Making a case against a computational theory of consciousness, the mathematical physicist Sir Roger Penrose claims that humans have an intuitive grasp of mathematics, even of mathematical theorems. Stanislas Dehaene, a mathematician turned cognitive neuropsychologist, shares Penrose's musings of the claim that the mind and brain are super supercomputers, but where Penrose argues by assertion as if mathematical intuition were self-evident, Dehaene builds his thesis from varied sources of evidence -- cultural, psychological and biological.
Many non-human animals have been shown experimentally to have a number sense. Mammals and birds can distinguish one from two, three, four or many. For a living species to distinguish and classify objects around them, is probably a necessary trait to survive long enough to reproduce. Single celled organisms have internal clocks that estimate the passage of time. The genes that form part of the cellular counting machinery in fruit flies have their homologues in mammals.
The brain itself communicates in numerical codes. Nerve cells (neurons) convey messages by electrical pulses that pass down the nerve fiber (axon) until they reach the junction with the next neuron or an effector such as a muscle. There they release little packets of a chemical transmitter that trigger the next cell to respond. The response depends on the frequency of electrical pulses and the quantity of transmitter released. Given this framework, counting is thus an embedded feature of brain work.
That the capacity to manipulate numerical symbols is dependent on brain processes is self-evident. But until recently the only way of exploring these processes has been by studying the circumstances in which people have lost their number sense as a result of brain damage. Some, for instance, can readily call out strings of numbers like 1, 2, 3, 4 but cannot add 2 and 2 or decide whether 6 is more or less than 8. Neurologists describe this as ''acalculia,'' and look for the holes in the brain that may cause it. The brain, however, is an integrated, coherent system, not a collection of independent modules. Many different regions are engaged in even the simplest calculations.
A number sense, according to Dehaene, is profoundly shaped by culture and technologies, the crucial technologies being the invention of spoken words and written symbols for numbers. Today we take for granted the logic of Arabic numerals and, in our use of computers, the binary symbols of programming codes. But the Arabic numeral system, including the essential zero, was an invention. With Arabic, the logic of adding, subtracting, multiplying and dividing is immediately apparent. Just try subtracting the Roman numeral XIV from LXXII, or, multiplying the two. Even the names we give our numbers affect our understanding of them (compare the French quatre-vingt and the English 80 for ease of manipulation). Dehaene points out that the superior linguistic logic of Chinese number terminology enables the greater speed of mental calculation shown by Chinese speakers over those using European languages. Brain scans of people raised in different mathematical cultures might help reveal what is fixed, what is developmental, and to identify brain regions involved in calculations of our neuronal machinery.
Source: The New York Times Book Review, February 8, 1998 pg. 16