Irreducible complexity is a term coined by Michael Behe in "Darwin's Black Box" as a proof for the necessity of a designer in earthly organisms. Unfortunately, Behe's definition of the term is intuitive rather than formal, and his examples of irreducible complexity are therefore not provably irreducibly complex. I read "Darwin's Black Box" a number of years ago with great interest. Since reading it, I have been on the lookout for a formal definition of "irreducible complexity" and have been unable to find one (at least one that satisfied my mathematical/computer science background). Here is my attempt to provide some candidate formal definitions of irreducible complexity in the context of evolution.(1 - please read this note!)
First a comment: this is an attempt to define an irreducible biological system while ignoring the issue of complexity. The issue of complexity in the question of a proof of intelligent design is irrelevent. The reason it is irrelevent is because complexity is a relative measurement; some things are more complex than others. Therefore any proofs involving complexity would also produce relative statements such as biological system X is more complex than biological system Y. As well, there does not seem to be an upper bound on complexity. Thus there is no way to determine if a biological system is maximally complex. Finally, complexity may have no relationship to the question of design since often (human) designed systems are appreciated for their simplicity whereas complex designs have a notorious side: the Rube Goldberg devices.(2)
That said, here is my first candidate definition of an irreducible system:
Define U as the universe of all possible genotypes
Note that by genotype I am meaning the complete set of inheritable biological data that defines the phenotype of a specific organism. For the purposes of this definition, I can ignore the whole notion of a species.
Define set O = { all S such that S is axiomatically an original genotype }
The set O contains the genotypes of all the first organisms (possibly the size of the set is 1). These first organisms were created through some other means than the following fit and fmutate natural process. They may have been created through chemical processes such as existed in the theorized "primordial soup".
Define function fit: U × U —> U
The function fit chooses the most "fit" of two genotypes in some given environment. It models natural selection.
Define a relation(3) fmutate: U —> U for all S (element of U) such that (1) for all S' = fmutate(S) we have fit(S, S') = S' and (2) there exists F (an element of O) such that S = fmutaten(F) for some value of n >= 0
The relation fmutate always produces a more "fit" genotype. Of course, this is not to say that less fit modifications of a genotype cannot be produced in nature. We are simply defining fmutate in such a way so that it can be used to formally define irreducibility. We also state that we are really concerned with the transitive closure of fmutate since we only want to consider genotypes that have their origin in set O. In the above definition, n represents the number of generations a genotype is from the original genotype.
Define predicate freducible on U such that freducible(R) if there exists R' such that fmutate(R') = R
Define predicate firreducible on U such that firreducible(I) if for all I', fit(I, I') = I implies fmutate(I') != I
In other words, a system is irreducible if we cannot find any less fit predecessor genotype that can be mutated to the genotype in question. This definition is fairly close to Behe's definition in that he is always concerned with genotypes of increasing fitness (his dicussion is actually about biological molecular systems that are part of the phenotype, but I think the point stands).
Behe's assumption of monotonically increasing fitness is not technically correct. As he has said in a personal communication: "Theoretically one can have the situation where a mutation is harmful but survives by luck and goes on to lead to a new, even better feature. However, as the mutation becomes more harmful and the population size becomes large, the probability of that occurring becomes very very small." (Behe, 2003) Must we consider the possibility that a mutation can produce a less fit system as a legitimate successor? Seemingly, yes, even though the probability is small for such a mutation to survive in a population. Irreducibility becomes much more difficult to find (in practice) since a predecessor to a system may be either more or less fit than the system and fitness no longer plays a role in irreducibility. Irreducibility then simply becomes a question of reproductive viability: can a system be proven to have a viable predecessor? I personally think the following is a more accurate model for defining irreducibility.
This changes the definition as follows:
Define predicate viable on U
The predicate viable decides if a genotype is reproductively viable and therefore if it succeeds in becoming a candidate operand in the following relation:
Define relation(3) vmutate: U —> U for all S (element of U) such that (1) viable(S) implies viable(vmutate(S)) and (2) there exists F (an element of O) such that S = vmutaten(F) for some value of n >= 0
The relation vmutate always produces a viable genotype if given a viable genotype.
Define predicate vreducible on U such that vreducible(R) if there exists R' such that viable(R') and vmutate(R') = R
We don't care about non-viable genotypes.
Define predicate virreducible on U such that virreducible(I) if for all I' such that viable(I'), then vmutate(I') != I
The simple definition above should be sufficient if we can find appropriate viable and vmutate that can be formally mapped to natural selection processes and genetic modification processes respectively. We can then either prove that no genotype I such that virreducible(I) can exist with the given natural processes or prove, probably by counter-example, that such a genotype I does exist. Behe proposes some examples of I from molecular biology, but only provides intuitive reasoning to show their irreducibility.
This leads nicely to the consideration of the following:
Define IRREDUCIBLE for some viable and vmutate such that IRREDUCIBLE is true iff there exists some S an element of U such that virreducible(S)
In other words, is there even one (possibly theoretical) example of a system which is irreducible?
. . .
Obviously, in order to make these definitions worthwhile, we need to find O, viable, and vmutate that are accurate models of life. If we do not have accurate models, then any formal proof we could do would be extremely weak in that it would simply not apply to the real world. It is actually easy to show that if we are willing to ignore issues of probability, then an accurate vmutate must allow for the possibility of "massive" mutation.(1) Massive mutation is a mechanism by which we could easily move into an intuitively irreducible genotype. It should be noted again, there may be some place for an empirical investigation of irreducibility if we find numerous examples of genotypes that can only be explained by massive mutations.
If we discard a perfectly accurate vmutate or make other "unrealistic" assumptions, we may consider much more theoretical questions such as a meta-proof: that such proofs are impossible along lines similar to Goodel's Theorem or the Halting Problem. The basis of such a proof could rest on proving that biological processes at a molecular level are equivalent to Turing machines and then show that IRREDUCIBLE is undecidable.
Disclaimer:
Unfortunately my background in molecular biology is limited to personal interest that probably runs at about the freshman university level. I would appreciate any feedback on these definitions at mishkin-irreduciblecomplexity@berteig.com.
Notes:
(1) In researching this article, I came across a related paper at: http://www.bioinf.uni-leipzig.de/Publications/PREPRINTS/03-003.pdf (local copy) which discusses the topology of the genotype and phenotype. The authors state:
From the mathematical point of view it is natural to consider the collection U of all accessibility relations on a given genotype space. . . . What are the natural properties of U? We propose: (U0) X x X [element of] U (ergodicity). . . . The ergodicity hypothesis says that at some [emphasis in original] level everything is accessible from everywhere, if we just wait long enough or if we are content with sufficiently small probabilities.
In my opinion, this is a viable hypothesis by the simple fact that it is possible for a massive random mutation to occur which spontaneously converts a single-celled organism's genotype into the genotype of a human. Incredibly unlikely, true, but still calculatably possible. This simple fact puts a fatal hole in the idea of formally proving any sort of intelligent design along the lines of irreducible complexity. However, there is still a miniscule chance of an empirical proof if sufficient analysis was done and we found extremely frequent examples of genotypes that could only be produced through such massive random mutations. If this sort of discovery were to occur, we would have to consider alternative theories to Darwinian evolution. After all, Darwinian evolution rests partly on a probabilistic mechanism.
(2) the mathematical definition of entropy may be a suitable stand-in for the notion of complexity
(3) fmutate and vmutate are relations instead of functions because for any given system in their domain, there may be an arbitrary number of resulting systems. (Thanks to an anonymous reviewer for pointing out that using functions is inaccurate to the reality of the biological world.)
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