Emergent Properties

15 Sep 2023 15:45

In complexity theory. Logical analysis of the concept.

In Plato, e.g., the virtues of the ideal city in the Republic (doesn't Popper talk about this in his discussion of Plato?).

This is of course part of the continual argument about reductionism, and those most enamoured of emergent properties tend to be anti-reductionists. (I freely confess to being a reductionist, and thinking my opponents wooly-headed on this issue. For what follows, no warranty, express or implied, etc.) This term is used in a couple of senses, only one of which should trouble reductionists.

The weakest sense
Is also the most obvious. An emergent property is one which arises from the interaction of "lower-level" entities, none of which show it. No reductionism worth bothering with would be upset by this. The volume of a gas, or its pressure or temperature, even the number of molecules in the gas, are not properties of any individual molecule, though they depend on the properties of those individuals, and are entirely explicable from them; indeed, predictable well in advance.
As above, but now we add the caveat that "the new property could not be predicted from a knowledge of the lower-level properties." Note that we cannot know that something is an emergent in this sense; we can only know that it cannot be predicted by us, with our current abilities. But "predict" is, here, ambiguous. It could mean foresee or prognosticate (i.e., make a statement about a future event which proves to be true), or it could mean deduce or explain --- what is sometimes called retrodiction. The "foresee" sense doesn't seem very important, because we typically invent a micro-level theory to explain an already-observed macro-level phenomenon. It is of course very nice indeed if the micro-theory predicts a new macro-phenomenon, which on investigation is found to happen; but this makes emergence an accidental result of what we happened to notice first.
Retrodiction or Explanation
"An emergent is a higher-level property, which cannot be deduced from or explained by the properties of the lower-level entities." This is almost troubling. The key is in "properties." Reductionists --- sane ones, anyhow --- don't deny that things interact; we spend a great deal of time worrying about those interactions. If by "properties" is meant just properties in the logical sense, then of course there are emergents, but so what? In this sense, pressure and volume are emergents.

On the other hand, if we are allowed both our properties and our relations, then "emergence" is a notion with teeth. The existence of any emergent properties, in this strong sense, would mean that universal reductionism is false. (Though it might be true locally, or for all other properties, or still be the most useful means of guiding inquiry, etc.) But, as above, I don't see how "X is an emergent property (strong sense)" could be established. At best we could say "X may be an emergent, since we have been unable to deduce it from the lower-level properties Y."

Does anyone know of any good candidates for this kind of emergent?

Addendum, April 2001: I'm pretty sure I know how to define "emergence," in the first, weakest, and most intelligble sense, in a quantitative, operational and objective way. One set of variables, A, emerges from another, B if (1) A is a function of B, i.e., at a higher level of abstraction, and (2) the higher-level variables can be predicted more efficiently than the lower-level ones, where "efficiency of prediction" is defined using information theory. See the concluding chapter of my dissertation, which unfortunately needs the previous chapters to be fully understood, but where I prove that, in a simple case (monoatomic ideal gas) thermodynamics emerges from statistical mechanics. [2023: Or, better yet, see my paper with Cris Moore.]

Addendum, December 2004: Since last updating this notebook, I have spent some time reading about the philosophers' notion of "supervenience", and feel either I'm very confused, or they are. A higher-level property is supposed to "supervene" on lower-level properties if differences at the higher level imply differences at the lower level, but not necessarily vice-versa. This is generally felt to let us admitting that the high-level, edifying properties are, indeed, related to the unedifying low-level ones, without having to embrace outright reductionism. But it seems to me to take about few lines of set theory to show that "supervenes on" means the same thing as "is a function of", and really that's all that reductionists ask for. So, like I said, I'm puzzled.

(For the record: Let's write \( lSu \) for the relationship "higher-level property \( u \) supervenes on lower-level property \( l \)". The defining property of supervenience is that if \( u_1 \neq u_2 \), then we cannot have \( lSu_1 \) and \( lSu_2 \). So, if \( lSu_1 \) and \( lSu_2 \), then \( u_1 = u_2 \). But this establishes a many-one relationship mapping each low-level property to a unique high-level one, which is to say the high-level property is a function of the low-level property. Conversely, it is obvious that functional dependence implies supervenience. Hence, the two concepts are identical. QED.)