Bee has a blog post:

"What is emergence? What does “emergent” mean?" For whatever reason, 99% of my comments simply don't go through, so discussion over there seems impossible. My thoughts on the subject are newly formed, and I'm putting them down here, so that I can get unstuck, and not because these ideas are right or have any merit. So this post will likely be revised or even deleted.

This from Bee is as good a description of "emergent" as you can find:

Something is emergent if it comes about from the collective behavior of many constituents of a system, be that people or atoms. If something is emergent, it does not even make sense to speak about it for individual elements of the system.

There are a lot of quantities in physics which are emergent. Think for example of conductivity. Conductivity is the ability of a system to transport currents from one end to another. It’s a property of materials. But it does not make sense to speak of the conductivity of a single electron. It’s the same for viscosity, elasticity, even something as seemingly simple as the color of a material. Color is not a property you find if you take apart a painting into elementary particles. It comes from the band structure of molecules. It’s an emergent property.

It is in the discussion of weak and strong emergence that I drift away. I think I get stuck on the "can be/cannot be derived".

Weak emergence means that the emergent property can be derived from the properties of the system’s constituents and the interactions between the constituents.....In physics the only type of emergence we have is weak emergence. With strong emergence philosophers refer to the hypothetical possibility that a system with many constituents displays a novel behavior which cannot be derived from the properties and the interactions of the constituents. While this is logically possible, there is not a single known example for this in the real world.

(Perhaps it is because I'm stuck on the notion of derivation as is done in mathematical logic.)

Consider the Second Law of Thermodynamics. Pick any set of microscopic laws - the Standard Model of particle physics; or the Standard Model modified in any way, or one of the 10^500 universes of superstring theory. Or make heat a fluid (Lavoisier's "caloric" was the context in which Carnot did his work). The Second Law remains true in all these cases. While as students of physics, we are indoctrinated with statistical mechanics as underlying the Second Law of Thermodynamics, the Law actually arises from very general considerations, with no assumptions at all about the microscopic physics, i.e., from the mathematical properties of Pfaffians and some mapping of mathematical concepts into physical concepts. (Zemansky's 1965

*Kelvin and Caratheodory--A Reconciliation* indicates what I'm talking about).

The way I see it, the Second Law is true if heat is an invisible fluid and it is also true if matter is made of atoms and heat is simply the energy of random molecular motion. So in what sense can the Second Law of Thermodynamics be said to derived from the properties of the system's constituents? The Second Law is not only an emergent law, in some sense of the term "strongly emergent", it is so. It is a law that is true no matter what the underlying microscopic physics is, and thus can be "derived" from some axiom set including axiom A with some mapping of mathematical to physical concepts, but can equally well be derived from some other axiom set including the axiom (not A) with some other mapping of mathematical to physical concepts.

In this regard, I'm not sure the concepts of "weak emergence" and "strong emergence" are particularly useful. An example is the never-ending debate of whether the phenomenon "consciousness" is strongly emergent, or is explainable ultimately in terms of the brain and its cells. Let's imagine humans, electronic/compute devices, Fred Hoyle's

solar-system sized cloud all exhibit "consciousness". Taking the Second Law of Thermodynamics as our exemplar, and assuming that a mathematical description of "consciousness" is feasible, one has to concede the possibility that such a description is rather independent of the microscopic details. If that turns out to be true, then the same problematic situation (at least to me) arises - how can such a description be said to be "derived" from the properties and interactions of the constituents of the conscious system?

The chemistry of hydrogen, carbon, etc., are a particular way because of the properties of their constituents, and would be different if e.g., the electron/proton mass ratio was different, or if the radius of the proton measured electron Compton wavelengths was much larger. Derivation of the chemistry crucially depends on these properties. That is one kind of emergence. I'd place all the things described by the Wilsonian renormalization group in this category too.

A second kind of emergence is where the behavior of the system can be described independent of its constituents, e.g., as with the Second Law of Thermodynamics. These are perhaps two useful types of "emergence", especially if we can find more laws of physics like the Second Law.