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Numerical cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural bases of numbers and mathematics. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes researchers in cognitive psychology, developmental psychology, neuroscience and cognitive linguistics. This discipline, although it may interact with questions in the philosophy of mathematics is primarily concerned with empirical questions.
Topics included in the domain of numerical cognition include:
Similarly, researchers have set up hidden speakers in the African savannah to test natural (untrained) behavior in lions (McComb, Packer & Pusey 1994). These speakers can play a number of lion calls, from 1 to 5. If a single lioness hears, for example, three calls from unknown lions, she will leave, while if she is with four of her sisters, they will go and explore. This suggests that not only can lions tell when they are "outnumbered" but that they can do this on the basis of signals from different sensory modalities, suggesting that numerosity is a multisensory concept.
Because of the numerous controls that were in place to rule out non-numerical factors, the experimenters infer that six-month-old infants are sensitive to differences between 8 and 16. Subsequent experiments, using similar methodologies showed that 6-month-old infants can discriminate numbers differing by a 2:1 ratio (8 vs. 16 or 16 vs. 32) but not by a 3:2 ratio (8 vs. 12 or 16 vs. 24). However, 10-month-old infants succeed both at the 2:1 and the 3:2 ratio, suggesting an increased sensitivity to numerosity differences with age (for a review of this literature see Feigenson, Dehaene & Spelke 2004).
In another series of studies, Karen Wynn showed that infants as young as five months are able to do very simple additions (e.g., 1 + 1 = 2) and subtractions (3 - 1 = 2). To demonstrate this, Wynn used a "violation of expectation" paradigm, in which infants were shown (for example) one Mickey Mouse doll going behind a screen, followed by another. If, when the screen was lowered, infants were presented with only one Mickey (the "impossible event") they looked longer than if they were shown two Mickeys (the "possible" event). Further studies by Karen Wynn and Koleen McCrink found that although infants' ability to compute exact outcomes only holds over small numbers, infants can compute approximate outcomes of larger addition and subtraction events (e.g., "5+5" and "10-5" events).
There is debate about how much these infant systems actually contain in terms of number concepts, harkening to the classic nature versus nurture debate. Gelman & Gallistel 1978 suggested that a child innately has the concept of natural number, and only has to map this onto the words used in her language. Carey 2004, Carey 2009 disagreed, saying that these systems can only encode large numbers in an approximate way, where language-based natural numbers can be exact. One promising approach is to see if cultures that lack number words can deal with natural numbers. The results so far are mixed (e.g., Pica et al. 2004); Butterworth & Reeve 2008, Butterworth, Reeve & Lloyd 2008.
Single-unit neurophysiology in monkeys has also found neurons in the frontal cortex and in the intraparietal sulcus that respond to numbers. Andreas Nieder (Nieder 2005; Nieder, Freedman & Miller 2002; Nieder & Miller 2004) trained monkeys to perform a "delayed match-to-sample" task. For example, a monkey might be presented with a field of four dots, and is required to keep that in memory after the display is taken away. Then, after a delay period of several seconds, a second display is presented. If the number on the second display match that from the first, the monkey has to release a lever. If it is different, the monkey has to hold the lever. Neural activity recorded during the delay period showed that neurons in the intraparietal sulcus and the frontal cortex had a "preferred numerosity", exactly as predicted by behavioral studies. That is, a certain number might fire strongly for four, but less strongly for three or five, and even less for two or six. Thus, we say that these neurons were "tuned" for specific quantities. Note that these neuronal responses followed Weber's law, as has been demonstrated for other sensory dimensions, and consistent with the ratio dependence observed for non-human animals' and infants' numerical behavior (Nieder & Miller 2003).
Modification of the usual decimal representation was advocated by John Colson. The sense of complementation, missing in the usual decimal system, is expressed by signed-digit representation.
Topics included in the domain of numerical cognition include:
- How do non-human animals process numerosity?
- How do infants acquire an understanding of numbers (and how much is inborn)?
- How do humans associate linguistic symbols with numerical quantities?
- How do these capacities underlie our ability to perform complex calculations?
- What are the neural bases of these abilities, both in humans and in non-humans?
- What metaphorical capacities and processes allow us to extend our numerical understanding into complex domains such as the concept of infinity, the infinitesimal or the concept of the limit in calculus?
Comparative studies
A variety of research has demonstrated that non-human animals, including rats, lions and various species of primates have an approximate sense of number (referred to as "numerosity") (for a review, see Dehaene 1997). For example, when a rat is trained to press a bar 8 or 16 times to receive a food reward, the number of bar presses will approximate a Gaussian or Normal distribution with peak around 8 or 16 bar presses. When rats are more hungry, their bar pressing behavior is more rapid, so by showing that the peak number of bar presses is the same for either well-fed or hungry rats, it is possible to disentangle time and number of bar presses.Similarly, researchers have set up hidden speakers in the African savannah to test natural (untrained) behavior in lions (McComb, Packer & Pusey 1994). These speakers can play a number of lion calls, from 1 to 5. If a single lioness hears, for example, three calls from unknown lions, she will leave, while if she is with four of her sisters, they will go and explore. This suggests that not only can lions tell when they are "outnumbered" but that they can do this on the basis of signals from different sensory modalities, suggesting that numerosity is a multisensory concept.
Developmental studies
Developmental psychology studies have shown that human infants, like non-human animals, have an approximate sense of number. For example, in one study, infants were repeatedly presented with arrays of (in one block) 16 dots. Careful controls were in place to eliminate information from "non-numerical" parameters such as total surface area, luminance, circumference, and so on. After the infants had been presented with many displays containing 16 items, they habituated, or stopped looking as long at the display. Infants were then presented with a display containing 8 items, and they looked longer at the novel display.Because of the numerous controls that were in place to rule out non-numerical factors, the experimenters infer that six-month-old infants are sensitive to differences between 8 and 16. Subsequent experiments, using similar methodologies showed that 6-month-old infants can discriminate numbers differing by a 2:1 ratio (8 vs. 16 or 16 vs. 32) but not by a 3:2 ratio (8 vs. 12 or 16 vs. 24). However, 10-month-old infants succeed both at the 2:1 and the 3:2 ratio, suggesting an increased sensitivity to numerosity differences with age (for a review of this literature see Feigenson, Dehaene & Spelke 2004).
In another series of studies, Karen Wynn showed that infants as young as five months are able to do very simple additions (e.g., 1 + 1 = 2) and subtractions (3 - 1 = 2). To demonstrate this, Wynn used a "violation of expectation" paradigm, in which infants were shown (for example) one Mickey Mouse doll going behind a screen, followed by another. If, when the screen was lowered, infants were presented with only one Mickey (the "impossible event") they looked longer than if they were shown two Mickeys (the "possible" event). Further studies by Karen Wynn and Koleen McCrink found that although infants' ability to compute exact outcomes only holds over small numbers, infants can compute approximate outcomes of larger addition and subtraction events (e.g., "5+5" and "10-5" events).
There is debate about how much these infant systems actually contain in terms of number concepts, harkening to the classic nature versus nurture debate. Gelman & Gallistel 1978 suggested that a child innately has the concept of natural number, and only has to map this onto the words used in her language. Carey 2004, Carey 2009 disagreed, saying that these systems can only encode large numbers in an approximate way, where language-based natural numbers can be exact. One promising approach is to see if cultures that lack number words can deal with natural numbers. The results so far are mixed (e.g., Pica et al. 2004); Butterworth & Reeve 2008, Butterworth, Reeve & Lloyd 2008.
Neuroimaging and neurophysiological studies
Human neuroimaging studies have demonstrated that regions of the parietal lobe, including the intraparietal sulcus (IPS) and the inferior parietal lobule (IPL) are activated when subjects are asked to perform calculation tasks. Based on both human neuroimaging and neuropsychology, Stanislas Dehaene and colleagues have suggested that these two parietal structures play complementary roles. The IPS is thought to house the circuitry that is fundamentally involved in numerical estimation (Piazza et al. 2004), number comparison (Pinel et al. 2001; Pinel et al. 2004) and on-line calculation (often tested with subtraction) while the IPL is thought to be involved in overlearned tasks, such as multiplication (see Dehaene 1997). Thus, a patient with a lesion to the IPL may be able to subtract, but not multiply, and vice versa for a patient with a lesion to the IPS. In addition to these parietal regions, regions of the frontal lobe are also active in calculation tasks. These activations overlap with regions involved in language processing such as Broca's area and regions involved in working memory and attention. Future research will be needed to disentangle the complex influences of language, working memory and attention on numerical processes.Single-unit neurophysiology in monkeys has also found neurons in the frontal cortex and in the intraparietal sulcus that respond to numbers. Andreas Nieder (Nieder 2005; Nieder, Freedman & Miller 2002; Nieder & Miller 2004) trained monkeys to perform a "delayed match-to-sample" task. For example, a monkey might be presented with a field of four dots, and is required to keep that in memory after the display is taken away. Then, after a delay period of several seconds, a second display is presented. If the number on the second display match that from the first, the monkey has to release a lever. If it is different, the monkey has to hold the lever. Neural activity recorded during the delay period showed that neurons in the intraparietal sulcus and the frontal cortex had a "preferred numerosity", exactly as predicted by behavioral studies. That is, a certain number might fire strongly for four, but less strongly for three or five, and even less for two or six. Thus, we say that these neurons were "tuned" for specific quantities. Note that these neuronal responses followed Weber's law, as has been demonstrated for other sensory dimensions, and consistent with the ratio dependence observed for non-human animals' and infants' numerical behavior (Nieder & Miller 2003).
Relations between number and other cognitive processes
There is evidence that numerical cognition is intimately related to other aspects of thought – particularly spatial cognition.[1] One line of evidence comes from studies performed on number-form synaesthetes.[2] Such individuals report that numbers are mentally represented with a particular spatial layout; others experience numbers as perceivable objects that can be visually manipulated to facilitate calculation. Behavioral studies further reinforce the connection between numerical and spatial cognition. For instance, participants respond quicker to larger numbers if they are responding on the right side of space, and quicker to smaller numbers when on the left—the so-called "Spatial-Numerical Association of Response Codes" or SNARC effect.[3] This effect varies across culture and context,[4] however, and some research has even begun to question whether the SNARC reflects an inherent number-space association,[5] instead invoking strategic problem solving or a more general cognitive mechanism like conceptual metaphor.[6][7] Moreover, neuroimaging studies reveal that the association between number and space also shows up in brain activity. Regions of the parietal cortex, for instance, show shared activation for both spatial and numerical processing.[8] These various lines of research suggest a strong, but flexible, connection between numerical and spatial cognition.Modification of the usual decimal representation was advocated by John Colson. The sense of complementation, missing in the usual decimal system, is expressed by signed-digit representation.
