[FOM] Formalization Thesis

Vaughan Pratt pratt at cs.stanford.edu
Mon Jan 21 01:14:22 EST 2008

Bill Taylor wrote:
> Note the fiddling with the string, and record the closest any two
> separate pieces of string ever get to one another.  (We must never let
> the string touch itself, but that is trivial.)  Whatever this distance,
> let the distance between string and sleeve remain always at most 1/3 of
> this distance, then two bits of cellophane can never get closer to touching.
> Am I overlooking something simple?

This would work if both knots were sleeved.  However because one knot is 
bagged, such moves as passing the sleeved knot through the bagged knot, 
or (as in my commutation example) sliding the whole string including the 
snugged sleeve knot along the shape formed by the bagged knot, drags the 
bag into the bagged knot in a way that (a) may put indefinitely many 
layers of cellophane between two points on the string depending on the 
number of moves, making 1/3 too large, and worse (b) may entangle the 
string in the cellophane in a way that obstructs a move that would 
otherwise have unknotted the string.

As an example of (b), namely an unknotted string entangled in the 
cellophane despite being unknotted, consider the situation where the 
string including its attachment points is entirely outside the 
cellophane cylinder and each on its own is unknotted, with the 
cellophane amounting to a straight pole.  The string may run straight 
across (no entanglement), or may form a single (aka simple) hitch around 
the cylinder (shown as a pole in the diagrams at 
http://www.realknots.com/knots/hitches.htm), or a clove hitch, or a 
rolling magnus hitch, or a cow hitch.  These five configurations are 
distinct (non-interconvertible) links but become the same knot (namely 
the trivial knot) when the cellophane cylinder is removed.

While the attachment points for these configurations are different from 
those in Conway's construction, these examples make the point that 
situations can exist in which the trivial knot is nontrivially entangled 
with the cellophane cylinder.  What is missing from your account of 
Conway's proof is an argument that, no matter how many layers of 
cellophane may get trapped between two neighboring pieces of string, the 
cellophane does not block some move along the way to untying the string 
to show it is the trivial knot.

The relevance of all this to FOM is that geometric intuition in general 
and "proofs by diagram" in particular can be misleading, a (the?) 
message in Lakatos's "Proofs and Refutations."  Paradox is sometimes 
credited as the principal driver of formality and rigor in mathematics, 
but fallacy surely deserves no less credit given the ease with which 
fallacies can be smuggled undetected into proofs, witness the monthly 
"Fallacies, Flaws, and Flimflam" section in The College Mathematics 
Journal some of whose fallacies can be quite challenging to track down.

Vaughan Pratt

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