Editing Specified complexity (section)

==Specificity==

In a more recent paper,<ref>William A. Dembski (2005).   [http://www.designinference.com/documents/2005.06.Specification.pdf ''Specification: The Pattern that Signifies intelligence''] {{Webarchive|url=https://web.archive.org/web/20070728121523/http://www.designinference.com/documents/2005.06.Specification.pdf |date=2007-07-28 }}</ref> Dembski provides an account which he claims is simpler and adheres more closely to the theory of [[statistical hypothesis testing]] as formulated by [[Ronald Fisher]]. In general terms, Dembski proposes to view design inference as a statistical test to reject a chance hypothesis P on a space of outcomes Ω.

Dembski's proposed test is based on the [[Kolmogorov complexity]] of a pattern ''T'' that is exhibited by an event ''E'' that has occurred. Mathematically, ''E'' is a subset of Ω, the pattern ''T'' specifies a set of outcomes in Ω and ''E'' is a subset of ''T''. Quoting Dembski<ref>(loc. cit. p. 16)</ref>
<blockquote>Thus, the event ''E'' might be a die toss that lands six and ''T'' might be the composite event consisting of all die tosses that land on an even face.</blockquote>
Kolmogorov complexity provides a measure of the computational resources needed to specify a pattern (such as a DNA sequence or a sequence of alphabetic characters).<ref>Michael Sipser (1997). ''Introduction to the Theory of Computation'', PWS Publishing Company.</ref> Given a pattern ''T'', the number of other patterns may have Kolmogorov complexity no larger than that of ''T'' is denoted by φ(''T''). The number φ(''T'') thus provides a ranking of patterns from the simplest to the most complex. For example, for a pattern ''T'' which describes the bacterial [[flagellum]], Dembski claims to obtain the upper bound φ(''T'') ≤ 10<sup>20</sup>.

Dembski defines specified complexity of the pattern ''T'' under the chance hypothesis P as

:<math> \sigma= - \log_2 [R \times \varphi(T) \times \operatorname{P}(T)], </math>

where P(''T'') is the probability of observing the pattern ''T'', ''R'' is the number of "replicational resources" available "to witnessing agents". ''R'' corresponds roughly to repeated attempts to create and discern a pattern. Dembski then asserts that ''R'' can be bounded by 10<sup>120</sup>. This number is supposedly justified by a result of Seth Lloyd<ref>{{cite journal | last=Lloyd | first=Seth |author-link=Seth Lloyd| title=Computational Capacity of the Universe | journal=Physical Review Letters | volume=88 | issue=23 | date=2002-05-24 | issn=0031-9007 | doi=10.1103/physrevlett.88.237901 | page=237901| pmid=12059399 |arxiv=quant-ph/0110141| bibcode=2002PhRvL..88w7901L | s2cid=6341263 }}</ref> in which he determines that the number of elementary logic operations that can have been performed in the universe over its entire history cannot exceed 10<sup>120</sup> operations on 10<sup>90</sup> bits.

Dembski's main claim is that the following test can be used to infer design for a configuration: There is a target pattern ''T'' that applies to the configuration and whose specified complexity exceeds 1. This condition can be restated as the inequality
:<math> 10^{120} \times \varphi(T) \times \operatorname{P}(T) < \frac{1}{2}. </math>

===Dembski's explanation of specified complexity===

Dembski's expression σ is unrelated to any known concept in information theory, though he claims he can justify its relevance as follows: An intelligent agent ''S'' witnesses an event ''E'' and assigns it to some reference class of events Ω and within this reference class considers it as satisfying a specification ''T''. Now consider the quantity φ(''T'') &times; P(''T'') (where P is the "chance" hypothesis):
[[Image:TargetReplicationalRsources.png|thumb|350px|Possible targets with complexity ranking and probability not exceeding those of attained target ''T''. Probability of [[union (set theory)|set-theoretic union]] does not exceed φ(''T'') &times; P(''T'')]]

<blockquote>
Think of S as trying to determine whether an archer, who has just shot an arrow at a large wall, happened to hit a tiny target on that wall by chance. The arrow, let us say, is indeed sticking squarely in this tiny target. The problem, however, is that there are lots of other tiny targets on the wall. Once all those other targets are factored in, is it still unlikely that the archer could have hit any of them by chance?

In addition, we need to factor in what I call the replicational resources associated with ''T'', that is, all the opportunities to bring about an event of ''T'''s descriptive complexity and improbability by multiple agents witnessing multiple events.
</blockquote>

According to Dembski, the number of such "replicational resources" can be bounded by "the maximal number of bit operations that the known, observable universe could have performed throughout its entire multi-billion year history", which according to Lloyd is 10<sup>120</sup>.

However, according to Elsberry and Shallit, "[specified complexity] has not been defined formally in any reputable peer-reviewed mathematical journal, nor (to the best of our knowledge) adopted by any researcher in information theory."{{sfn|Elsberry|Shallit|2003|p=14}}

===Calculation of specified complexity===

Thus far, Dembski's only attempt at calculating the specified complexity of a naturally occurring biological structure is in his book ''No Free Lunch'', for the [[bacterial flagellum]] of [[E. coli]]. This structure can be described by the pattern "bidirectional rotary motor-driven propeller". Dembski estimates that there are at most 10<sup>20</sup> patterns described by four basic concepts or fewer, and so his test for design will apply if

:<math> \operatorname{P}(T) < \frac{1}{2} \times 10^{-140}. </math>

However, Dembski says that the precise calculation of the relevant probability "has yet to be done", although he also claims that some methods for calculating these probabilities "are now in place".

These methods assume that all of the constituent parts of the flagellum must have been generated completely at random, a scenario that biologists do not seriously consider. He justifies this approach by appealing to [[Michael Behe]]'s concept of "[[irreducible complexity]]" (IC), which leads him to assume that the flagellum could not come about by any gradual or step-wise process. The validity of Dembski's particular calculation is thus wholly dependent on Behe's IC concept, and therefore susceptible to its criticisms, of which there are many.

To arrive at the ranking upper bound of 10<sup>20</sup> patterns, Dembski considers a specification pattern for the flagellum defined by the (natural language) predicate "bidirectional rotary motor-driven propeller", which he regards as being determined by four independently chosen basic concepts. He furthermore assumes that English has the capability to express at most 10<sup>5</sup> basic concepts (an upper bound on the size of a dictionary). Dembski then claims that we can obtain the rough upper bound of

:<math> 10^{20}= 10^5 \times 10^5 \times 10^5 \times 10^5 </math>
for the set of patterns described by four basic concepts or fewer.

From the standpoint of Kolmogorov complexity theory, this calculation is problematic. Quoting Ellsberry and Shallit{{sfn|Elsberry|Shallit|2003|p=12}} "Natural language specification without restriction, as Dembski tacitly permits, seems problematic. For one thing, it results in the [[Berry paradox]]". These authors add: "We have no objection to natural language specifications per se, provided there is some evident way to translate them to Dembski's formal framework. But what, precisely, is the space of events Ω here?"