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MrMind comments on Wrong however unnamed - Less Wrong
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There is an algebraic analogy that might be useful (or confusing, if you don't know the math involved). In a tensor space, there are elements called pure tensors that can be described by a simple term (rank 1 tensors), while others cannot be described by a such item. In a similar fashion, entangled systems in quantum mechanics cannot be described considering the simple sub-costituents in isolation.
Another, perhaps simpler, analogy is a probability distribution of two dependent variables.
Whenever I read "more than the sum of its parts", I do not imagine the literal sum, but rather that some variables or observables in the system are tensorial in nature.
...I will look up tensors:) I have already tried to understand the piece before, but could not.
Actually, I would appreciate it if you could write a post and expand your analogy to other matters within your interests; it would be great if people grew used to sharing their heuristics without thinking them biases from the outset.