`  : I agree that some of the examples here are somewhat fluffy.  But emergence means slightly different things in different fields.  For example, while physicists use the term emergence, it doesn't have the same hype attached to it as the term does in the behavioural sciences.  It's simply used to mean a phenomenon or concept which only makes sense, or behaves in an expected way, at macroscopic scales (in space or time) but not at microscopic scales.  Thus questions like ``is this electron hot or cold?  blue or green?  soft or hard?  smooth or rough?  solid or liquid?`` have little or no useful meaning, because these concepts are emergent and thus do not apply in this setting (or if they do apply, they behave unexpectedly and so one's intuition on the concept may be very misleading here).  That's a helpful thing to keep in mind, it lets one avoid any number of elementary errors when trying to understand particle physics, and thus does have a non-trivial amount of explanatory power.  As for mathematics (which is my own profession), I've almost never seen the term ``emergence`` used... the closest analogues seem to be the distinction between ``local`` and ``non-local`` (or ``global``) concepts, or between ``asymptotic`` and ``non-asymptotic`` behaviour.  It is true, I would classify most of the mathematical examples here as examples of non-local concepts rather than emergent ones.  If I have some time I might try to rework some of the mathematical material (which I was not involved in earlier).   `