  We have two aims. The main one is to expound the idea of renormalization in
quantum field theory, with no technical prerequisites (Sections 2 and 3). Our
motivation is that renormalization is undoubtedly one of the great ideas, and
great successes, of twentieth-century physics. Also it has strongly influenced
in diverse ways, how physicists conceive of physical theories. So it is of
considerable philosophical interest. Second, we will briefly relate
renormalization to Ernest Nagel's account of inter-theoretic relations,
especially reduction (Section 4).
  One theme will be a contrast between two approaches to renormalization. The
old approach, which prevailed from ca. 1945 to 1970, treated renormalizability
as a necessary condition for being an acceptable quantum field theory. On this
approach, it is a piece of great good fortune that high energy physicists can
formulate renormalizable quantum field theories that are so empirically
successful. But the new approach to renormalization (from 1970 onwards)
explains why the phenomena we see, at the energies we can access in our
particle accelerators, are described by a renormalizable quantum field theory.
For whatever non-renormalizable interactions may occur at yet higher energies,
they are insignificant at accessible energies. Thus the new approach explains
why our best fundamental theories have a feature, viz. renormalizability, which
the old approach treated as a selection principle for theories.
  We also maintain that universality, a concept stressed in renormalization
theory, is essentially the familiar philosophical idea of multiple
realizability; and that it causes no problems for reductions of a Nagelian
kind.
