So much for the philosophy now, but what is the actual evidence for these claims? Psychologists are beginning to realize that much of our mental life rests on the operation of dedicated, biologically-determined mental modules that are specifically attuned to restricted domains of knowledge, and that have been laid down in our brains by evolution (cf. Steve Pinker's How the Mind Works). For instance, we seem to have domain-specific knowledge of animals, food, people, faces, emotions, and many other things. In each case - and number is no exception -, psychologists demonstrate the existence of a domain-specific system of knowledge using the following four arguments:

• one should prove that possessing prior knowledge of the domain confers an evolutionary advantage. In the case of elementary arithmetic, this is quite obvious.
• there should be precursors of the ability in other animal species. Thus, some animals should be shown to have rudimentary arithmetic abilities. There should be systematic parallels between their abilities and those that are found in humans.
• the ability should emerge spontaneously in young children or even infants, independently of other abilities such as language. It should not be acquired by slow, domain-general mechanisms of learning.
• the ability should be shown to have a distinct neural substrate. My book The Number Sense is dedicated to proving these four points, as well as to exploring their consequences for education and for the philosophy of mathematics. In fact, solid experimental evidence supports the above claims, making the number domain one of the areas in which the demonstration of a biologically determined, domain-specific system of knowledge is the strongest. Here, I can only provide a few examples of experiments.
  1. Animals have elementary numerical abilities. Rats, pigeons, parrots, dolphins, and of course primates can discriminate visual patterns or auditory sequences based on number alone (every other physical parameter being carefully controlled). For instance, rats can learn to press one lever for two events and another for four events, regardless of their nature, duration and spacing and whether they are auditory or visual. Animals also have elementary addition and subtraction abilities. These basic abilities are found in the wild, and not just in laboratory-trained animals. Years of training, however, are needed if one wants to inculcate number symbols into chimpanzees. Thus, approximate manipulations of numerosity are within the normal repertoire of many species, but exact symbolic manipulation of numbers isn't � it is a specifically human ability or at least one which reaches its full-blown development in humans alone.
  2. There are systematic parallels between humans and animals. Animals' numerical behavior becomes increasingly imprecise for increasingly large numerals (number size effect). The same is true for humans, even when manipulating Arabic numerals: we are systematically slower to compute, say, 4+5 than 2+3. Animals also have difficulties discriminating two close quantities such as 7 and 8. We too: when comparing Arabic digits, it takes us longer to decide that 9 is larger than 8 than to make the same decision for 9 Vs 2 (and we make more errors, too).
  3. Preverbal human infants have elementary numerical abilities, too. These are very similar to those of animals: infants can discriminate two patterns based solely on their number, and they can make simple additions and subtractions. For instance, at 5 months of age, when one object is hidden behind a screen, and then another is added, infants expect to see two objects when the screen drops. We know this because careful measurements of their looking times show that they look longer when, a trick makes a different number of objects appear. Greater looking time indicates that they are surprised when they see impossible events such as 1+1=1, 1+1=3, or 2-1=2. [Please, even if you are skeptical, don't dismiss these data with the back of your hand, as I was dismayed to discover Martin Gardner was doing in a recent review of my book for The Los Angeles Times. Sure enough, "measuring and averaging such times is not easy", but it is now done under very tightly controlled conditions, with double-blind video tape scoring. I urge you to read the original reports, for instance Wynn, 1992, Nature, vol. 348, pp. 749-750 � you'll be amazed at the level of detail and experimental control that is brought to such experiments.]

     Like animals and adults, infants are especially precise with small numbers, but they can also compute more approximately with larger numbers. In passing, note that these experiments, which are very reproducible, invalidate Piaget's notion that infants start out in life without any knowledge of numerical invariance. In my book, I show why Piaget's famous conservation experiments are biased and fail to tell us about the genuine arithmetical competence of young children.

  4. Brain lesions can impair number sense. My colleagues and I have seen many patients at the hospital that have suffered cerebral lesions and, as a consequence, have become unable to process numbers. Some of these deficits are peripheral and concern the ability to identify words or digits or to produce them aloud. Others, however, indicate a genuine loss of number sense. Lesions to the left inferior parietal lobe can result in a patient remaining able to read and write Arabic numerals to dictation while failing to understand them. One of our patients couldn't do 3 minus 1, or decide which number fell between 2 and 4! He didn't have any problem telling us what month fell between February and April, however, or what day what just before Wednesday. Hence the deficit was completely confined to numbers. The lesion site that yields such a number-sense deficit is highly reproducible in all cultures throughout the world.
  5. Brain imaging during number processing tasks reveals a highly specific activation of the inferior parietal lobe, the very same region that, when lesioned, causes numerical deficits. We have now seen this activation using most of the imaging methods currently available. PET scanning and fMRI pinpoint it anatomically to the left and right intraparietal sulci. Electrical recordings also tell us that this region is active during operations such as multiplication or comparison, and that it activates about 200 ms following the presentation of a digit on a screen. There are even recordings of single neurons in the human parietal lobe (in the very special case of patients with intractable epilepsy) that show specific increases in activity during calculation.

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