Date: Jan 30, 2013 2:50 PM Author: GS Chandy Subject: Re: Proving a definition of multiplication (wrong) by induction Jonathan Crabtreee posted Jan 30, 2013 4:19 AM (GSC's remarks follow):

>

> > Jonathan Crabtree posted Jan 29, 2013 6:47 AM

> (GSC's

> > remark interspersed):

> > >

> > > 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.

> > >

> > To the best of my understanding, the definition

> does

> > NOT fail proof by induction (see attachment, using

> > "n" instead of "b", for convenience, and showing

> > start of proof only for integers). [My 'formal

> > statements' in the document won't pass muster with

> > teachers demanding a high degree of rigor and

> > precision, but I'm unable to do anything about that

> > at this point of time].

> >

> > In my opinion, the Collins dictionary definition

> > fails mainly because it is rather poorly

> > articulated.

> >

> > [I've not done anything with your P.S.].

> >

> > GSC

> > > So what other proofs can be used to prove ab does

> > not

> > > equal a added to itself b times?

> > >

> > > Thanks

> > > Jonathan Crabtree

> > > P.S. Apart from proof by common sense. Eg.

> > >

> > > let a = 1 and b = 0

> > > so 1 x 0 = 1 + 0 (by definition)

> > > and 1 x 0 does not equal 1

> > >

> > > or

> > >

> > > let a = 1 and b = 1

> > > so 1 x 1 = 1 + 1 (by definition)

> > > and 1 x 1 does not equal 2

> > >

> > > *

> > >

> >

> http://www.collinsdictionary.com/dictionary/english/mu

> >

> > > ltiplication

> >

> >

> > Message was edited by: GS Chandy

>

> Dear GSC

>

> Thank you for your interest and taking the time to

> document your thoughts.

>

> Rather than repeat the logic, please refer to my

> extended response to Dave.

>

> Jonathan

> P.S. Best wishes from Australia, where like India, we

> also celebrated 26th January as our national day.

>

Thanks for your kind response.

Several aspects of the logic in your extended response to Dave Renfro are flawed. I believe Dave, in his http://mathforum.org/kb/message.jspa?messageID=8191372 , has pointed out some of the logical inadequacies in your response.

I have some other issues with the logic in your response

- but cannot now pause to write them up in detail, as I'm in the midst of preparing for a major series of workshops. (But I'm sure you will see all of these inadequacies as you ponder the matter a bit).

Thanks for your kind wishes on the occasion of Indian Republic Day (26th January) and I heartily reciprocate with my good wishes for your Australia Day on the same date.

My own major project is to try to contribute enough to change mindsets and then realities on the ground here so that we could justifiably be proud of India in, say, 5-10 years' time. We have far too many things to be ashamed of right now (despite our growing economic clout, military might and middle-class wealth that we so proudly showcased at the Indian Republic Day parades in Delhi and other cities around the country).

[This is, of course, only my personal opinion (with which most Indians belonging to the middle- and upper-economic strata - and a few of those belonging to the lower economic strata - would most vehemently disagree). Doubtless those disagreeing would be able to find plenty of powerful arguments to justify their disagreement with my opinions on these matters].

With kind regards

GSC

Message was edited by: GS Chandy