A Brain for Numbers: The Biology of the Number Instinct, by Andreas Nieder
Introduction
Life without numbers is inconceivable for us. How else would we count objects, tell time, calculate prices, and so on? Our scientifically and technically advanced culture simply would not exist without numbers. Numbers are fascinating because they constitute very abstract units of thoughts. When assessing the number of items in a set, the sensory appearance of the elements is meaningless. Three fingers, three calls, and three hand movements can all be classified by the cardinal number “three.” Numbers are also intriguing to examine because, once extracted from sensory input, we use them to calculate and transform numerical information according to abstract principles. This is what arithmetic and mathematics are all about. Numerical operations therefore provide a “window of opportunity” to study the biology of numerical cognition. Ultimately, it gives rise to our number theory, a full-blown symbol system that parallels the language faculty in humans.
Comprehending and processing numbers is a form of behavior, and questions of behavior—how and why it emerges—can be addressed from different biological angles and at different levels. This was most vividly recognized by Nobel laureate Nikolaas Tinbergen, who formulated the four fundamental different types of problems in biology in his influential article, “On aims and methods of ethology,” from 1963: the problems of “causation,” “survival value,” “ontogeny,” and “evolution.”
With respect to number processing, these problems can be expressed as four questions: How does the brain give rise to our understanding of numbers and arithmetic? What is the utility to possessing number skills? How does numerical competence develop during the lifetime of an individual? And, finally, how did numerical competence evolve over the history of life on Earth? My aim in writing this book was to elucidate and attempt to answer all four of these questions from a biologist’s point of view. However, the emphasis of this book is on how the brain and its neurons process numbers. This field of research has made significant progress over the past two decades and has yielded insights that gave rise to a new understanding of the evolution and the neural foundation underlying our grasp on numbers.
Naturally, each of the different disciplines contributing to our understanding of numbers has different research methods at its disposal and provides scientific knowledge at different, yet complementary, levels of explanation. In order for the reader to be able to appreciate the scope and limits of these various methods, as well as the different levels of explanations, these different methods will be briefly introduced along the way.
At the most microscopic scale, this book will show how studies with twins have begun to decipher the genetic basis of mathematical functions and disorders. To understand the workings of the brain, the book explores the neuronal code for numbers by recording the electrical activity of single neurons. These recordings, mainly in experimental animals, are complemented by brain imaging scans of human brains using positron emission tomography (PET) and magnetic resonance imaging (MRI). Much is to be learned about the causal role of neurons and brain areas when they are inactivated. While chemical inactivation and transcranial magnetic stimulation (TMS) transiently shut down neurons to influence calculation behavior, neuropsychological research studying patients with brain injuries historically also provided rich insights into the location and workings of number-processing brain areas.
An arsenal of behavioral methods is available to investigate how the minds of adults, infants, and animals represent numbers. These can broadly be classifiesd into methods exploring spontaneous behavior and those exploring behavior in trained individuals. Developmental psychologists mainly rely on explorations of the spontaneous behaviors in newborns and infants when they discover what the human mind is capable of without number experience. Exploitations of spontaneous behaviors are also important when learning about the ecological relevance of numbers for animals in the wild. Training experiments, on the other hand, inves- tigate the behaviors of animals under controlled laboratory conditions to show the full range of numerical skills. As will be shown, working with trained animals offers the advantage of combining behavioral research with brain research. Comparing behavioral findings across the animal kingdom reveals how numerical competence emerged during the course of biological evolution. It turns out that processing numbers offers a significant benefit for survival, which is why this behavioral trait is present in many animal populations in the first place.
One of the key findings over the past decades is that our number faculty is deeply rooted in our biological ancestry, and not based on our ability to use language. This number instinct in its non-symbolic humble beginnings can therefore be traced through evolution and development. Numerical competence does not emerge de novo in humans, but builds on ancient biological precursors. Therefore, understanding the biological mechanisms of this evolutionarily and developmentally fundamental number instinct is instrumental to understanding our symbolic mathematical capabilities. Any scientific theory of number competence requires a mechanistic and functional, and therefore biological, explanation. The quest for this explanation is exactly the topic of this book.
Chapter 1
Thinking about Numbers
1.1 Mathematical Reality
Where do numbers come from? What sounds like a simple enough question is a fundamental controversy in the philosophy of mathematics, one that was first pondered by the Ancient Greeks. Since then, this question more than ever remains unresolved. There is overwhelming scholarly work devoted to the topic, but this is not the place to touch upon its philosophical underpinnings and challenges. The overall “big picture” of this question will, suffice to say, be just enough to cover the conceptual basis for this book.
Numbers have something to do with mathematics; that much is obvious to anyone. For the purpose of this book, we will be using Phillip J. Davis’s and his co-authors’ definition of mathematics (a definition that they call “naïve, but adequate for the dictionary and for an initial understanding”):
Mathematics is the science of quantity and space, plus the symbolism relating to quantity and to space.
The sciences of quantity and of space are generally known as arithmetic (from the Greek arithmos, “number”) and geometry. They continue:
Arithmetic, as taught in grade school, is concerned with numbers of various sorts, and the rules for operations with numbers—addition, subtraction, and so forth. And it deals with situations in daily life where these operations are used.
Easily said, arithmetic, a major branch of mathematics, involves numbers. In this vein, this book is about how the brain deals with arithmetic, or simply numerical quantities and the operations involving them.
But what kind of entities are numbers? To start, we must ask ourselves, do we discover numbers as objective realities, or do we invent them as products of our mind? This simple question has deeply troubled mathematicians to the core. As Mario Livio, author of Is God a Mathematician? expresses:
If you think that understanding whether mathematics was invented or discovered is not that important, consider how loaded the difference between “invented” and “discovered” becomes in the question: Was God invented or discovered? Or even more provocatively: Did God create humans in his own image, or did humans invent God in their own image?
Livio’s comparison concerning the relationship between God and humans illustrates how charged this question is, and how entire believe systems depend on how it is answered. Mathematicians are torn on how to respond to this question and, subsequently, adopt two radically different philosophical positions: The first position, termed mathematical platonism or mathematical realism, is the view that abstract mathematical objects, including numbers and sets, exist independently of us and our thoughts. Numbers have objective properties. We therefore discover numbers, just as we discovered the law of gravity and other laws of physics. The British mathematician Godfrey Harold Hardy (1877–1947), a proponent of mathematical realism, writes:
I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe gran- diloquently as our “creations,” are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.
The second position, called non-platonism or anti-realism, comprises all opponents of mathematical platonism that include formalism, fictionalism, and logicism. Each is united by the overall idea that numbers and other mathematical objects don’t exist as real entities and are independent from our mind. According to this position, we therefore invent numbers, just as we invent arbitrary games. American mathematicians Edward Kasner (1878–1955) and James Roy Newman (1907–1966) express this position when they write:
Mathematics is man’s own handiwork, subject only to the limitations imposed by the laws of thought. ... We have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed.
Despite the metaphysical significance of the question and the heated debates among mathematical figureheads, a solution to this problem is not in sight. Even individual mathematicians have a hard time deciding which side they are on. They are doomed to a somewhat contradictory attitude, which is revealed in this excerpt from Davis and co-authors:
The typical working mathematician is a Platonist on weekdays and a formalist on Sundays. That is, when he is doing mathematics he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.
One would most certainly not expect such a vague stance from mathematicians devoting their professional lives to exact sciences. But be that as it may, one thing is certain: Our survival depends on whether we successfully interact with our environment, which in turn is determined by how well we perceive the outside world, including the number of objects and everything around us. If we ignore the physical reality with numbers out there, we will definitely pay for it. Just as our ancestors might have fallen prey to the notorious saber-tooth tiger, today we just as well might get hit by a car or experience some other terrible accident. Surely, there is some sort of physical reality out there that we shouldn’t ignore. And the organ that is responsible for preventing such accidents and ensuring our species’ survival is the brain. The brain, in turn, is connected to sensory organs that provide us with perceptions as input, as well as bones and muscles that allow us to respond appropriately as output.
This book tries to portray the fascinating feats of a working brain that has had to deal with an objective reality during the course of hundreds of millions of years of biological evolution. Even if much of pure mathematics may just be a “meaningless game”, my fundamental conviction and the basis of this book are that number is a property of real objects and events. By representing numerical quantity and arithmetic operations, the brain is gathering and processing information about the outside world to help its carrier survive in a hostile and competitive world.
1.2 Cardinal Numbers as Objective Properties of a Set
Numbers come in many varieties: natural numbers, rational numbers, real numbers, complex numbers, and others. This book is dedicated only to natural numbers. Just as Platonists and their opponents cannot settle the question of mathematical reality, philosophers of mathematics also can’t agree on what numbers actually are.
A number may be conceived as “multitudes of units”, “nothing but names” or numerals, a mental entity or projection, and as several other ideas. All of these interpretations have their respective advantages and problems.
In this book, however, I am adopting yet another position, namely the set-size view of numbers. It posits that a cardinal number is a real and objective property of a set. This view offers the most consistency with a biological conception of numbers. However, this view hardly originates with me. In fact, the English philosopher and empiricist John Locke (1632–1704) included numbers in his list of real (or primary) properties in his influential book An Essay Concerning Human Understanding (II.VIII.17) dating back to the year 1690. As such, a cardinal number is independent of any observer, in contrast to subjective properties or sensations such as color or pain. For instance, the cardinal number two of protons and neutrons in the nucleus of elemental helium is a real property of its set; it determines what the element helium is, irrespective of our thinking about helium. Similarly, spiders and insects can be distinguished by the former having eight legs, whereas the latter possess six legs. And we know about cardinal numbers because of their instances—the number of siblings in the family, the number of syllables in a song—that we perceive. And from these instances, we infer an abstract category: “number of items”.
This view of cardinal number as a real and objective property of a set presupposes that there is an external world that is independent of us (a philosophical position called ontological realism), and that the cardinal number is part of it. I, for one, consider this to be true. The physical world we inhabit is an objective reality. The universe, according to current knowledge, is ca. 13 billion years old; the planet earth only ca. 5 billion years old. In contrast, our species, Homo sapiens, has only existed on this planet since about 300,000 years ago. The physical nature we interact with has therefore existed long before man, and it will continue long after man has disappeared. If we consider the temporal and spatial dimensions of cosmic history relative to our own evolutionary history, it would be absurd to deny the existence and properties of physical nature as independent of human existence and experience.
Physical facts are objective and support realism in relation to an external world. It is the same with numbers: if a set contains three items, this set contains three items irrespective of whether I watch it or whether I judge it to contain two items. My subjective experience of set size may be erroneous or illusory, but the set size exists objectively.
The notion of cardinal numbers as an objective and perceptually accessible property is eloquently expressed by the Austrian-born mathematician and philosopher Kurt Gödel (1906–1978), one of the greatest mathematicians of the twentieth century and probably the most important logician since Aristotle. Gödel shook the world of logic when in 1931 he published his two incompleteness theorems (Unvollständigkeitssätze), which demonstrate the inherent limitations of formal axiomatic systems such as arithmetic. His intellectual skills deeply impressed another titan of mathematics, Albert Einstein. The two geniuses became close friends during their time at the Institute for Advanced Sciences at Princeton University after World War II. One anecdote states that the elderly Einstein told people he went to work “just to have the privilege of walking home with Kurt Gödel.” The following excerpt exhibits Gödel’s ideas on humankind’s interaction with mathematical objects:
But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perceptions will agree with them. ...
One can distill three important ideas from Gödel. First, he assumes that sets are real objects, thus expressing mathematical realism. Second, he assumes that we have an intuitive understanding of mathematical objects, a flair for mathematics, just as we have an intuition for the nature of sensory objects. Such a number instinct was also recognized by mathematician Tobias Dantzig (1884–1956) almost two decades earlier. In his book Number: The Language of Science, first published in 1930, he referred to this instinct as a “number sense”. According to Dantzig, this number sense, which will play a dominant role in this book, is already present in some “brute species.” Dantzig emphasized that this number sense should not be confused with counting (“an attribute exclusively human”), but is an evolutionary precursor for counting. Right at the beginning of his book, he explains:
Man, even in the lower stages of development, possesses a faculty which, for want of a better name, I shall call Number Sense. This faculty permits him to recognize that something has changed in a small collection when, without his direct knowl- edge, an object has been removed from or added to the collection.
Today, the idea of an intuitive ability to assess numerical quantity is inextricably associated with the work of Stanislas Dehaene, a French mathematician and neuroscientist. Dehaene, whose studies will be mentioned throughout this book, is professor of experimental cognitive psychology at the Collège de France and director of the Neurospin facility near Paris. In his book from 1997 The Number Sense: How the Mind Creates Mathematics, Dehaene substantiated and popularized the idea of a number sense with an unprecedented wealth of empirical evidence both from his own work and the newly emerging field of mathematical cognition. In the introduction to his book, Dehaene posits:
This ‘number sense’ provides animals and humans alike with a direct intuition of what numbers mean.
Similar to our perception of time and space, he argues, humans also perceive numbers as a spontaneously accessible feature of the world. We don’t have to learn to estimate quantity; we come born with a fundamental understanding, an innate instinct, of what numerical quantity is. This innate intuition of numbers is one of several systems of “core knowledge” human cognition is built on. Human numerical cognition is built on this innate intuition both in ontogeny—that is, during the development of an individual—and in phylogeny, the evolutionary history of humans.
The third profound insight in Gödel’s statement is his reference to perception. He calls mathematical intuition “a kind of perception.” Nowadays, the study of perception is indeed firmly anchored in experimental neuroscience; much of what we know about behavior and the brain stems from an analysis of perception. With this reference, Gödel therefore predicts that the experimental sciences will have dominant authority regarding the nature of numbers. This idea reverberates throughout the chapters of this book, in which insights from evolutionary biology, neurobiology, and psychology are woven together to elucidate the biology of numbers.
1.3 Knowledge of Numbers
Earlier in the book, I claimed that the natural world and cardinal numbers are independent of us, thereby confessing ontological realism. But how can we even know about the world and numbers? We can only know that the world and numbers exist if we have the capacity to experience objective facts. A philosophical position that postulates such a capacity to experience objective facts is called epistemological realism, and I admit to this idea.
How can we be certain to know objective facts? This question leads us to the philosophical discipline of “theory of knowledge,” also termed “epistemology.” Its fundamental question is “What can I know?” While epistemology was done from a philosopher’s armchair in ancient times, today any noteworthy epistemological framework has to reflect on what science tells us about our relation to the world. A theory of knowledge that emphasizes the role of natural scientific methods is called “naturalized epistemology,” a term coined by American philosopher Willard V. O. Quine (1908–2000). For Quine, similar to Descartes, the sciences first involved in epistemology were psychology and the physiology of perception.
In the following years, the Darwinian theory of evolution became another cornerstone of naturalistic epistemologies. This resulted in the rise of “biological” or “evolutionary epistemology.” Evolutionary epistemology is the attempt to address questions of “how can I know” from an evolutionary point of view. Philosopher Paul Thomson at John Carroll University even claimed that “Darwin’s theory is potentially much more important for epistemology than psychology, information theory, or cognitive science.” The main theses of evolutionary epistemology are summarized by German philosopher Gerhard Vollmer:
Thinking and knowing are capabilities of the human brain, and this brain arose during biological evolution. Our cognitive structures fit (at least partly) to the world because—phylogenetically—they emerged through adapting to the real world and because—ontogenetically—they have to cope with the environment of each individual.
The biologist George Gaylord Simpson (1902–1984) used the following, more radical and provocative statement to summarize this stance:
The monkey that had no realistic perception of the branch he was jumping for was soon a dead monkey—and did not belong to our ancestors.
In other words, we owe our superb three-dimensional vision to our arboreal ancestors for whom this constituted a survival advantage. Since then, we are not only able to exploit three-dimensional vision for seeing branches, but also for monitoring manipulations of objects with our dexterous hands. In the same way, we owe our symbolic mathematical abilities to the non-symbolic number instinct that provided a survival advantage for our primate ancestors. Applying Simpson’s logic to the concept of knowing numbers, it is clear that the monkey that had no realistic grasp of the number of food items was soon a starving monkey—and did not belong to our ancestors.
Being able to assess the number of objects in a set was always a survival benefit for animals, and I will provide evidence for this claim in a later chapter. Animals with a realistic assessment of set size had a better chance to survive and were therefore selected for in a competitive environment, which in turn led to our ability to assess quantities in the course of evolution.
The same principle continues in our lineage. We are certain that 1 + 1 = 2 because those previous hominids who believed in 1 + 1 = 2, rather than 1 + 1 = 1, survived and reproduced, and those who did not, were not evolutionarily successful. A hominid ancestor who had a ready grasp of elementary mathematics would be better suited for life’s struggles than one who did not. Consider, for example, two prehistoric men, one of whom assesses number and the other who does not. They both sneak through a dense forest outside of their territory. Suddenly, they hear the war cries of several men. One prehistoric man exclaims, “Ah, it looks as though the inhabitants of this forest are here, but it is unclear how many.” The other says nothing, but rapidly makes off. Which one of these two prehistoric men was more likely to be our ancestor?
Adaptations to the sense organs and the brain occur in the natural world, and in response to it. If the theory of evolution is correct (which is undisputed among biological scientists based on a plethora of evidence) and we possess innate and heritable “organs of knowledge,” then these organs of knowledge are subject to the same major driving forces—genetic variation and natural selection—as all other organs and their resulting faculties. Like an instinct, the principles of mathematics are reflections of the innate dispositions that are wired into the brain of every mature, healthy human being. This is why understanding the evolution and the physiology of the brain is the key to unlocking the secrets of arithmetic and mathematics.
Chapter 2
Numerical Concepts, Representations, and Systems
… Cognitive scientists discuss two separate mental systems that may allow animals, infants, and adults to represent number without symbols: the object tracking system and the approximate number system. The evidence for both of these mental systems will be discussed in more detail [later] in the book. The object tracking system (OTS), also known as object file system, allows grasping only small set sizes, from one to about four, in an unconscious but relatively precise way. The workings of the OTS have been deduced from the behavioral characteristics that emerge when pre-verbal infants are confronted with small sets of objects. The idea is that the visual system singles out (or individuates) objects that it wishes to track by assigning “pointers” to them, or by storing objects in “files”. Each file can store precisely one object, and all files can store their items in parallel, or simultaneously, which is why the reaction time during discrimination from one to four items hardly increases. In addition, this process works without attending to individual objects; the objects are stored automatically and unconsciously. However, because it is thought that the visual system only contains three or four of such files, a maximum of only four objects can be stored. This is referred to as the “set-size limit” of the OTS.
As a result of this, larger numbers cannot be tracked.
Importantly, however, the OTS is not dedicated to number representations; numbers are only implicitly, or unconsciously, represented because of filled-up files.
The OTS is thought to be responsible for a “subitizing” effect, from the Latin word subitus (“suddenly”). This effect describes an effortless, fast, and accurate process to judge a small number of items.
While real counting of larger sets of items takes about 200–350 ms per item, one to four items can be judged with reaction times of only about 40–100 ms per item. This rather mild increase in reaction time per additional item is taken as evidence that the items of a set are assessed more or less at the same time, and with a mechanism that processes each item in a parallel manner. The rather sudden increase in reaction time beyond numerosity four has been interpreted as a switch from simultaneous non-verbal judgments to a serial symbolic counting strategy for higher numbers.
Originally, subitizing has been interpreted as a recognition of sets of dots as reoccurring figural patterns: one is a singleton, two a line, three a triangle, four a square. This idea is supported by evidence that rapid assessment could be extended up to seven or eight items if the patterns were familiar. Moreover, participants rate different patterns of the same numerosity as more similar within the subitizing range (≤4 items) than outside of it (>5 items). However, when all points are arranged collinearly or when objects are not simple dots but complex household items, the subitizing effect is still present. Because of this contradiction, the pattern-recognition account for subitizing has largely fallen out of favor. More recently, subitizing is explained by the OTS mechanism.
The second mental system for non-symbolic number judgments is termed the approximate number system (ANS; formerly known as analog magnitude system). The ANS is a quantity estimation system for an unlimited amount of set sizes.
It represents the number of items consciously. However, as the name already indicates, it can only represent the number of items inexactly, or approximately. The ANS has two important behavioral characteristics, called the “numerical distance effect” and the “numerical size effect”.
Numerical distance effect refers to the finding that numerically distant numbers are easier to discriminate than numerically closer numbers. For instance, if the discrimination between 5 and 6 is very defective, the discrimination between 5 and 7 is more accurate and the discrimination between 5 and 8 is even better.
On the other hand, the numerical size effect says that, at a given numerical distance, it is easier to discriminate numbers with low values than numbers with high values. For example, it is easier to discriminate 2 versus 3 than 8 versus 9, even though the numerical distance is 1 on both cases. In order to reach a similarly good discrimination performance as present in 2 versus 3, the numerical distance would have to increase in proportion with the magnitudes of the numbers, in this example to 8 versus 12. In other words, the ability to discriminate between quantities varies as a function of ratio, and is therefore considered “ratio-dependent.”
It is easy to see that the discrimination of numerical values of sets follows systematic relationships. The German physician Ernst Heinrich Weber (1795–1878) was the first to discover such systematic relationships, not for number judgments, but for the sensation of weights. He realized that we do not perceive the absolute weight difference between two objects, but the ratio of the weight difference. If one object weighs 100 g, the comparison object needs to weigh 110 g for us to just be able to discriminate the weights; the so-called just-noticeable difference is 10 g. However, for an object of 200 g, the comparison object has to weigh 220 g to tell the difference in weight. In this second case, the just-noticeable difference is 20 g.
Weber realized that this systematic relationship could be captured mathematically for all possible weights by a constant: the just-noticeable difference, ΔI, divided by the reference weight, I, is a constant, c. More generally, the Weber fraction reads ΔI/I = c. For the weight example above, the constant is 0.1, because 100 g / 10 g = 0.1 and 200 g / 20 g is also 0.1. This relationship is nowadays called Weber’s law, and it not only applies to weight discrimination or other sensory intensities, but also to numerical discrimination. For example, if an animal is just able to discriminate 5 from 10 in 50% of the cases (ΔI = 5; I = 10), then we can predict it will just be able to discriminate 10 from 20 (ΔI = 10; I = 20). In both cases, the Weber fraction is 0.5 and therefore constant, because 5 / 10 = 0.5, and 10 / 20 = 0.5.
In fact, Weber’s law is a hallmark of the ANS. The lack of a ratio effect is a distinctive signature that allows experimental differentiation of the ANS from the OTS.
This pattern of a constant Weber fraction value across a broad range of numerosities emerges in studies with humans, non-human primates, and other animals, suggesting that the ANS is a basic mechanism available for representing numerosity.
The precise value of the Weber fraction differs between species, and to some extent also between individuals. Small Weber fractions indicate the capacity to discriminate small numerical differences, whereas large Weber fractions suggest inferior number discrimination ability. The Weber fraction therefore is an objective measure for the precision with which numbers can be discriminated from other numbers.
If we continue to increase the number of visual items within a given area, we will reach a point where we can no longer resolve the items, and they will merge into what is commonly termed texture. Texture-density estimation is also discussed by some as a potential third mechanism of numerosity assessment. David Burr, professor of physiological psychology at the University of Florence in Italy, and coworkers suggest that texture-density estimation becomes active if a visual scene becomes too crowded, for example as with a large flock of birds. In this case, the density becomes too great to segment the items from each other, and the scene is perceived as a texture rather than an array of elements. However, it is unclear whether this process deals with the number of items in a set or rather relates to sensory discriminations independent of number.
In contrast to non-symbolic, or analogical, number representations, symbolic number representation is characterized by the reflection of cardinality via different number notation symbols, such as Arabic numerals (“8”) or numerical words (“eight”).
Number symbols are signs that refer to cardinality, and importantly are part of a symbol system that has a compositional syntax and semantics. As part of a symbol system, number symbols are processed and transformed in a logical way based on rules in arithmetic. Because we usually use the language system to express number symbols, symbolic number representations are often called linguistic or verbal number representations. However, since the linguistic and the number systems are not the same, as will be emphasized in this book, I prefer the term symbolic.
Symbolic number representations can be based on different number notations, such as the written Arabic numeral “8” or the numerical name “eight.” At the same time, symbolic number notations can address different sensory modalities, such as seeing a written numeral versus hearing a spoken numerical name. How these sensory modalities and number notations are represented will also be one of the topics of this book.
The transition from non-symbolic to symbolic number representations is one of the big mysteries in cognitive neuroscience.
Even though hotly debated, I will later present several lines of evidence suggesting that our symbolic number representations—both ontogenetically and phylogenetically— actually build on non-symbolic number representations. Comparative psychologists have showed that animals can discriminate numerical information, and developmental psychology experienced a breakthrough when tapping into numerical cognition in human infants of only few months or even days of age. These developments indicated that numerical competence does not emerge de novo in humans, but builds up on a biological precursor system. It is also obvious that our language faculty seems to play a role early in development, as it is conspicuous that children learn to count once they have mastered speaking. I will discuss some of the ideas in depth in Part V, which deals with development specifically.
Whatever the foundations, one thing is clear: once children do arrive at this symbolic stage, number representations are qualitatively transformed and massively enhanced to become the constituents of a full-blown and uniquely human number theory. Symbolic number representations provide us with the most exact quantity representations. Five is precisely five all the time, never four or six. Number symbols become part of a combinatorial symbol system and can be processed and transformed according to arithmetic rules.
Symbolic number representations are the constituents of our science and technology. Number theory can be considered the other symbol system in addition to natural language. Without number theory, we would all still live in hunter-and-gatherer societies.
To sum up, the empirical property of “number of items” is represented in the mind by (non-symbolic or symbolic) mental representation. Evidently, all mental events are caused by the workings of neurons, or in other words observable physical processes in the brain. Therefore, a different type of cardinality representation needs to be addressed, one that tells us how cardinality is reflected at the neuronal level. This is indeed the fascinating “neuronal representation.” Because neuronal representations give rise to the mind, there are of course no mental representations without neuronal representations, but mental and neuronal representations are far from the same. As I will show in the later parts of the book, the neural machinery giving rise to number representations has been deciphered over the past two decades down to the level of single neurons.
Chapter 4
The Utility of Number for Animals
4.1 A Matter of Fitness
Considering the multitude of situations in which we humans use numerical information, life without numbers would be difficult to imagine for us. But what was the benefit of numerical competence for our ancestors, before they became Homo sapiens? In other words, why would animals crunch numbers in the first place? This question seeks to understand the benefit of numerical competence for individual animals and entire populations of species. Only if numerical capabilities are beneficial (or at least not harmful) for the individuals will this faculty be maintained in a population across generations and sometimes conserved over millions of years in large taxonomic groups of animals.
As outlined in the previous section about the theory of evolution, beneficial traits from a biological point of view are those that help their carrier stay alive and reproduce. An individual must survive long enough to pass on its genes directly to the next generation by successfully mating. An animal that meets this demand has high “individual fitness”. But there is a certain degree of sophistication in this process. An individual can also increase its fitness indirectly by helping relatives to reproduce, because relatives share a substantial proportion of the same genes. Taking this opportunity into account results in “inclusive fitness”, the ultimate type of fitness.
How can an animal increase its overall fitness and become a winner in this respect? Well, by possessing genetically inherited traits that ensure its success in this evolutionary race against the ever-changing environment. An animal that masters these challenges well is adapted to its overall environment. Beneficial inherited traits that serve this need are therefore said to be of “adaptive value”.
It is important to remember one aspect: While the genotype determines which capabilities are available, it is the phenotype that is under close and life-long scrutiny. The phenotype is the access point of selection pressures, and it is at this level that beneficial traits have to stand the test to be passed on via genes into the next generation.
To accomplish this goal of staying alive and passing on genes, two classes of behaviors are necessary. First, behaviors are needed for an individual to survive across the lifespan until reproduction is possible in a mature adult, and for some species, to later pamper offspring to make sure they survive long enough. For an individual, this means first and foremost to find food and avoid becoming food, but also to pick one’s way through a cluttered environment and to count on help from friends in daily business. The following section of this chapter demonstrates that numerical cognition serves this purpose. Several studies examining animals in their ecological environments suggest that representing number enhances an animal’s ability to exploit food sources, hunt prey, avoid predation, navigate in its habitat, and persist in social interactions.
In addition, a second set of skills are necessary that directly increase the chances of passing on the right genes during reproducing. Animals always compete for mating partners, and even when such a partner is found, the next challenge is to make sure that their own offspring succeed rather than a competitor’s progeny. In the last section of this chapter [MGH: not included here for lack of space], I will demonstrate that numerical competence plays a major role in this endeavor, from monopolizing a receptive mate to increasing the chances of fertilizing an egg and finally promoting the survival chances of offspring.
4.2 Staying Alive
Quorum sensing. Before numerically competent animals evolved on the planet, single-celled microscopic bacteria—the oldest living organisms on earth—already exploited quantitative information. The way bacteria make a living is through their consumption of nutrients from their environment. Mostly, they grow and divide themselves to multiply. However, in recent years, microbiologists have discovered they also have a social life and are able to sense the presence or absence of other bacteria; in other words, they can sense the number of bacteria.
Take, for example, the marine bacterium called Vibrio fischeri. It has a special property that allows it to produce light through a process called bioluminescence, similar to how fireflies give off light. If these bacteria are in dilute water solutions, in other words when they are alone, they make no light. But when they grow to a certain cell number of bacteria, all of them produce light simultaneously. Therefore, these bacteria can distin- guish when they are alone and when they are together. Somehow they have to communicate cell number, and they do this using a chemical language. They secrete communication molecules, and the concentration of these molecules in the water increases in proportion to the cell number. And when this molecule hits a certain amount, called a quorum, it tells the other bacteria how many neighbors there are, and all bacteria glow. This behavior is called “quorum sensing”: the bacteria vote with signaling molecules, the vote gets counted, and if a certain threshold (the quorum) is reached, every bacterium responds. This behavior is not just an anomaly of Vibrio fischeri; all bacteria use this sort of quorum sensing to communicate their cell number in an indirect way via signaling molecules.
Quorum sensing is not confined to bacteria; animals are using it to get around, too. Japanese ants (Myrmecina nipponica), for example, decide to move their colony to a new location if they sense a quorum. In this form of consensus decision making, ants start to transport their brood together with the entire colony to a new site only if a defined number of ants are present at the destination site. Only then is it safe to move the colony. Quorum threshold increases with colony size—that is, the larger the colony, the more individuals are required at the new site to reach the quorum. With larger colonies, the quorum threshold becomes slightly larger, but not proportionally. Despite some ratio dependency, the discrimination behavior of the ants does not fully obey Weber’s law. Of course, one also has to keep in mind that it is virtually impossible in free-ranging ant colonies to control for non-numerical factors that could supporting the ants’ decision process.
Navigation. Enumerating landmarks can play an important role for animals to find their way in their daily goal to survive. The honeybee, for example, relies on landmarks to measure the distance of a food source to the hive. In an early experiment by Lars Chittka and Karl Geiger mentioned above, bees were trained to collect sugar water at a feeder between the third and fourth tent in a row of four tents. Changing the number of tents and distances between them caused a performance pattern that indicated a compromise between conflicting distance information, namely the absolute distance flown and the number of landmarks passed. Even though the number of landmarks cannot explain entirely how honeybees measure distance, numerosity is nonetheless an important factor.
In an elegant and thoroughly controlled experiment using flying tunnels, Marie Dacke and Mandyam Srinivasan provided clear evidence for bees’ aptitude to sequentially enumerate landmarks. Bees were trained to forage in a 4-meter-long tunnel equipped with five yellow stripes distributed on the tunnel wall as landmarks. Different cohorts of bees were trained to find the feeder at landmark 1, 2, 3, 4, or 5. During training, bees were prevented from relying on distance information by varying the separation between the landmarks and thus the position of the rewarded landmarks. When the bees mastered this task, they were tested in new tunnels without being rewarded. The bees indeed searched in the vicinity of the learned landmark number. In other words, bees trained to the first landmark searched mainly near landmark 1, bees trained to the second landmark hovered close to landmark 2, and so on. Trained bees found the correct landmark even when the spatial layout was changed; that is, when landmarks were closer together or more spaced apart, or even irregularly spaced. Correct performance even remained when the landmarks changed from the original yellow stripes to disks or even overlapping baffles they had to fly through. Bees sequentially enumerate the number of variable landmarks, and in doing so, show signs of abstraction. For honeybees, assessing numbers is vital to finding their way between a nectar source and the hive.
Choosing between food patches. Numerical cognition helps animals develop efficient foraging strategies. The theory of optimal foraging states that animals, when faced with two or more food options, would benefit if they can choose the one that provides the greatest energetic gain, and more food items obviously are more nutritious than just a few. Since food is a natural incentive, it allows for testing even cognitively less advanced vertebrates, such as amphibians, which can hardly be trained on complex numerosity discrimination tasks. Motivated animals are expected to spontaneously approach the larger patch of food items during such tasks. And indeed, amphibians go for more. When showing red-backed salamanders (Plethodon cinereus) in forced-choice situations two transparent test tubes that contain different numbers of flies, they reliably choose the tube containing more flies by moving toward it and touching it with their snouts. They were able to select two flies over one, and three over two, but failed to discriminate three versus four and four versus six. A later study, however, showed that salamanders are also able to discriminate larger numerosities, such as 16 versus eight crickets. All they need is a sufficiently large numerical ratio between the test sets. Similarly, frogs (Bombina orientalis) preferred more mealworm larvae in free-choice experiments and discriminated one versus two, two versus three, three versus six, and four versus eight prey items.
“Going for more” is a good rule of thumb in most cases, but sometimes the opposite strategy is favorable. The field mouse (Apodemus agrarius) loves live ants, but ants are dangerous prey because they bite when threatened. When a field mouse is placed into an arena together with two ant groups of different quantities, it surprisingly “goes for less.” Mice that could choose between five versus 15, five versus 30, and 10 versus 30 ants always preferred the smaller quantity of ants. The field mice seem to pick the smaller ant group in order to ensure comfortable hunting and to avoid getting bitten frequently.
Food items are probably the most popular stimuli to test the spontaneous numerical capabilities of animals. Thus, a variety of animal species, such as robins, crows, coyotes, dogs, elephants, and different species of monkeys and apes, have been shown to differentiate number of food items. As mentioned earlier, food items are particularly hard to control for non- numerical parameters, such as the amount of food items, the overall space many food items will cover, or hedonic value. It can therefore never be excluded that the animals pay attention to such parameters rather than the actual number. Still, many studies suggest that animals in the wild are sensitive to number of food items, and foraging is a particularly important domain in which numerical competence pays off.
Hunting prey. Numerical cues play a role in communal spider-eating spiders, known as araneophagic spiders. One such spider-eating spider is Portia africana in Kenya (hereafter Portia). Small Portia juveniles adopted an especially intricate predatory strategy when preying on Oecobius amboseli (hereafter oecobiid), a small spider species that builds tent-like silk nests on boulders, tree trunks, and the walls of buildings. Portia often practice communal predation, especially when the prey is an oecobiid. In typical sequences, two Portia settle alongside each other at an oecobiid’s nest, and when one Portia captures an oecobiid, it is joined by the other to feed alongside it. Portia base their decision of settling near a prey spider nest on the number of conspecifics already present. They prefer one spider being present over zero, two, or three spiders. The reason why these spiders prefer to hunt in pairs rather than in larger groups may have to do with increasing numbers of freeloaders in large groups. The more members a hunting party has, the more likely it becomes that some members won’t cooperate. Therefore, larger groups are often worse at capturing prey than smaller groups.
The probability that wolves capture elk or bison varies with the group size of a hunting party. This is at least what research with wild wolves at Yellowstone National Park suggests. Wolves often hunt large prey, such as elk and bison, and large prey can kick, gore, and stomp wolves to death. Therefore, there is a lot of incentive to “hold back” and let others go in for the kill, particularly in larger hunting parties. As a consequence, wolves have an optimal group size for hunting different prey. For elks, capture success levels off at two to six wolves. However, for bisons, the most formidable prey, nine to 13 wolves are the best guarantor of success. Therefore, for wolves, there is “strength in numbers” during hunting, but only up to a certain number that is dependent on the toughness of their prey.
Avoiding predation. Animals that are more or less defenseless often seek shelter among large groups of social companions. By joining large groups, the probability of becoming prey for each individual member is decreased. For many fish, becoming social is therefore the main anti-predatory strategy. The bigger the shoal in which to seek shelter, the better for the fish. Indeed, individual fish that are inserted in an unfamiliar and potentially dangerous environment tend to join other conspecifics. If two shoals are present, they usually join the larger shoal, which means that they are able to distinguish the larger shoal from the smaller one. The ability to compare the number of conspecifics can therefore be vital in such cases as a life or death situation.
Biologists recognize at least three distinct advantages of such behavior. First, an individual fish reduces the risk of being caught as the quantity of individuals in the group increases, a phenomenon termed the “dilution effect.” Second, a predator has a harder time singling out an individual that lives in larger shoals, a phenomenon known as the “confusion effect.” And finally, many individuals together have a higher chance of detecting a predator, a consequence termed the “many eyes effect.” Evidently, hiding within larger groups has undeniable advantages.
But hiding out is not the only anti-predation strategy involving numerical competence. Black-capped chickadees (Poecile atricapilla) in Europe found a way to announce the presence and dangerousness of a predator. Like many other animals, chickadees produce alarm calls when they detect a potential predator, such as a hawk, to warn their fellow chickadees. For stationary predators, these little songbirds use their namesake “chick-a-dee” alarm call. It has been shown that the number of “dee” notes at the end of this alarm call indicates the danger level of a predator. A call such as “chick-a-dee-dee” with only two “dee” notes may indicate a rather harmless great gray owl. Great gray owls are too big to maneuver and follow the agile chickadees in woodland, so they aren’t a serious threat. In contrast, maneuvering between trees is no problem for the small pygmy owl, which is why it is one of the most dangerous predators for these small birds. When chickadees see a pygmy owl, they increases the number of “dee” notes and call “chick-a-dee-dee-dee-dee.” Here, the number of sounds serves as an active anti-predation strategy.
Social territory defense. Groups and group size also matter a lot if resources cannot be defended by individuals alone. Many animals therefore live in social groups that claim and defend territories against intruders. Defending a territory usually means that grave and potentially lethal conflicts with rivaling groups can occur. The ability to assess the number of individuals in one’s own group relative to the opponent party is therefore of clear adaptive value. The doctrine “strength in numbers” applies in particular for social territory defense. It serves as the basis for smart decisions on whether to attack and risk damaging fights, or retreat and lose territory.
Several mammalian species have been investigated in the wild, and the common finding is that numerical advantage determines the outcome of such fights. In a pioneering study, Karen McComb and coworkers at the University of Sussex investigated the spontaneous behavior of female lions (Panthera leo) at the Serengeti National Park when facing intruders. The authors exploited the fact that wild animals respond to vocalizations played through a speaker as though real individuals were present. If the playback sounds like a foreign lion that imposes a threat, the lionesses would aggressively approach the speaker as the source of the enemy. In this acoustic playback study, the authors mimicked hostile intrusion by playing the roaring of unfamiliar lionesses to residents. Two conditions were presented to subjects: either the recordings of single female lions roaring, or of groups of three females roaring together. The researchers were curious to see if the number of attackers and the number of defenders would have an impact on the defender’s strategy. Interestingly, a single defending female was very hesitant to approach the playbacks of a single or three intruders. However, three defenders readily approached the roaring of a single intruder, but not the roaring of three intruders together. Obviously, the risk of getting beaten and hurt when entering a fight with three opponents was foreboding. Only if the number of the residents was five or more did the lionesses approach the roars of three intruders. In other words, lionesses decide to approach intruders aggressively only if they outnumber the latter. This clearly shows their ability to take quantitative information into account.
Our closest cousins in the animal kingdom, the chimpanzees (Pan troglodytes), show a very similar pattern of behavior. Chimpanzees live in groups of 20–150 individuals. Males defend the group’s territories and sometimes kill members of neighboring groups. Chimpanzee inter-group fights include both “gang attacks,” in which multiple fighters concentrate attack on a single victim, and “battles,” with multiple opponents on each side. Primatologists think that this aggressive behavior, which reduces the strength of the victim’s group, enhances the attacker’s chances of success in future battles. This, in turn, results in increases in territory size, better access to food and females, as well as reduced danger from neighboring groups. Gang attacks are a nasty but readily understood case in which some of the attackers immobilize the victim while others beat, bite, and otherwise injure the victim. From a numerical cognition point of view, “battles” with multiple fighters on each side are much more interesting, because the numbers of individuals in the confronting parties matter a lot.
Michael Wilson and coworkers from Harvard University used a similar playback approach for chimpanzees as previously described for the lionesses. In each trial, a single vocalization of a single foreign male was played through a speaker. These playbacks elicited cooperative responses, with the nature of the response depending on the number of adult males in the resident party. Parties with three or more males consistently joined in a chorus of loud vocalizations and approached the speaker together. Parties with fewer adult males usually stayed silent, approached the speaker less often, and traveled more slowly if they did approach. The chimpanzees behaved like military strategists. They intuitively follow equations used by military forces to calculate the relative strengths of opponent parties. In particular, chimpanzees follow predictions made in Lanchester’s “square law” model of combat. This model predicts that, in contests with multiple individuals on each side, chimpanzees in this population should be willing to enter a contest only if they outnumber the opposing side by a factor of at least 1.5. And this is what wild chimps do. Parties with one male do not approach a single caller until joined by allies, while parties with two males approached a single caller in four out of seven cases.
Using a similar playback technique, numerical assessment of hostile callers was also demonstrated in wild spotted hyenas (Crocuta crocuta). Just like lions and chimpanzees, hyenas live in social clans that contain up to 90 individuals, and they cooperatively and fiercely defend their territories. The playback experiments demonstrated that they respond with increasing levels of vigilance to calls produced by one, two, and three unknown intruders played back via speakers. Hyenas also took more risks approaching the speaker when they outnumbered the calling intruders. Numerical comparison thus guides their decisions to enter social contests, and they only do so when their group outnumbers their opponents. This has nothing to do with cowardliness. On the contrary, this calculated behavior saves the lives of clan members. …
[The remaining section in this chapter is titled 4.3 Benefits for Reproduction]
[You have now reached the end of this Social Science Files exhibit.]
The Source of today’s exhibit has been:
Andreas Nieder, A Brain for Numbers: The Biology of the Number Instinct, The MIT Press 2019
[MGH: From my nomad days 2018: counting the threat, counting the brood.]
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