Math without Words

Numerical reasoning seems independent of language

Nineteenth-century German mathematician Carl Friedrich Gauss used to joke that he could calculate before he could talk. Maybe it was no joke. Recent work casts doubt on the notion that language underlies mathematical ability and perhaps other forms of abstract thinking.

Writing in the March 1 Proceedings of the National Academy of Sciences USA, scientists from the University of Sheffield in England describe impressive mathematical abilities in three middle-aged men who had suffered severe damage to the language centers of their brains. "There had been case studies of aphasics who could calculate," says study co-author Rosemary Varley. "Our new take was to try to identify roughly parallel mathematical and linguistic operations."

Varley and her colleagues found that although the subjects could no longer grasp grammatical distinctions between, say, "The dog bit the boy" and "The boy bit the dog," they could interpret mathematical formulas incorporating equivalent structures, such as "59 - 13" and "13 - 59."

The researchers found ways to pose more abstract questions as well. For instance, to investigate the subjects' understanding of number infinity, they asked them to write down a number bigger than 1 but smaller than 2, using hand motions for "bigger" and "smaller" and a flash of the eyebrow, indicating surprise, for "but." Then they asked the subjects to make the number bigger but still smaller than 2 and to reiterate the procedure. The subjects got the answer by various means, including the addition of a decimal place: 1.5, 1.55, 1.555 and so forth.

Although subjects easily answered simple problems expressed in mathematical symbols, words continued to stump them. Even the written sentence "seven minus two" was beyond their comprehension. The results show quite clearly that no matter how helpful language may be to mathematicians--perhaps as a mnemonic device--it is not necessary to calculation, and it is processed in different parts of the brain.

The idea that language shapes abstract thought was most forcibly propounded 50 years ago in the posthumously published writings of American linguist Benjamin Lee Whorf. He argued, among other things, that the structure of the Hopi language gave its speakers an understanding of time vastly different from that of Europeans. Although Whorf's hypothesis continues to inspire research, a good deal of his evidence has been discredited. Much more widely respected is the proposal, associated with linguist Noam Chomsky of the Massachusetts Institute of Technology, that language, mathematics and perhaps other cognition all depend on a deeper quality, sometimes called "mentalese."

Chomsky suggested that the key part of this deeper quality might be a quite simple and uniquely human power of "recursive" calculation. Recursion, he and his colleagues argue, may explain how the mind spins a limited number of terms into an infinite number of often complex statements, such as "The man I know as Joe ate my apple tree's fruit." Recursion could also generate mathematical statements, such as "3 (4/6 + 27)/4."

Chomsky's theory may, perhaps, be reconciled with the new evidence. Some scholars have argued that the brain may build its mathematical understanding with language and that the structure may still stand after the scaffolding is removed. Indeed, the one subject in the Sheffield study who had had doctoral-level training in a mathematical science did no better than the others in arithmetic, but he outperformed them at algebra.

Rochel Gelman, co-director of the Rutgers University Center for Cognitive Science, says that the brain-lesion studies offer much clearer evidence than can be obtained from the more common technique of functional brain scanning. "Pop someone in a scanner and ask a question, and you may get a lot of activation in language areas," she points out. "But it could be just because the subject is talking through the problem--recruiting language, although it's not a crucial component."

The recent work, together with studies of animals and of children, strongly supports the independence of language and mathematics, Gelman says. "There are cases of kids who are bad with numbers and good with words and bad with words and good with numbers, a double dissociation that provides converging evidence."

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