poincaré resonances, the standard equations which seek to define the hard-and-fast trajectories of motion and change would serve scientists just fine ... and those equations show no arrow of time. So they don't really show how dynamical systems that operate far from thermodynamic equilibrium evolve unidirectionally with respect to the flow of time.

All living organisms, as well as certain physical and chemical systems that don't qualify as living, are far-from-equilibrium dynamical systems. These systems, which Prigogine labels "dissipative" because they take in and dissipate energy and matter as they hold themselves far from equilibrium, are prone to crises of instability — "bifurcation points" — at which their trajectories fork. Which fork they take depends on minute fluctuations that are seemingly due to chance. Each time a branch is taken, there is new, more complex order in the system.

Reading between the lines, it looks as if bifurcation points in dissipative systems correspond to Poincaré resonances in the more general theory of dynamical systems.

In my earlier posts in this series, I called these bifurcation points "seeds of surprise" because it can't be foretold with certainty which of two or more forking branches a system will follow as it finds its way out of an evolutionary instability. What if each bifurcation point corresponds to a Poincaré resonance? According to the interpretation at this web page, resonances are points on a graph "where the equations [being graphed] intersect in such a way that one of the members is divided by zero." Here, "the equations" could be taken as those which describe a path leading to or away from a bifurcation point. At the bifurcation point itself, there is a mathematical discontinuity such that the various equations have different solutions — another way of saying that crucial terms or "members" in them turn out to be "divided by zero." As a result, the standard equations can no longer be used to predict what the system will do.

Events transpiring at bifurcation points qualify as "divergences due to resonances," at least in my understanding of Prigogine (see p. 40). They also qualify as divergences due to fluctuations (see pp. 68ff.). That would seem to imply that fluctuations are due to resonances. Writ large, that would seem to mean that key events in the evolution of life on this planet, wherein new forms of order arose over time, came about by virtue of the intrinsic behavior of the stuff of music!

Seeds of Surprise, Part 2

The End of Certainty by Ilya Prigogine

I'll take another go at setting forth some of the ideas in Ilya Prigogine's excellent 1997 book The End of Certainty: Time, Chaos, and the New Laws of Nature — which I first discussed in Seeds of Surprise, Part 1 — along with my ideas about those ideas.

One of Prigogine's notions is that, inasmuch as atoms are made up of particles and these particles are in motion, they are dynamical systems. As such, the motion of electrons and other particles in an atom possesses an "arrow of time." So there ought to be a way to mathematically model the quantum behavior of subatomic particles such that time appears in the equations as an irreversible variable.

In standard quantum theory, time is a reversible variable. The equations normally used apply equally well to forward-directional time as to reverse-directional time.

In fact, the forward and reverse directions of the flow of time are not apparent in the standard quantum equations. Accordingly, at least as far as quantum phenomena are concerned, many theorists attribute the arrow of time that we experience, to the effects on a quantum system of its being observed, as scientists study it from the outside.

When scientists try to measure the position of an electron, they can pinpoint that position only if they abandon hope of also knowing the electron's momentum: its velocity in a particular direction, times its mass. Hence measuring the electron's position is incompatible with measuring its velocity, and observing its velocity makes it impossible to know its position precisely.

Think of trying to photograph the end of a horse race. If a flash photo is taken right at the finish line, it freezes the horses' motion. The stop-motion photo thereby pinpoints which horse's nose crossed the line first — i.e., the relative positions of the horses at the finish of the race. On the other hand, if a non-flash, time-exposure photo is taken, using a long exposure time, the lead horse's nose will be but a blurred streak in the picture. The form and length of the streak will be a function of how fast the horse was moving — its speed, which relates to its velocity.

Therefore, how you set up the camera determines whether it captures position or momentum. Either alternative can be measured with certainty only if the other alternative is sacrificed.

The stop-action alternative which pinpoints the horses' relative positions has no time dimension whatever. The blurry-streak alternative which records the horses' speed does not, meanwhile, tell you in which direction the horses really moving. Common sense tells us, it is quite true, that horses run "forward," i.e., with their noses out in front of the rest of them. But strictly speaking, the camera does not know this. The blurry streak formed by the moving equine noses would look just the same if the horses were being transported backward!

So either act of photographic commemoration, stop-action or blurry-streak, lacks an arrow of time. We, of course, supply one when we look at and interpret the blurry-streak photo.


Similarly, the standard equations of quantum theory lack an arrow of time. Yet various interpretations of the theory exist in which time irreversibility is imposed by virtue of acts of external observation. I discussed one such interpretation, which exalts the importance in quantum phenomena of outside observers' choices as to what to observe and how, in my earlier post Genesis by Observership.

In a well-known thought experiment in quantum physics, that of Schrödinger's Cat, a live cat is imagined to be put in a closed box with a mechanism that may (or may not) indirectly cause the cat's death by at some moment releasing (or not releasing) a radioactive particle as a trigger to the breaking open of a vial of poison gas. In our imagination, we let the experiment run unobserved for a period of time. Then, after a while, we open the box and see whether the cat is alive or dead.

It is only at the moment we do so, Schrödinger showed mathematically, that the outcome becomes etched in stone ... even though the release of the fateful particle and the resulting death of the cat happened (if at all) earlier in time! Inconceivable as it seems, apparently no other understanding is consistent with the laws of standard quantum mechanics. Our observation of the event alone is what makes that (earlier) event certain.

Thus, in this interpretation of the mathematics of quantum theory there can be no arrow of time, except under conditions of external observation.


Prigogine objects to that approach. He seeks (and finds) an alternative mathematical formulation of quantum theory which builds in the arrow of time from the get-go. In return, it trades in the (limited) certainty which the standard equations can yield and replaces those equations with probabilistic formulas that are unable to pinpoint which event, out of a multitude of possibilities, is actually so.

The standard equations can yield certainty in the limited way a photo of a horse race's finish can: one variable can be pinpointed with certainty by sacrificing another. But those standard equations lack an arrow of time. That leads to, among other paradoxes, the one concerning Schrödinger's imaginary cat in which a seemingly later "photo" — a visual observation, that is — determines a seemingly earlier event.

Prigogine's equations remove that proneness to paradox in quantum theory at the expense of making certainty fundamentally impossible at the level of subatomic phenomena. But his equations do successfully account for the arrow of time.

More than that, they generalize to other types of systems involving physical "stuff" in dynamical motion. Such systems include thermodynamic systems, with their vaunted entropy, and dissipative systems which self-organize and evolve in seemingly the opposite direction. They also include systems whose futures are in thrall to "deterministic chaos."

Dissipative systems are quite important to us. After all, all living things are dissipative systems. So, too, are various kinds of non-living physical and chemical systems, at least under certain circumstances.

Prigogine's equations involving "probability distribution functions" are basically the same for all these kinds of dynamical systems.


My thought about all this is that those other, non-quantum kinds of dynamical systems might, like quantum systems, have alternative mathematical models in which acts of external observation would successfully change probabilities into certainties — just as they do in the study of quantum phenomena, at least according to some interpretations of the relevant theory.

For brevity, I'll refer to such hypothetical alternative mathematical models as AMMs. If an AMM exists for, say, a dissipative system, certain things would likely result. One of them is that the choice of which new stable state a dissipative system enters at a dynamical "bifurcation point" is not a coin flip but reflects an act of external observation. The event which results from the bifurcation in turn can depend on an act of choice made by the outside observer (see Genesis by Observership for more on the astounding consequences of observational choices).

If an AMM exists for a living, evolving, dissipative system, it just might be that the bifurcation events of its evolutionary pathways are determined by choices made by God, who sees all worldly events from outside the context of irreversible time. Hence, such an AMM would be expected to represent time as reversible, just as do the equations of standard quantum theory. Accordingly, such an AMM would violate one of Prigogine's foundational assumptions: his preference for a mathematical model in which time is irreversible.

Moreover, if such an AMM exists for a living/evolving/dissipative system, that AMM would very likely allow the chance-based alternatives intrinsic to a mathematical model such as Prigogine's to be supplanted by descriptions of events which are knowable (at least in limited fashion) as certainties.

At the same time, such an AMM would not be wholly deterministic. There would be types of systems experiencing types of situations in which the future is not set. There would be an element of surprise in these systems' future evolutionary paths that could not even in concept be eliminated by our gaining more thorough knowledge of their present states.


Such an alternative mathematical model would not invalidate Prigogine's, any more than Riemannian geometry invalidates Euclidean. Prigogine's model succeeds in accounting in a very general way for the arrow of time as it affects many different types of systems which are usually treated as separate. If you are interested in a purely physical-world approach, it looks to me (as admittedly a layman) that Prigogine's mathematics are spot on target when it comes to explaining time's irreversible flow.

The alternative approach I am suggesting seems to require that there be an observer outside the world system whose observational choices we experience as the workings of chance.

Prigogine's original description of bifurcation points in dissipative systems is a case in point. It involves chance fluctuations, which "choose" which path the systems will take out of states of instability. His probabilistic, statistical mathematical model replaces this description by one in which chance (another name for probability) is involved at a fundamental level, at all times, not just at discrete points of instability.

The Prigogine model accepts both chance (or probability) and the arrow of time as brute facts or bottom-level realities in our world. The AMM I seek sees them both as artifacts of the choices made by God in his "observation" of our otherwise freely evolving world.

Seeds of Surprise, Part 1

The End of Certainty by Ilya Prigogine

Fritjof Capra's The Web of Life — see my recent Organicity series — has philosophical implications that chime well with Ilya Prigogine's excellent 1997 book The End of Certainty: Time, Chaos, and the New Laws of Nature.

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.

The Ordained and the Organic

The Web of Life by Fritjof Capra

Fritjof Capra's book The Web of Life, as I said in earlier installments in this Organicity series, has spiritual implications.

The spiritual mind, I suggested in The J.K. Rowling Effect, can be likened to a type of self-organizing system with a complex "phase portrait." Accordingly, it is prone to critical "points of instability" or "bifurcation points." I imagine that one of these spiritual bifurcation points corresponds to being born again, in evangelical Christian terms, or to the Dark Night of the Senses spoken of by Catholic mystics like St. John of the Cross.

The person who has experienced this crisis and crossed into new (for him or her) spiritual territory can easily believe that whatever is not adulterated by man's original sin is expressly ordained by God. The idea that all good things are divinely ordained chimes with the story of creation in the Book of Genesis, chapter 1, in which every living creature appears on the scene at God's immediate behest. No wonder those who live in this particular spiritual frame of mind have trouble with Darwin's theory of evolution.

Another point of spiritual bifurcation can follow later on: the Dark Night of the Soul, in which the seeker eventually ceases to be able to maintain that mistrustful, standoffish, all-things-must-pass attitude toward the events of this world. Now the material world exerts its own pull upon the spiritual mind, leading, in famous cases like that of Thomas Merton, to a Zen awakening.

I cannot claim a Zen awakening, but I have developed a deeper appreciation for the organic, as opposed to the ordained, aspects of this world. In fact, I think that the ordained/organic dichotomy is one of the key differences between two major basins of attraction of the spiritual mind.


A basin of attraction in a self-organizing system, such as I hypothesize the human mind to be, is like a drainage basin in the geography of a continent. In such a drainage basin, a river, lake, or some other prominent body of water toward which rainfall naturally gravitates, corresponds to an "attractor" in a self-organizing system.

For example, any rain that falls in the United States east of the Continental Divide and west of the Appalachians winds up in the Great Lakes or Mississippi River. Rain to the west of our mid-continent drainage basin heads for the Pacific Ocean. Rain that falls on the East is Atlantic-bound.

Likewise, in self-organizing systems there is a "virtual landscape" in which multiple basins of attraction exist, each comprising a discrete set of the totality of available system states. In any particular basin of attraction, some of the possible system states in that basin form an "attractor" toward which all the other system states in that same basin of attraction inexorably flow. If the system happens to be operating within a certain basin of attraction, but not on that basin's attractor, it will move to a state on the attractor itself. Once a state on the attractor is arrived at, all successive system states are also on the attractor.

Small or large perturbations can cause the system to jump unpredictably from one state to another. The new system state may or may not be in the same basin of attraction as the old. If the system is a reasonably stable one, small perturbations will usually leave it in the same basin. Larger perturbations can put even a stable, non-chaotic system in a wholly different basin of attraction.


The attractor in that new basin may be a "strange" (i.e., chaotic) one. Strange attractors are associated with mathematical "chaos." Some or all of the attractors in the abstract topography of a system at any moment in time can be "strange" attractors. In fact, some systems have just one single attractor: a "strange" one.

Systems whose attractor is "strange" ans whose future path in moving from one state to the next is chaotic will cycle back around and repeat an already-visited state only after visiting every other state available to the system. In chaotic systems of great complexity in which the number of available states is great, the time needed to cycle around and repeat a state can be a large multiple of the age of the universe.

Systems in chaos are extremely sensitive to initial conditions. Any imprecision, no matter how tiny, in measuring the initial state of a system will render predictions about future states of the system increasingly unreliable as time goes on. What's more, the degree of predictive unreliability increases exponentially. This is why it is impossible to predict the weather more than a few days in advance.

Although, it is impossible to predict chaotic systems' long-term behavior, this behavior is nonetheless deterministic. That is, the system's initial conditions do in fact determine its entire future path. It's just that there is no computer in the world capable of doing the necessary computations precisely enough (or, for that matter, swiftly enough) to arrive at an accurate prediction.

Exogenous changes in the system's parameters can put an otherwise non-chaotic system in a basin of attraction associated with a strange attractor: chaos can come out of order, seemingly. But Capra makes clear that a chaotic, strange attractor is in fact orderly. Just because we can't predict the system's future doesn't mean that its future is random. It is in fact as deterministic as that of a system in a basin of attraction whose guiding attractor is a "point attractor" (a single destination state) or "periodic attractor" (an endless loop of repeating states). Barring external perturbations or changes to the fundamental parameters of the system, the future path of the system is set in stone.

Moreover, a self-organizing system's virtual landscape can exhibit change over time. Its configuration of basins and attractors can be revised, sometimes greatly, by even small changes in the numerical parameters of the nonlinear mathematical equations that describe the system (see Web of Life, pp. 135-137). This is another reason why systems that start in a stable and orderly basin of attraction can end up in another which may be chaotic, or vice versa.


But there is yet another source of indeterminacy and unpredictability in dissipative structures such as all living systems represent. This one interests me more than that associated with "chaos" and "strange attractors." The indeterminacy that occurs at the system's "bifurcation points" is apparently one that couldn't go away even if our computer models were to become infinitely fast and infinitely precise.

Capra discusses this topic in chapter 8, which deals primarily with the late Nobel laureate Ilya Prigogine's theory of "dissipative structures." A dissipative structure is one through which there is a continual flow of matter and energy. The strength of the matter-energy flow determines whether the system needs nonlinear equations to model it, as opposed to just linear equations. In nonlinear equations, corresponding to a strong or swift flux of matter and energy through the system, all of the equations' variables are raised to the power of two or higher. Hence there are multiple possible solutions to the set of equations by which the dissipative system is modeled.

A system whose modeling equations are nonlinear maintains itself far from thermodynamic equilibrium, the state of maximum entropy. "Entropy" is the degree to which a system has "run down" toward the minimum possible level of order, as dictated by the Second Law of Thermodynamics. For example, if you have a container with two compartments, one containing nitrogen and one oxygen, and you remove the barrier separating the compartments, the molecules will tend to mix into a single uniform distribution of nitrogen and oxygen.

Given sufficient time, that expected result — minimum order, maximum entropy — is almost certain to eventuate. It is the result of the motion of the gas molecules, and it happens faster when the gases are heated than if they are cool.

Although it may not be technically correct to put it this way, I think of the behavior of far-from-equilibrium dissipative systems, including living organisms, as "exporting" entropy: the heat energy they give off adds more to the disorder of the external environment than the amount by which the system itself reverses its own entropy. Accordingly, dissipative systems manage to take themselves in the opposite, anti-entropic, direction, at the expense of increasing the entropy in their surroundings. The further from thermodynamic equilibrium the system is, to more entropy it exports to its environment.

But a dissipative structure is typically able to arrive at any of multiple possible stable states far from thermodynamic equilibrium. The farther from equilibrium it is, the more nonlinear it is, and the greater the number of possible solutions there are to its set of equations (see Web of Life, p. 182). Capra describes it this way: "New situations may emerge at any moment. Mathematically speaking, the system encounters a bifurcation point in such a case, at which it may branch off into an entirely new state."


Here is where things get really fascinating ... and a bit confusing. At each bifurcation point, the system must "choose" — metaphorically speaking, since the system generally lacks intentionality — one of two or more possible paths. Which path it will choose is not knowable by an outside observer. The "choice" cannot be predicted, much less controlled, by an external agent. As Capra says, "There is an irreducible random element at each bifurcation point" (p. 183).

Even so, the branch of the path which the system will "choose" for its future trajectory at any given bifurcation point "depends on the previous history of the system," Capra says (p. 182). Unlike systems that can be modeled by linear equations, dissipative systems operating in the nonlinear range do not "forget their initial conditions."

This is, admittedly, a point which confuses me. My confusion comes to a head on page 191. In one place on the page, Capra says:

A bifurcation point is a threshold of stability at which the dissipative structure may either break down or break through to one of several new states of order. What exactly happens at this critical point depends on the system's previous history. Depending on which path it has taken to reach the point of instability, it will follow one or another of the available branches after the bifurcation.

At another point on the same page he says:

At the bifurcation point, the dissipative structure also shows an extraordinary sensitivity to small fluctuations in its environment. A tiny random fluctuation, often called "noise," can induce the choice of path. Since all living systems exist in continually fluctuating environments, and since we can never know which fluctuation will occur at the bifurcation point just at the "right" moment, we can never predict the future path of the system.

So, which is it? Does the system's history "choose" which branch the system will take, or does a random fluctuation make the crucial choice? Perhaps it is a combination of both, involving some probability function. All I can really say is that Capra is, to my mind, uncharacteristically unclear about the interplay of history and chance as they affect dissipative systems at critical bifurcation points.


An example of a linear dynamical system is that of a planet orbiting the sun. The shape of its elliptical orbit never varies, which means that we can run the imaginary "movie" of its motion backwards and forwards in time, with perfect predictability as to where the planet was on any given date in history and as to where it will be on a given date in the future. Its "initial conditions" — such as where the planet came from in the first place — make no difference.

Not so with nonlinear dissipative systems operating far from equilibrium. At their bifurcation points they take this branch or that as a result of the details of their unique histories and/or small random fluctuations in their environment. The most minute differences in their initial conditions or in their environmental circumstances will bias them in favor of branch A and away from branch B.

To me, their imperviousness to external prediction and control and the absolute autonomy of their unique histories and futures make their behavior trajectories organic rather than ordained.


Yet common sense tells us that we living beings — however autonomous, however unique, however unpredictable — exhibit an awful lot of expectability. Sons and daughters tend to look like their mothers and fathers. Identical twins tend to look like each other. Swans don't come from ducks' eggs.

We must all be something like fingerprints. Every human finger can be expected to possess a fingerprint. Yet each fingerprint is unique. Even identical twins have different fingerprints.

Fingerprints show up on fingertips, obviously, at some stage of human embryonic development. A human being in utero develops from a fertilized egg cell. During the process of development, the zygote — the fertilized egg — divides and makes two cells, then four, then eight, and so on. In the early embryonic stages, there are no detectable differences among the cells. Soon enough, however, the cells start to differentiate and form distinctive tissues and organs.

Presumably, if Prigogine's ideas about dissipative systems and their bifurcation points hold for embryos, the embryo is continuously "choosing" certain branches along its own developmental trajectory and discarding other equally available branches. Somehow, some combination of these "choices" results in the formation of ten unique fingerprints ... while at the same time (if the embryo is normal) guaranteeing that there will be ten fingers and ten prints.

Thus is there a mix of unpredictability (the distinctive loops and whorls the individual fingerprints will wind up having) and expectability (that there will indeed be fingers, and fingerprints).


I imagine that, as with other aspects of emergence in living dissipative systems, the human embryo has its associated "phase portrait," its own abstract topography of attractors and basins of attraction which account for its potentialities as it develops. This topography produces both comforting expectability and unpredictable uniqueness.

I also imagine that the same principles apply to the spiritual mind within each unique human being. Our spiritual paths are indeed individual and unique. Yet, at the same time, there are certain expectable patterns in our spiritual life. I suggest the commonality and expectability in our spiritual paths are due to certain basins of attraction universally present in the human spiritual mind.

Let me suggest three such basins of attraction. The first is the one characterized by not being "spiritual" at all, by having no particular religious beliefs, etc. This, canonically, is the state one starts out in and then leaves behind when (say) one is "born again," after which the former, non-spiritual state is typically thought of as having been a "godless" one.

The second basin of attraction of the spiritual mind is that in which there is a profoundly God-oriented (shall we call it?) "steady state" toward which the experiencer is pulled by an "attractor" which is, quite naturally, the experiencer sees as God or Christ or the Holy Spirit. (I am speaking in Christian terms; other religions have their corresponding equivalents.) The spiritual mind that is in thrall to this attractor wants more than anything else to experience God as ordaining every blessed outcome in this world.

The third basin of spiritual attraction corresponds to Merton's Zen awakening following his Dark Night of the Soul ... and (I hope) to my own spiritual embrace of the organic nature of life. Once one's mind is in that third basin, or so my feeling goes, it becomes important to see the things of this world as emerging out of organic, self-organizing principles that guide the evolution of dynamic, ever-changing, living systems. The question of worldly outcomes ordained directly by God becomes much less important when one is in this third spiritual basin of attraction.

Some questions arise: where do these three spiritual basins of attraction come from? By virtue of what occurrences do we move from one to another? Is the progression from seeing the world as "godless" to seeing it as "divinely ordained" to seeing it as "organically emergent" typical, and if so, why? Finally, is there any way to bridge the dichotomy between the last two, such that God can be seen as influencing or even directing the emergence of organic novelty in this world?

More about these questions in later posts ...