Skip to content
Events

The origins of numerical abilities

Discussion meeting

Location

The Royal Society, London, 6-9 Carlton House Terrace, London, SW1Y 5AG

Overview

Scientific discussion meeting organised by Professor Brian Butterworth FBA, Professor Charles Gallistel and Professor Giorgio Vallortigara.

Chickens show numerical abilities

Humans possess numerical abilities and have invented cultural tools to use them efficiently; but many other creatures, from insects to primates, have in the past decade been found to possess numerical abilities, possibly descended from a common ancestral form. Therefore, when we acquire numerical competence, for example in school, we build upon a specific inherited cognitive foundation.

Speaker abstracts and biographies are available below. Recorded audio of the presentations are also available below. Meeting papers will be published in a future version of Philosophical Transactions B.

This meeting was followed by a related satellite meeting 'The origins of numerical abilities: the future' at the Royal Society at Chicheley Hall on 22-23 February 2017.

Enquiries: Contact the Scientific Programmes team

Event organisers

Select an organiser for more information

Schedule of talks

20 February

Session 1 09:00-16:45

The adaptive value of numerical abilities

8 talks Show detail Hide detail

Chairs

Professor Brian Butterworth FBA, University College London, UK

Professor Sarah Benson-Amram, University of Wyoming, USA

09:05-09:50 How does the brain code quantity?

Professor Charles Gallistel, Rutgers University, USA

Abstract

The representation of numerosity must be embedded in a system for representing both discrete and continuous quantities. The most basic question then is the coding question: how does the brain encode a quantity? Neuroscience has nothing to offer in the way of an answer. Computer science does. The code is probably not unary, which rules out analog codes, like rate codes, because it must support the implementation of addition and multiplication over many orders of magnitude. Also analog codes have reading noise; the build up of purely computational error makes dead reckoning impossible. The representation must be able to approximate a huge range of the computable numbers (negative, positive, |n|>> 1 and  |n|<<1). Twos-complement fixed point with settable offset (bias) and slope (scale) and a 2-bit to 8-bit integer portion has much to recommend it: 1) it obeys Weber’s Law. 2) It is the most computationally efficient of the known codes (least amount of hardware and least energy use). 3) It is particularly advised for Archimedean computations like dead reckoning, where a great many additions of small quantities may produce a quantity orders of magnitude larger. 4) It converts subtraction to addition. 5) The bit flipping that is the key to this conversion has a natural chemical realisation. 6) Division and multiplication by 2 are particularly simple and known to be psychologically easy and fast. 7) It can produce an exact code for large integers. 8) It solves the exact equality problem.

Show speakers

09:50-10:35 Numerical assessment in the wild: insights from social carnivores and other mammals

Professor Sarah Benson-Amram, University of Wyoming, USA

Abstract

Playback experiments have proven to be a useful tool to investigate the extent to which wild animals understand numerical concepts and the factors that play into their decisions to respond to different numbers of vocalising conspecifics. Professor Benson-Amram will review a series of playback experiments conducted with wild social carnivores and other mammals, including African lions, spotted hyenas, and chimpanzees, which demonstrate that these animals can assess the number of conspecifics calling and respond based on numerical advantage. Additionally, she will discuss the key role that individual discrimination and cross-modal recognition can play in the ability of animals to assess the number of conspecifics vocalising nearby. For example, a listener hearing three vocalisations would benefit from being able to assess whether the vocalisations were emitted by the same individual or three different ones, and whether the identity of the callers match individuals they have recently seen in the area. Because the costs and benefits associated with approaching conspecifics change depending on the callers’ age, sex, and relatedness, listeners will likely adjust the level of their behavioural response to playback experiments where the identity of the callers change even when the number of callers is held constant. The listener’s sex, age, social rank and social system will also help determine their behavioural response to varying number of competitors. Finally, Professor Benson-Amram will discuss the implications of these findings for understanding how carnivores and other animals may have evolved a concept of ‘one’.

Show speakers

10:35-11:00 Coffee

11:00-11:45 At the roots of numerical cognition: insights from the day-old domestic chick (Gallus gallus)

Dr Rosa Rugani, University of Padova, Italy

Abstract

The ability to represent number and to use numerical concepts, such as real numbers, logarithms, and square roots, is a prerogative of a subset of human beings who have received specific mathematical instruction. In the last few decades, however, it has been demonstrated that non-verbal numerical abilities (i.e., those calculations that can be solved in the absence of words) are widespread within the animal Kingdom. To investigate the ontogenetic origins of numerical knowledge Dr Rugani used the domestic chick (Gallus gallus) as animal model. Unlike previous studies on adult animals, this model can be tested very early in life, allowing a precise control of sensory experience.

Dr Rugani will discuss evidence revealing that day-old domestic chicks can: (i) discriminate between different numbers of artificial social companions (i.e., objects they were exposed to soon after hatching); (ii) solve rudimentary arithmetic calculations, such as 1+1+1 vs. 1+1; and (iii) use ordinal information, identifying a target element, (e.g., the 4th), in a series of identical elements, on the basis of its numerical position in the series. 

These studies suggest that non-verbal numerical comprehension can be observed in animals in the absence of (or with very reduced) experience, indicating that numerical competences may not have emerged ‘de novo’ in our species together with language, but that they could be based on an evolutionary-ancient precursor system.


Show speakers

11:45-12:30 The primacy of numerical information

Professor Elizabeth Brannon, University of Pennsylvania, USA

Abstract

The ability to use numbers is one of the most complex cognitive abilities that humans possess and is often held up as a defining feature of the human mind. Alongside the uniquely human symbolic system for representing number we possess an approximate number system (ANS) that is evolutionarily ancient and developmentally conservative. In this talk Professor Brannon will illustrate the signatures of the ANS with experimental data from human babies and nonhuman primates. She will describe behavioural and neurobiological data that demonstrates how the human and nonhuman primate mind privileges numerical information over other types of quantitative information. She will argue that this numerical privilege implicates the biological importance of number in our evolutionary history.

Show speakers

12:30-13:30 Lunch

13:30-14:15 Numerical abilities in fish

Dr Christian Agrillo, University of Padova, Italy

Abstract

While there is a well established tradition of studying numerical abilities in mammals and birds, it is only in the last decade that some studies have proposed that teleost fish possess similar capacities. There is substantial evidence showing that fish integrate numerical information and continuous quantities (such as cumulative surface area or convex hull) when assessing which group of fish or objects is larger/smaller. Typically, their performance is more accurate when both pieces of information are simultaneously available, although different fish species were also shown to use pure numerical information when prevented from using continuous quantities. The ability to discriminate small numbers of social companions seems to be already displayed at birth while large number discrimination develops later as a consequence of maturation and experience. The similarities among species (i.e., Gambusia holbrooki, Poecilia reticulata, Danio rerio, Pterophyllum scalare and Xenotoca eiseni) appear greater than the differences, and in general, the numerical capacities of fish partially match those reported in mammals and birds, raising the intriguing idea that our non-symbolic numerical abilities are more ancient than previously thought and date back at least as far as the divergence between fish and land vertebrates. Dr Agrillo will summarise the current state of art in the literature, focusing on three main topics: the relation between discrete (numerical) and continuous quantities, the ontogeny of numerical abilities in fish and the comparison of numerical abilities of fish and other vertebrates.

Show speakers

14:15-15:00 Neural correlates of the numerical abilities of anurans: neurons that count

Professor Gary Rose, University of Utah, USA

Abstract

Acoustic communication plays important roles in the reproductive behaviour of anurans (frogs and toads). The acoustic repertoire of most species consists of several call types, but some anurans gradually increase the complexity of their calls during aggressive interactions between males and when approached by females. Observations of natural behaviour, as well as experimental studies, have revealed the numerical abilities of anurans in their acoustic communication. In particular, anurans are able to discern the number of properly timed pulses (notes) in their calls. The temporal intervals between successive pulses provide information about species identity and call type. A neural correlate of this numerical ability is evident in the responses of neurons that show ‘tuning’ for mid to fast pulse rates. These ‘interval-counting’ neurons respond only after at least a threshold number of pulses have occurred with the correct timing. A single interpulse interval that is 2-3 times the optimal value can reset this interval-counting process. Whole-cell recordings of the membrane potentials of midbrain neurons, in vivo, have revealed that complex interplay between activity-dependent excitation and inhibition contributes to this counting process. Single pulses primarily elicit inhibition. As additional pulses are presented with optimal intervals, cells are progressively depolarised and spike after a threshold number of intervals have occurred. Similarly, pulses that are presented at long intervals (slow rates) elicit primarily inhibition. As interpulse intervals are shortened, however, depolarisation progressively increases. The mechanisms that underlie this apparent shift in the balance of excitation and inhibition during a pulse sequence are under investigation.

Show speakers

15:00-15:45 Counting insects

Professor Lars Chittka, Queen Mary University of London, UK

Abstract

When counting-like abilities were first described in the honeybee in the mid 1990s, many scholars were sceptical, but such capacities have since been confirmed in a number of paradigms and also in other insect species, though curiously not in a solitary bee species in a natural foraging task. Counter to the intuitive notion that counting is a cognitively advanced ability, neural network analyses indicate that they can be mediated by very small neural circuits, and we should therefore perhaps not be surprised that insects and other small brained animals such as some small fish exhibit such abilities. One outstanding question is how bees actually acquire numerical information. Recent work on the question of whether bees can see ‘at a glance’ indicates that bees must acquire spatial detail by sequential scanning rather than parallel processing. This is confirmed for a numerosity task in the bumblebee.

Show speakers

15:45-16:00 Tea

16:00-16:45 Comparative cognition of space and number: the case of the mental number line

Professor Giorgio Vallortigara, University of Trento, Italy

Abstract

Evidence will be discussed about encoding of geometry and forming associations between space and numbers in non-human animals. A variety of vertebrate species are able to reorient in a rectangular environment in accord with its metrical and sense relations, i.e. using simple Euclidian geometry. There seems to be a primacy of geometric over non-geometric information and, possibly, innate encoding of the sense of direction. Moreover, the hippocampal formation plays a key role in geometry navigation in mammals, birds and fish. Although some invertebrate species show similar behaviours, it is unclear whether the underlying mechanisms are shared. A disposition to associate numerical magnitudes onto a left-to-right-oriented mental number line appears to exist independently of cultural factors, and can be observed in animals with very little numerical experience, such as three-day old chicks. This evidence supports a nativistic foundation of such orientation. Preliminary evidence suggests that the same is observed in human neonates and in zebrafish.

Show speakers

17:00-18:15

Poster session

21 February

Session 2 09:00-12:15

The numerical abilities of primates

4 talks Show detail Hide detail

Chairs

Professor Giorgio Vallortigara, University of Trento, Italy

09:00-09:45 Numerical abilities of chimpanzees

Professor Tetsuro Matsuzawa, Kyoto University, Japan

Abstract

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.

Show speakers

09:45-10:30 Brain-imaging studies of the relationship between language and mathematics

Professor Stanislas Dehaene, Collège de France, France

Abstract

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.

Show speakers

10:30-10:45 Coffee

10:45-11:30 Number representations in the convergently evolved brains of primates and corvid birds

Dr Andreas Nieder, Tübingen University, Germany

Abstract

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. 

Show speakers

11:30-12:15 Principle before Skill or Skill before Principle? Both

Professor Rochel Gelman, Rutgers University, USA

Abstract

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’.

Show speakers

12:15-13:10 Lunch

Session 3 13:15-17:00

Numerical abilities, symbolism, education and mathematics

5 talks Show detail Hide detail

Chairs

Professor Elizabeth Brannon, University of Pennsylvania, USA

13:10-13:55 From number sense to systems of numerical notation

Professor Francesco d'Errico, Bordeaux University, France

Abstract

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.

Show speakers

13:55-14:40 The deep history of counting words

Professor Mark Pagel FRS, University of Reading, UK

Abstract

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.

Show speakers

14:40-15:25 Cognitive access to numbers: the philosophical significance of empirical findings about basic number abilities

Professor Marcus Giaquinto, University College London, UK

Abstract

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.

Show speakers

15:25-15:45 Tea

15:45-16:20 The implications of an innate numerosity-processing mechanism for education

Professor Brian Butterworth FBA, University College London, UK

Abstract

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.

Show speakers

16:20-17:00 Discussion, chaired by Professor Charles Gallistel

Professor Cédric Villani, Institut Henri Poincaré and Université Lyon I, France
Professor Marcus du Sautoy OBE FRS, University of Oxford, UK
Professor Charles Gallistel, Rutgers University, USA

Show speakers

Event downloads

Related events

The origins of numerical abilities The Royal Society, London 6-9 Carlton House Terrace London SW1Y 5AG UK