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!