Many people believe that mathematics provides a model of what thinking is, or should be. They imagine that mathematical thinking always proceeds in a logically rigorous, step-by-step fashion from one truth to another, like a formal proof or a computer program. In fact, insights in mathematics -- whether they are the scholar's breakthroughs or the student's leap to a new level of understanding -- involve a different mode of thinking that is essentially nonlogical. Ambiguity and its cousins, contradiction and paradox, are everywhere in mathematics, both in content and thinking. Strangely, the subject that appears to be the very paradigm of reason, and that is therefore the model for many other disciplines, contains as an irreducible element exactly what reason ostensibly does away with. Mathematical thought has nonrational, though not irrational, components. Here, the author expands on the concept of ambiguity in mathematics.