Date: Feb 4, 2013 5:51 PM
Author: Dave L. Renfro
Subject: Re: Proving a definition of multiplication (wrong) by induction

Jonathan Crabtree wrote:

> In the following;
> "ab = a added to itself b times" or
> "ab = a added to a b times"
> ...the ambiguity about what 'times' refers to is really the
> essence of any confusion with definitions involving multiplication.
> So to clarify, are you stating that in formal definitions,
> the word 'times' refers to the count of addends, or binary additions?

In a formal treatment one would clear up any possible ambiguity
that arises in the use of a term such as this. To me, it's a
simple matter of writing

a + a + a + ... + a

and then saying exactly how many a's are intended, or equivalently,
saying exactly how many +'s are intended. If the phrase "b times"
could be ambiguous for the intended audience, then one shouldn't
use it without additional clarification. I consider this
analogous to giving driving directions to someone by saying
something like "up there a ways you want to turn left".
How far? Are there any roads to the left before getting to
the road I'm supposed to turn left on?

Dave L. Renfro