This brief essay is a reflection on two recent articles in the online magazine Aeon. The first is an article by Margaret Wertheim on “How to play mathematics” (headlined under the intriguing title “Embodied mathematics and the sea slug”!)
In it the author notices that “the world is full of mundane, meek, unconscious things embodying fiendishly complex mathematics” and she makes the point that some very complex bits of mathematics literally grow out of dumb animals and plants. How come? From these little things some big bits of mathematics can actually grow.
Whereas it might have taken us humans centuries to come up with concepts like hyperbolic surfaces – and complexity – such things can be found embodied in simple, algorithmic growth patterns. Indeed Margaret Wertheim shows how knitting patterns can deliver the same outcomes!
So while we may discover some complex mathematical representations using high level concepts and arcane formulae, it is possible to get the same results “bottom-up” using an algorithmic approach. I am reminded of Stephen Wolfram’s (2002) magnum opus “A new kind of science” in which he showed how simple algorithms could reproduce many complex outcomes.
The bottom line is fundamental: simple embodied, enactive and recursive processes can produce some amazingly complex patterns, processes and outcomes that can only be summarised with difficulty; if at all.
A similar message is contained in the Aeon article “How Europe became so rich” by Joel Mokyr which makes the point that the history of Europe is complex and highly contingent – it could have come out very differently – but it was driven by a combination of competition between states and the free flow of ideas and talent. Mokyr argues that the ascendancy of Europe was not predetermined. “It was rather what is known as a classical emergent property, a complex and unintended outcome of simpler interactions on the whole. The modern European economic miracle was the result of contingent institutional outcomes. It was neither designed nor planned.”
So, much of the living world is not as predictable as we might like to think: we have a habit of imposing “top-down” post-hoc frameworks on a contingent world in a search for mastery and control. This physics envy extends widely across the biological and the social sciences. (Now this is not to say that mathematical and Newtonian approaches do not work in situations where they are applicable. It is just that life is different: it is complex, embodied, recursive and evolving. I’ll have more to say about the importance of embodiment in the next blog.)
Edgar Morin was the first to distinguish between two approaches to complexity. He distinguished between “general complexity” – which approaches complexity largely through natural language – and “restricted complexity” which uses mathematical and model systems to attempt to reveal the more predictable and universal properties of complex systems.
General complexity “draws its epistemological implication from the point of view of the subject who knows” . As Gregory Bateson pointed out the observer inside a system of complex reflexive interactions must experience a degree of ignorance of the evolving whole. These ideas go back to the Macy Conferences of 1946-53 and the concept of 2nd order cybernetic “observing systems” that I have written about before. General complexity lies more in the realm of philosophy at present and requires trans-disciplinary approaches.
Restricted complexity – the kind we are more familiar with – represents the attempt by science and engineering to find generalisations that would make general complexity more tractable by seeking hidden regularities. Formal languages are used to model complex systems using computational techniques. Such approaches tend to deal in fractals, power laws and multi-agent models and lie very much within the ambit of science (or those disciplines that would be sciences).
The very concept of a “system” is a form of restricted complexity  that seeks mastery. But as Michel Serres has argued  the system paradigm requires us to master, not nature, but the desire for mastery itself – requiring both self-consciousness and self-control. Time and time again we seek simplification and universal generalisations that would, we hope, provide predictability and control. As we have seen from recent events, when dealing with life such a desire for mastery can lead us astray. (see for example )
[I am grateful to Humberto Mariotti for helping me to find the literature about general complexity that is not in the Anglophone literature and culture but is, rather, in Romance languages and cultures. These have, I find, a more sophisticated approach to such knowledge and epistemology. Having been brought up and educated in the Western Anglo tradition, this has been an eye opener for me. Much of the work on complexity in the Romance languages has not been translated into English, so it is not widely known in the Anglo world.]
One major work that has only been partially translated is La Méthode (1977) in which Edgar Morin  discussed the general properties of living systems: these “systems of systems of systems” that are interdependent and interlocked in recursive ways. And, as I have pointed out, given the presence of distributed robustness and other properties of life, the nature of nature is not reducible to universal simplifications and predictability. The militant realism and rationalization of the Anglophone sphere is a limited case of a more general and sophisticated way of knowing.
Morin’s arguments take us back to the 2nd order cybernetics of the Macy Conferences and to the 1st, 2nd and 3rd thoughts of Terry Pratchett that I have written about before . In La Méthode, Morin writes of the need for a trinary dialogue between philosophical reflexive knowledge, empirical scientific knowledge and epistemological (2nd order) “knowledge of the value of knowledge”. This dialogue is generally lacking in the Anglophone world and often leads to overreach and the failure of prediction and grand strategies. As Malaina (ref 1) noted, a more balanced effort to reconcile the narrative and formal languages would be profitable.
 Alvaro Malaina (2015) “Two complexities: the need to link complex thinking and complex adaptive systems science”. Emergence, Complexity and Organization. 2015 Mar 31. Edition 1. doi: 10.emerg/10.17357
 Edgar Morin (1977) in an inaugural address to the Congres de l’AFCET, Versailles. Published in (1982) “Science avec conscience”, Paris, Fayard, pp. 172-189. English translation by Sean Kelly
 Michel Serres (1982) “Knowledge in the classical age: La Fontaine and Descartes” Ch 2. Pp. 15-28. In “Hermes; Literature, science, philosophy” eds. Josué Harari and David Bell, Johns Hopkins University Press, Baltimore
 Andrew Lo and Mark Mueller (2010) “WARNING: physics envy may be hazardous to your wealth” arXiv:1003.2688v3
 Volume 1 translated into English by J.L. Roland Bélanger (1992) as “Method: towards a study of humankind; Vol 1, The nature of nature”, American University Studies, Series V, Vol III, Peter Lang, New York.
 Terry Pratchett (2004) “A hat full of sky”, A story of discworld, Discworld novels #34, Doubleday.