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 breaking the system apart into its component parts, analyzing the behavior of each part, and then taking the system to be (in some sense) the “sum” of that behavior should yield the same prediction as considering the gas as a whole.

It’s worth briefly considering some of the technicalities behind this condition. Strictly speaking, the additvity condition on linearity makes no reference to “parts,” as it is a condition on equations, not physical systems being modeled by equations. Rather, the condition demands that given any set of valid solutions to the equation describing the behavior of the system, the linear combination of those solutions is itself a solution. This formal statement, though more precise, runs the risk of obfuscating the physical (and philosophical) significance of linearity, so it is worth thinking more carefully about this condition with a series of examples.

Linearity is sometimes referred to as “convexity,” especially in discussions that are grounded in set-theoretic ways of framing the issue. In keeping with our broadly geometric approach to thinking about these issues, this is perhaps the most intuitive way of presenting the concept. Consider, for instance, the set of points that define a sphere in Euclidean space. This set is convex (in both the ordinary sense and the specialized sense under consideration here), since if we take any two points that are inside the sphere, then the linear combination—the weighted average of the two points—is also inside the sphere. Moreover, the line connecting the two points will be inside the sphere, the triangle defined by connecting any three points will lie entirely inside the sphere, and so on. More formally, we can say that a set of points is convex if

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