*Alice laughed. “There’s no use trying,” she said: “one* ca’n’t *believe impossible things*.”

*“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”*

Like the White Queen in Lewis Carroll’s *Through the Looking-Glass*, mathematicians are called upon to believe in things that, at first glance, defy common sense and appear impossible.

Indeed, the language of mathematics is strewn with mystical (even pejorative) terms that reflect the struggles that mathematicians have faced over centuries when coming to grips with these fantastic notions: irrational, transcendental, and imaginary numbers; higher dimensions, curved spaces, and finite universes; and much more. Yet, the mathematics underlying these concepts is as real as any other mathematics—and has amazingly broad applications.

In his book, *Yearning for the Impossible: The Surprising Truths of Mathematics* (A K Peters), mathematician John Stillwell of Monash University in Australia argues that “there is no doubt that mathematics flirts with the impossible, and seems to make progress doing so.”

He notes, “Mathematics is a discipline that demands imagination, perhaps even fantasy.”

In answering why this is a reasonable supposition, Stillwell quotes Russian mathematician A.N. Kolmogorov (1903–1987), who wrote in 1943 that “at any moment there is only a fine layer between the ‘trivial’ and the impossible. Mathematical discoveries are made in this layer.”

As Stillwell puts it, “Mathematics is a story of close encounters with the impossible because *all its great discoveries are close to the impossible*.”

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Stillwell provides an illuminating introduction to this mathematical neverland, beginning with the notion of irrational, then venturing into the realm of imaginary numbers, parallel lines and points at infinity, the infinitesimal, curved space, the fourth dimension, the ideal (along with prime factorization), periodic space, and the infinite.

Many of these ideas “seem impossible at first because our intuition cannot grasp them, Stillwell writes, “but they can be captured with the help of mathematical symbolism, which is a kind of technological extension of our senses.”

Common sense by itself is too limiting for making progress in mathematics. New concepts arise out of leaps of imagination. And such out-of-the-box thinking puts mathematics into a rich intellectual landscape that it shares with physics, philosophy, literature, and art.