  This is a companion to another paper. Together they rebut two widespread
philosophical doctrines about emergence. The first, and main, doctrine is that
emergence is incompatible with reduction. The second is that emergence is
supervenience; or more exactly, supervenience without reduction. In the other
paper, I develop these rebuttals in general terms, emphasising the second
rebuttal. Here I discuss the situation in physics, emphasising the first
rebuttal. I focus on limiting relations between theories and illustrate my
claims with four examples, each of them a model or a framework for modelling,
from well-established mathematics or physics. I take emergence as behaviour
that is novel and robust relative to some comparison class. I take reduction
as, essentially, deduction. The main idea of my first rebuttal will be to
perform the deduction after taking a limit of some parameter. Thus my first
main claim will be that in my four examples (and many others), we can deduce a
novel and robust behaviour, by taking the limit, N goes to infinity, of a
parameter N. But on the other hand, this does not show that that the infinite
limit is "physically real", as some authors have alleged. For my second main
claim is that in these same examples, there is a weaker, yet still vivid, novel
and robust behaviour that occurs before we get to the limit, i.e. for finite N.
And it is this weaker behaviour which is physically real. My examples are: the
method of arbitrary functions (in probability theory); fractals (in geometry);
superselection for infinite systems (in quantum theory); and phase transitions
for infinite systems (in statistical mechanics).
