June 29, 2004
Department of Mathematics,Statistics and Computer Science
University of Illinois at Chicago
Chicago, Illinois 60607-7045

To Whom It May Concern:
I am writing this letter in behalf of Tom Mandel and his work
on an insight of startling simplicity, and the notational model that he
developed from this insight. I shall describe in my own terms the
structure of this insight, but the reader of this letter is encouraged to
look directly at Tom Mandel's constructions and to encourage him to
communicate concerning them.

The question is the nature of a system, or a description in the widest
possible sense. We find "things" in the world and are most concerned with
how those "things" are related to one another. Tom's favorite example is
the two sides of a coin, separate and yet related (and brought into
existence) by the body of the coin itself. Any two apparently separate
things are related, but the story can become quite complex. Contemplate
the relationship between the pen and one's ability to write, or the
relationship between the electron and the operations of the internet.
A fundamental notation would be one that leaves room, given two
entities, for the relationship that goes between them. This relationship
can seem to arise from the individuals (as a conversation arises among its
discussants), or it can appear that the individuals arise from the
relationship itself (as electron and positron arise from
space/energy/photon, or as world and observer arise from awareness).

One should make one's own notations. Each such notation is itself
an instance of the ideal of relationship in one its many manifestations.
Tom uses

A -------

where C stands for
the arising from A and B // the connection/relation of A and B.

Such notation is simple, yet
insistent, calling
for the aticulation of C along with A and B.

Why is this important?  The answer is: Because such notation and the
attitude behind it continually call the question of relationship and the
nature of relationship. All descriptions, all systems are built this way.
But we keep forgetting the glue and putting it into the background. Here
all three fundamentals in any distinction are brought into the foreground,
and we want to know them and their dialogue. (Powerful Guru, Da Free John,
lead his students again and again with the question "Avoiding
Relationship?". There is no place to hide.)

There is an important symmetery here. If we bring forth the
relationship, then we can ask both how the parts create the whole,
and how the whole comes to be divided into parts.

We are all too familiar with
myriad ways of building from parts, but it is all too easy to lose sight
of the whole. To turn around into the whole, to ask how does this unity
become a multiplicity is the essential question of science. It is the
question that leads to theory, understanding, unification, clarification
and practical understanding. Without the question of the whole, we have
only recipes for the building of structures from known parts.

The essential insight that we all have and lose, again and again, is that
sense of the whole and an intuition of how the multiplicity arises.
There is no multiplictity!
In relationship, all parts cohere into the whole.
There is a multiplicity!!
It is obtained by ignoring the relationships the
return to the whole.

To the degree that systems are seen to evocations of
a whole or larger structure, there is an opening for plasticity,
creativity and new knowledge.

Having found this insight and his notation, where will Mandel go from
here?  Really, the question is, where will we go in our relationship to
him and to each other in this quest for unity in multiplicity.

Yours truly,

Lou Kauffman

P.S. There are many existing and possible mathematical, scientific and
philosophical continuations of this discussion. Prominent among them, for
me, is "Laws of Form" by G. Spencer-Brown, a book with a notation and
insight very close to the above remarks and to Tom's essential notions.
Ludwig Wittgenstein said "The limits of my language are the limits of my
world." We have the opportunity to expand our language, to expand
beyond imagination our world. Mathematics is all about
relationship, and the formal images of its patterns. Graph theory in
particular is concerned about the diagramming of entities in such a way
that significant relationships are indicated. Topology is the attempt to
formulate the notion of a continuous whole (a topological space) and to
find the appropriate ways of dividing the space into parts that reveal its
global strucuture. The themes are always there: mathematics, logic, music,
biology, physics, astronomy/cosmology, linguistics, philosophy, ...

All reflections of
the possibility
of saying anything at all
to anyone,
about the