The collectivist illusion
A good example of this is the theological arguments from design. These arguments start from the premise that there is a lot of complexity around us, an obvious fact. They then assess that chance could not possibly bring about complexity of such a high level. Having established this, they conclude that there must be an intelligent agency behind such complexity, which they call "god".
What this line of reasoning fails to understand is that chance is not the complement to intelligent agency. Rather, we know that the interaction between parts of nature, called natural law, can produce complexity. Since non-agents can produce the complexity that we observe, the supposed necessity of a designer is disproven. The parts, which in this case are the entities that compose the universe, do not necessitate a whole, a god, to be explained.
Slightly more sophisticated theologians then try to show that parts cannot bring about attributes of the whole, that complexity could not exist naturally without some intelligence being present in the parts. Of course, our assumption that intelligence is necessary for complexity is a simple assumption that comes from the fact that human agency is the one we observe most often.
But all arguments of this sort also commit the fallacy of composition, by transposing an attribute of the parts to the whole without justification. Attributes that are not self-contained are specifically problematic. We cannot posit that a natural universe could not have complexity based on its parts, simply because complexity is a relational property.
What this means is that things are complex because they have a certain kind of arrangement in their parts, therefore complexity is a relation between parts in the whole. This is why it is a relational property. Because of this, there is no reason to posit that wholes cannot be complex because of properties of their parts, since complexity arises by definition from assemblages of parts, not the parts themselves.
The fallacy of composition is important in understanding why assumptions of central source or control are wrong. It is not necessary to posit a god as source of complexity since complexity is a relational attribute of natural entities. It is not necessary to posit a soul as source of the mind since the mind is a relational attribute of brain parts. Government is not necessary to bring about economic or social benefit since such benefits arise naturally from the relations between social agents.
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