There once was a man who said, “Damn!
It is borne in upon me I am
            An engine that moves
            In predestinate grooves,
I’m not even a bus, I’m a tram.”

            – Maurice Evan Hare (1905)

PQR shows how we can reconcile two diametrically opposed philosophical concepts: free will and determinism.

Free will refers to the concept that the self (e.g., “you” or “me” – whatever that means) has the ability to make its own choices in the present moment. Although our choices may be influenced by pre-existing factors (such as our physical bodies, our past experiences and our environment) these factors do not determine our actions: in the last analysis we as individuals make our decisions for ourselves.

On the other hand, determinism (or predestination) refers to the concept that everything that happens in the Universe (including the choices its inhabitants make) is ordained in advance. In a deterministic Universe, someone armed with an exact knowledge of its present state and its rules of operation could, in theory, predict its future state precisely at any given time.

Thus, determinism would appear to be incompatible with free will. Our thoughts and lives are based on the belief that we have freedom of choice, yet how can we have this freedom if our actions are predestined? No one wants to accept the idea that we are just cogs in some vast machine. Furthermore, our human society requires us to take personal responsibility for our actions, and a belief in determinism would, in effect, represent a denial of our responsibility. (“Officer, the Universe made me do it.”)

Our intuitions and our daily experience strongly suggest that we do indeed have free will. Although someone who knows me closely may often be able to predict my actions, I am a being with free will and so I always retain the capacity to surprise them. For instance, in a sandwich shop I can ask for anything that takes my fancy. In practice, of course, my choices are limited as I will tend to avoid foods I dislike, and there would be no point in my ordering, say, a helicopter (although I might well ask for a submarine). Thus my regular sandwich maker may be able to predict my order with 90% accuracy, but this prediction is, in the last analysis, really no more than a guess. However predictable I may seem, I can always surprise the sandwich maker – even if in practice I rarely do so.

Thus, empirically, free will can be viewed as our ability to surprise others (and perhaps, on occasion, even ourselves). Without this ability, we would indeed be mere robots carrying out a predefined program, just like the man who said “Damn!” in the 1905 poem above.

Curiously enough, 1905 was also the year of Albert Einstein’s famous paper theorizing that light consists of a finite number of indivisible packets (or quanta) of energy. This paper led to the development of quantum theory, which has largely supplanted the concept of determinism. Quantum theory says that we can never obtain an exact knowledge of the state of the Universe, and that its rules of operation are not deterministic but probabilistic. In other words, events at the quantum level cannot be measured or predicted precisely, but exist only as a range of statistical probabilities. Quantum theory says that our Universe is not deterministic (or at least, that we can never know what happens at its most fundamental levels).

PQR theory, on the other hand, says that our Universe is deterministic, being completely defined by the natural numbers. As I will explain elsewhere, PQR can be reconciled with the well-established results of quantum mechanics by redefining (a euphemism for abandoning) our existing ideas of the space-time continuum. But here I will explain how PQR is consistent with both the concepts of free will and determinism discussed above.

To restate the problem: how can we experience free will in a deterministic universe? In other words, how can we be free to choose our actions if they are predetermined for us? In principle, PQR gives us an exact knowledge of the Universe – past, present and future – so with a powerful enough computer we could, in theory, calculate the Universe’s configuration at any given point in time. In fact, to make the calculations a bit simpler, we need only compute the photons of visible light that impinge on a small area in a brief interval of time, and we now have a “virtual camera” that can take photographs of the past, present and future.

Supposing we project this camera a week forward into the Earth’s future, and it sends back a picture of me, sitting on the beach in Hawaii. That now means I have to go to Hawaii and sit on the beach next week, whether I want to or not. Of course, you could argue that the camera would not send back such a picture unless I was already going to go to Hawaii – but that does not solve the problem. Either way, I no longer have the freedom to avoid the beach in Hawaii.

The resolution of this seeming paradox lies in Gödel’s Theorem, a very subtle and powerful result in mathematical logic. Without getting too technical about the details, Gödel’s Theorem says that any system of logic that is powerful enough to include the rules of arithmetic must either contain a contradiction (a statement that is provably both true and false at the same time), or it must contain a statement which, although true, cannot be proven using the system. (In practice, there will be lots of these “true but unprovable” statements, not just one.)

Applied to the natural numbers, this theorem tells us that there are some true statements about the natural numbers that can never be proven mathematically. And since PQR theory says that the Universe is equivalent to the natural numbers, we can also apply Gödel’s Theorem to the Universe (a logical system which is powerful enough to include not only the rules of arithmetic but also the mathematicians who formulated them). This tells us that there are true statements about the Universe (i.e., facts) that are unprovable from within it. (Either that, or the Universe contains contradictory facts, in which case I can prove that you are not reading these words.)

Thus PQR tells us that we cannot know every true fact about the Universe, because we cannot prove every true statement about the natural numbers. (In fact, mathematicians believe that the vast majority of statements about the natural numbers cannot be proved either true or false.) In practical terms, this means that the calculations necessary to predict even the simplest facts about the future would probably be too voluminous and complex, even for all the world’s most powerful computers working together. And this problem can’t be solved by building bigger and better computers: although the future may be fixed, the Universe is simply not big enough or complex enough to hold a computer powerful enough to predict it. We may know some facts about the future with relative confidence (the sun will rise tomorrow), but don’t expect to compute which horse will win tomorrow’s race. (At least, don’t expect results before the race is over. I fear, or rather I hope, that PQR will turn out to be like the early numerical models for predicting the weather, which needed three days in order to compute a reasonably accurate two-day forecast.) We may be able to compute some precise facts about the Universe, but I believe these will always relate to the past, and probably only to the very distant past.

We can now see the fallacy in the virtual camera idea discussed above. It assumed the existence of a “powerful enough computer,” and Gödel’s Theorem strongly suggests that we cannot build such a computer in our Universe. Although our futures may be fixed, they are unknowable to us. From where we stand, the future is not immutable, as we can change it by our actions. So for all practical purposes we can pretend the future does not yet exist. In other words, even if our free will is only an illusion, it’s a very convenient one.

So viewed from within, the Universe appears to grant us free will. On the other hand, if we could view it from outside, it would appear deterministic. But of course we can’t do this, because there’s no way to get there, no place to stand, and nothing to see anyway (as no photons go there). The bottom line: Within the Universe, our free will is real. It only becomes an illusion when we try to consider the Universe from the outside. That is the ultimate paradox of reality.

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