Information Theory and Biology

Hi all,

What we proved in a previous discussion was that when examined only with respect to Information Theory, the debunking of Information Theory vis-a-vis RNA and as an applied apologetic, is much simpler. But, to be fair, the full treatment infra will be required. So, in my previous post I gave a quick and dirty scholium on Information Theory as applied to Biology. However, I deliberately neglected a closer examination of the interface between Biology and Information Theory for fear it would just confuse a discussion in which the participants were already confused enough.

I’d now like to remedy that omission.

The problem faced in this scenario is remarkably similar to the problem posed by the well-known fallacy of Teleology I’ve previously disproven. I will need to reintroduce that proof here as a necessary ingredient of the propaedeutics.

Scholium: Positively Disproving “Intelligent Design”

We imagine a bucket of marbles, say a few billion of them, and we paint each one with a unique number so that each can be uniquely identified. We choose integer multiples in the form of a count, beginning with 1 and going in order to the highest value we have. Now, suppose we take a number of marbles much smaller than that total, say 10. Now, let us randomly pick those 10 marbles and place them in another bucket, call it the bucket of Intelligent Designs. Most people can clearly see that the probability of those 10 marbles having values within an interval of, say, 100, is exceedingly slim. This is the Intelligent Design argument. It is nonsense. And the reason why is that we have no way of knowing how many marbles were placed in the Intelligent Designs bucket. So, if we placed every marble from the starting bucket into the Intelligent Designs bucket the probability would then be exactly 100%; that is 100% that human beings were created by natural events only.

Let us tighten this up.

Let us begin a causality train whose program will be to generate a set of dependent variables, effects, from a set of independent variables, the causes. To get the set of effects A and the set of causes B which caused the set A, A and B must contain members that are not strictly arbitrary. We can define a rank n order m metric tensor, , of “causality” generators; each denoted ϕ11, ϕ11 , … , ϕnm. Then for each ϕij we can define a domain and range for each; corresponding to the sets B and A respectively. Now, let it be observed empirically that there exists a set a and b such that a ∈ A and b ∈ B; and both a and b contain one or more elements, that is:

r ∈ a ∈ A, s∈ b ∈ B and we guarantee that an enumeration of elements exists such that:

u < v.

where u is the enumeration ru ∈ a and v is the enumeration rv ∈ A.

If the generators ϕnm meet the definition of a function, that is, a rule that assigns to each element b ∈ B exactly one element ϕnm(b) ∈ A, then it is likewise possible to find a set of generators which also meet the definition of a function, δnm(s) ∈ a.

Now, we let the generators ϕnm and δnm be functions that strictly assign each element of its corresponding domain randomly to exactly one element in its corresponding range.

Then the probability that there exists a generator δnm is ∝ u / (v – u). However, the probability that there exists a generator ϕnm = 1.

QED.

In other words, appeals to beauty, order and the appearance of an intelligent design, as just one example of this proof’s application, to suggest, imply or otherwise provide evidence for an intelligent actor as the cause thereof is nonsense. Let’s go back to the marbles to explain why and try to state this proof in English since we promised no advanced math here.

In the marble example the starting bucket represents all the possible independent variables of any nature or “universe”; that is, the environment variables in which, for example, something like Deoxyribonucleic Acid was first “created”. What Intelligent Design incorrectly assumes is that we are selecting only a subset of all possible independent variable values (physical properties of an environment) and placing them in Intelligent Design bucket (the thing so ordered and “intelligently designed”) for which we require the filling of a similarly narrow band of marble number values. This is not reality. The reality is that we must consider the full range of values of independent and dependent variables, which results in a 100% percent chance that intelligent design can generate something of any complexity. The probability of producing something that we subjectively call complex is proportional to the dependent variables associated with that complex object (to be exact, the dependent variables that are responsible for the complex character of the object) and all independent variables, which are essentially infinite, and thus also 100%. So, it is 100% probable that nature generated those complex structures.

Now, we can resume the discussion where we left off.

Let an “Information System”, A, be defined as Information Theorists might define it; namely, a system of events, ai, ∈ A whose position on a causality train K is sufficiently well defined. Let this position, Kk, be a “Logical event”; specifically, a Logical event strictly involving a change Δq in a base pair sequence Q. By change we mean a change (the generation of a “symbol”) in the base pair ordering of pairs of Adenine to Guanine and Cytosine to Thymine in an arbitrarily chosen segment of DNA or RNA.

Now, let an “Information System”, B, be defined as Biologists might define it; namely, a system of events, bj ∈ B whose position on a causality train L is sufficiently well defined. Let this position, Ll, be a “Biological event”; specifically, a Biological event offset from an event ai ∈ A exactly one step to the “left” context of B and in its causality train such that there exists an i and j such that:

bj => ai

and whose empirical standard of definition is any statistically normalized resultant of environmental events offset from bj to the right context of L; all evaluated over some closed time interval t ∈ T. We next require that L ≡ K.

Then we can define the former relation as an interface between Mathematically treated Information Theory and Biologically treated change over time (Evolution) as an “Information” transference point, call it W:

bj => ai

which is a critical interface between two qualitatively different treatments of logic and fact (two disciplines; Mathematics and Biology) because it is required that the entropy H ∈ B, call it HB > HA; where HA is the entroy, H ∈ A. The condition HB > HA .

Then a “complex” system can be defined for any i sufficiently large. However, it is noted that i is guaranteed to be finite if the system A is amenable to direct empirical observation.

But ∞ =  j ∈ ϕnm (B)

Q.E.D.

The fallacy inherent in the attempt to apply Information Theory to Biology, once fully analyzed, can be seen to be the same fallacy afflicting the overall “Intelligent Design” argument: the assumption that the set of all possible inputs is finite is wrong. And this fact is not easy to see at first glance. This likens the “Intelligent Design” and “Information Theory” approach to two sides of a counterfeit coin; but a very sophisticated one nonetheless. Because j is infinite (it is possible that the universe, to include the Earth and its environment that resulted, is but one outcome of an infinite number of possibilities, as far as human beings can possibly know) any degree of complexity in the set of all i’th effects is possible.

Therefore, when treated alone, Information Theory appears to conclude that information appears ex nihilo as uncertainty in A is reduced. However, this is an artifact of the interface W (the division of disciplines) since we have noted that it is required that HB > HA. Thus “Information”, or what will become “Information”, in A is causally entrained beginning in B, and the origin of Information is not ex nihilo; which is what I sought to show.

A final point to make – and what I wanted to clarify – is that this only means that nature is front-loading the mathematical events which reduce uncertainty with substantial “Information” that is not apparent when the problem is treated solely within the Information Theorems of Mathematics. And this “Information” comes from an environment (to include familiar constructs such as Selection Pressures) located at Earth over a time interval sufficiently large to define it. But in considering this set of “Information” sources, we cannot consider only the universe that now exists, but we have to consider all possible universes due to our ignorance of the true range of possibilities. So, there is no claim that Information originates ex nihilo.

Ignorance and superstition never quits.

- kk

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