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):

http://mathforum.org/kb/message.jspa?messageID=8248950

> 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 [1] for some of

my other favorite 1800s algebra texts.)

Charles Smith

"A Treatise on Algebra" (1896)

http://books.google.com/books?id=BagXAAAAIAAJ

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

http://mathforum.org/kb/message.jspa?messageID=6862438

Dave L. Renfro