Let the great British mathematician G.H. Hardy have the final say: "I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations,' are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onward, and I shall use the language which is natural to a man who holds it. A reader who does not like the philosophy can alter the language: it will make very little difference to my conclusions."
Stanislaw Dehaene, a neuroscientist at a Paris medical research institute, is, I regret to say, as down on mathematical realism as Reuben Hersh. "The Number Sense" is subtitled "How the Mind Creates Mathematics." However, instead of seeing numbers as social constructs, he thinks they are present at birth inside each person's head where the brain is genetically wired to "recognize" them. How the brain can recognize numbers unless in some sense they are "out there" is not made clear.
Dehaene is convinced that many animals--mice, raccoons, dolphins, pigeons, parrots and, of course, apes--also are capable of understanding very small integers and to perform simple arithmetic with them. His first chapter is a fascinating survey of recent experiments with animals that seem to support this view. Rats, for example, can be trained to take the fourth entrance to any maze. They can be trained to press a lever exactly n times if n is a small number.
In one experiment, a rat was conditioned to press a certain lever only after it heard two tones followed by two flashes of light. Dehaene takes this as evidence that a rat somehow knows that two plus two equals four. A chimp named Sheba, after finding the numeral two under a table, and the numeral four inside a box, was trained to select the numeral six from a set of cards. To Sarah Boysen, who designed this and similar sensational experiments, it shows that chimps can add two and four to get six. If true, it proves that mathematics is not restricted to human cultures.
"Never had an animal come any closer," Dehaene writes, "to the symbolic calculation abilities exhibited by humankind." I'm inclined to doubt this. Assuming Sheba was not responding to unconscious cues from her trainers, I think it is possible to believe that she was simply associating a combination of two meaningless symbols with a third symbol without the slightest inkling of their arithmetic meanings.
In his next chapter, "Babies Who Count," Dehaene describes experiments with babies that he is convinced show that they can do simple arithmetic long before they can speak. Most of this research fails to impress me. For instance, 4 1/2-month-old infants are shown two Mickey Mouse toys. A screen is placed to conceal the toys. Behind the screen, two red balls are substituted for the toys. When the screen is removed and the babies see the two balls, they are not in the least surprised. But if they see only one red ball or three red balls, they appear shocked.
"Mickey Mouse turning into a ball. . . is an acceptable transformation as far as the baby's number processing system is concerned," Dehaene writes. "As long as no object vanishes or is created de novo, the operation is judged to be numerically correct and yields no surprise reaction in babies." Twoness is preserved. But if two objects turn magically into one or three, the baby is startled. "The demonstration is irrefutable," the author says. "Babies know that one plus one makes neither one nor three, but exactly two."
How did psychologist Tony Simon, who supervised this test, decide when a baby is surprised? By measuring the average amount of time a baby takes when looking at objects. If it takes a few seconds longer when it sees that two objects have changed to one or three, this is taken to mean the baby is shocked. Measuring and averaging such times is not easy because babies look here and there and seldom keep a fixed gaze on anything. There is so much room here for an experimenter's expectations to bias statistics that I find it hard to accept this measurement and similar tests as proof that very young babies, like chimps, have an inherited ability to do simple arithmetic.
Jean Piaget's famous experiments with infants showed that children have to be several years old before they have any grasp of numbers. Dehaene believes Piaget was wrong. "Babies are much better mathematicians than we thought only 15 years ago," he assures us. "When they blow the first candle on their birthday cake, parents have every reason to be proud of them, for they have already acquired, whether by learning or by mere cerebral maturation, the rudiments of arithmetic and a surprisingly articulate 'number sense.' "