There is a continuum between mathematics at one extreme, language, poetry and music at the other extreme.      

     Mathematics is a precise language with symbols and statements that have precise meaning. It is most precise, but can typically be applied only to well-defined situations we understand well.

     What we ordinarily call language (e.g. English) is more vague and ambiguous than mathematics. This ambiguity corresponds to the ambiguity in perception and knowledge. When we say dogs pee on trees while lifting a leg, there is ambiguity in the fact that dogs is an ill-defined concept, as is trees, as is the generalization that they pee while lifting a leg. This ambiguity serves a purpose: what we know is that one or more particular dogs have urinated on the trunk of a few particular trees in a few occasions. However, this knowledge is not very useful: those times are in the past, and we care about the future; those dogs are mine, and others care about their dogs, those trees are in my neighborhood, and others care about their own. If we expressed our experience using math, we might say something more along the lines of 'My dog Rafi peed on a tree that appeared to be an Eucalyptus, two Oak trees and ten Pine trees on sixty two occasions between 1980 and 1989'. By generalizing, our knowledge is less accurate, but more useful. Since we have not tested the exact limits of the generalization, we are unable to use exact language to describe it, and hence our words are designed to tolerate some ambiguity.

      Poetry takes this ambiguity a step further. The best poetry can often be interpreted in several ways, and authors are usually careful not to specify which interpretation they had in mind. This allows each reader to interpret it in the way that suits their frame of mind best, or in the way they like best. This ambiguity allows a personalization in the message that would otherwise be impossible, allowing a poem to speak to what is relevant to each reader.

     Music, and art in general, goes even further in the ambiguity continuum: unlike words, music carries a very ambiguous meaning, with the same music meaning different things (if anything) to different listeners. As such, it allows us to convey or share moods, regardless of the specifics of what words would elicit that mood in different listeners. An ambiguous language, such as music, is like an elastic key that fits many locks; every listener gets to adapt the key to fit his mind best.

     Humor is also related to this ambiguity continuum, but falls outside of it: humor is the disambiguation of a previously ambiguous situation which results in the diffusion of tension. More on the evolution of humor here.

(Dogs pee on trees while lifting a leg. Art courtesy of Joshua Eisenberg. See also The Adult Male Dog Peeing Behavior.)

      [This essay is derived from a conversation between Demetri Spanos and I during a coffee break we both took at Caltech in 2007.]