Prigozhin's premise is that the arrow of time as we perceive it is due to what he describes as a lack of reversibility, i.e., irreversibility.
The thinking goes something like this.
Suppose I have Newton's law were F = ma (force = mass times acceleration). Now acceleration is a change in velocity over some period of time.
(Think of it this way: You are driving along at 55 mph and you decide to press the accelerator to the floor for ten seconds - during that ten seconds your velocity will increase given a reasonably, non-junk car; that's your acceleration. The mass of your car is, for purposes here, its weight. So during the ten seconds the force with which you might hit something is increased, say as you go from 55 mph to 70 mph.)
Now given our scenario the laws of physics don't say whether time with regard to acceleration is moving forward or backward.
That's right - if time moves "forward" (positively) then our car goes faster.
But if time goes "backward" (negatively) then our car slows down.
Now as vaunted as Newton's law is no where does it require time to move only forward.
This is not only true of Newton's law but also Quantum Mechanics and virtually all the rest of physics.
The direction of time is not constrained to move forward.
So Prigozhin asks and presumably answers why is this? Certainly in terms of observations we never see time moving "backward" (negatively). In fact, as far as I know, no experiment has ever shown time operating in reverse.
Yet as best as today's science, physics and mathematics can describe there is no reason it can't.
Prigozhin describes time's perceived behavior as "irreversiblity."
So what does this mean?
We can take a simple example: a billiard ball being rolled on a table. Someone picks it up, rolls it, it rebounds off the sides a few times and rolls to a stop.
Now we can "model" this with a computer quite accurately. In fact, there are billiard programs available that work just like the real thing - replete with friction, physics and all the rest.
The problem is that the models used by these programs are by and large "reversible" (save for the injection of "randomness" in the program which, for billiard balls is not necessarily significant or observable so we will assume for this discussion that its not present).
So I can run the program backwards (either by design or by hand with the debugger). In any case the result will be that the ball will return to its starting point (the equivalent of time running backward).
But in reality this is not possible.
One reason is that there is a change in the thermodynamic entropy of the billiard ball "environment." Picking up the ball, rolling the ball, bouncing the ball off the edges all involve the transfer of heat.
Again this might be simulatable on the computer but Prigozhin claims in part that the thermodynamic aspects of this contribute to the irreversibility.
So what does this say about human "models" of things that simulate physical aspects, e.g., the billiard ball model or something much larger, say a climate model?
For one thing it says that these models simulate our equations for the physical aspects of these things - not the things themselves.
So we as human's have created these equations to describe things. Its little wonder then when we use them to model the actual physical world the equations operation exactly as we expect.
Unfortunately, even in the world of Quantum Mechanics, this match-up between equations and observed reality don't offer a description or insight into the true nature of what's being simulated. While the equations might predict what we see they don't tell us why we see what we see.
And clearly, through Prigozhin work, the equations must be incomplete because they are reversible and reality is not.
I think that there is also a connection here between Wolfram's "Computational Irreducibility" and something that is physically "irreversible". Almost as if Wolfram has discovered that, given certain equations and algorithms, there is a point at which you cannot peer back beyond the "irreversibility" of a given physical situation.
So we can think about the following experiment. Inside a hermetically sealed, closed, shielded box which is totally isolated from the rest of the world I place my billiard table, a ball and a some means to cause the ball to roll on the table. While I may be able to calculate exactly where the ball will end up my calculations, at least given our level of knowledge today, will remain reversible while the situation inside the sealed room will not. Not really different from the Large Hadron Collider, for example.
So what does all this say?
For one thing that "modeling," particularly of large-scale physical systems, is basically a bogus non-scientific endeavor because without some notion of irreversibility the models only simulate the researchers equations and not physical reality. In turn this means that our models don't offer understanding of why something is by definition.
For another it says that something like the pressure of a gas in a sealed container is likely governed by Wolfram's "Computational Irreducibility" because the behavior of the gas (and the molecules that make it up, their energy, momentum, and so on) is "irreversible."
What inspires me to think about this and to study authors like Prigozhin are articles like this one at Wired about human's causing global warming. From the source article "Large-scale increases in upper-ocean temperatures are evident in observational records1. Several studies have used well-established detection and attribution methods to demonstrate that the observed basin-scale temperature changes are consistent with model responses to anthropogenic forcing and inconsistent with model-based estimates of natural variability..."
Science studying science-created, fully reversible models.
Models based as well on admittedly incomplete data.
There is far more here than I can cover in a single post, but at least I have tried to scratch the surface of what it is that I am thinking. Its also difficult to present this material without all the mathematical complexities to motivate it.
Prigozhin's work is now more than a quarter century old and "The End of Certainty" was written nearly 15 years ago. It seems clear to me that his work reaches down to a fundamental level upon which all of human understanding pivots.
I am personally interested in the relationship between Wolfram's work on "Computational Irreducibility" and how it works with Prigozhin's ideas - particularly in the area of what can and cannot be "known" about computational systems.
I guess in some way's what is most "profound" about all this is how little we can know about something as simple as the output of one of Wolfram's basic automata even though we have a full and complete description of it.
If we cannot understand something that simple how can we understand something as complex as physical reality?