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.

Chronicle of Higher Education
From the issue dated August 3, 2007
By WILLIAM BYERS

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.

Let me be more precise about what I mean by ambiguity: It is the existence of multiple, possibly inconsistent, points of view. In other words, an ambiguous situation is a single situation or idea that can be seen from two or more conflicting viewpoints.

A good metaphor for ambiguity is binocular vision. When you look at things out of one eye, the world seems flat and two-dimensional. However, when you use both eyes, the inconsistent viewpoints registered by each eye combine in the brain to produce a unified view that includes something entirely new: depth perception. In the same way, the conflicting points of view in an ambiguous situation may give birth to a new, higher-order understanding.

The metaphor reveals that ambiguity refers not only to an objective description of a situation, but also to the manner in which situations of inconsistency and even conflict can be resolved. Thus, ambiguous situations contain the potential for change; they are dynamic and can be creative. Ambiguity points to a valid way of thinking involved in mathematics, one that needs to find its place alongside logic if we are to account for the power and effectiveness of the discipline.

Ambiguity, as I use the term, is no great mystery; it is present in every joke. A joke typically has two conflicting points of view, one of which is not explicit. The conflict is resolved by "getting the joke." A joke is analogous to a moment of creativity precisely because you have to "get it"; an explanation will not do. When you do get the joke, the tension that arose because of the initial conflict dissolves in laughter.

Not only is ambiguity essential to humor, it is common in poetry and the fine arts. Leonard Bernstein said that it "is one of art's most potent and aesthetic functions. The more ambiguous, the more expressive." To go even further afield, ambiguity is the heart of Zen Buddhism. Koans, those verbal puzzles that are the basis of Zen training, always contain an ambiguous core. For example: "What is the sound of one hand clapping?" In contrast with our normal tendency to avoid the paradoxical, the point of a koan is to focus on its ambiguities.

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