[bdr]

"in fact, it would not be a bad definition of math to call it the study of terms that have precise meanings."

What meanings? I think that math is made of structure, not meaning. The latter is a human phenomenon. For example, a proof using geometry and one using algebra might be mathematically identical, but have different meanings (created when they are perceived).

[kc5tja]

The symbology used to form each proof is universally accepted and agreed upon. The symbol "=", for example, is taken not as a verb (a becomes b), but as a statement of truth (a IS b). Likewise, "+" has a specific meaning. The juxtaposition of terms is implicitly understood (e.g., all mankind agrees to interpret it as) as multiplication of some form (which is generalized in higher mathematics to function composition, which makes sense once it's explained). It is the ultimate language, where one symbol has one, and only one, meaning.

It is the structure which is a human phenomenon. The fact that you can express any algebraic expression in reverse polish notation is a great example of this. 1 2 + 3 4 <asterisk> / has the same meaning as (1+2)/(3*4). (Sorry for the <asterisk> thing -- the markup system of this blog is positively broken since it doesn't provide a means of escaping the markup characters.) The symbols, and hence the meaning behind them, remain the same.

[Goladus]

I think the question is whether you can say the terms used in either proof have precise semantics. Any of the terms in a correct geometry proof can be traced back precisely to the initial axioms, which are terms with meanings defined as precisely as possible.