The following appeared originally in a writers discussion forum, in response to a question on whether a sufficiently powerful computer could predict everything a human would do in advance, assuming everything was deterministic–and whether quantum interdeterminacy would interfere with that.
What many people tend to forget is that “deterministic” doesn’t equal “predictable.” Let’s start with the classical formalism and then consider the quantum one.
The equations of classical gravity are deterministic. Once you have the initial coordinates and velocity vectors for all bodies involved, the state of the world at any time T is uniquely determined. But is it actually computable?
Try this experiment. Model a very simple star system on your computer: just one massive planet, like Jupiter, turning around the star, like our Sun. Then throw in a tiny comet on an eccentric trajectory. The question is: after how many turns around the star will the comet be ejected out of the system?
Try this several times: when rounding all computations down to the 4th place after the period, to the 5th place, to the 6th place…, to the 17th place, and so on. For most of the initial positions, the answer will be all over the board, jumping up and down and performing pirouettes: a sequence of numbers with no pattern to them whatsoever. It can jump from 5 to 113 and then dive to 47 and stay there for a while only to rise to 73 very easily. No rhyme or reason.
The three-body problem is as simple as it gets, but it already shows the butterfly effect. A tiniest deviation anywhere–and the end result diverges vastly over time. It is completely deterministic, yet it is unpredictable. Even with exact, God-like knowledge of the initial conditions and even with infinite computing power, it is incomputable.
Do you think the human brain is any simpler? In fact, it is the ultimate butterfly effect machine, the most complex one that we know of. In our brain, many billions of proverbial needles are standing on end every microsecond, “deciding” which way to fall.
That doesn’t mean that human behavior cannot be predicted–it often can be. Even though the butterfly effect rules in the brain, the state manifold can develop very strong “attractors” (a term in the science of dynamic systems), to which many trajectories of the system’s state converge from different initial states. For example, this is true when the subject is in the grip of strong emotions, or if some trauma is so impressed upon the psyche that it keeps influencing the entire life, or if someone is in the situation when they must strategize very logically and therefore predictably (actually, this is also a case of strong emotion–when the desire to think everything through very logically is exceptionally strong).
But most of the time, predicting human behavior is very, very hard. Even in the most cut-and-dry cases, there is always a chance that some “needle” will fall another way, with wildly divergent results. In contrast, a system that is entirely digital, however huge, is always predictable–for the very reason that it is discrete and therefore completely reproducible.
This theme is a subject of my science fiction novella Pink Noise: A Posthuman Tale, where the pink noise in our brains (meaning the noise with the fractal dimension of 1, the most complex noise in the universe) defends us from the kind of mind control that posthumans, who have uploaded their minds into the digital format, are vulnerable to–a heavy price for their immortality.
If this is true even in the classical treatment, what about the quantum one? It is actually a different side of the same phenomenon: the irreversibility of time. This would take a while to explain adequately, but here’s a gist.
Quantum mechanics has two components: very deterministic equations, which are linear and time-reversible, and the “quantum collapse” phenomenon. If you observe something, its wave function collapses, according to some probabilistic rule. The creators of quantum mechanics were very uneasy about this, since it feels like a foreign add-on to the main formalism. It doesn’t organically “join” with it but must be postulated. It even led to some superstitions about the observer’s “consciousness” being involved.
Since then, many different attempts have been made to reconcile quantum collapse with the main deterministic formalism. I follow Prigogine’s school of thought, which gave rise to the Belgian school of quantum mechanics. It’s not as well known as some of the more sensationalist ones, but I believe it’s right, because it also reconciles quantum mechanics with the laws of thermodynamics (which is a huge pain in any sort of physics where the equations are time-reversible).
A brief digression first on the nature of science in general. When we do science, we create models with the explicit purpose of approximating the nature. We test them experimentally to see if they hold up. But the key word here, for our current purposes, is “approximating.” In order to create a model that’s tractable, we always abstract some minor things away: for example, assuming that the fluid is incompressible for a theory of incompressible fluid. Granted, we can extend that particular model to arrive at a theory of compressible fluid, but then we ignore something else. And so on.
What we ignore defines the theory’s “applicability domain.” The Newtonian mechanics continues to be used within its applicability domain because its results approximate the real world very faithfully, within acceptable error margins–not because it’s some sort of metaphysical Truth. In general, we should hold no expectations of any theory to be absolutely true. They’re all approximations, valid to a certain error margin within their specific applicability domains.
The way that we abstract minor details away can be likened to approximating a curve by a straight line (the tangent) in the immediate vicinity of a point on the graph. In general, whenever you see linear equations, it should ring a warning bell in your mind: we’re looking at an approximation in a tiny neighborhood.
It is often thought that quantum mechanics has extended the classical mechanics’ applicability domain. That’s because, in the situations in which quantum mechanics applies, the limit of h->0 (Planck constant) yields the classical formalism. But this is myopic. Quantum mechanics is linear (and the more recent attempts to develop non-linear quantum mechanics are still in an embryonic stage). So, it’s only an approximation that’s bound to break in some limit, when we leave its applicability domain.
In fact, although many quantum effects have been discovered that are very strange and very useful, they always manifest only under conditions of extreme simplicity. Either the number of interacting particles is very small, or the temperature is very low, or some constraint is imposed that forces everything to stay in two dimensions, or some regularity in the environment (a crystal lattice is a high-temperature superconductor, for example) forces the particles into aggregate quantum states, and the like.
Prigogine’s opinion is, quite simply, that quantum mechanics breaks in the limit of complexity: specifically, for complex macro systems that are far from equilibrium. In other words, systems that have on the order of the Avogadro number of interacting particles that exhibit the butterfly effect.
Many people implicitly believe that any macro system can be reduced to an aggregate of quantum states it is composed of. If only we had a powerful enough computer, we could solve the Schrodinger equation for an Avogadro number of particles, no prob. And maybe that would work for a system near equilibrium (like cold and hot water quietly mixing at rest). But for a system far from equilibrium (like a brain or even a weather system), no one will ever accomplish that. If they try, they’ll run into the kind of problems that render even infinite computing power useless.
So, to Prigogine, quantum collapse occurs when a quantum system interacts with a complex system far from equilibrium (the observer), which breaks the time-reversible, linear quantum formalism. The world is time-reversible only under conditions of extreme simplicity (and the recently-in-vogue time crystals are of that sort, too)–and very probably only in approximation, anyway. Interaction with the observer makes time irreversible again.
The seeming discreteness of quantum mechanics is likely an artifact of the same sort, only holding within its applicability domain. The spectra of Hamiltonian operators in non-linear quantum mechanics don’t have to be discrete at all. Indeed, discreteness would have precluded the butterfly effect if not for quantum collapse (ironic, isn’t it?).
This approach to quantum collapse isn’t very popular, because people don’t like to reduce the applicability domains of their theories (and least of all, quantum physicists). But this doesn’t make it wrong.
In short, quantum mechanics does NOT contain the classical formalism entirely. They have overlapping but non-trivially intersecting applicability domains. That Venn diagram is far more complex that normally believed. And the probabilistic indeterminism of quantum mechanics is actually a different side of deterministic unpredictability of classical one.
For more on this, see a companion paper to my Pink Noise novella.