Mathematical Science Writers without PhD

Martin Gardner is a prototypical example, I think.

http://en.wikipedia.org/wiki/Martin_Gardner

William Poundstone has written some excellent books addressing mathematical topics. One is called "Gaming the Vote," which addresses Arrow's Theorem and its implications for voting. (Ken Arrow won the 1972 Nobel in Economics for his theorem and his work in social choice.) Another is "Prisoner's Dilemma" which addresses Game Theory.

John Derbyshire has written a book about the Riemann Hypothesis called "Prime Obsession," which is a heroic attempt to explain the topic to amateurs. He has also written "Unknown Quantity," about the history of algebra.