The meta-system tower and its grounding

There is more to come on this topic.

This will refer to my PhD thesis, and to the Principia Cybernetica, in particular to its MetaSystem Transition Theory.

In systems that hold and manipulate information (such as computers, organisms, societies and evolving populations) it is possible for a system to hold and manipulate information that represents the system itself, in such a way that there is a causal links in both directions between the system and the information; if the system changes, so does the information; if the system changes the information, the system itself changes accordingly. These two forms of self-reference are called reification and reflection respectively; sometimes they are referred to collectively a reflection, and such a system is called a reflective system.

A system that holds and manipulates information (whether reflective or not), does so in a structural field [Smith] (or just field) in which the information is represented. (For example, an electronic calculator stores and manipulates information in the field of numbers.) A reflective system is part of its own structural field... thus, the self-referential nature of the system is also part of the structural field.

For information in a field to mean something, an interpretation must be put onto the field.... another way to regard this is that the field must be taken in a context. For many fields, there is a natural interpretation -- for example, for the information represented on a clock-face (the angle of the hands) the natural interpretation is a time of day. A structural field and its interpretation, and the link between them, are tied together in another structural field, which I shall refer to as the metafield of that field.

Of course, the metafield, being itself a field, needs an interpretation, an so on, so we see an infinite tower of fields appearing. By convention, in this field of study, we consider the metafield of a field to be above the field.

This tower is itself a representational system, and so is represented in a structural field that contains not only the representation of such an infinite tower, but also of the level shifts between individual fields -- it contains not only self-referential systems, but also the means to supply a meaning to each of these systems (by applying to it the system above it) -- and so is able to give meanings to the self-references within it. This field, containing the tower, is the hyperfield of each of the fields within the tower. The hyperfield may itself be reflective.

But if this tower-containing hyperfield can give a meaning to the way each of the self-referential fields within it gives meaning to the self-referential fields below it in the hyperfield, what gives meaning to the hyperfield? The meaning of the hyperfield comes, of course, from the metafield of the hyperfield... and the whole embedding of field in metafields, and field-metafield relationships in hyperfields, continues as a new tower (the metatower) in the next pair of metafieldness/hyperfieldness dimensions... and this tower of towerness is itself at the base of another tower...

Now we can see the full structure of the tower system... we can include it all in any or every one of the fields within it. But for a field to include structure that is indirectly above it, the immediate metafield of that field must also include it. Thus, each field has complete control over what may be done in the fields below it.

Thus, the metatower system is asymmetric; each field can include fields indirectly above it only through their inclusion in the field immediately above it, but it automatically includes all fields below it.

Let us now consider a structural field that is (at least) one level outside all those that can be reached by metafield and hyperfield transitions. This field, the ground field, contains all other fields (and, presumably, itself... it contains all fields)... and yet can do so while being itself flat -- all other fields are simply directly part of it. Note that while the ground field is uncountably infinitely far away from any of the fields it contains (and so we may refer to it as the omega-hyperfield (omega is conventionally used to represent the first number that cannot be reached by starting at zero and adding one any number of times), all the fields it contains are immediate parts of it.

Now consider the similarity between the relationship between the ground field and a field under it, and that between a meta-field and a field under it... the former is one beyond the limit of the latter... and yet any field, given suitable contents, could appear to be the ground field to fields under it... the suitable contents being those fields. Thus, there are relative ground fields and an absolute ground field; any field with the ability to create within it fiels to which it is the metafield or hyperfield can make itself into a relative ground field.

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