[FOM] Re: Shapiro on natural and formal languages
Rob Arthan
rda at lemma-one.com
Wed Dec 8 17:21:51 EST 2004
On 1 Dec 2004, at 14:47, Timothy Y. Chow wrote:
> On Tue, 30 Nov 2004 JoeShipman at aol.com wrote:
>> Chow proposes two examples:
> [...]
>> Unfortunately, example 1 is too easy to prove non-visually.
>> Example 2 doesn't really qualify
>
> All right, let me try again:
>
> - The fundamental group of the punctured torus is the free group on two
> generators.
>
> Proof: Stick the fingers of both hands into the puncture and pull back
> the outer skin all the way around the torus, as if you're opening a
> curtain.
> This shows that the punctured torus is homotopic to a wedge of two
> circles.
So the punctured torus and the wedge of two circles have the same
fundamental group. But what's the evident visual proof that the
fundamental group of the wedge of two circles is free on two
generators?
> There are lots of similar "rubber sheet" arguments with a similar
> flavor. Here's a fun one: Can a two-holed torus where the two loops
> are "linked"
> be unlinked in R^3?
That's either a tautology or it needs you to define "linked" and
"unlinked" so that your statement has some mathematical content.
>
> - pi_3(S^2) is nontrivial.
>
> Proof: This amounts to visualizing the Hopf fibration.
and to realising that the Hopf fibratiion doesn't split (which is
"visually" obvious if you realise that a splitting of the Hopf
fibration would make S^3 homeomorphic with S^1 x S^2 which it can't be
because the two spaces have different fundamental groups).
I actually think you're looking in the wrong place for the examples
you're interested in. Obviously geometry and topology are informed by
visual thinking, but much of the point of these subjects is to help out
when simple visual imagination fails to deliver, cf., the Jordan Curve
theorem - which is visually obvious and true, buit took 40 years or so
to prove properly; and the proposition that the complement of a
homeomorphic image of a 2-sphere in 3-space is simply connected - which
is visually obvious and false (Alexander's horned sphere providing a
counter example).
A better example in elementary topology might be the scissors-and-paste
part of the classification of compact surfaces. But this has a very
algebraic flavour to it. Even better in my opinion are purely algebraic
things like:
1 + 2 + .. + n = 1/2n(n+1)
proved visually by thinking of an n x (n+1) rectangular array of points
divided into two similar triangles. In examples like this, visual
insight bypasses an inductive argument very directly.
Regards,
Rob.
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