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Three Mechanisms for Randomness
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Posted by: Gunnar Tomasson
WOLFRAM'S THESIS
Stephen Wolfram introduces the concept of “three basic mechanisms” for “apparent randomness” on pp. 299-300 of A New Kind of Science as follows:
“In nature one of the single most common things one sees is apparent randomness. And indeed, there are a great many different kinds of systems that all exhibit randomness. And it could be that in each case the cause of randomness is different. But from my investigation of simple programs I have come to the conclusion that one can in fact identify just three basic mechanisms, as illustrated in the pictures below.
[Three pictures.]
[Comments on the three pictures.] Three possible mechanisms that can be responsible for randomness. The diagonal arrows represent external input. In the first case, there is random input from the environment at every step. In the second case, there is random input only in the intial conditions. And in the third case, there is effectively no random input at all. Yet despite their different underlying structure, each of these mechanisms leads to randomness in the column shown on the left. The first mechanism corresponds to randomness produced by external noise, as captured in so-called stochastic models. The second mechanism is essentially the one suggested by chaos theory. The third mechanism is new, and is suggested by the results on the behavior of simple programs in this book. I will give evidence that this third mechanism is the most common one in nature.
In the first mechanism, randomness is explicitly introduced into the underlying rules for the system, so that a random color is chosen for ever cell at each step.
This mechanism is the one most commonly considered in the traditional sciences. It corresponds essentially to assuming that there is a random external environment which continually affects the system one is looking at, and continually injects randomness into it.
In the second mechanism shown above, there is no such interaction with the environment. The initial conditions for the system are chosen randomly, but then the subsequent evolution of the system is assumed to follow definite rules that involve no randomness.
A crucial feature of these rules, however, is that they make the system behave in a way that depends sensitively on the details of its initial conditions. In the particular case shown, the rules are simply set up to shift every color one position to the left at each step.
And what this does is to make the sequence of colors taken on by any particular cell depend on the colors of cells progressively further and further to the right in the initial conditions. Insofar as the initial conditions are random, therefore, so will the sequence of colors of any particular cell be correspondingly random.
In general, the rules can be more complicated than those shown in the example on the previous page. But the basic idea of this mechanics for randomness is that the randomness one sees arises from some kind of transcription of randomness that is present in the initial conditions.
The two mechanisms for randomness just discussed have one important feature in common: they both assume that the randomness one sees in any particular system must ultimately come from outside that system. In a sense, therefore, neither of these mechanisms takes any real responsibility for explaining the origins of randomness: they both in the end just say that randomness comes from outside whatever system one happens to be looking at.
Yet for quite a few years, this rather unsatisfactory type of statement has been the best that one could make. But the discoveries about simple programs in this book finally allow new progress to be made.
The crucial point that we first saw on page 27 is that simple programs can produce apparently random behavior even when they are given no random input whatsoever. And what this means is that there is a third possible mechanism for randomness, which this time does not rely in any way on randomness already being present outside the system one is looking at.
If we had found only a few examples of programs that could generate randomness in this way, then we might think that this third mechanism was a rare and special one. But in fact over the past few chapters we have seen that practically every kind of simple program that we can construct is capable of generating such randomness.
WOLFRAM'S THESIS RESTATED
His references to “apparent randomness” and “apparently random behavior” indicate that Wolfram remains of two minds with respect to the very concept of “randomness” in the context of empirical science – that, while disputing the conventional approach thereto, he has yet to work out the full implications of “this third mechanism” for randomness.
As here construed, these implications may be stated succinctly as follows:
1. Let “apparent randomness” in a system's evolution through time denote something which is not implied by structural factors specified up front.
2. Thus, the first mechanism for randomness reduces to the proposition that a system's evolution is a function of non-structural factors not specified up front impacting at every stage the results of interaction between the system's up-front structural factors and subsequent inputs of non-structural factors.
3. Also, the second mechanism for randomness reduces to the proposition that a system's evolution is a function of structural factors specified up front, the details of whose path through time is uniquely determined by whatever set of non-structural factors may have set the system itself in motion along some given evolutionary path.
4. If “apparent randomness” is something which is not implied by structural factors defined up front, then the system's “apparently random behavior” in the case of Wolfram's third mechanism for randomness is aptly labeled - for it is apparent rather than real.
5. That is to say: The system's behavior is NON-random.
This was the position taken by Albert Einstein with respect to the epistemological aspects of Quantum Mechanics – a position which was derided and dismissed by his (philosophically challenged?) peers, including Niels Bohr, Werner Heisenberg, Max Born, and - in our own time, by Stephen Hawking.
Gunnar
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