If some day we receive an information-rich signal from another star, no one expects it to be written in English, Chinese, or Swahili. Instead, researchers engaged in the Search for Extraterrestrial Intelligence (SETI) often suggest that mutual comprehension will come through the language of math.

Imagine for a moment an extraterrestrial civilization that can build radio telescopes and transmitters, and thus signal its existence across interstellar space. Wouldnt such a civilizations knowledge of the physical universe overlap at least in part with our own?

Surely, many have argued, the ability to build a radio telescope requires a capacity to count and to recognize that 2 + 2 = 4. The Dutch mathematician Hans Freudenthal started with this assumption to create an interstellar language called Lingua Cosmica the language of the cosmos, published in 1960. Freudenthals step-by-step tutorials begin with basic counting, then progress through arithmetic to increasingly complex forms of mathematics.

If extraterrestrial savants can follow our descriptions of long division, shouldnt they be able to follow a refresher course in hyperbolic geometry as well? Or might there be extraterrestrials, in some ways even more technologically advanced than we, who never ventured beyond the three dimensions of Euclids conceptual world?

Sarukkai, a researcher at the National Institute of Advanced Studies in India, suggests that mathematics on other worlds may differ considerably from ours. He is not convinced by the argument that something as basic as counting will lead to the convergent evolution of mathematics on Earth and on distant planets: "Even if numbers or counting can be a common genesis, who is to say that calculus is a universal, necessary consequence of mathematical thought?"

Sarukkai explains his skepticism: "Let us say we accept that numbers or representations of numbers will occur in extraterrestrials." By his analysis, this assumption gets us only part of the way to a shared understanding of mathematics as a whole. On Earth, Sarukkai says, math is "a specific human activity which discovers particular structures such as algebra, calculus, topology, group theory, and so on." Though extraterrestrials may be very good with numbers, they might do things with them that humans had never imagined. Similarly, parts of human math may be incomprehensible to extraterrestrials.

Having radio telescopes in common, Sarukkai claims, says more about shared patterns of thinking than about any underlying mathematics or technology: "If we begin with the assumption that the extraterrestrial folks have radio telescopes, then we are making an assumption about processes of their thought more than their language or even their technology. That is, what their having radio telescopes most importantly tells us is that these creatures reason in some particular way."

In Sarukkais view, the attempt to identify universal languages reflects humankinds long-standing uneasiness with ambiguity. Typically, ambiguity is seen as an obstacle to understanding the world, as reflected in the eighteenth-century philosopher Thomas Reids view that "there is no greater impediment to the advancement of knowledge than the ambiguity of words." As an antidote, scholars have been preoccupied over the centuries with finding languages that have fixed and definite meanings. "The search for 'universal' language or 'pure' language," say Sarukkai, "is part of human history in all civilizations. In part, this reflects an enormous distrust of ambiguity in meaning." Ironically, it is exactly the imprecision of any language that makes it work so well. As Sarukkai notes, "it is semantic ambiguity that allows individuals and societies to develop and flourish!"

If we cannot count on the universality of mathematics for interstellar communication, is there any hope of comprehending at least some of the meaning an extraterrestrial is attempting to convey? "Definitely yes," according to Sarukkai. But he doesnt think we will stumble across a pre-existing universal language. Rather, we will need to invent languages for interstellar communication: "We always construct languages based on our needs, our capacities, and our traditions."

"I doubt we will find a language ready for use," says Sarukkai. The key, in his view, is to expect some ambiguity as we attempt to bridge the vast distances that separate humans and extraterrestrials: "In looking for a language for interstellar communication, we should be looking not for one-to-one matching, but for some kind of mapping which allows us to understand vaguely rather than with certainty."

Even gaining a vague understanding of the universe as seen by another civilization might help us expand beyond our parochial view of the world. And given the tremendous challenges of interstellar communication, living with ambiguity may be our only choice.