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Signal to Noise: A Critical Analysis of Active Information

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The following is a guest post by Aurelio Smith. I have invited him to present a critique of Active Information in a more prominent place at UD so we can have a good discussion of Active Information’s strengths and weaknesses. The rest of this post is his.


My thanks to johnnyb for offering to host a post from me on the subject of ‘active information’. I’ve been following the fortunes of the ID community for some time now and I was a little disappointed that the recent publications of the ‘triumvirate’ of William Dembski, Robert Marks and their newly promoted postgrad Doctor Ewert have received less attention here than their efforts deserve. The thrust of their assault on Darwinian evolution has developed from earlier concepts such as “complex specified information” and “conservation of information” and they now introduce “Algorithmic Specified Complexity” and “Active information”.

Some history.

William Demsbski gives an account of the birth of his ideas here:

…in the summer of 1992, I had spent several weeks with Stephen Meyer and Paul Nelson in Cambridge, England, to explore how to revive design as a scientific concept, using it to elucidate biological origins as well as to refute the dominant materialistic understanding of evolution (i.e., neo-Darwinism). Such a project, if it were to be successful, clearly could not merely give a facelift to existing design arguments for the existence of God. Indeed, any designer that would be the conclusion of such statistical reasoning would have to be far more generic than any God of ethical monotheism. At the same time, the actual logic for dealing with small probabilities seemed less to directly implicate a designing intelligence than to sweep the field clear of chance alternatives. The underlying logic therefore was not a direct argument for design but an indirect circumstantial argument that implicated design by eliminating what it was not.*

[*my emphasis]

Dembski published The Design Inference in 1998, where the ‘explanatory filter’ was proposed as a tool to separate ‘design’ from ‘law’ and ‘chance’. The weakness in this method is that ‘design’ is assumed as the default after eliminating all other possible causes. Wesley Elsberry’s review points out the failure to include unknown causation as a possibility. Dembski acknowledges the problem in a comment in a thread at Uncommon Descent – Some Thanks for Professor Olofsson

I wish I had time to respond adequately to this thread, but I’ve got a book to deliver to my publisher January 1 — so I don’t. Briefly: (1) I’ve pretty much dispensed with the EF. It suggests that chance, necessity, and design are mutually exclusive. They are not. Straight CSI [Complex Specified Information] is clearer as a criterion for design detection.* (2) The challenge for determining whether a biological structure exhibits CSI is to find one that’s simple enough on which the probability calculation can be convincingly performed but complex enough so that it does indeed exhibit CSI. The example in NFL ch. 5 doesn’t fit the bill. The example from Doug Axe in ch. 7 of THE DESIGN OF LIFE (www.thedesignoflife.net) is much stronger. (3) As for the applicability of CSI to biology, see the chapter on “assertibility” in my book THE DESIGN REVOLUTION. (4) For my most up-to-date treatment of CSI, see “Specification: The Pattern That Signifies Intelligence” at http://www.designinference.com. (5) There’s a paper Bob Marks and I just got accepted which shows that evolutionary search can never escape the CSI problem (even if, say, the flagellum was built by a selection-variation mechanism, CSI still had to be fed in).

[*my emphasis]

Active information.

Dr Dembski has posted some background to his association with Professor Robert Marks and The Evolutionary Informatics Lab which has resulted in the publication of several papers with active information as an important theme. A notable collaborator is Winston Ewert Ph D, whose master’s thesis was entitled: Studies of Active Information in Search where, in chapter four, he criticizes Lenski et al., 2003, saying:

[quoting Lenski et al., 2003]“Some readers might suggest that we stacked the deck by studying the evolution of a complex feature that could be built on simpler functions that were also useful.”

This, indeed, is what the writers of Avida software do when using stair step active information.

What is active information?

In A General Theory of Information Cost Incurred by Successful Search, Dembski, Ewert and Marks (henceforth DEM) give their definition of “active information” as follows:

In comparing null and alternative searches, it is convenient to convert probabilities to information measures (note that all logarithms in the sequel are to the base 2). We therefore define the endogenous information IΩ as –log(p), which measures the inherent difficulty of a blind or null search in exploring the underlying search space Ω to locate the target T. We then define the exogenous information IS as –log(q), which measures the difficulty of the alternative search S in locating the target T. And finally, we define the active information I+ as the difference between the endogenous and exogenous information: I+ = IΩ – IS = log(q/p). Active information therefore measures the information that must be added (hence the plus sign in I+) on top of a null search to raise an alternative search’s probability of success by a factor of q/p. [excuse formatting errors in mathematical symbols]

They conclude with an analogy from the financial world, saying:

Conservation of information shows that active information, like money, obeys strict accounting principles. Just as banks need money to power their financial instruments, so searches need active information to power their success in locating targets. Moreover, just as banks must balance their books, so searches, in successfully locating targets, must balance their books — they cannot output more information than was inputted.

In an article at the Pandas Thumb website Professor Joe Felsenstein, in collaboration with Tom English, presents some criticism of of the quoted DEM paper. Felsenstein helpfully posts an “abstract in the comments, saying:

Dembski, Ewert and Marks have presented a general theory of “search” that has a theorem that, averaged over all possible searches, one does not do better than uninformed guessing (choosing a genotype at random, say). The implication is that one needs a Designer who chooses a search in order to have an evolutionary process that succeeds in finding genotypes of improved fitness. But there are two things wrong with that argument: 1. Their space of “searches” includes all sorts of crazy searches that do not prefer to go to genotypes of higher fitness – most of them may prefer genotypes of lower fitness or just ignore fitness when searching. Once you require that there be genotypes that have different fitnesses, so that fitness affects their reproduction, you have narrowed down their “searches” to ones that have a much higher probability of finding genotypes that have higher fitness. 2. In addition, the laws of physics will mandate that small changes in genotype will usually not cause huge changes in fitness. This is true because the weakness of action at a distance means that many genes will not interact strongly with each other. So the fitness surface is smoother than a random assignment of fitnesses to genotypes. That makes it much more possible to find genotypes that have higher fitness. Taking these two considerations into account – that an evolutionary search has genotypes whose fitnesses affect their reproduction, and that the laws of physics militate against strong interactions being typical – we see that Dembski, Ewert, and Marks’s argument does not show that Design is needed to have an evolutionary system that can improve fitness.

I note that there is an acknowledgement in the DEM paper as follows:

The authors thank Peter Olofsson and Dietmar Eben for helpful feedback on previous work of the Evolutionary Informatics Lab, feedback that has found its way into this paper.

This is the same Professor Olofsson referred to in the “Some Thanks for Professor Olofsson thread mentioned above. Dietmar Eben has blogged extensively on DEM’s ideas.

I’m not qualified to criticize the mathematics but I see no need to doubt that it is sound. However what I do query is whether the model is relevant to biology. The search for a solution to a problem is not a model of biological evolution and the concept of “active information” makes no sense in a biological context. Individual organisms or populations are not searching for optimal solutions to the task of survival. Organisms are passive in the process, merely affording themselves of the opportunity that existing and new niche environments provide. If anything is designing, it is the environment. I could suggest an anthropomorphism: the environment and its effects on the change in allele frequency are “a voice in the sky” whispering “warmer” or “colder”. There is the source of the active information.

I was recently made aware that this classic paper by Sewall Wright, The Roles of Mutation, Inbeeding, Crossbreeding and Selection in Evolution, is available online. Rather than demonstrating the “active information” in Dawkins’ Weasel program, which Dawkins freely confirmed is a poor model for evolution with its targeted search, would DEM like to look at Wright’s paper for a more realistic evolutionary model?

Perhaps, in conclusion, I should emphasize two things. Firstly, I am utterly opposed to censorship and suppression. I strongly support the free exchange of ideas and information. I strongly support any genuine efforts to develop “Intelligent Design” into a formal scientific endeavor. Jon Bartlett sees advantages in the field of computer science and I say good luck to him. Secondly, “fitness landscape” models are not accurate representations of the chaotic, fluid, interactive nature of the real environment . The environment is a kaleidoscope of constant change. Fitness peaks can erode and erupt. Had Sewall Wright been developing his ideas in the computer age, his laboriously hand-crafted diagrams would, I’m sure, have evolved (deliberate pun) into exquisite computer models.

References

History: Wm Dembski 1998 the Design inference, explanatory filter ( Elsberry criticizes the book for using a definition of “design” as what is left over after chance and regularity have been eliminated)

Wikipedia, upper probability bound, complex specified information, conservation of information, meaningful information.

Elsberry & Shallit

Theft over Toil John S. Wilkins, Wesley R. Elsberry 2001

Computational capacity of the universe Seth Lloyd 2001

Information Theory, Evolutionary Computation, and
Dembski’s “Complex Specified Information”
Elsberry and Shallit 2003

Specification: The Pattern That Signifies Intelligence by William A. Dembski August 15, 2005

Evaluation of Evolutionary and Genetic
Optimizers: No Free Lunch
Tom English 1996

Conservation of Information Made Simple William Dembski 2012

…evolutionary biologists possessing the mathematical tools to understand search are typically happy to characterize evolution as a form of search. And even those with minimal knowledge of the relevant mathematics fall into this way of thinking.

Take Brown University’s Kenneth Miller, a cell biologist whose knowledge of the relevant mathematics I don’t know. Miller, in attempting to refute ID, regularly describes examples of experiments in which some biological structure is knocked out along with its function, and then, under selection pressure, a replacement structure is evolved that recovers the function. What makes these experiments significant for Miller is that they are readily replicable, which means that the same systems with the same knockouts will undergo the same recovery under the same suitable selection regime. In our characterization of search, we would say the search for structures that recover function in these knockout experiments achieves success with high probability.

Suppose, to be a bit more concrete, we imagine a bacterium capable of producing a particular enzyme that allows it to live off a given food source. Next, we disable that enzyme, not by removing it entirely but by, say, changing a DNA base in the coding region for this protein, thus changing an amino acid in the enzyme and thereby drastically lowering its catalytic activity in processing the food source. Granted, this example is a bit stylized, but it captures the type of experiment Miller regularly cites.

So, taking these modified bacteria, the experimenter now subjects them to a selection regime that starts them off on a food source for which they don’t need the enzyme that’s been disabled. But, over time, they get more and more of the food source for which the enzyme is required and less and less of other food sources for which they don’t need it. Under such a selection regime, the bacterium must either evolve the capability of processing the food for which previously it needed the enzyme, presumably by mutating the damaged DNA that originally coded for the enzyme and thereby recovering the enzyme, or starve and die.

So where’s the problem for evolution in all this? Granted, the selection regime here is a case of artificial selection — the experimenter is carefully controlling the bacterial environment, deciding which bacteria get to live or die*. [(* My emphasis) Not correct – confirmed by Richard Lenski – AF] But nature seems quite capable of doing something similar. Nylon, for instance, is a synthetic product invented by humans in 1935, and thus was absent from bacteria for most of their history. And yet, bacteria have evolved the ability to digest nylon by developing the enzyme nylonase. Yes, these bacteria are gaining new information, but they are gaining it from their environments, environments that, presumably, need not be subject to intelligent guidance. No experimenter, applying artificial selection, for instance, set out to produce nylonase.

To see that there remains a problem for evolution in all this, we need to look more closely at the connection between search and information and how these concepts figure into a precise formulation of conservation of information. Once we have done this, we’ll return to the Miller-type examples of evolution to see why evolutionary processes do not, and indeed cannot, create the information needed by biological systems. Most biological configuration spaces are so large and the targets they present are so small that blind search (which ultimately, on materialist principles, reduces to the jostling of life’s molecular constituents through forces of attraction and repulsion) is highly unlikely to succeed. As a consequence, some alternative search is required if the target is to stand a reasonable chance of being located. Evolutionary processes driven by natural selection constitute such an alternative search. Yes, they do a much better job than blind search. But at a cost — an informational cost, a cost these processes have to pay but which they are incapable of earning on their own.

Meaningful Information

Meaningful Information Paul Vit´anyi 2004

The question arises whether it is possible to separate meaningful information from accidental information, and if so, how.

Evolutionary Informatics Publications

Conservation of Information in Relative Search Performance Dembski, Ewert, Marks 2013

Algorithmic Specified Complexity
in the Game of Life
Ewert, Dembski, Marks 2015

Digital Irreducible Complexity: A Survey of Irreducible
Complexity in Computer Simulations
Ewert 2014

On the Improbability of Algorithmic Specified
Complexity
Dembski, Ewert, Marks 2013

Wikipedia, upper probability bound, complex specified information, conservation of information, meaningful information.

A General Theory of Information Cost Incurred by Successful Search Dembski, Ewert, Marks 2013

Actually, in my talk, I work off of three papers, the last of which Felsenstein fails to cite and which is the most general, avoiding the assumption of uniform probability to which Felsenstein objects.

EN&V

Dietmar Eben’s blog

Dieb review “cost of successful search

Conservation of Information in Search:
Measuring the Cost of Success
Dembski, Marks 2009

The Search for a Search: Measuring the Information Cost of
Higher Level Search
Dembski, Marks 2009

Has Natural Selection Been Refuted? The Arguments of William Dembski Joe Felsenstein 2007

In conclusion
Dembski argues that there are theorems that prevent natural selection from explaining the adaptations that we see. His arguments do not work. There can be no theorem saying that adaptive information is conserved and cannot be increased by natural selection. Gene frequency changes caused by natural selection can be shown to generate specified information. The No Free Lunch theorem is mathematically correct, but it is inapplicable to real biology. Specified information, including complex specified information, can be generated by natural selection without needing to be “smuggled in”. When we see adaptation, we are not looking at positive evidence of billions and trillions of interventions by a designer. Dembski has not refuted natural selection as an explanation for adaptation.

ON DEMBSKI’S LAW OF CONSERVATION OF INFORMATION Erik Tellgren 2002

Comments
Joe, EL is speaking what she at minimum SHOULD know is false as a reasonable glance at the head of this thread will show. And that is in a situation where, days ago I corrected AS on that, as may be seen at 304 above. But then the whole OP to begin with is itself revealed to be a strawman caricature set-up and knock down, they do not seem to get the message that the needle in haystack search that sets the context for identifying, defining and using active information is directly tied to FSCO/I setting up narrow zones T in large config spaces W, that are to be found by blind needle in haystack search. And of course there is a series of papers that takes this up into general discussions but such is patent root context. All of this uncomfortably echos the transparent denials in the teeth of the vampire clergyman video smearing and scapegoating incident three years past. And BTW the opening salvos attacking me were on a personal blog post where I had spoken to porn issues with the Internet where -- though I didn't know it then -- alluded to a line in the atheist's anthem that spoke to churches having coffers of hate. In short the same circle full well exemplified the matter at hand, expressing preference for Sunday porn viewing over Sunday church attendance viewed as contributing to houses of hate. Something is seriously wrong with that sort of hate speech and bigoted view of Judaeo-Christian Theism. Then, imagine when we are back full circle to that same bigotry song when some of the circle tried to smear me at a local newspaper site where I stood up against abuse of parliamentary privelige by a local pol. And of course that is where by accusing me falsely of abuse and suggesting a venue not connected online the implicit threat was dropped: we know you, we know where you are, we know those even remotely connected and where they are. If that is not a manifestation of stalking and ugly, ugly threats, Mafioso style, nothing is. Those who insistently cossett such need to realise what they are enabling. KF PS: So far as I know KS is not outright stalking [the connexions online go other ways and the on the ground aspects point other ways unless his handle connects to people that would be most surprising . . . ], though obviously he seems to be part of the enabling. The circle of objectors needs to take a serious timeout and think about what they have been participating in and/or enabling. John Surratt's 1870 speech -- though of course it should be taken with a grain or two of salt -- might help them wake up. (But it should have been obvious that between a kidnapping, hostage taking conspiracy and murder lay the thinnest of partitions. Trying to pretend or imagine otherwise is one of the most telling things about that historical case.)kairosfocus
April 29, 2015
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Elizabeth's just a barrel of laughs. More later.Mung
April 29, 2015
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Joe Felsenstein @ TSZ:
DEM call every process a “search”, even crazy ones.
Well now we know that's just false. Unless every process can be represented as a probability distribution.Mung
April 29, 2015
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Zachriel says that ID is better science:
No, in science, the one that has testable entailments is the better hypothesis.
ID has the testable entailments while evolutionism does not. Thank you, Zachriel.Joe
April 29, 2015
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Zachriel: …no one had any idea how such an *irreducibly complex* structure {mammalian middle ear} could have evolved [via blind Darwinian mechanisms] incrementally. mike1962: Thank you again So the answer is "design"? mike1962: One bare possibility is as good as another. No, in science, the one that has testable entailments is the better hypothesis.Zachriel
April 29, 2015
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Zachriel: It’s part of the chain of entailments. While we might not have to explain everything about the designer, if there is no evidence of the designer, or the designer just pops in and out, then it will tend to undermine the hypothesis.
One bare possibility is as good as another. My implied challenge to Popperian stands.mike1962
April 29, 2015
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Zachriel: current science has not been able to reconstruct the chain of events in any detail.
Thank you
...no one had any idea how such an *irreducibly complex* structure could have evolved [via blind Darwinian mechanisms] incrementally.
Thank you againmike1962
April 29, 2015
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Yes, kairisfocus, I haz it: She said:
So nobody seems to want to defend Active Information as an ID argument. Oh well.
And right after that keiths attacks you. And you wonder why I just want to fight and get it over with...Joe
April 29, 2015
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Joe, have you got that EL clip? KFkairosfocus
April 29, 2015
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Mung, Just you wait as they are preparing a load of diatribe response to Ewert. ;) Seriously, Felsenstein may be good at his profession but he sucks at analyzing and critiquing IDJoe
April 29, 2015
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F/N: Reminds me of the story of rocket science. First, take a broomstick and support it on your fingertip as it stands vertically. Next, turn it into a rocket body that has to be light and will be broken it it tilts out of control. Replace the finger tip with a controlled explosion, shooting gases that you have to throttle and direct. Don't forget the pumps to run the fuel together. Where of course if things break enough they go boom. Now, put a capsule on top, with people in it who need to be protected from many things that can go wrong. Next, integrate orbital dynamics where you want to go somewhere rather specific and need to bring back the people in one piece. Functionally specific, highly constrained complex organisation that is information rich and is on a balance point that would be all but impossible to achieve by blind chance and necessity. Now, add to this, that your rocket needs to be self replicating as an automaton -- probably, by nanotech swarm . . . see significance of building things up from cells? What does this tell you? KFkairosfocus
April 29, 2015
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The use of the term "search" has led some to argue over whether or not Darwinian evolution can be considered a search. After all, Darwinian evolution is not a teleological process and does not search for a goal. However, all that we mean by search is a process that can be represented as a probability distribution. That means that all processes reducible to chance and necessity qualify as searches.Mung
April 29, 2015
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"Entropy is perpetually trying to break birds apart, but they remain alive." ok, now that made me laugh.Mung
April 29, 2015
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Thanks Aurelio. Felsentein and English raise a number of objections. However, their objections are directed against an argument we do not make. In fact, regarding the conclusion of conservation of information, they are in agreement with us. Imagine that.Mung
April 29, 2015
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Bob O'H, That is all I have been saying this entire thread. And that is what "The whole point is that GAs do NOT simulate natural selection." refers to. You lose.Joe
April 29, 2015
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Joe - you didn't write that natural selection is not a search in 519. Now you're putting words into your own mouth!Bob O'H
April 29, 2015
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Bob O'H- you wrote "One possible source of confusion is that GAs are used as search algorithms in computer science. But this does not necessarily mean that evolution is a search itself. The idea of a GA has been re-purposed." And that leads to my response as natural selection is not a search, let alone a search algorithm. However if you think that GAs do model natural selection then you are out to lunch and should not be teaching biology to anyone.Joe
April 29, 2015
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Joe - you wrote "The whole point is that GAs do NOT simulate natural selection. Thank you for admitting that they do not.". But I made no such admission - I made no comment on whether they do or not.Bob O'H
April 29, 2015
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Bob O'H, What words did I put in your mouth? Please be specific as what I posted in 519 appears to follow directly from what you posted in 517.Joe
April 29, 2015
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Mung (attn, Popperian), Active Information is both precisely conceptualised and a metric is developed for it. Cf 313 above and the onward linked. KFkairosfocus
April 29, 2015
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mike1962: However, (again), we do not need to explain the origin of the builder. It's part of the chain of entailments. While we might not have to explain everything about the designer, if there is no evidence of the designer, or the designer just pops in and out, then it will tend to undermine the hypothesis. While there will always be gaps in science—that's the nature of knowledge—, the process is to continue to test the limits of the hypothesis. So, with Stonehenge, archaeologists look for evidence of the builders, the use of the monument, the connection to surrounding peoples, the place in history, the origin of the people, what they ate, where they slept, anything and everything that can be learned, always testing each and every hypothesis. So far the evidence points to a peculiar species of tool-using amniote that evolved on the third rock from the Sun.Zachriel
April 29, 2015
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Popperian:
However, the term “Active Information” is incredibly vague.
Active information is precisely defined. Read the OP.Mung
April 29, 2015
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Joe @ 519 - you're putting words into my mouth. Please don't - I have enough in there as it is.Bob O'H
April 29, 2015
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mike1962: Thank you. Just to clarify, that doesn't mean the genetic transition will always be obscure, just that current science has not been able to reconstruct the chain of events in any detail. mike1962: You have provided no evidence that complexity “emerged” by undirected Darwinian evolution which is what he is defending as having “withstood an overwhelming amount of criticism.” Actually, we have significant evidence of the evolution of the mammalian middle ear. The original data came over a century ago from studies of embryonic development. Those researchers showed that the same structures which led to reptilian jaw bones led to auditory ossicles in mammals. This strange finding meant that there should have been an evolutionary transition between these structures, but no one had any idea how such an *irreducibly complex* structure could have evolved incrementally.Zachriel
April 29, 2015
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Zachriel:
The evidence indicates it was a peculiar species of tool-using ape that inhabits the third rock from the Sun, already known for building such structures.
The only tools ape use are sticks. And only an imbecile would think that sticks were the tool used to build Stonehenge. Enter Zachriel.Joe
April 29, 2015
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Popp:
You mean positively prove it?
No, I said TEST IT.
The core of Darwin’s original idea has withstood over 150 years of significant criticism.
Natural selection, the core of Darwin's idea, has proven to be impotent.
Neo-Darwinism is an alternative, non-authoritative source of knowledge, as defined by Popper, which genuinely creates new non-explanatory knowledge that did not exist before and was not present at the outset.
Blah, blah, blah- all meaningless drivel. Non-sequitur.
However, the term “Active Information” is incredibly vague.
Not really. However Darwinism and evolutionism are incredibly vague. Can you please reference this evolutionary theory you keep referring to?Joe
April 29, 2015
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Zachriel: If the posited designer appears out of nowhere leaving no evidence, then it would undermine the hypothesis. In the case of Stonehenge, we have evidence of the peculiar species of ape over millions of years, including in the vicinity of Stonehenge during the period in question, along with evidence of their building in stone.
However, (again), we do not need to explain the origin of the builder. Moreover, only in your opinion has the "builder(s)" of life left no evidence. Some people disagree. But that's beyond the scope of this thread. As for humans being "apes", you are free to put them in that category if you choose, but I think that is an unfair characterization. There are many profound differences.mike1962
April 29, 2015
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Zachriel: There’s no way to reconstruct the genetic mutations that occurred,
Thank you.
...though we have some knowledge of the genes involved. However, we can largely reconstruct the incremental adaptive steps.
So then, my implied challenge to Popperian's statement stands:
Popperian: However, Darwin has since provided such a theory: complexity emerges from [undirected] variation and [undirected] selection. And that theory has withstood an overwhelming amount of criticism.
You have provided no evidence that complexity "emerged" by undirected Darwinian evolution which is what he is defending as having "withstood an overwhelming amount of criticism." He's wrong. And you're reply doesn't help him/her. Care to reply to the other questions I posed to Popperian?mike1962
April 29, 2015
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mike1962: When one inquire about the proximate cause of an action, it is not necessary to inquire about the non-proximate causes. If the posited designer appears out of nowhere leaving no evidence, then it would undermine the hypothesis. In the case of Stonehenge, we have evidence of the peculiar species of ape over millions of years, including in the vicinity of Stonehenge during the period in question, along with evidence of their building in stone. mike1962: What kind, how many, and in what order did the “blind” molecular mutations occur in order to produce it? There's no way to reconstruct the genetic mutations that occurred, though we have some knowledge of the genes involved. However, we can largely reconstruct the incremental adaptive steps.Zachriel
April 29, 2015
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"Zachriel": The evidence indicates it was a peculiar species of tool-using ape that inhabits the third rock from the Sun, already known for building such structures.
Which is my point. When one inquire about the proximate cause of an action, it is not necessary to inquire about the non-proximate causes. Thanks for your agreement.
A canonical example is the mammalian middle ear.
Thank you. What kind, how many, and in what order did the "blind" molecular mutations occur in order to produce it?mike1962
April 29, 2015
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PDT
1 2 3 4 5 20

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