Eigenform and Eigenbehavior

Introduction to Eigenform

An eigenform is a fixed point for a transformation. In this context an arbitrary transformation is allowed. Tranformation means change. To look for an eigenform is to look for something that does not change in the presence of change. We work with this notion all the time, making objects of ourselves, always changing and yet considered to be a single person, making constant objects of the moving display of our perceptions.

This usage of the notion of fixed point is an extension of some technical uses of the term, but we mean it to be taken both informally and formally. In the next paragraph, I give a specific example of how an eigenform can arise as the fixed point of a transformation that is simple and syntactical. For our purposes an eigenform is the analog of an eigenvector in analysis or linear algebra, but it is much more general and includes the fixed points that occur in reflexive domains as will be explained below.

I define T(x) = [x]. Then I can apply T again and again to an arbitrary x as shown below:

x
[x]
[[x]]
[[[x]]]
[[[[x]]]]
[[[[[x]]]]]
...

If you do this for a long time it begins to look like

E = [[[[[[[[[[...]]]]]]]]]]

and this expression has the form of a something that does not change if you put one more set of brackets around it. Thus E (above) is an eigenform for the transformation T where

T(x) = [x].

The entity E appears in our perception due to the recursive action of the transformation, and it is a consequence of how one deals with E, seeing it as invariant under T, that makes it into an object for our perception. That E appears as an object is part and parcel of being an eigenform. Thus Heinz von Foerster spoke of eigenforms in the phrase “objects as tokens for eigenbehaviours” (Foerster 2003). Heinz, in a wonderful turn of perception, turned the mathematical idea on its head. He pointed out that ordinary objects are tokens for eigenbehaviours. Ordinary objects are invariances of processes performed in the space of our experience. The ‘space of our experience’ is the context in which we have our experience and it is the experience itself. I make objects by finding fixed points in the recursion of my interactions. Eigenforms are a touchstone for the relationship of circular and recursive processes and the ground of our apparent worlds of perception. ...