**the limits of observation or computation; and strongly emergent properties are ones that cannot be derived**

*given***from full knowledge of the properties and states of the components. They must be understood in their own terms.**

*in principle*In the previous sections we used the term critical point to describe the presence of a very narrow transition domain separating two well-defined phases, which are characterized by distinct macroscopic properties that are ultimately linked to changes in the nature of microscopic interactions among the basic units. A critical phase transition is characterized by some order parameter φ( μ) that depends on some external control parameter μ (such as temperature) that can be continuously varied. In critical transitions, φ varies continuously at μc (where it takes a zero value) but the derivatives of φ are discontinuous at criticality. For the so-called first-order transitions (such as the water-ice phase change) there is a discontinuous jump in φ at the critical point. (10)

**phase of the substrate of dispersed individuals; rather, it is an occasional abnormal state of brief duration. It is as if water sometimes spontaneously transitioned to steam and then returned to the liquid phase. Solé treats “percolation” phenomena later in the book, and rebellion seems more plausibly treated as a percolation process. Solé treats forest fire this way. But the representation works equally for any process based on contiguous contagion.**

*equilibrium*Although it might seem very difficult to design a microscopic model able to provide insight into how phase transitions occur, it turns out that great insight has been achieved by using extremely simplified models of reality. (10)

In social insects, while colonies behave in complex ways, the capacities of individuals are relatively limited. But then, how do social insects reach such remarkable ends? The answer comes to a large extent from self-organization: insect societies share basic dynamic properties with other complex systems. (157)

*per se*. Fundamentally it demonstrates that the aggregation dynamics of complex systems are often non-linear and amenable to formal mathematical modeling. As a critical variable changes a qualitatively new macro-property “emerges” from the ensemble of micro-components from which it is composed. This approach is consistent with the generativity view — the new property is generated by the interactions of the micro-components during an interval of change in critical variables. But it also maintains that systems undergoing phase transitions can be studied using a mathematical framework that abstracts from the physical properties of those micro-components. This is the point of the series of differential equation models that Solé provides. Once we have determined that a particular system has formal properties satisfying the assumptions of the DE model, we can then attempt to measure the critical parameters and derive the evolution of the system without further information about particular mechanisms at the micro-level.