In my opinion, one of the most important concepts to discuss in a liberal arts math course is the notion of mathematical proof—what it is, why mathematicians put so much emphasis on it, whether it is overrated, and how the concept has evolved over time. There are several ways to approach this subject.
Give examples of proofs. This MO question as well as this one provide some nice examples. I'd also recommend the MAA series of books on Proofs Without Words.
Discuss the role of computers and experiment in mathematics. Jonathan Borwein has co-authored several books on experimental mathematics, e.g,. The Computer as Crucible. Though much of the mathematical content may be too advanced, the introductions to these books are extremely lucid and valuable.
Discuss the foundations of mathematics. My top recommendation in this category would be Torkel Franzen's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Again, some sections of the book may be too technical, but there are plenty of extremely well-written and valuable non-technical sections.
Even among the educated public, one frequently encounters people who have no concept of how unique mathematical proof is, who think that computers have put mathematicians out of business, and who have heard just enough about Gödel's Theorem to be dangerous. The above readings should go a long way towards dispelling these common misconceptions.
