Introduction
The de facto Intelligent Design community make claim to the notion that
information is conserved, at least as far as so called “natural processes” are
concerned. In particular, one of the founding gurus of the movement, William
Dembski, has stated the conservation of
information in mathematical terms. The aim of this paper is to investigate
this claim and its limitations.
What is information? There is more than one valid answer to that
question, but this paper will be looking at the kind of information defined as
the negative of the logarithm of a probability p; that is, -log p
As an example, consider a configurational segment of length n taken from a series of coin tosses.
Given n it follows that the number of
possible configurations of heads and tails is 2n. If we assume
that each of these possible configurations is equally probable then any single
configuration will have a probability of 2–n. For large n this is going to be a very small value. In this case our knowledge of which
configuration the coin tossing will generate is at minimum; all possibilities
from amongst a huge class of possibilities are equally likely. Consequently,
when the coin tossing takes place and we learn which configuration has appeared
it is highly informative because it is just one amongst 2n equally likely possibilities. The measure of just how
informative that configuration is, is quantified by I, where:
I = -log 2-n = n log 2
(0.0)
Conveniently this means that the length of a sequence of coin tosses is proportional
to the amount of information it contains.
Information, as the very term implies, is bound up with observer
knowledge: When calculating
probabilities William Dembski uses the principle of indifference across mutually excluding outcomes; that is, when
there is no information available which leads us to think one outcome is more
likely than another then we posit a priori that the probabilities of
the possible outcomes are equal. I’m inclined to follow Dembski in this
practice because I hold the view that probability is a measure of observer
information about ratios of possibilities. In my paper on probability I defined probability recursively as follows:
Probability of case C = Sum of the probabilities of cases favoring C / Sum of probabilities of all cases
(0.1)
This definition is deliberately circular or, if you want the technical
term, recursive. For a recursively
defined probability evaluating it depends on the recursion terminating at some
point. Termination will, however, come about if we can reach a point where the
principle of indifference applies and all the cases are equally probable; when
this is true the unknown cancels out on the right hand side of (0.1) and the
probability on the left hand side can then be calculated. From this calculation it is clear that
probability is a measure of human information about a system in terms of the
ratio of possibilities open to it given the state of human knowledge of the
system. This means probabilities are ultimately evaluated a priori in as much as they trace back to an evaluation of human
knowledge about a system; the system in
and of itself doesn’t possess those probabilities.
It follows then that once
an improbable event has occurred and is
known to have occurred, the self-information of that event is lost because
it no longer has the power to inform the observer; the probability of a known event is unity and therefore of
zero information. But for an observer who has yet to learn of the event, whether the event has actually happened of
not, the information is still “out there”. Information content, then, is
observer relative.
Observer relativity means that self-information is not an intrinsic property of a physical system
but rather an extrinsic property. That is, it is a property that comes about
through the relation the system has with an observer and that relation is to do
with how much the observer knows about the system. Therefore a physical system loses
its information as the observer learns about it, and yet at the same time there
is no physical change in that system as it loses this information; where then has
the information gone? Does it now reside in the observers head? But for another observer who is still
learning about the system that information, apparently,
remains “out there, in the system”.
Given the relativity of self-information, then treating it as if it were
an intrinsic property of a physical system
can be misleading. The observer relativity of self-information makes it a very
slippery concept: As an extrinsic
property that relates observers to the observed, a system can at once both
possess information and yet at the same time not possess it!
This observer information based concept of probability is very relevant
to the subject in hand: that is, to the cosmic generation of life. Given that
the configurations of life fall in the complex ordered category it follows that
as a class life is a very rare case
relative to the space of all possible configurations. So, assuming the
principle of indifference over the total class of possible configurations we would
not expect living configurations to exist; this is because their a priori probability must be extremely
small and therefore their self-information or surprisal value is very high: Living configurations are very special
and surprising configurations.
But the fact is life does exist
and therefore the a posteriori probability of instantiated life is unity. It is this intuitive contradiction
between the a priori probability and
the a posteriori probability of life
that constitutes one of the biggest of all scientific enigmas. In this paper I will attempt to disentangle
some of the knots that the use of self-information introduces when an attempt
is made to use it to formulate a conservation law. I also hope to throw some
light on the a priori/a posteriori
enigma of life.
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