in reply to MOPT-02 - substitution and formal systems
And past reading of that book (An Eternal Golden Braid) has me immediately objecting to the assignment of meaning to your symbols. I think you may have just shortchanged the audience here by skipping right over the idea that just because a system produces results that have behaviour corresponding to a meaning that the system itself is supposed to mean that. I don't have your experience and wouldn't have been able to write your meditation on my own but I've really benefitted by being able to divorce an ascribed meaning from a system.
In fact, that may be an important skill on it's own for use when reverse engineering an application. Part of the task involves observing a system in operation and attempting to assign meaning. The problem is that frequently your observed meaning is violated if you let the system generate more and more theorems. I had a practical application of that last year when reversing a COINSERV database and then the Xerox Metacode format. I had to continually revise my assumptions regarding the meaning of various things as I ran into more data. Eventually I got things right but it was a long, slow process.
So it's useful to look at a system and have an "aha!" moment and describe it as a binary counter. It's also useful to step back and remember that it may have a different meaning which only partly intersects with a binary counter.