The origins of numerical abilities

This talk summarises numerical capacities in chimpanzees focusing on three topics: cardinal and ordinal aspects, decimal number system, and photographic memory for numerals. The Ai project started in 1977 when a 1-year-old female chimpanzee named Ai arrived the Primate Research Institute of Kyoto University. She mastered logico-mathematical skills including the use of Arabic numerals to label sets of items in addition to their colour and constituent objects. Next, we tested Ai on ordinal aspects of number through a sequencing task that required her to touch numerals in ascending order. These two related skills were later verified in six other chimpanzees. After introducing the numeral 0 we tested acquisition of the decimal number system from 0 to 19. A major problem for chimpanzees in these tasks was the time needed to make a judgement. Chimpanzees have a strong tendency to make quick decisions, and this led to the estimation, rather than one-by-one counting, of the items displayed. They mastered the ordinal aspect of numbers beyond 10, although two-digits numerals greater than 10 led to lower accuracy. However, the tendency for quick decisions has advantages in situations that need rapid comprehension, such as memorising numerals at a glance. Three young chimpanzees showed better performance than human adults. Eye-tracking revealed the chimpanzee-unique way of watching the world: a quick scan of the whole image, jumping from one place to the next. This contrasts with humans, who direct their gaze at specific points as they try to grasp the meaning of the image.

Scientists since Galileo have insisted that mathematics is structured as a language – but is this language similar to spoken language? Do mathematicians use classical language areas when doing mathematics? Converging evidence from several laboratories suggests that the left posterior superior temporal sulcus (pSTS) and inferior frontal gyrus (IFG, pars triangularis and orbitalis) play a central role in the syntax of written, spoken, or even sign language. The group used fMRI to investigate whether these brain areas also contribute to various aspects of mathematics. For the syntax of complex verbal numbers, such as ‘three hundred forty-nine’, which form a subdomain of natural language, a major contribution of the left IFG is observed. However, for the syntax of algebraic expressions such '((3+2)-4)+1', or when professional mathematicians reflect upon high-level mathematical concepts in algebra, analysis, geometry or topology, the activation spares language areas. Instead, high-level mathematics involves bilateral intraparietal areas involved in elementary number sense and simple arithmetic, and bilateral infero-temporal areas involved in processing Arabic numerals. The evidence suggests that the acquisition of mathematical concepts recycles areas involved in elementary number processing.

Findings in animal cognition, developmental psychology and anthropology indicate that numerical skills are rooted in non-verbal biological primitives. To decipher the neuroethological and evolutionary foundations of number representations, we study behaving monkeys and crows in combined psychophysical/neurophysiological studies. Monkeys and corvids are particularly interesting with respect to their distinguished cognitive capabilities based on independently and distinctly evolved endbrain structures. In primates, the circuitry in the six-layered neocortex endow them with high-level cognition. Interestingly, this six-layered neocortex is found to be lacking in non-mammalian vertebrates, such as birds, given that the last common ancestor of mammals and birds lived 300 Mio. years ago. Still, corvid intelligence rivals primates. 

Recently collected behavioural and neuronal data show an impressive correspondence of neuronal mechanisms found in the corvid brain with those described earlier in the primate brain: neurons are tuned to individual preferred numerosities, and neuronal discharges prove to be relevant for both species’ correct performance. Both the neuronal and the behavioural tuning functions are best described on a logarithmic number line, arguing for a non-linearly compressed coding of numerical information, just as predicted by the psychophysical Weber-Fecher Law.

Our comparative approach suggest that this way of coding numerical information has evolved at least twice during the course of evolution, irrespective of the precise origin and anatomical structures found in intelligent vertebrate brains, based on convergent evolution. This suggests that the realised neuroethological representations constitute a superior solution to a common computational problem. 

Learning to count and to find answers to addition and subtraction problems begins in infancy. Verbal and symbol re-representations build on the preverbal system for representing and reasoning about quantity. These develop without formal schooling in most cultures, especially ones with generative rules for ordered count terms that represent sequentially repeated additions. Reliance on this Natural Number system is a major and continuing obstacle to mastering more advanced externalised symbols for reasoning about quantity, that is, to learning ‘mathematics’. A startling example is the mistake that a large fraction of college students make in reasoning about oppositely directed percent changes. Although examples of these changes are omnipresent in modern culture, many students reason about them incorrectly. They process them as simple additions and subtractions, believing therefore that an X% increase is reversed by an X% decrease. Success on some problems but not many, favours a ‘skill before principle’ account of learning ‘mathematics’.

Establishing how numeracy has evolved in our lineage is a largely unexplored field of study. This is mainly due to the fact that archaeologists have not designed specific heuristic tools to infer numeric knowledge from past material culture. However, based on a cursory survey of the archaeological record, it can be reasonably argued that constitutive elements of number sense such as approximate representation of numerical magnitude (approximate number system), and precise representation of the quantity of few individual items (parallel individuation system), must have been already largely mastered by early hominins. Nevertheless, a threefold challenge remains, i.e., when did verbal or gestural counting systems, shared at present by all human cultures, arise in the history of humankind, when were exosomatic devices (Artifical Memory Systems, AMS), conceived and created to store, process and/or transmit numeric information, and how did they evolve to reach the symbolic systems of graphic marks that most of us uses for recording numbers. Upper Palaeolithic (40,000-10,000 years BP) archaeological sites from Europe have yielded numerous objects carrying sets of marks, incised with a variety of techniques, and interpreted as systems of notation recording numerical information. This interpretation has been proposed anew on novel theoretical and analytical grounds. On the one hand, a survey of AMSs presently in use in different human cultures individuates four distinct factors to code information: (i) the number of marks, (ii) the accumulation of marks over an extended periods of time, (iii) the spatial distribution and arrangement of the marks, and (iv) their morphology (or their different morphologies). On the other hand, it has been shown that a technological analysis of the marks, based on criteria established experimentally, is needed to interpret them as AMS, and to discard alternative functions. Recently, research on Late Upper Palaeolithic objects bearing sequential notches was conducted to test the hypothesis that these alignments comply to the Weber-Fechner law. Such compliance would entail that Upper Palaeolitic modern humans were able to distinguish between differences in magnitude of a particular stimulus, such as the difference in width between sequential notches, and the intensity of its perception. We present results obtained from the application of these two approaches to both published and unpublished objects bearing sequential markings uncovered from Lower Palaeolithic, Middle Palaeolithic, and Middle Stone Age contexts. The sample includes a notched baboon fibula from Border Cave, known as the Lebombo bone, which was interpreted as one of the earliest known evidence of numerical notation. This leads us to propose hypotheses about the mechanisms that have stimulated the transition from numerical somatosensory systems to exosomatic counting systems in some human cultures.

English speakers use the counting word ‘two’, the French say deux, Greeks say δύο (duo), Russians say два (dva), Sanskrit speakers say dve, and Caesar would have said duo. English, French, Greek, Russian, Sanskrit and Latin are all members of the Indo-European family of languages. The various forms of the word two that all these languages use are cognate – that is, they descend from a common shared form reconstructed by linguists as duoh – spoken at the time of the origin of the Indo-European languages perhaps 6000 to 8000 years ago. By comparison, each of these languages uses a different and non-cognate form of the word bird. Professor Pagel presents evidence to show that the remarkable conservation in the Indo-European languages of a stable pairing lasting thousands of years between an otherwise arbitrary sound (two, duo, deux, and etc) and the meaning two is repeated in other language families and is typical of the deep histories of the simple counting words (taken here as the words representing ‘one’ to ‘five’). He reflects on several reasons for the conservation of counting words, including speculation about numerical abilities having a hard-wired neurological basis.

We cannot see, hear, touch, taste or smell (finite cardinal) numbers; they do not emit or reflect signals; they leave no traces from which their existence and nature can be inferred. So how can we have cognitive access to them? A rapid review of the major views about the nature of numbers shows that each view faces the problem of cognitive access or is unsatisfactory for some other reason. Professor Giaquinto will try to show how we might unblock the impasse by questioning a common tacit assumption. Then he will try to indicate how, given the view he favours, the problem of cognitive access can be solved, drawing on empirical findings from the cognitive sciences.

Evidence presented at this conference suggests that many species possess a mechanism for extracting numerosity from the environment. If we humans inherit a similar mechanism, this could be the foundation for our numerical competencies, rather than, as has been often claimed, reasoning, linguistic or spatial skills. Professor Butterworth will show that indeed humans possess an innate core capacity for numerosity processing. He can also show that individual differences in this capacity contribute to explaining differences in arithmetical attainment and correlated neural structures. On the basis of this, he will then go on to suggest ways in which the education of individuals should be shaped.