Subject: Re: [HM] "President Garfield's Proof"
From: Michael Lambrou (lambrou@itia.math.uch.gr)
Date: Tue Jan 18 2000 - 10:17:43 EST
Ed raised an interesting point, as to the subjectiveness of what is
rendered "a similar proof"
>
> Allow me to make a case that disagrees somewhat with Michael Lambrou,
> who wrote of President Garfield's proof:
>
> "Well, to criticize the proof, I feel that it does not have enough new
> ideas to qualify it as a genuinely new proof: ..."
>
> Some people are "splitters", and find differences between things.
> Loomis, who collected over 300 "different" proofs of the Pythagorean
> theorem was such a style of thinker. Among those, there are many pairs
> that seem to me more closely related than the Garfield proof and the
> Chinese proof, though, indeed, these two are very similar.
>
> Other people are "lumpers", and find similarities among things. A
> determined lumper might find as few as three really different central
> ideas among the 300 proofs Loomis collected, though most of us would
> divide them into two main families, similarity proofs and dissection
> proofs, and perhaps 6 to 10 smaller families.
>
> If I were trying to demonstrate that the Pythagorean theorem has a
> variety of different proofs, I would certainly not choose the
> Garfield/Chinese pair to support my contention.
>
> On the other hand, if I were trying to demonstrate that diverse
> cultures solve the same problems and discover very similar solutions,
> despite their separation in time, space and culture, then I might
> choose this pair.
>
> That said, let me paraphrase two Williams, Gates and Clinton, and
> say "It depends on what you mean by 'different.'"
Here is, however, why I think THESE two proofs are not different. Remember
Garfield's proof divides by 2 and then multiplies by 2. That's all.
To make an analogy (and add humor to our wonderful list), if you ask
me for a dollar I may put my hand in my right pocket and give you one.
But there is another way. I will transfer the dollar from my right pocket
to my left, then take it back to the right, and finally give you the
dollar.
Is this a different way? No, I don't think so. OK, I agree, it depends
on what you mean by "different".
Michael Lambrou
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