Date: Feb 5, 2013 10:56 AM
Author: Dave L. Renfro
Subject: Re: Proving a definition of multiplication (wrong) by induction

Jonathan Crabtree wrote (in part):

> 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

For what it's worth, here's how one of my favorite 1800s
algebra texts deals with this matter. (See [1] for some of
my other favorite 1800s algebra texts.)

Charles Smith
"A Treatise on Algebra" (1896)

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 [...]

[1] 6 October 2009 math-teach post

Dave L. Renfro