I think that the frontiers and challenges of complexity as a discipline has been very well highlighted by Steven Strogatz in his book Sync on page 287 (Ref 1). Strogatz is a mathematician whom I admire for his intuitive approach to maths, which I believe might make maths popular amongst the masses.
I quote from his book the two relevant paragraphs, which I feel are important.
“…. I don’t want to leave you with a false impression. Sync is just a small part of a much larger body of thought. It is by no means the only approach to the study of complex systems. The chemist Ilya Prigogine and his colleagues feel that the key to unlocking the mysteries of self organization lies in a deeper understanding of thermodynamics. They see the emergence of order as a victories uphill battle against entropy, as a complex system feeds itself on energy flowing in from the environment. The community of physicists interested in pattern formation see fluid mechanics as its paradigm, where the rolling of a turbulent fluid gives rise intermittently gives birth to coherent structures like helices and plumes, rather than degenerating into a bland, uniform smear. The physicist Hermann Haken and his colleagues view the world as a laser, with randomness and positive feedback conspiring to produce the organized forms that occur all around us. Researchers at Santa Fe Institute are struck by the ubiquity of evolution through natural selection, not only in biological population, but in immune systems, economics, and stock markets. Others conceive the universe to be a giant computer, running a cryptic program whose discovery would constitute the end of science.
But for now, these are mostly pipe dreams. We’re still waiting for a major breakthrough in understanding, and it could be a long time in coming. I think we may be missing the conceptual equivalent of calculus, a way of seeing the consequences of the myriad interactions that define a complex system. It could even be that this ultracalculus, if it were handed to us, would be forever beyond human comprehension. We just don’t know.”
That is quite a grim reminder that not only reveals a quick glimpse of the unchallenged frontiers of the discipline of Complexity but also throws at us a challenge at the same time.
But I think a very likely discipline that has been missed out or the practitioners of the discipline hasn’t yet explicitly joined in is Engineering, especially the wing that practically deals with non-linear dynamics — the discipline called Condition Monitoring. Having my roots in that discipline I think that the new maths of complexity is mostly likely to be worked out or generated from this field. It is not that the maths doesn’t exist. One thing that is quite mature in the field of Condition Monitoring is “prediction of emergence in complex systems”. That is how the field got its name. The prediction, as it should be, is always done in the short term taking into account the ensemble as a whole. Admittedly, most of the maths is graphical. But I don’t see any issue with the graphical maths since it does the job so well indeed, which is a) short term prediction b) understanding and interpreting the interactions at play at any given moment along with interpretation of how new orders are created.
The most interesting thing is that it does not stick to one world-view as most streams, outlined above, have done. It sees the world from multiple perspectives – both Newtonian and Non-Newtonian — waves & vibrations, dynamics and non-linearity, sync & resonance, thermodynamics & fields, flow & fluid mechanics, randomness and feedback, wear and electrons, chaos and evolution, determinism and probabilistic. Undoubtedly it is tall order. A true master in this field (though such masters are difficult to spot since they mostly live like recluses) can easily flow from one perspective to another or hold multiple perspectives together at the same time while observing a phenomenon (I am referring to only one rather secretive school; with a handful of practitioners having not more than 10 masters). There are no inhibitions or ideological hold ups. That is where the masters draw their strength from.
But the maths that might be created wouldn’t look the same as we are used to. And it must not be so. It must not be ‘calculus’ that can predict the past, present and future for all times to come from a single observation. But at the same time it must not overlook the role of ‘differences’ and ‘integration’. It also must not be so universal that it can be applied to all or many frames of references. It must not look like ‘laws’. However, at the same time it must be simple enough for people to make sense to gain insights that would help them to model, adapt, innovate, re-design and predict. What more is needed?
The good news is that Nemetics (a branch-out from that secretive school of practice) research is rather close to creating that practical maths. At least, as of now, it can predict an emergence in the short term for the most complex of all cases close to 85% accuracy. For relatively simpler cases of complex systems the success of prediction is now close to 100%. And that is quite an achievement.
It is based on interactions of three vital components — Energy, Damping and Constraints, without which no real life complex system or transforming process can exist.
1. Sync, Steven Strogatz, Penguin Books, 2003