The structure of systems at different scales is often referred to as a hierarchy, and traditional thinking implies that this is so. Molecules make up organelles, which make up cells, which make up tissues, which make up organs, which make up organ systems, which make up bodies, and beyond. I refer to any scale at which recognizable coherent patterns in behavior are observable and describable as a level of organization. This is an epistemic notion because it relates to what we can capture in models, not to the structure of reality. Hierarchies of scale are a mixed ontological and epistemic notion: the levels are still based on what people can discern, but there is an added ontological assumption that a higher level exhaustively includes the elements of a lower level. Now I will show why levels of organization, not hierarchies, are the domains and ranges of scientifically useful reduction and emergence relationships.

One way to think about this is to separate strict hierarchies from roughly hierarchal organization. If one goes strict than phenomena at any level can be broken down into phenomena at any lower level. We can loosen this requirement by only requiring that the part-whole relationships follow an ordering even if not every level has the same descriptive power with respect to every other level. That's an improvement because it allows for cases like the following: A is higher than B which is higher than C. A cannot be reduced to B, but both A and B can be reduced to C.

Although, consider that we may conclude that this relationship among A, B and C implies that A isn't really higher than B, though they are both higher than C. If we find some level D that is between A and B such that A can be reduced to D, and D to B, then doesn't that also imply that A can be reduced to B via D? Seems so. When people/scientists consider this issue, their notion of levels seems to be limited to constitution: what parts make up the whole in a physical sense. And as long as we are talking about physical objects in the world, the part-whole relationships keeps looking strictly hierarchal.

But now consider this: bodies are objects, cells are objects; and since bodies are higher up on the hierarchy, bodies are made up of cells. But bodies are not made of just cells. There is water and nutrients and other chemicals in a body that are not parts of (or even inside) cells. And these molecules are at a completely different scale than cells in fact cells are made up of (reducible to) these very objects. So this is a case in which bodies and cells can both be reduced to molecules, but bodies cannot be reduced (in the ontological sense) to only things at the cellular level. This is so even though a body contains all the molecules that its cells are made of...the cells are a proper subset of the body. But we can pull this same trick again and point out that there are parts necessary for the body that are not part of any molecule, but are rather in the domain of submolecular physics. We can always do this, and there is always going to be something missing.

This leads us to the issues that there is no lowest level and that no science is complete, but these are not the current topic. I'm more interested in taking this toward part-whole relationships that are not constitution. Translating phenomena from one level to another is typically a lossy compression, some of the details are left out. This is true whether one is translating macro to micro or micro to macro. But here comes the real kicker, the phenomena we are translating are not always objects.

A flock (e.g. of birds) is an object at the macrolevel, and flocks are made up of birds (which makes it look like an ontological hierarchy), but the flock object is clearly not reducible to its constituent bird-object parts alone. The behaviors of the birds are a crucial feature of the flock object-phenomenon, but behaviors are not part of ontology. Thus when we are talking about emergence and reduction of phenomena we must go beyond just ontology and therefore beyond the notion of constitutional hierarchy. (Note: I wonder if this is all that is supposed to be meant by the slogan "More than the sum of its parts.")

When we develop a translation between phenomenon A and phenomenon B, it is sometimes the case that some of the objects of B are some of the parts of A. In this case a translation from A to B is called a reduction and a translation from B to A is called an emergence. I identify/define the levels of organization as the ranking of all ontologies that figure into a reduction or emergence relation. Thus the levels will not be strictly hierarchical. The levels will be refined and filled in as new models/theories demonstrate new reduction/emergence relationships. And although any particular translation is between two levels, there may actually be a continuum of levels.

(Afterthought: I want to be clear that some people may use the word 'hierarchy' to mean what I refer to as a level of organization. That is, they may not subscribe to the ontological requirement, or the demand that each layer can be losslessly described at each other, but refer to the levels as hierarchies. Fine. I don't care about which words people use. This isn't a semantic issue. The truth is that some very good scientists are not comfortable with these concepts, and have never even considered the differences. It is likely that many people's concepts of reduction and emergence include the ontological requirements of complete hierarchies, and they do not distinguish between applying them to the world or to models. Disabusing people of this sloppy usage is part of the point.)