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Thanks for your opinion .... jrcramer

I would advise against crescendi using swell pedal in baroque music. Unless you like fancy things like bach on saxophone, or monteverdi on theremin...

I disagree. It is a proof in the sense that it is a valid numerical statement or equation, and were it invalid, it would simply not be a proof. It would be an invalid statement within the syntax of mathematical language, otherwise known as garbage. This is flawed in oh so many ways. We must account for the development of this number system in our understanding. It is the pattern (within nature, in the most general sense) that led to the eventual development of this number system. Here is how most believe this began... from observing patterns in nature, expectations formed regarding these patterns. They did not become RULES until a system was developed around those expectations. In effect, when the number system was created, rules were born from it. Prior to those rules, all we had were expectations based on observing patterns. I sincerely fail to understand the point you are trying to make here. If you're attempting to say that an expectation cannot exist without a rule, I'm saying the exact opposite. I'm doing so with clear and concise examples. Patterns in nature/environment lead, first, to expectations, which may later lead to "rules" about something. It's perfectly clear to me that this is the process. Perhaps you are applying the usage of the word "rule" liberally without isolating rule from expectation or proof.