Let me see if I can add some context. One of Nick's obsessions is that of emergence, and in particular, he concerns himself with a possible relationship between emergence and Simpson's paradox. I started to wonder what there is generally to say about the paradox. As a first approximation, I think that there is nothing especially "statistical" about the paradox, rather, it appears to be a consequence of geometry and mereology. I can imagine a few worms each stretched to the southeast corner of a graph and yet their aggregate appears to stretch northeast. This leads me to wonder about the qualities of quantities. When can one expect there to exist such a disconnect between individuals and wholes? This leads me to think about intensive versus extensive quantities as they are conceived in thermodynamics (and now everywhere).
Extensive quantities like magnitudes, counts, volumes, and mass have the property that when you put two or more individuals together with the same value as one another, that same property wrt the whole will likely be different. This case should be especially likely when the number of dimensions where the extensive quantity manifests is high. On the other hand, for intensive quantities (density, color, temperature, hardness, viscosity...) this is not the case. I feel that there may be something here worth connecting. To reframe Simpson's paradox in terms of what kinds of quantities remain invariant when moving from the individual to aggregate *level*.
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