I have referred to 'emergence' numerous times over the years in talks and papers, usually describing the phenomenon by means of some simple examples. This new article (replacing the original 2002 article) is a definitive account of the concept, covering varieties of emergent properties. According to the article, emergence covers a range of properties that aree neither exactly dependent on underlying phenomena, nor completely autonomous from them. "Consider, for example, a tornado. At any moment, a tornado depends for its existence on dust and debris, and ultimately on whatever micro-entities compose it; and its properties and behaviors likewise depend, one way or another, on the properties and interacting behaviors of its fundamental components. Yet the tornado's identity does not depend on any specific composing micro-entity or configuration, and its features and behaviors appear to differ in kind from those of its most basic constituents." What I would say is that the thoughts, ideas, perceptions and beliefs we have are all like that tornado - not exactly the underlying physical instantiation (there's no specific set of neural states that is 'this idea'), but not autonomous from it either.
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