In considering the elusive “information with meaning”, which certainly exists because this is some, it seems as if it’s impossible to distinguish signal from noise without considering the sender and the receiver: in some way a message must be seen to convey a communication between a sender and a receiver.
We watch a telegraph operator (or a spy, or an alien) receiving messages which appear as gobbledygook to us. But if certain messages appear to provoke some significant (maybe repeated) response (the operator raises a flag, or sends a reply), we suspect that message to contain meaning, even if we can’t translate it.
Similarly a recording of the last speaker of a language may be an actor speaking gibberish, but if that last speaker is still alive, and mimes the meaning of his words, we know it’s speech and can even learn the language. We can, and do, work out the significance of animal signals in this way, though they are not trying to teach us, and though there is a qualitative difference between language and mere signals.
Merv’s prime number example is based on a shared assumption by the receivers and the aliens (or at least the scriptwriter!) that maths is a universal language: the message itself is statistically indistinguishable from noise, but its effect on us, producing a conviction of a similar effect on the sender, tells us there are folks out there.
A “black box” machine, reading code, is surely accessible on the same criteria, being an active, if not conscious, receiver of information. If it sits there doing nothing until a certain combination of characters appears, when it whirrs and spits out a widget, then the response is the evidence for something other than noise. The response might be complicated: for example, like Chris’s example, only every 11th character or message might be heeded, but further understanding would be gained not from studying the code alone, but the machine’s responses to the code.
As far as I’m aware, information theory does not have much to say about receiver reponses, and it’s in those that “meaning” resides (whether a simple instruction or all the nuances of a human poem). If there’s an application to DNA, surely it’s in matching coding to the organism’s responses.
Beyond that there must come a “universal language” inference like that of Merv’s prime numbers: in this case the inference that the messaging between generations of organisms involves teleology at the biological level, or at a higher level of creation above that.
Without that, we are indeed talking about faces in the clouds, pareidolia: the mere illusion of meaning. But then we have to explain why nature’s “receiver” responds to an illusion as if to real information. I’m not sure even Shannon envisaged information being transmitted in a medium apart from teleology.
