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An Excerpt from "Through the Looking Glass"

The Sciences - May/June 1997

The resolution to the reverse-engineering problem came in 1943, in a theorem proved by the former Soviet mathematician Israel M.Gelfand, now at Rutgers University in New Brunswick, New Jersey. Gelfand's method of reconstructing space was both elegant and ironic. In a world where verbs are the objects, he pointed out, the nouns become actions. In a sense, Gelfand gave a mathematical answer to the question with which the Irish poet William Butler Yeats ended his poem "Among School Children." "How can we know the dancer from the dance?" The dancers are points; the dances, functions. In Gelfand's approach (contrary to the usual way of thinking about such things) the dance comes first and summons dancer into existence. To discover a dancer's identity - "to know the dancer", in Yeat's words - one need only watch dancer perform: not just one dance, though, but every possible dance.

According to Gelfand's theorem, the idea of reconstructing space will work provided the shadow land of functions (which mathematicians call by the uninformative name of "commutative C-star algebras") obeys a short list of specifications, or axioms. Foremost on the list is commutativity: the multiplication of functions is just as commutative as the multiplication of real numbers."

By Dana Mackenzie

All material is Copyright 2012 to T. V. Gelfand and T. I. Gelfand(unless otherwise noted)