> In short, mathematicians must get definitions right so the pedagogy > can follow. Using ill-defined definitions such as ab = a added to > itself b times just confuses children and their teachers. > > All the above supports the case for math professors (who should > both know better and lead by example) being sent to the naughty > corner for repeating, without thinking, ab equals 'a added to > itself b times'.
I think you're confusing "formal presentation" with what one mathematician might informally say to another person in a situation where it's clear the other person knows what is going on. As for pedagogy, extremely little of this is done by mathematicians, at least regarding the literature I imagine you're finding this in. Your welcome to give some precisely cited quotes of this ambiguity in the mathematics literature, by the way. In any event, your main point is an English editing issue, not something that suggests (let alone proves) that mathematical induction cannot be applied to a certain situation (your original claim).
For what it's worth, here's how one of my favorite 1800s algebra texts deals with this matter. (See  for some of my other favorite 1800s algebra texts.)
The following is from p. 14, the beginning of Article 28:
* In Arithmetic, multiplication is first defined to be the taking * one number as many times as there are units in another. Thus, to * multiply 5 by 4 is to take as many fives as there are units in four. * As soon, however, as fractional numbers are considered, it is found * necessary to modify somewhat the meaning of multiplication [...]