[FOM] Formalization Thesis
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.
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