Aesthetics / Cognitive Science / Computer Science / Creativity / Incompleteness / Neuroscience / Philosophy / Science

Formal Systems and Creative Systems

File:Mosaico alhambra.jpg

Each formal system, be it an algorithm, a formal theory made up of axioms and rules of inference, a formal grammar describing a set of strings of characters, or whatever kind of formalism, is a finite length text, so it contains only a finite amount of information. So even if a very large, infinite or at least unlimited amount of data can be generated from it or can be parsed by it, that data can only be structured according to a limited range of patterns. The formal system can then be viewed as a compressed version of all that data. The compression is only possible because the data contains redundancy, regularity or patterns.

Any data structured according to other patterns is not covered by the particular formal system. Since it is always possible to construct data following different patterns, each single formal system is limited. It has blind spots. It can be extended into a more extensive formal system, but that one would be limited as well.

Such an extension is always possible, but a formal system cannot describe or achieve its own extension. Describing a limited multiplicity of patterns, it simply does not describe any pattern it does not describe, so it cannot transcend itself. It does not contain the information required to create something novel that is not already contained in its genotype. It is uncreative.

A creative system can therefore not be an algorithm. It can contain a formal system or algorithm as its component, but it must in addition contain a mechanism capable of creating a novel pattern and adding this new pattern, or rather the finite rule describing it, to the formal system. By doing so, the formal part or knowledge of the system is changed and extended. This creative mechanism, however, must be applied to the formal system from the outside. If it were under its control, it would be part of that algorithm or formal system and the result would be a limited or incomplete formal system again, producing only a limited multiplicity of patterns.

While formal systems can be thought of as abstract mathematical objects that can be described without referring to any physical embodiment or implementation, a creative system – being capable of performing processes that cannot be described inside a single formal system – needs to be a physical system. While formal systems are timeless, closed, describing something once and for all, creative systems are developing and historical, existing in physical time and embodied in matter.

For a related article, see here.

(The picture is from https://commons.wikimedia.org/wiki/File:Mosaico_alhambra.jpg)

9 thoughts on “Formal Systems and Creative Systems

    • No, but it is not possible the way the way the AI research community has tried for 60 years. You need more than algorithms, but it is possible. Algorithms are formalisms and formalisms are always limited. If you think of an AI system as a self-programming system, each algorithm would only contain a limited amount of programming know-know-how (patterns of how to put programs and other information together) and so there are places in the “knowledge-scape” it cannot reach. But it is always possible to extend it and that is a constructive process. However, if you try to integrate that extension process into the algorithm, what you get is a limited algorithm again, so it must be outside of it, not under its controll. You cannot do that with algorithms but physical systems cappable of doing it are possible (and I think the human brain can be viewed as such a systems).

  1. nannus, interesting post. I hope you’re okay with a few questions.

    What are your thoughts on how we could distinguish a creative system from a formal system that has patterns with which we are not yet familiar?

    I’m a bit confused by the requirement that a system be physical. What would you see as an example of a non-physical system?

    What about physical systems do you think enable them to transcend the limitations of formal systems?

    Sorry if any of this is getting too far ahead in your series. No worries if you plan to cover them later.

    • Sorry it took so long for me to answer; I do not have much time at the moment. I would like to address these question in longer articles in the future, but at the moment I can only give you a short answer to each.

      1. Distinguishing a formal system from a creative system can indeed be difficult. If you have a creative system that contains a formal system F1 and transforms this into an extended formal system F2 by applying a creative mechanism (think of the diagonalization + modification operation describe in my other article as an example), it should be possible to construct a larger formal system F3 in which this transition will happen inside the formal system. I think you can always formalize these transitions in hindsight. But there is no all-encompassing formal system that would describe all of these extensions. There is no way to express all possible developments of a creative system inside a single formalism, but distinguishing the two can indeed be difficult.

      2. To run an algorithm or the derivations inside a formal system in practice, you always need a physical implementation. However, algorithms or other formalisms can be treated as abstract mathematical entities that “exist” in a timeless manner. An algorithm’s properties can be understood in abstract terms independently of any physical embodiment. As a result, mathematicians normally abstract away from the physical implementation. You can describe an algorithm without any reference to an actual physical system.

      On the other hand, I think this separation of the abstract system from its physical embodiment is no longer possible for creative systems. You need to include the physics of the system in the description, while for algorithms, you can leave it out of the description. The creative mechanism that can change the formal component of the system from one stage of development to the next (think of the diagonalization + modification operation described in my other article as an example) must not run under the control of the formal system or algorithm, it must be asynchronous to it. I think you need physical time for this. A physical system can change, it can evolve (like organisms are doing).

      I admit this requires more explanation and I will try to provide it in further articles (or at least point out directions in which I see the necessity of research, since I do not have the complete picture). For the time being, just a comparison with biology: The genes of an organism do not describe it completely. What they cannot capture is the evolution of the genes themselves. While some processes of genetic change (for example, crossing-over between chromosomes) are happening at least partially under the control of the genes, some parts of the mutational processes are outside the genome’s control. And this is necessary for the organism to be able to evolve. Completely controlled evolution would be no evolution at all but ontogenesis involving some controlled gene editing process. It could not generate anything new. For the physical system in terms of which the organism is implemented, a mutation of the genes is just a normal physical process like any other, but it is not part of what the genome of the organism describes. So the organism as a formally describable entity is embedded into a system (or emulated by a system) that has more properties than what is covered in the formal description. One can try to build computer simulations of organisms and this is actually being done, and one could describe these simulations as abstract algorithms independent of their physical implementation, i.e. the algorithm can be understood “completely” as far as it is determined by its genes, but this description is incomplete since it does not include all of the possible evolutionary changes the organism may go through. In the development of cognitive systems, I see a similar division into aspects covered by a formal description and a creative process that, while being normal for the physical system, is outside the formal description.

      • Nannus,
        I appreciate the thoughtful reply. No worries on the wait. These discussions always happen when we have the time.

        On 1, I guess my follow up question would be, how can we be sure creative systems (in the manner you describe them) exist? That is, how do we know that systems we intuitively see as “creative”, aren’t simply enormously complex formal systems? In other words, how can we be sure creativity, in the manner you conceive it, isn’t an illusion? (I’m not asking whether creativity exists, but whether it isn’t what it intuitively seems.)

        On 2, certainly any system, to have any actual effect in the world, must be physical. Even what we call “abstract” systems have some physical reality, in brains if nowhere else, although that physical instantiation generally doesn’t allow them to actually execute.

        The difficulty is that no formal system that we can actually create is ever a complete description of the physical system that instantiates it. That means that a fully complete mathematical model of the physical system would be a superset of the mathematical model of the implementation of the formal system. There would be numerous formal systems that we could interpret the physical system to be implementing. (I recently did a post on this titled: “Are rocks conscious?”) But still, in principle, a fully complete mathematical model of the physical system could itself be said to be a formal system.

        I guess maybe the question is, are there aspects of physical systems that can’t ever be modeled mathematically? Individual quantum effects come to mind as a possibility, although it’s not clear how relevant they are above the atomic level in non-isolated systems. And even quantum effects can be mathematically modeled en-mass. Still, we can’t develop a fully complete mathematical model of any physical system yet, so the possibility remains, although humanity has a habit of inventing new mathematical tools when it runs into problems not currently conducive to mathematics. (See Newton’s invention of calculus for example.)

  2. Reblogged this on Creativistic Philosophy and commented:

    An intuitive interpretation of what was described in the previous article in more mathematical terms. The diagonalization operation described in the previous article is an example of a process that goes beyond one formal system, yet if you try to integrate it into the formal system, you get a limited formal system again and you can apply it again from the outside.

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