The "information" if DNA is not information like writing down your randomly generated bank pin on a PostIt note. The information contained within DNA is a record of positive and negative feedback loops operating for millions of years, two things humans in general find difficult to process mentally and AIG Creationists specifically reject.
Feedback loops. Millions of years. DNA is literally a palimpsest recording of those happening.
Excellent way of stating it.
In some senses DNA has information in it. But this was refined from randomness by natural selection: imagine in essence the base sequences began "random" and the sequences that don't work well were eliminated by natural selection, so the sequences that remain are the ones that do work. Therefore the "information" is only what remains from the initial random "non-information" after natural selection. You don't need a mind, but you do need something that can select based on some parameter present within only some of the initially random sequences: in this case totally non-self-aware none intelligent natural selection.
This "selection" of specific "information" from a wider possible range by natural forces happens all the time but we tend not to think of it in this fashion. If we take a flat rock, then toss it 100 times and it always comes to rest on a face and not on a edge, does the rock encode "information" inserted by an intelligence so it "knows" to not come to rest on an edge? In a sense it possess information, but this information is the stability imposed on gravity on the shape of the rock- it doesn't involve a consciousness at all. Did a mind have to insert in water the information to form a 6 sided snowflake? Only if you insist on believing, counter to all evidence, that the mind of god (or gods) created the entire universe.
As with most Creationist "insights" I suspect that the people who actually devise these kinds of arguments know that they are lying, but just rely on more ignorant people to believe and exponentially propagate them. I am often struck by how 99% of Creationists arguments against the TOE are absolute strawmen and rely on a complete lack of understanding of what the TOE really states.
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