In the remainder of this chapter, I will attempt to demonstrate that this scenario for the origins of mathematic truths has some validity, using recent results from cognitive neuroscience studies. In spite of important advances in psychology and brain imaging techniques, and of a few recent forays into the brain processes underlying algebra (Anderson et al. 2004; Qin et al. 2004), it remains exceedingly difficult to examine the cerebral bases of higher mathematical objects. Thus, research in my laboratory has focused on one of the most basic objects, one that lies at the foundations of mathematics: the concept of number. Our recent data suggest that this concept is universally shared and is rooted in a long evolutionary history, but also that is has been refined thanks to cultural inventions whose acquisition can be investigated in both human children and isolated human cultures.
Consider one of the simplest arithmetic abilities: deciding which of two numbers is the largest. Many experiments have now shown that this elementary arithmetic operation is accessible to laboratory animals and even to untrained animals in the wild (Brannon and Terrace 2000; Hauser et al. 2000; Hauser et al. 2003;
McComb et al. 1994; Rumbaugh et al. 1987; Washburn and Rumbaugh 1991). For instance, macaque monkeys spontaneously choose the larger of two sets of food items (Hauser et al. 2000), and lions spontaneously estimate whether their group is more numerous than another group (McComb et al. 1994).
In a particularly impressive laboratory demonstration, macaque monkeys were trained to order a set of cards by the number of objects that they bore (Brannon and Terrace 2000). After training with cards bearing from one to four objects, monkeys generalized their performance to untrained numbers five through nine. This experiment, like many others, included stringent controls for non-numerical parameters such as density, object size, shape, etc. Thus, it can be concluded that the performance of the animals genuinely reflects an elementary competence to perceive numbers.
In most such tasks, animal performance improves as the distance between the two numbers to be compared increases (distance effect), and also improves as the numbers get smaller (size effect). Altogether, these two effects are well captured by stating that comparison performance depends on the ratio of the compared numbers (Weber's law). Thus, number seems to be represented only approximately, on an internal continuum comparable to that used for other perceptual dimensions, such as weight or height.
What is the evidence, however, that the ability to compare sets in animals has anything to do with the human arithmetic ability? A remarkable finding is that, when humans compare two Arabic numerals, one can observe distance and size effects similar to those found in animals with non-symbolic stimuli (Dehaene et al. 1990; Moyer and Landauer 1967). Thus, human subjects are slower and make more errors when deciding which of eight and nine is the larger, than when comparing five and nine. Similar effects are observed when comparing two-digit numerals such as 78 and 65: the response time curve shows a continuous distance effect, as would be observed if those numbers were presented as sets of objects. One could have thought that numbers presented in symbolic format would be compared with digital precision, by an exact algorithm. However, the tight parallel between comparisons of symbolic and non-symbolic numbers (Buckley and Gillman 1974) rather suggests that humans continue to use an analog representation, comparable to that available to non-human primates, even when processing Arabic symbols. I have suggested that the quantity representation serves as a "core quantity system" towards which new cultural symbols such as digits and words are quickly translated, and which provides the meaning of those symbols (Dehaene 1997).
Recently, with Philippe Pinel, we have pursued the neural basis of the distance effect during number comparison using functional magnetic resonance imaging (fMRI; Pinel et al. 1999, 2001, 2004). Our results indicate that, during number comparison, the activation of the left and right intraparietal sulci (IPS) shows a tight correlation with the behavioral distance effect: it too varies in inverse relation to the distance between the numbers to be compared. Based on a meta-analy-sis of many fMRI studies of arithmetic tasks, including comparison, calculation (Chochon et al. 1999), approximation (Dehaene et al. 1999), or even the mere detection of digits (Eger et al. 2003), we have suggested that the bilateral horizontal segment of the IPS (HIPS) may play a particular role in the quantity representation (Dehaene et al. 2003).
Indeed, investigations of brain-lesioned patients indicate that a lesion of this region, at least in the left hemisphere, can cause severe deficits of number comprehension and calculation (acalculia). More recently, this region has also been pinpointed as a possible anatomical locus of impairment in children with dyscal-culia, a lifelong impairment in arithmetic that cannot be attributed to global mental retardation or to environmental variables. In premature children with dyscal-culia, and in a genetic disease called Turner's syndrome, loss of gray matter in the depth of the IPS has been observed with magnetic resonance imaging (Isaacs et al. 2001; Molko et al. 2003, 2004). These results support the argument that the availability of an intact quantity representation in IPS plays an essential role in guiding arithmetic development in children. The IPS may provide children with a foundational intuition of "number sense:" what is a number and how numerical quantities can be compared or combined. If the IPS is impaired, either genetically or accidentally, then number sense is deteriorated and dyscalculia may ensue.
Further support for this argument is provided by the oft-replicated observation that infants already possess elementary numerical abilities comparable to those of other animal species (for review, see Feigenson et al. 2004). Thus, knowledge of numerical quantities predates by many years the acquisition of number words and number symbols. I suggest that it does not merely precede it but actually plays an active role in making it possible.
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