Mathematical Truth?

I have always been intrigued by math and the brain (primarily because I am so bad at math). I was intrigued not just by the mathematics themselves, or how you do mathematics, but also why they take the form they do; this seems to me to be the foundation problem at the heart of mathematics. It turns out that I was in luck: in recent years, the cognitive neuroscience of numeracy, or ‘numerical cognition’, has emerged as an important area where the interaction between brain structure and human culture can be studied empirically. Number can be defined as the only property of sets that remains invariant under substitutions of any items in the set (1). Thus, we talk about 4 chairs, 4 people, 4 events. Why is it that even small children demonstrate some degree of mathematical understanding, yet so many adults view mathematics as a purely academic domain?

Whereas certain numerical concepts, such as the real numbers, are only ever represented by a subset of human adults, there is evidence to suggest that animals, young infants and adult humans possess a biologically determined representation of number and of elementary calculation abilities. Lesion and brain-imaging studies in humans indicate that a specific neural substrate, located in the left and right intraparietal area, is associated with knowledge of numbers and their relations (‘number sense’). (1) Recent evidence supporting a cognitive approach to this knowledge includes its spontaneous emergence at a young age during development; its presence in animals (albeit in a simpler form); and its association to a specific substrate that can be identified reproducibly in different individuals.

Evidence that animals possess number discrimination, cross-modal numerosity perception and elementary arithmetic abilities comparable to those of human infants has been reviewed in detail in recent years. Like human infants, various animal species including rats, pigeons, raccoons, parrots, monkeys and chimpanzees can discriminate the numerosity of various sets, including visual objects presented simultaneously or sequentially. Experiments of symbolic ‘language’ training also provide evidence for abstract numerical representations in animals. For example, monkeys and chimpanzees have been taught to recognize the Arabic digits 1-9 and to use them appropriately to refer to sets of objects. (3) However, symbolic labeling abilities such as these are exceptional and require years of training and are never found in animals in the wild. Thus, such experiments cannot be taken to indicate that exact symbolic number processing is within the normal behavioral repertoire of animals. However, they do indicate that abstract representations of number are available to animals and can, under certain circumstances, be mapped on behaviors that can serve as numerical symbols.

There is slightly more research that has been conducted with infants to see whether they inherently possess the concept of number. Previously, the accepted wisdom in developmental psychology was based on the Piagetian framework and suggested that infants are devoid of numerical competence. Piaget’s studies of young children’s numerical representations suggested that abstract knowledge of arithmetic requires considerable learning and does not appear before 4-7 years of age. (4) However, more recent, less-demanding and non-verbal tests have indicated repeatedly that children between 1 and a half and 4 years of age have mastered number conservation. Discrimination in visual numerosity was first demonstrated in 6-7 month old infants using the classic method of habituation-recovery of looking time. (5) Infants watched as slides with a fixed number of dots (for example, 3) were presented to them repeatedly until their looking time decreased, indicating habituation. At that point, the presentation of slides with a different number of dots (for example, 2) was shown to yield significantly longer looking times, indicating dishabituation and therefore discrimination between 2 and 3. In general, human infants have been found to possess elaborate world knowledge, for instance about objects, colors, faces and language. This knowledge would seem to validate the hypothesis that humans have been endowed by evolution with biologically determined predispositions to represent and acquire knowledge of specific domains, including mathematics.

Stanislas Dehaene, in his essay in the Neurobiology of Human Values, argues that a primate’s brain has been built by evolution to entertain the number concept; however, largely unique to humans is the ability to connect our imprecise concepts of time, space and number to categorically defined words and written symbols. (6) Mathematical reality is therefore a cultural and mental construction, but one which draws upon the constraints imposed on humans by millions of years of brain evolution. He argues that although the concept of number is universally shared and is rooted in a long evolutionary history, it is refined due to cultural inventions. He tests his hypothesis by describing research that he conducted on the Mundarukú, an Amazonian group whose language includes very few words for numbers; they essentially have numbers for 1 through 5, plus quantifiers such as few or many, but nothing else. (6 p.151). In spite of their linguistic limitations, the group gave evidence of an excellent understanding of large numbers. They were able to decide which of two sets of dots was the more numerous, even with numbers ranging up to 80, and even in the presence of considerable variation in parameters such as size and density. Through these observations, he concluded that linguistic labels are not necessary to master the major concepts of arithmetic (quantity, addition, subtraction) and to perform approximate operations. Linguistic codings of numerals, however, is essential to go beyond this ancient system and to perform exact calculations. If this interpretation is correct, what limits the Mundarukú is not a lack of conceptual knowledge. Rather, the linguistic coding of numbers is a ‘cultural tool’ that augments the cognitive strategies permitting us to resolve concrete problems. This refutes the hypothesis that language determines conceptual structure.

The book The Mathematical Brain by Brian Butterworth also focuses on a the concept of number. (7) Butterworth’s central hypothesis is that our brain is ‘born to count’: our genes contain instructions that specify how to build a number module (a set of neural circuits specialized for processing numbers). Those circuits, which are associated in part with the left inferior parietal lobe, makes us sensitive to numerosities in our environment and allows us to understand and to manipulate numbers mentally. Loss of those circuits would result in a selective inability to grasp the meaning of numbers.

This makes a lot of sense; If there is a genetic plan for a number module, then one might expect to find an occasional child who is born without it, either due to a genetic defect of to pre- or perinatal cerebral damage. One of the results of this damage is what has become known as ‘developmental Gerstmann’s Syndrome’, or developmental dyscalculia. This condition has been primarily reported in children; these children show a highly selective deficit for number processing although they have normal intelligence, normal language acquisition and a standard level of education. However, Butterworth claims to have identified another patient, Charles, who is ‘born blind to numerosities’.(7) Although Charles is a very bright student, he has experienced profound, lifelong difficulties in mathematics, to the point of still having to count on his fingers in order to solve single digit addition problems. Chronometric tests reveal at least 2 major impairments. First, Charles cannot “subitize”: he cannot decide how many items are presented on a computer screen, even if there are only two or three, unless he painstakingly counts them one by one. Second, he has an abnormal intuition of number size: whereas we normally take less time to decide which of two numbers is larger as the distance between them gets larger, Charles takes more time for more distant numbers, presumably because he is using a very indirect counting strategy. What are conditions such as these a result of? Are they neurobiological? Recently conducted research that has suggests that dyscalculia does in fact have it’s roots in damage to a particular area of the brain.

Knowledge of different categories of words and objects such as persons, tools, animals and actions can be associated with distinct patterns of brain activation. If the elementary understanding and manipulation of numerical quantities is part of our biological evolutionary heritage, it would seem to follow that it also has a neural substrate. Two arguments suggest that number processing is associated with a specific cerebral network located in the inferior intra-parietal area of both hemispheres.  First, neuropsychological studies of human patients with brain lesions indicate that the internal representation of quantities can be impaired selectively by lesions to that area. (8) Second, brain-imaging studies reveal that this region is activated specifically during various number processing tasks. Roland and Friberg were the first to monitor blood flow changes during calculation as opposed to rest. (9) When subjects repeatedly subtracted three from a given number, activation increased bilaterally in the inferior parietal and prefrontal cortex. These localizations were later confirmed using functional magnetic resonance imaging (fMRI). This evidence leads us to speculate that the inferior parietal cortex holds a biologically determined representation of numerical quantities. When analyzed along with other research evidence, it follows that this numerical representation is available to animals and to infants before language acquisition.

There are of course many more questions that remain to be answered, as well as some refining that needs to be done in order to most convincingly defend this hypothesis. For example, although the role of the intraparietal cortex in number processing is supported by considerable evidence in adult humans, its involvement in infants and animals remains speculative. In addition, the idea that a single brain area underlies an entire domain of competence such as arithmetic is unfathomable. Presumably, only the core of number meaning- knowledge of numerical quantities and their relations- is encoded in the intraparietal cortex. There are undeniably many other regions of the brain that deal with other dynamics of mathematics and number processing, such as digit identification or multi-digit calculation.

In conclusion, the research that is currently being conducted by scientists such as Dehaene and Butterworth aims at understanding the cerebral origins of one of the foundations of mathematics, the concept of number. Studies show that the concept of approximate number is independent of language and universally shared, but that exact calculation is not. Most mathematicians, and indeed most people, believe that mathematics is a domain of pure truths that predate the human mind and have an abstract existence, independent of the human ability to discover them. But the more we learn about neurobiology, the more we tell ourselves that this mathematical reality must be made of something, that it must be the creation of complex, interlocking assemblies of neurons within our brains. More research is most definitely needed in order to better understand the dynamics of our conception of mathematics and its origins, but it is very possible that we will soon be able to answer the question of whether mathematics is a system of clear truths or whether it is a tangle of relations between obscure entities.

Works Cited

1. Abstract representations of numbers in the animal and human brain.

2. Ordinal judgements of numerical symbols by macaques (Macaca mulatta)

3. Piaget, J. (1952) The Child’s Conception of Number, New York: Norton, 1952

4. Perception of numbers by human infants. 

5. Dehaene, S. How a Primate brain comes to know some mathematical truths. Neurobiology of Human Values, J.P. Changeux, A.R Damasio, W. Singer & Christen (Eds.) Springer, 2005. pp 143-155

6. Butterworth, B. The Mathematical Brain. London: Macmillan, 1999

7. Cerebral pathways for calculation: double dissociation between rote verbal and quantitative knowledge of arithmetic.

8. Localization of cortical areas activated by thinking