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 doesnt have the same hype attached to it as the term does in the behavioural sciences  Its 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 ones intuition on the concept may be very misleading here  Thats 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 nontrivial amount of explanatory power  As for mathematics which is my own profession Ive almost never seen the term emergence used the closest analogues seem to be the distinction between local and nonlocal or global concepts or between asymptotic and nonasymptotic behaviour  It is true I would classify most of the mathematical examples here as examples of nonlocal 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