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3. Conclusions



With the present paper I discussed a further example of a quite typical method of Klee's. It essentially consists of building complex structures by means of iterative procedures, by adding destabilising elements in order to achieve highly irregular and unpredictable behaviours. Though the iterative procedures are deterministic, the final outcome is irregular and quite unpredictable; we really face something quite close to deterministic chaos.

In fact Klee's way of seeing chaos is close to the corresponding concept in contemporary sciences.

Meaningfully, according to Klee's way of thinking, the chaos theme is connected with other important themes, which in turn characterise complexity sciences, and have relevant epistemological consequences.

Reiner Hedrich [25] illustrates some salient characteristics of the development of complexity sciences. Based on some of these characteristics, he shows that the development of complexity sciences didn't follow the historical course depicted in Kuhnís model of scientific revolution. Here we are interested in two of those characteristics.

First, the development of the new theories took a very long time, about one hundred years. This long latency period is due to the research of a solid mathematical theory useful for a better and extensive description of dynamic systems behaviour. In fact, such a long period covers abundantly the entire lifetime of Klee (1879-1940, like to say: from Poincaré to Birkhoff).

Secondarily, Hedrich stresses that the central kernel of the ideas in the new paradigm does not concern a specific disciplinary field: instead it is a sort of conceptual foundation, a shared background for every empirical discipline. This fact makes it conceivable that there was a widespread and (to some extent) unconscious emergence of such ideas, even though in purely qualitative and intuitive terms.

Mainly due to the phenomena of both deterministic chaos and sensitive dependence on the initial conditions (here not discussed) which characterises the dynamic systems, there are two important epistemological consequences of the introduction of the complexity paradigm (discussed in Hedrich's essay too): these are weaker versions of both the causality principle and determinism. In the essay Exact experiences in the field of the art [26] Klee seems to anticipate this theme. He argues that the artist has to discover and learn the laws that govern the natural world, in order to reach a better understanding and hence a more functional representation of things. But, inside these laws, there is a wide space of variability and unpredictability, that the artist has to exploit to avoid lifeless representations of the world. In the gap between law and unpredictability, the freedom of the artist works, but he always has to remember the necessity of both: the law and its limits. Ever since the early years of the Diaries, Klee expressed a full awareness about that, with concise and evocative words:

"Don't let the eternal spark become completely smothered by law's measure! Take steps in time! But don't go away from this world completely" [27]

Klee expressed the same concept many times during his lessons at Bauhaus. Indeed he was deeply intrigued by the creative power of irregularity:

"Irregularity means more freedom without lawbreaking" [28]

I showed elsewhere [29] that this conception of Kleeís is possibly connected with his ideas about the evolutionary mechanism acting in Nature. However there is something more and deeper.

Once again we must stress that Klee spoke about irregularity inside a law ("without lawbeaking"), which makes us think to something deterministic, but in the same time free. In addition there is the idea of pushing the system to irregular behaviours, which make us think to the edge of chaos slogan in complexity sciences.


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