In article ,
wrote:

For that, we'd need a generalization of finite sets whose cardinality
can be be complex.

Has anyone since done anything with the idea
http://groups.google.co.uk/group/sci...d3ff8198196ace
that the set of Motzkin paths has cardinality i?

Maybe you meant Motzkin *trees*. In case anyone is wondering,
these are rooted planar trees where each node has one or two
daughter nodes. The set M of Motzkin trees is equipped with
an obvious isomorphism

M = 1 + M + M^2

since every Motzkin tree is either a one-node tree, a node
connected by an edge to another Motzkin tree, or a node connected
by two edges to two Motzkin trees.

Using the techniques of Schanuel, Gates, Leinster and Fiore,
the "generalized cardinality" |M| of the set of Motzkin trees
satisfies

|M| = 1 + |M| + |M|^2

so

|M| = +-i

This sort of reasoning seems completely insane at first, but it
leads to many valid and interesting results; for details see

http://math.ucr.edu/home/baez/week202.html

ANYWAY:

Jeff Morton and I put a lot of work into this idea when we were
trying to categorify the quantum harmonic oscillator. The Motzkin
trees are a categorification of the Gaussian integers; the
"+-i" hints that Galois theory is relevant. We figured out how
to categorify the algebraic integers in any algebraic extension of
the rationals, getting an "algebraic extension" of the category
of finite sets. We figured out the beginnings of a theory that
associates a "Galois 2-group" to any such algebraic extension.
I was pretty excited about this, but Jeff was eager to reach ideas
connected to physics, and this seemed like a long way around.
In particular, one needs not just algebraic numbers but also
transcendentals to make sense of the "exp(-itH)" in quantum mechanics.

So, we dropped this project and came up with a much simpler
category of "U(1)-sets" whose "cardinalities" are complex:
a U(1)-set is simply a finite sets of points ("quanta") labelled
by phases. Here we are putting the phases in "by hand" instead
of seeing them emerge from category-theoretic considerations.
This is a bit unfortunate, but the advantage is that everything
works quite quickly and smoothly, and there's a clear physical
meaning to it all.

If anyone who knows categories, combinatorics and Galois theory
wants to become a math grad student at UCR and work on the project
Jeff and I dropped, they should contact me.