In this article, I want to start introducing some of the basic ideas underlying the work of Kurt Ammon. Ammon is a mathematician and computer scientist whose work I consider relevant for philosophy in general. Since there has not yet been a sufficient reception of his ideas in philosophy, I want to introduce some of his basic ideas in this and subsequent articles to make them more accessible.
In this article, I want to quote some important sections from “The Automatic Developments of Concepts and Methods“ by Kurt Ammon (University of Hamburg, 1987), specifically the section where the concept of “Analytical spaces” is introduced. I consider this a very useful philosophical concept.
On page 73, Ammon writes:
Roughly speaking, analytical spaces form the components of creative systems. They consists of consistent but incomplete knowledge and the objects this knowledge refers to. The boundary regions of analytical spaces contain anomalies such as inconsistencies, inefficiencies, and gaps. Different analytical spaces are complementary in the sense that they cannot be unified into a single analytical space within a limited space of time.
A more concise definition is given on page 74:
An analytical space of a creative system consists of a limited amount of consistent knowledge such as a few concepts and methods and the objects this knowledge refers to.
It is important to understand that here the knowledge and the objects it refers to are viewed together. The knowledge develops by its interaction with the objects it refers to. These objects have more properties than the knowledge covers, so the creative system (a human being or a group of humans, for example, or possibly a computer program, may generate new knowledge through interaction. Ammon gives an extended example that I want to cite here in its entirety (page 76):
If a team of engineers has developed a new product such as a car or an airplane which is put on market, weak points of such a product are recognized after some time and the product is revised and improved. This process is repeated many times. The knowledge of the engineers about the product and the product itself form analytical spaces. At any given point in time, the engineers have only incomplete knowledge about the product, which is revised and extended through experience. The product serves as the basis for the construction of new knowledge. The accumulated knowledge forms the basis for revising and improving the product. Thus, there is a coevolution of the knowledge and the product, i.e. the knowledge, the product and the interaction between the knowledge and the project run through an evolution process. This cyclic structure also applies to earlier stages of the product development process.
Unlike classical epistemological approaches, here the knowledge and the objects it refers to are treated as belonging together. The system starts with simple knowledge and evolves by interacting with the objects which, at any time, have some unknown properties.
On page 80, Ammon additionally defines the concepts of division and unification of analytical spaces:
The generation of analytical spaces and the bifurcation of existing analytical spaces in creative systems is called division and the integration of different analytical spaces is called unification.
He then presents two hypotheses (in Ammon’s text, definitions, hypotheses etc. are numbered, I have left out those numbers here):
(Creativity Principle) Cognitive structures evolve by division and unification processes of analytical spaces. An important question concerning creative systems is whether there is an explicit and general theory about cognitive structures. The following principle states that such a theory does not exist:[…](Division Principle) The cognitive structures of creative systems cannot be unified into a single analytical space.
This means that any explicit (formal or algorithmic) theory of creative systems is incomplete.
As a result, there is a trade-off between explicitness (or exactness) of descriptions and completeness. Ammon calls this “indeterminacy principle” because it resembles the indeterminacy principle of quantum mechanics in that you cannot achieve both goals at once. You can have explicit or exact theories that are very special or very general, but less explicit or vague theories, but not both at the same time. If you try to make your descriptions of cognition (or culture, or mathematics…) general, they will become vague or break up into different analytical spaces. If you try to make them exact, they will lose their generality. However, this is a topic to be explored in a separate article.