The essay comments on Klaus Mainzer's book 'Thinking in Complexity. The Computational Dynamics of Matter, Mind and Mankind' (Springer 2004). The way of perceiving the world called 'thinking in complexity' proves revolutionary in science and philosophy as it radically changes our world perspective. This is Mainzer's point which is reported in some details. Besides such reports, the essay poses some related questions among which are the following. The revolutionary initiative is due to mathematical logic at this point from which theoretical computer science has emerged, that is, the discoveries concerning computability as due to Gödel, Church, Turing, Post etc. This software complexity is entangled with the complexity of hardware, the latter meaning dynamic systems changing in time as studied by physics, technology, biology etc. The essay distinguishes two uses of 'complexity'. (1) The most...

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The essay comments on Klaus Mainzer's book 'Thinking in Complexity. The Computational Dynamics of Matter, Mind and Mankind' (Springer 2004). The way of perceiving the world called 'thinking in complexity' proves revolutionary in science and philosophy as it radically changes our world perspective. This is Mainzer's point which is reported in some details. Besides such reports, the essay poses some related questions among which are the following. The revolutionary initiative is due to mathematical logic at this point from which theoretical computer science has emerged, that is, the discoveries concerning computability as due to Gödel, Church, Turing, Post etc. This software complexity is entangled with the complexity of hardware, the latter meaning dynamic systems changing in time as studied by physics, technology, biology etc. The essay distinguishes two uses of 'complexity'. (1) The most general one regarding any system, i.e. any set having elements interrelated with one another, which are either (A) abstract and static, as are the domains of mathematical theories etc. or (B) dynamic, that is, changing in time as are bodies, ecosystems, minds, societies, etc. (2) A more specific notion refers only to those dynamic systems which behave in a way we call nonlinear. There is a historical parallel between hardware complexity and the ancient Atomists' notion of complexity on the one hand, and between sofware complexity and the Stoic idea of 'rational seeds' as programming reality. Against the background of the old atomic notion, there appears the striking novelty of the modern definition of complexity in terms of feedbacks and non-linear behaviour. Special attention is paid to the problem of insights which produce new algorithms to reduce complexity of some computational procedures, esp. mathematical proofs. This is perfectly exemplified with Gödel's statement to the effect that: passing to the logic of the next higher order has the effect, not only of making formally (i.e. algorithmically) provable certain propositions that were not formally provable before, but also of making it possible to shorten, by an extraordinary amount, infinitely many of the formal proofs already available. Such a shortening means reducing complexity.

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