Lizzie,
You continue to astonish.
In the first sentence of your reply to my prior post you wrote: “I know that it is possible for intelligent life-forms to send radio signals, because we do; my priors for the a radio signal to have an intelligent origin are therefore substantially above zero.”
As I demonstrated earlier, the issue is not whether nature or intelligent agents can cause radio signals. We know that both can. The issue is whether we have any warrant to distinguish this particular signal from a natural signal.
Then you write: “I know of no non-intelligent process that might generate prime numbers (presumably expressed as binary code), and so my priors on that are low.”
Upon a moment’s reflection I am certain you will agree that this is not, strictly speaking, correct. It is easy to imagine such a process. Imagine (as you suggested) a simple binary code that assigns two “dots” to the number “two” and three “dots” to the number “three” and five “dots” to the number “five” and so on, and also assigns a “dash” to delimit each number (a cumbersome code to be sure, but a conceivable one). In this code the series “dot dot dash dot dot dot dash” denotes the first two prime numbers between 1 and 100. Surely you will agree that it is well within the power of chance and mechanical necessity to produce a radio signal with such a simple sequence.
So what do we now know? We know that nature sends out radio signals. But that is not all we know. We know that it is entirely within the realm of reason to suppose that nature could send out a radio signal that denotes the first two prime numbers between 1 and 100 given a particular binary code.
From this information we must conclude that if the signal we received were only the first two prime numbers, we would have no warrant to assign a high probability to “intelligent cause.”
Nevertheless, we both know that your calculation (and it is a very good calculation for which I commend you) that the probability that this particular signal has an intelligent source is for all practical purposes “one” is correct.
Nature can send out a radio signal.
Nature can embed a pattern in that signal that appears to generate prime numbers under the binary protocol we have designated.
Why, then, are we warranted to infer intelligent agency and not the work of nature as the cause of this particular signal?
The answer has nothing to do with your or my “intuition” about the signal.
The answer is that we both know that nature can do two things. (1) It can generate highly improbable patterns. Imagine ANY 500 bit long series of dots and dashes, and you will have a pattern that could not reasonably be replicated by chance before the heat death of the universe. And (2) it can generate specified patterns (for example, the two prime numbers we saw above).
We also know something about what nature cannot do. You said, “I know of no non-intelligent process that might generate prime numbers.” You were almost right. As I have already demonstrated, you should have said “I know of no non-intelligent process that might generate A COMPLEX PATTERN OF prime numbers.”
In other words, you and I know that while nature can do “specified,” and nature can do “complex,” it cannot do “specified and complex at the same time”! This is not your intuition speaking Lizzie. Without seeming to know it, you have made an inference from the universal experience of the human race.
Here’s the most important “take away” for purposes of the discussion we have been having: As much as you have bucked against the idea, you were able to make this design inference based upon nothing more than the character of the embedded signal (i.e., that it contained complex and specified information at the same time, that is to say, complex specified information).
Welcome to the ID camp Lizzie!