***** To join INSNA, visit http://www.insna.org ***** Dear colleagues, In a previous email I sepeficied the mechanism of double contingency in terms of anticipatory systems as follows: x(t) = a (1 - x(t+1)) (1 - x(t+1)) The next state of the system is determined by the selective operation of expectations upon each other in a dyadic interaction. The simulations are robust and show that the system can move erratically from the one to the other side. (If one wishes, one can play with the parameters in the excel sheet and follow the consequences; at http://www.leydesdorff.net/temp.doublcont.xls ). In Chapter 3 of Soziale Systeme, Luhmann (1984) discussed "double contingency" as central to the emergence of social systems. Borrowing the concept from Parsons, he provides it with a completely new solution. In my opinion, the simulations in terms of expectations accord with this solution. Paul Hartzog (cc) sent me a short piece in which he explains Luhmann's solution in English. (Can you bring it online, please, Paul?). It made me aware that Luhmann moves fast in this chapter from "double contingency" towards the emergence of social systems without a specification of the mechanism. (In footnote 12, p. 157, Luhmann warns against Von Foerster's too fast movement.) The social system "emerges" from double contingency (in the singular!). I guess that a double contingency can go on forever when no third party comes into play. Piet Strydom used the term "triple contingency" for explaining the emergence of a modern communication society in 16th and 17th century. The third party becomes abstracted as a public. In priniciple, one could model a triple contingency analogously using: x(t) = a (1 - x(t+1)) (1 - x(t+1)) (1 - x(t+1)) This leads to a cubic equation of x(t+1) as a function of x(t). Cubic equations have analytical solutions, and there is a (freeware) add-in in Excel for solving them. The solutions may imply i = sqrt(-1), and thus be in the complex domain. For all values of the bifurcation paramater a the system is highly unstable and quickly degenerates into a complex one. One interpretation would be that triple interactions provide a short-term window for organization (decision-making) to step into the system. The relation between interaction and organization would then be conditional for the emergence of the social system. An alternative formulation is: x(t) = a (1 - x(t)) (1 - x(t+1)) (1 - x(t+1)) x(t)/(1 - x(t) = a (1 - x(t+1))(1 - x(t + 1)) By replacing [x(t) / (1 - x(t))] with y, the solution is similar to the one for double contingency, but mutatis mutandis: x(t+1) = 1 ¡À sqrt(x(t)/ (a * (1 - x(t))) This formula is in the simulation as stable as double contingency for values of a ¡Ý 8, but I don't yet have an analytic solution for this. For lower values of a, the system vanishes. Using an internal degree of freedom the system might be able to change its value of a endogenously and thus alternate between double contingency and its disappearance. In summary, in the case of a triple contingency, the system can show the behavior of a window for organization to step in by using three incursive terms (based on expectations), or bring a double contingency to an end by bringing a historical contingency (modeled as a recursive term) into play. Using the internal degree of freedom for changing the value of a, the social system would also be able to generate double contingencies (interactions) endogenously. From entropy statistics, we know that a system with three dynamics can generate a negative entropy in the mutual information among the three (sub)dynamics. (I use this as an indicator of self-organization in other studies.) However, there is still a missing link between the above reasoning and the emergence of a social system as a possibility because the complex system is not yet generated. I suppose that I have to bring the social distributedness into play and not write x(t), but ¦²ixi(t). With best wishes, Loet ________________________________ Loet Leydesdorff Amsterdam School of Communications Research (ASCoR) Kloveniersburgwal 48, 1012 CX Amsterdam Tel.: +31-20- 525 6598; fax: +31-20- 525 3681 [log in to unmask] <mailto:[log in to unmask]> ; http://www.leydesdorff. net/ <http://www.leydesdorff.net/> Now available: The Knowledge-Based Economy: Modeled, Measured, Simulated <http://www.universal-publishers.com/book.php?method=ISBN&book=1581129378> . 385 pp.; US$ 18.95 The Self-Organization of the Knowledge-Based Society <http://www.universal-publishers.com/book.php?method=ISBN&book=1581126956> ; The Challenge of Scientometrics <http://www.universal-publishers.com/book.php?method=ISBN&book=1581126816> _____________________________________________________________________ SOCNET is a service of INSNA, the professional association for social network researchers (http://www.insna.org). To unsubscribe, send an email message to [log in to unmask] containing the line UNSUBSCRIBE SOCNET in the body of the message.