I suspect however that we are mixing apples and oranges - or more specifically, intelligent design, relational biology, irreducible complexity, information theory and molecular biology and complex systems theory.
Like you, I'm not aware of a mathematical proof of intelligent design.
But if tacticalogic would like to see a mathematical proof with regard to biological systems, that the whole is greater than the sum of the parts, I strongly recommend he read Rosen's Life Itself.
The book has nothing to do with the intelligent design hypothesis.
And concerning information theory and molecular biology, Yockey literally "wrote the book." But he also does not argue for intelligent design.
Concerning mathematical models, Shannon's "mathematical theory of communications" is the root of information theory per se. The model has survived well since the 1940's and is highly transportable to all kinds of applications, including molecular biology.
And Rosen doesn't leave the reader hanging in Life Itself - he offers an excellent circular model for relational biology. His model is new, though, and has not yet built a track record.
And finally, the subject of complexity is well, complex. To make sense of it, one ought to start at the root and build a sense of the terms involved including complexity, chaos, entropy, randomness, probability and order. The meanings of such terms in mathematics are not always the same as in science or in common usage.
betty boop wrote an outstanding essay contrasting combinatorics to Bayesian probability.
Frankly, people will often choose one or the other model depending on which one helps their presupposition. Creationists often use combinatorics to argue there was not enough time for biological evolution. Evolutionists use combinatorics to argue against the "just so" objections. And either side may toggle back to arguing for Bayesian probability if that is convenient.
It's quite subjective!
And there are many methods which can be used to analyze complexity which fall in two different categories: least description and least time. They include functional complexity, algorithmic complexity, Kolmogorov complexity, etc.
And then there are theories concerning the rise of complexity (or order depending on how one is arguing) - e.g. self-organizing complexity, cellular automata, chaos theory.
Irreducible complexity is yet another theory for analyzing complexity. Personally I don't care for it because it is backwards looking. I'd much rather deal with forward looking complex system theories.
The one thing they all have in common though is that order cannot arise in an unguided physical system. Period. There are always guides to the system - whether initial conditions, rules, space/time, physical causation, physical laws and constants, etc.
The intelligent design "hypothesis" essentially observes that the best explanation for "certain features" is that a guide was intelligent.
As I said before, I view the intelligent design hypothesis to be more of an observation - after all, many creatures are known to choose their mates thus affecting the inherited traits of their offspring.
Apples and oranges...
I'll agree there's some "apples and oranges" with regard to different theories.
This started off as a discussion that would (I hope) help clarify the issue of methodology. Looking at different theories for examles, it's easy to get caught up in the details of the theories themselves and end up with with "apples and oranges" comparisons. That wasn't my intent, and I've tried to avoid it but it can be a challenge when the theories themselves present so many opportunities for discussion aside from the original question.
[[And concerning information theory and molecular biology, Yockey literally “wrote the book.” But he also does not argue for intelligent design.]]
Wouldn’t information theory necessarily be3 ID (unless one tries to argue that nature somehow creates information/metainformation? I don’t see how one can escape at least inferring the possibility that an ID is needed for metainformation unless they appeal to the ‘intelligent nature’ of nature itself?
[[Creationists often use combinatorics to argue there was not enough time for biological evolution.]]
Some argue that it’s biologically imposible regardless of how much time is thrown at it- impossible no matter what stage you start the proposed process of macroevolution at
[[Irreducible complexity is yet another theory for analyzing complexity. Personally I don’t care for it because it is backwards looking.]]
I think it also looks at hte problem from ‘start to finish’- just as Miller tried and failed to ‘build complexity naturally’ from the ground up when he tried to describe how complex mammilian blood clotting ‘could have evolved’ naturally, ID looks at systems, examines the natural aspects and sees whether they could have evolved fro mthe ground up (at least soem of them ore indepth articles from lesser known ID scientists do- As well, Demski and others on his site go from ground up- ISCID (.org I beleive) also takes this approach on many issues- Sites like IRC and AnswersinGenesis and others mostly do take the back to front approach, and a lot of their articles are not very indepth, but they do have some pretty technical articles- you just have to really look for them on the site- but most of hteir articles are less technical and more general speaking for hte public
[[The intelligent design “hypothesis” essentially observes that the best explanation for “certain features” is that a guide was intelligent.]]
I have to point out here that that is a bit of a generalization- as mentioned, when you realyl start digging, you find many terrific articles describing whther nature is capable of what is being examined from the ground up or not, and they also point out the impossibilities that nature faces, and why they are impossible naturally, and describe scientificaly, why an intelligent designer was needed behind certain key constructions- Evos poo poo this as ‘appealing to the supernatural’ yet Evos ignore the impossibilities, and and must appeal to a ‘supernatural nature’ if these impossibilities are to be overcome naturally
haven’t really read through this site yet, but looks like it might be worth checking out- Cornelius discusses the fallicies used by some evolutionists, and explains the biological problems associated with their claims
http://darwins-god.blogspot.com/2009/10/graeme-finlay-evolution-is-true-part-ii.html
Me too, dearest sister in Christ! And for the same reason you indicated here —
...I view the intelligent design hypothesis to be more of an observation ....
That's how I tend to think of irreducible complexity: an inference from general observation that is directly opposed to the idea of reductionism (which itself is a general idea). I don't think of it as a "model" per se, and even wonder if it is model-able. But if it were, it would be like you say: "backward looking." Thus any idea of "inversely causal meta-information" [final cause], as Alex Williams termed it, for example, is ruled out in advance. Science shouldn't begin by ruling things out. Or so it seems to me, FWIW.
And that there are always "guides to the system" — i.e., "initial conditions, rules, space/time, physical causation, physical laws and constants" — seems so patently obvious to me I wonder why anyone would even question it....
Thank you so very much for your outstanding essay/post, dearest sister in Christ!