[FOM] Re: Shapiro on natural and formal languages

[Timothy Y. Chow]

On Wed, 8 Dec 2004, Rob Arthan wrote:
> 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?

Maybe this part isn't so clear; in that case, just rephrase my example to 
say, "The punctured torus and the wedge of two circles are homotopic."

>> 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.

I didn't mean to use the terms "linked" and "unlinked" generically, so to 
speak.  If drawing pictures were easier, I would have said, "Can [picture]
be continuously deformed to [picture] in R^3?" where the [picture]s are
specific examples of "linked" and "unlinked" two-holed tori.

> 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

Certainly that is "much" of the point, but it is not an accident that some 
of the most striking new advances in low-dimensional topology and geometry
have been made by mathematicians with a very strong visual intuition.  The
key ideas are often motivated by a simple, intuitive picture (that has to
be backed up later with proof, of course).

> 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.

Joe Shipman already mentioned something along these lines, or I would have 
mentioned it too...especially Conway's so-called "ZIP" version of the 
proof.  http://new.math.uiuc.edu/zipproof

>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.

These and other "proofs without words" were also mentioned previously.

Tim