Re: Cranks

From: Anthony Natoli (anthonynat_at_aol.com)
Date: 06/24/04 

Date: Thu, 24 Jun 2004 21:14:36 +0000 (UTC)

On 24 Jun 2004, H. Shinya wrote:
>On 24 Jun 2004, Chan-Ho Suh wrote:

(snip)

>>enough to understand a proof of the impossibility of angle trisection.
>
>I think any average person can understand the proof, if s/he tries
>(or wants to) I am assuming that anyone can understand any available
>proofs. UNDER THAT ASSUMPTION, I am talking.

Let's be careful here. I am the one who mentioned "angle
trisection" and "circle squaring". Any "average person"
would say "why couldn't I?".

However, these phrases are shorthand for well defined
and ancient geometric problems:

Contruct (goal) with a non-fixed compass and an unmarked
straightedge (that is, a ruler).

(I'm a bit forgetful of the term for the compass, but IIRC,
it is not the modern type circle-drawing compass with a
fixable spacing)

The three goals were: trisect a given angle, square a
given circle, and double a given cube, using only those
two ancient tools mentioned above.

The circle-squaring is impossible since pi is irrational.
The cube-doubling is impossible since 2^(1/3) is
irrational.

(I don't know the shorthand answer why angles can't be
trisected.)

And of course, CERTAIN angles can be easily trisected,
such as 90 degress. The problem was for any _given_
angle, besides the limited tools.

(There are no special circles or cubes which get around
the irrationality crux of their impossibility proofs.)

So it is not surprising if an amatuer mathematician
stumbles over the phrase "square the circle" or
"trisect an angle" and thinks "hey, that's possible...
how can anyone say it isn't?".

The amateur may not have read further for the extra
conditions on the problem, if the conditions are indeed
listed in the amateur's reading materials.

The inquisitive would say "why do we need such conditions?"
Well, that's the ancient problem, so don't question
the conditions.

The crank ignores the proof, or ignores its reasoning,
and/or fails to meet all of the conditions; for example,
by using a marked straightedge.

The crank may subtly fail to meet the conditions, but if
a person points out his error, the crank will inevitably
attack that person, his intelligence, his mother, and his
motives for pointing out the error.

A non-crank would either try to see the error, or would
accept the correctness of the person, and accept the
existence of the subtle error as being the case.

It sometimes all boils down to attitude. Cranks: bad
attitude. Non-cranks: good attitude.

Anthony Natoli

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