Emergent qualities in complex systems
There are some qualities of complex systems that can't be said to be present in any individual part.
The most notable example, I think, is life itself.
Living organisms are characterized by metabolism--energy ingestion and processing--and a process of self-replication through reproduction.
However, it can't be said that any atom--carbon, hydrogen, oxygen, etc--that makes up a living organism is alive. Atoms aren't alive and organic substances aren't alive.
Still, when organized just right as a living organism, the whole organism is alive. This is an example of a property that emerged from the organized whole, while not present in any of the parts.
Another example is language in the form of speech. Spoken language carries meaning from a speaker to an interlocutor, making themselves understood through the support of a shared societal and normative context.
It can't be said that any sound in particular carries meaning. A sound is merely a disturbance of air pressure, typically carried as sound waves.
In this sense, the verbal meaning happens to be reconstructed through the sound waves, but it really takes place in the mind of the listener.
I think we generally are overconfident in how well our intuition can predict how complex systems might behave just by knowing the parts alone. It seems to me that emergent qualities of systems, although they might be experienced and verified empirically, do not follow directly and clearly from a piecewise analysis of the parts, no matter how well the parts are understood.
Unforeseen interactions from the organization of the parts give rise to hard-to-predict dynamics that are only understood after the fact.