I have added a short sub-section to my booklet "The Meaning of Hegel's Logic", at the end of the section on Formal Logic, to sum up what I have learnt from the reading of the book Julio kindly sent me.

I would be very happy if anyone could suggest what might be a likely direction to go from here:

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Complexity, Mathematics and Dialectics

Over the past decade, a new science has emerged known as "Complexity". This science uses computer simulation to empirically investigate the outcome of very simple formal rules governing the interaction of a large number of component sub-systems, or individuals, in a system.

The need for an empirical science to study these phenomena is explained by a corollory of Gödel's Theorem discovered by Alan Turing, the founder of Computer Science: that in any computational system there at least one algorithm, the results of which cannot in priciple be predicted. This is a bit of truism for anyone who has progammed a complex computer system!

The implication for understanding the relation of formal and dialectical logic is profound. In any complex (i.e. real) system, the phenomena exhibited arise out of the relations of the component systems in a way which cannot in principle be deduced by formal logical analysis; and this is true even where the atomic relations are absolutely simple and static.

Such complex systems fall initially into two kinds: those where the atomic relations lead to a static or repetitive structure in the mass, where the mechanical application of formal logical analysis is valid, and those that lead to chaos, where the Law of Large Numbers and probability theory is valid. A simple measure derived from the atomic relations indicates whether the system is likely to be structured or chaotic. But there is also a third case, the border-line or transition case, where the atomic relations lead to "complex behaviour", rich in form, leaps, sudden transformations and dynamic structures, all in principle unpredictable from the standpoint of formal logic.

Furthermore, it has been empirically established that where the sub-systems have the capacity to replicate and mutate in some way (the normal condition of any significant "real" system), thus modifying the balance of the atomic relations, the system will in time either "die", or gravitate to a complex transition system, from a chaotic or structured state to a complex one, from a stable, predictable system to one giving rise to complex, "revolutionary" phenomena. Empirical computer simulation has reproduced analogues of systems (neurological or social) which learn, the origin of life from a "primeval soup", punctuated evolution, etc.

The scientists who are doing this work have expressed serious concerns about the state of the world economy, especially after the end of the relatively stable Cold War Period, and have tended to become fans of Heraclitus or Taoism; interestingly, none of them have so far recognised that a very precise science of the logic of complexity already exists in the Logic of Hegel.