[FOM] Formalization Thesis

[Vaughan Pratt]

Bill Taylor wrote:
> if the knot pair is somehow untied, the cellophane must continuously
> deform, so that a line in it, originally from the north point of one circle
> to the north point of the other, will still become a line in the vertical
> plane joining them.  But what was knotted before, has now become unknotted!
> Impossible, QED.
> 
> I thought this proof was brilliant, especially for a schoolboy.

Yes, great out-of-the-box line of reasoning, even for an adult.  However 
I have the following problem with it.  Consecutive knots commute, as can 
be seen by leaving one loose while snugging the other up and sliding the 
string along the shape formed by the loose knot until the snug knot has 
passed all the way through the loose knot.  Now with the cellophane in 
place it's clear that the cellophane need not be disturbed at all when 
you snug the bagged knot and thread it around the sleeved knot.  However 
if you try to snug the sleeved knot and thread it around the shape 
formed by the bagged knot all topological hell breaks loose in the 
cellophane.  How do you argue that the cellophane doesn't get snagged on 
itself somewhere in this process and block further progress?  Was there 
more to Conway's argument?

Vaughan Pratt