Date: Jan 29, 2013 10:02 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=8183575

> Multiplication* an arithmetical operation, defined initially in
> terms of repeated addition, usually written a × b, a.b, or ab,
> by which the product of two quantities is calculated: to multiply
> a by positive integral b is to add a to itself b times.
>
> i.e. ab = a added to itself b times
>
> This definition fails proof by induction.
>
> So what other proofs can be used to prove ab does not equal
> a added to itself b times?


I don't follow your argument. Assuming that something "fails proof
by induction" [1], it does not follow that the result is not true.
Maybe it can be proved by another method.

[1] By the way, this is not clearly phrased. Do you mean every
proof by induction must fail or some proof by induction must fail?
Also, I think you're using the term "proof" differently than
mathematicians use it (i.e. a logically correct argument), even
aside from the liberal use of "proof of a defintion" (by which
I assume you mean something like "a proof that something or
other fits the criterion for a definition").

Dave L. Renfro