Analytical Spaces

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.



  1. […] I have published an article on the concept of “Analytical Spaces” on my philosophy blog, see […]

  2. The concept of analytical spaces works very nicely with the creative process I use when composing electronic music or abstract 3D graphics.

    Each “track” that I create in an electronic composition is a separate “object”.

    I have limited knowledge inasmuch as I’m familiar with the object itself but I don’t know exactly what the results will be when I combine several tracks, or objects, together.

    This is even more obvious in the case of abstract 3D imaging, which literally combines various objects in sundry ways.

    Each individual object is “known” by itself but the results of combining them in various ways is only understood to a very limited degree, until it is tried and the results analysed.

    Analytical Spaces. I like it. Perhaps I should rename my gallery page.

    Good stuff nannus. Very thought provoking.

    1. Thank you, a great example.
      The examples in Kurt Ammons doctoral dissertation (that is what it is) are mostly from math and mathematical logic and computer science, which are also areas needing creativity, but it is relatively hard to read when you are not used to that kind of stuff. However, in such areas it is possible to formally prove the incompleteness of all theories and the necessety of creativity.
      However, these ideas are applicable everywhere and especially in the arts.

  3. […] created. This process can be thought of as the gradual development of increasingly sophisticated analytical spaces. Things that once where a challenge to our perception will become routine and the perceptive […]

  4. […] exact languages can only cover limited analytical spaces. They are useful for special purposes but they are always embedded into the normal language that is […]

  5. […] that medium-information structures are beautiful is that they lead to repeated extensions of the analytical spaces our perceptive system consists […]

  6. […] theories may be viewed as analytical spaces that contain a limited amount of knowledge about some aspects of reality, have a limited area of […]

  7. […] the calculations in all cases. In Ammon’s terminology, each of these limited methods forms an “analytical space” and there is no way to unify all analytical spaces into a single one covering all […]

  8. […] Creative systems develop by the division and unification of analytical spaces, i.e. finite connections of knowledge, together with the objects this knowledge refers to. This has been sketched in (Analytical Spaces) […]

  9. […] In a previous article I have described the concept of analytical spaces, developed by my friend Kurt Ammon, as a theory of knowledge. Amon views knowledge and the objects it refers to as developing together and he exemplifies this with an example from engineering: […]

  10. […] philosophy and related areas like classical artificial intelligence can only be viewed as limited analytical spaces. They cannot be […]

  11. I like it, and reminds me of when “Performance Art” became popular in the 1960’s. The “Performance artists” challenge the audience to think in new and unconventional ways, break conventions of traditional arts, and break down conventional ideas about “what art is”, taking place in situations where audiences may be merely coincidental, and it was free and “live”. But what you’ve written here is so interesting in that it reminds me that nothing is ever really finished, but it just adds to a space that already existed (creatively speaking).

    1. What Kurt described is quite general and I am thinking of an article about comparing it to the tradition of philosophical hermeneutics. A central idea of philosophical hermeneutics is the “hermeneutic circle”: particular ideas are interpreted in the light of more abstract generalization which in turn are derived from the special ones. It is like a circle or rather a spiral. And it cannot be formalized completely (or else it looses its generality).
      The idea of “breaking” existing structures is important here. One of Kurt’s central concepts is “structural break”.

  12. […] by the system. Each chunk of knowledge contained in the system is limited. We can view it as an analytical space. Analytical spaces have anomalies, i.e. the may encounter situations where the knowledge in them […]

  13. […] can be viewed as consisting of limited sets of knowledge covering limited parts of reality (see Analytical Spaces). Each of these sets of knowledge is incomplete. Taken together they are always incomplete as well. […]

  14. Reblogged this on The Asifoscope and commented:

    A basic concept underlying many of my thoughts…

  15. Life as a beta-tester? I wonder, does pure creativity per se issue forth from any ‘system’? In the physical sense, it must, yet if we speak of mind as the source of creativity . . . ? Is the integrated conscious and sub-conscious mind a system, and if so, how may it be described as such?

    1. Good question, hitting the nail right on its head. I cannot answer today (for family reasons, there is a life outside the internet 🙂 ) but I am going to write my thoughts about it as soon as I can (probably tomorrow).

    2. I think I am going to write a separate article on this issue, so just some short notes for now, since I am too tired today to write anything “systematic”.
      “System” is a vague concept and a “creative system” in the sense of a thing that can move out of the scope of validity of any systematic description of it would be an “unsystematic system” :-), so one can indeed argue if the term “system” applies here or in what sense. I need a little bit more time to write this.
      I remember (I think that was around 1985) when I was discussing with Kurt Ammon (we where talking about “Artificial Intelligence”) what he thought how thinking works. I remember us standing on the staircase on the outside of the computer science department in Schlüterstraße in Hamburg. His answer, just two words, was an enlightening moment for me. He said “Immer anders” (“always different” or “each time different”). There is no single formal theory describing it. Each instance can be described (so it is a system at each moment) but you cannot create a theory about it all (it is impossible to unify all analytical spaces). If you try to describe it in general, the description will diverge in all directions and somehow “splinter” or it will become very vague (both are manifestations of incompleteness). You can describe a special process exactly, but not how it works in general (and nothing has come out of AI research since they tryed to find the general structure of cognition, which does not exist).
      The discussion would also have something to do with the line of thought I have recently tried to develop about reductionism.
      The mind does not have a definite form (or it only has one at each moment, but not in general). In the mind, complex structures develop from simple ones, and the core of this development it is practically empty (Kurt has been speaking of the “Shunyata principle” here).
      We are always beta-testers. There is no final, complete version of us and we start from a very small or empty core (both as individuals, as cultures and as a species).
      Sorry, all very unsistematic stuff today that has to be sorted out a bit 🙂

  16. […] can view such a world-view as an analytical space, i.e. a limited body of knowledge about a limited section of reality. As long as you stay inside […]

  17. […] are using, either to interpret what we perceive, to think or to do something, defines a closed “space”. We can leave that space and take a vantage point outside that bit of knowledge. From this […]

  18. […] can be viewed as consisting of analytical spaces, i.e. limited chunks of consistent knowledge, together with the objects this knowledge refers to. […]

  19. […] of this creative and critical self-reflexivity of thought. Since they are operating in very large analytical spaces, the philosophical edges of their activity may disappear from their view, behind their horizon. On […]

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