The Reasoner - editorial on n-categories

The Reasoner - editorial on n-categories

[Vaughan Pratt]

In The Reasoner 2:12 (December 2008), freely downloadable as

http://www.kent.ac.uk/secl/philosophy/jw/TheReasoner/vol2/TheReasoner-2(12).pdf

from

http://www.thereasoner.org/

editor David Corfield writes about n-categories, giving an impressively
accessible short overview of the concept and then interviewing Tom
Leinster about his experience with n-categories.

The interview is followed by an article on the Paradox of Omnipotence by
Alex Blum, which addresses the frustration any omnipotent being must
surely experience at being unable to create a stone that she cannot
lift.  (Imagine the applications, such as blocking the fridge door when
on a diet: one paradox helping another.)  Many of us in moments of
temporary perceived omnipotence have experienced this accompanying sense
of impotence, which can occur so frequently in life that one learns to
suppress it subliminally in microseconds, becoming completely
unconscious of it at an early age (but not without much screaming before
then).

Properly understood, this so-called Paradox of Omnipotence is, as often
happens, really a principle, the Principle of Omniimpotence.
Paradoxical origins tend to potentize principles to a remarkable degree,
with potencies upwards of 200C (the fourth power of a googol).  As a
case in point the omniimpotence principle forms the basis of a useful
diagnostic.  Using all available tools, how would you go about creating
a stone you cannot lift?  If it seems impossible you may be suffering
from omnipotence.

The principle is modeled at a very elementary level by 0-1 matrices,
which cannot simultaneously contain a row of all 1's and a column of all
0's.  This is the zeroary case of a more general interference or
"uncertainty principle," the binary case of which is that any proper
meet in the rows of such a matrix precludes a proper join in the
columns.  (For this purpose a meet or join is held to be *proper* when
it is neither of its arguments.)  From this it follows that if a matrix
represents a semilattice by virtue of its rows having all meets then its
columns cannot have any proper join.

This and more can be found at

   http://boole.stanford.edu/pub/coimbra.pdf

as the dry edition of the notes from the course on Chu spaces I gave at
the School on Category Theory and Applications held at the University of
Coimbra in 1999.  (The wet edition, under the slogan "All wet all the
time," would have included the above wisdom on omniimpotence but wiser
heads prevailed, both mine.)

It is not immediately obvious to the untrained eye that there should be
any connection between the omnipotence of god and category theory.  It
should therefore be of some interest to both category theorists and
theologians that a more formal connection could exist beyond this mere
juxtaposition of articles in The Reasoner.

Vaughan

Re: The Reasoner - editorial on n-categories

[Vaughan Pratt]

Michael Barr wrote:
> It makes no more sense to
> ask what happens in the limit than it does to ask which way the fly was
> flying when it was crushed between the two locomotives.

What does "when it was crushed" mean?  Assuming an infinitely small fly,
i.e. a point, and modeling locomotives as line segments (since they're
much bigger than flies), if the locomotives are open at the front (the
better to absorb the shock of hitting a cow) then "when it was crushed"
cannot be when the frontiers of the locomotives met since there was room
for the fly then.  But at every other candidate for "when it was
crushed" we can ask in which direction it was flying.

The argument looks fine with at least one locomotive closed (cows be
damned).  With exactly one the fly is crushed before the locomotives
collide, while if both are closed the fly is crushed when they collide.
  In either case you're right.

Vaughan

Re: The Reasoner - editorial on n-categories

[Michael Barr]

What is missing from Vaughan's account is the "Principle of Asymptotic
Omnipotence".  Of course the supreme being can create a stone it cannot
lift.  But being omnipotent it can grow its strength to the point where it
can now lift it.  Omnipotence allows it to create a more massive stone
that it currently could not lift.  But then....  It makes no more sense to
ask what happens in the limit than it does to ask which way the fly was
flying when it was crushed between the two locomotives.

Michael

On Fri, 19 Dec 2008, Vaughan Pratt wrote:

> In The Reasoner 2:12 (December 2008), freely downloadable as
>
> http://www.kent.ac.uk/secl/philosophy/jw/TheReasoner/vol2/TheReasoner-2(12).pdf
>
> from
>
> http://www.thereasoner.org/
>
> editor David Corfield writes about n-categories, giving an impressively
> accessible short overview of the concept and then interviewing Tom
> Leinster about his experience with n-categories.
>
> The interview is followed by an article on the Paradox of Omnipotence by
> Alex Blum, which addresses the frustration any omnipotent being must
> surely experience at being unable to create a stone that she cannot
> lift.  (Imagine the applications, such as blocking the fridge door when
> on a diet: one paradox helping another.)  Many of us in moments of
> temporary perceived omnipotence have experienced this accompanying sense
> of impotence, which can occur so frequently in life that one learns to
> suppress it subliminally in microseconds, becoming completely
> unconscious of it at an early age (but not without much screaming before
> then).
>
> Properly understood, this so-called Paradox of Omnipotence is, as often
> happens, really a principle, the Principle of Omniimpotence.
> Paradoxical origins tend to potentize principles to a remarkable degree,
> with potencies upwards of 200C (the fourth power of a googol).  As a
> case in point the omniimpotence principle forms the basis of a useful
> diagnostic.  Using all available tools, how would you go about creating
> a stone you cannot lift?  If it seems impossible you may be suffering
> from omnipotence.
>
> The principle is modeled at a very elementary level by 0-1 matrices,
> which cannot simultaneously contain a row of all 1's and a column of all
> 0's.  This is the zeroary case of a more general interference or
> "uncertainty principle," the binary case of which is that any proper
> meet in the rows of such a matrix precludes a proper join in the
> columns.  (For this purpose a meet or join is held to be *proper* when
> it is neither of its arguments.)  From this it follows that if a matrix
> represents a semilattice by virtue of its rows having all meets then its
> columns cannot have any proper join.
>
> This and more can be found at
>
>  http://boole.stanford.edu/pub/coimbra.pdf
>
> as the dry edition of the notes from the course on Chu spaces I gave at
> the School on Category Theory and Applications held at the University of
> Coimbra in 1999.  (The wet edition, under the slogan "All wet all the
> time," would have included the above wisdom on omniimpotence but wiser
> heads prevailed, both mine.)
>
> It is not immediately obvious to the untrained eye that there should be
> any connection between the omnipotence of god and category theory.  It
> should therefore be of some interest to both category theorists and
> theologians that a more formal connection could exist beyond this mere
> juxtaposition of articles in The Reasoner.
>
> Vaughan
>
>
>