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.
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