Do humans have an innate capacity for mathematics?

—SHERRI CHASIN CALVO

Viewpoint: Yes, recent scientific studies suggest that we are born with at least some mathematical ability already "hardwired" into our brains.

John Locke (1632-1704), an English philosopher, famously argued in An Essay Concerning Human Understanding that "No man's knowledge here can go beyond his experience." Locke's position, known as empiricism, holds in essence that all human knowledge is acquired through experience and experience alone. Thus, Locke believed that humans possess no "innate ideas" at birth. As such, Locke likened the minds of humans at birth to a tabula rasa , or soft tablet (sometimes translated as a "blank slate"), upon which knowledge is written. Thus, for Locke and other empiricists, knowledge is a matter of nurture, not nature.

The complexities of mathematics seem to present an obvious and intuitive example of Locke's thesis against "innate ideas." No one, after all, is born knowing how to solve quadratic equations and other complex mathematical functions. Indeed, acquiring the ability to solve equations—let alone understand what a mathematical equation is—takes years of education and practice, and even then many continue to have enormous difficulty. Math, according to this view, is a matter of nurture, not nature: one has to learn how to "do math."

Some people, however, seem to have a knack for mathematics, even at a very young age. Every so often we hear reports in the media about children who are able to perform calculus and solve sophisticated mathematical equations. What accounts for the different mathematical abilities in these children? Do such prodigies indicate that math is more a matter of nature rather than nurture?

As with many issues, the answer appears to be somewhere in between. Indeed, despite the intuitive appeal of Locke's position, there now appears to be strong evidence that humans—and even higher-order mammals—possess an innate mathematical ability and can perform some rudimentary mathematical operations such as simple addition and subtraction. So, although we have to learn how to solve a quadratic equation, recent scientific studies suggest that we are born with at least some mathematical ability already hardwired into our brains. The purpose of this essay is to look at this "innate" mathematical ability that all humans share.

What Does It Mean for Math to be Innate?

At this juncture, it is worth commenting on the definitions of "innate" and "mathematics." According to Webster's Dictionary, something is innate if it exists "from birth," or is "inherent in the essential character of something." Likewise, mathematics is defined as "the systematic treatment of magnitude, relationships between figures and forms, and relationships between quantities expressed symbolically. " Certainly, no one contends that an infant has, at birth, the ability to express symbolically anything, let alone magnitudes, figures, forms, and quantities. Nurture, not nature, is required in order to understand the system of mathematics.

But does this mean that mathematics is not innate? The answer, of course, depends on how narrowly one interprets and defines mathematics. In a rather technical, but uninteresting, sense, mathematical ability is not innate; we cannot recognize the meaning of the symbols 2 + 2 = 4 at birth. We must, of course, learn the symbols of mathematics before we can express mathematical relationships, i.e., before we can "do math" as is commonly understood. This essay, however, adopts a broader view of mathematical ability. Although infants cannot express anything symbolically in terms of writing down mathematical or linguistic expressions, they nevertheless can, and do, reveal rudimentary understanding of such mathematical concepts as number, addition, and subtraction.

At this point it is important to recognize that there is a difference between understanding the meaning of—and being able to manipulate—the symbols 2 + 2 = 4, and the concepts underlying those symbols. In other words, although infants may not recognize that "2" means "two things" and + symbolizes the mathematical operation of addition, infants nevertheless can express behaviorally that they understand such concepts as quantity, addition, and subtraction. How they express this understanding is discussed below.

Infants and Mathematics

For many, just a glance at this WORD will reveal that it has four letters. One need not count the individual letters to arrive at a number—it seemingly is an automatic function of the brain. However, in the case of a word with NUMEROUS letters, it is not so easy to ascertain quickly the number of letters present—one may in fact be forced to count the letters in order to determine that "numerous" has eight letters in it.

This ability to discriminate among a collection of things, at least when the number of things in question is of a relatively small amount, is known as "subitizing." Although it is not clear how the brain operates when it subitizes, it is clear that this ability requires the brain to distinguish between quantities—in other words, to recognize that there is a difference between • and • and between • and •.

Research conducted throughout the 1980s and 1990s indicates that human infants as young as three or four days old have the capacity to subitize. In other words, they can recognize a difference in a collection of things, when the difference concerns the number of things present and not the things themselves. Thus, the concept of "number" appears to be innate. Within a few months, infants begin to exhibit behavior indicating that they understand that one plus one is equal to two, and that two minus one is equal to one. And as they become older, research indicates that they understand even more complex mathematical operations.

Researchers working on the arithmetic capabilities of infants are able to come to these conclusions by exploiting what is known in developmental psychology as the "violation-of-expectation paradigm." An example of an experiment that uses this paradigm to study the mathematical ability of infants is described below:

A researcher places an infant on her mother's lap facing a small stage. The researcher then sets a puppet on the stage immediately in front of the infant, making sure that she sees it. The researcher then places an opaque screen in front of the stage so that the infant can no longer see the puppet. While the screen is up, the researcher, in a manner completely visible to the infant, then places a second puppet behind the screen. After a moment, the researcher drops the screen to reveal one of two things: either two puppets or just one puppet (in the latter case, the researcher surreptitiously removes one of the puppets before dropping the screen).

What researchers have noticed is that when two puppets remain after the screen is dropped, infants tend not to take much notice. However, when the screen is dropped and there is only one puppet when there should be two, the infants tended to "startle" or stare longer at the stage than otherwise. Repeated experiments with various different objects and numbers of objects have tended to produce the same outcomes, providing support to the idea that not only do infants understand the concept of "number," they also have the capacity to understand that one thing added to another thing makes two things, and that one thing taken away from two things leaves one. This capacity seems to indicate that some mathematical abilities are hardwired in the human brain. The tablet of the brain, apparently, is not as "soft" as Locke presumed.

Mathematics and the Human Brain

Some neuroscientists believe they have identified the particular area of the brain responsible for our ability to understand mathematical concepts and perform arithmetic calculations. Neuroscientists have long been familiar with how lesions on the brain affect certain mental abilities of patients. In particular, neuroscientists have discovered that when legions occur on the angular gyrus within the inferior parietal cortex—a rather small area on the side of the brain toward its rear—mathematical ability is severely affected. Thus, mathematical ability appears to be localized in a different area of the brain than speech or language.

Another piece of evidence for a special "mathematics" area of the brain comes from studying perhaps the greatest mind of all time: that of Albert Einstein. When Einstein died in 1955 at the age of 76, his brain was removed and preserved in order to study it. In June of 1999, neuroscientists concluded studies of the first-ever detailed examinations of Einstein's brain, comparing the structure of Einstein's brain to the brains of 35 men and 56 women of normal intelligence. What they discovered was amazing. Although Einstein's brain was, for the most part, no different from "normal" brains, the inferior parietal lobe regions of his brain—the supposed seats of mathematical ability—were significantly larger, about 15% wider than normal. As a result of this finding, researchers have proposed studying the brains of other exceptionally intelligent people in order to better understand the relationship between "neuroanatomy" and intelligence.

The human brain is not a "soft palette" or a blank slate upon which all knowledge, including mathematics, is written. Just as the processor chips in computers have basic arithmetical operations hard-wired into them, there appears to be software already preprogrammed into the brain that allows it to acquire more knowledge. As discussed in this essay, humans do posses innate mathematical abilities, which, in turn, prepare us to understand more abstract mathematical concepts and problems.

—MARK H. ALLENBAUGH

Viewpoint: No, mathematics involves not just counting or simple arithmetic but also abstraction, which can only exist in the presence of language skills and symbolic representation.

What Is Mathematics?

Before we can address the question of whether or not our mathematical abilities are innate or acquired, we must define exactly what we mean by mathematics. This is not an easy task, but we can start by looking at what mathematics is not. It is not comprehension of quantity, which, as we will see, is a skill innate not only to humans but also to many animals. Nor is mathematics counting or simple arithmetic. These abilities are found in most human cultures, whereas higher mathematics is relatively rare. The Oxford English Dictionary defines mathematics as "the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra." This raises two important points: the abstract nature of math (see below) and the idea of math as relationships between numbers. We might also say that math focuses on changes in numbers, and on the operations that allow us to perform these changes.

Mathematics involves abstraction. The number 5, for example, can be used as an adjective to describe a group of objects, but it does not become a mathematical concept until we are able to abstract it—that is, separate it from its connection to the physical world. This is not easy to do. Although we have developed a sophisticated system for representing and manipulating large numbers, we have no real feeling for their magnitude, unless we visualize it as something concrete, say the population of Los Angeles, or the number of people in a sports stadium. We have to learn abstraction. Small children learning arithmetic begin by using real objects, often their fingers. The next step is to imagine the number applied to reality—a child may struggle to, say, subtract 1 from 4, until prompted by being asked to imagine a situation with real, familiar objects, such as pieces of candy. Children do not naturally use numbers in an abstract sense.

Does mathematics exist independently of the human mind? The Platonic view holds that it does, that it is a body of concepts existing in an external reality. This implies that math is gradually discovered and mapped out, much as an unknown country might be. The opposing view is that math is internal, created by humans as the product of our minds and the way they function, so that we would describe mathematical innovations as inventions. This view is advocated by George Lackoff and Rafael Nunez in their Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (2001).

This article draws on what might be termed an intermediate view, which comes from the mathematician Reuben Hersh. He holds that math is neither entirely physical nor entirely mental but, like law or religion, a social or cultural phenomenon. It is external to the individual but internal to society. This is known as the humanist philosophy of mathematics and has much in common with the views of another famous mathematician, Ian Stewart, who describes math as a "virtual collective." Each individual mathematician has his or her own system of mathematics inside his or her head. These systems are all very similar because all mathematicians have similar training, and they communicate their ideas with one another.

The Evolution of Mathematics

In order to see how mathematics has developed as a cultural phenomenon, rather than as an innate capability, we must look at its development in human history.

Number Sense.

Humans have the ability to subitize—this means we can see two or three objects and "know" how many there are, without having to count them. We can also add and subtract these small numbers. Experimenters have shown that pre-verbal human babies have these same skills, which suggests they are hardwired in the brain. Many other species, including chimps, raccoons, rats, and pigeons, also appear to be able to subitize, suggesting that the ability has ancient evolutionary origins.

In all species, subitization works only up to 3. Above this, a counting system becomes necessary. However, there are societies that have managed to survive without counting above 3 or 4. Many aboriginal societies lack words for numbers over 3. The Warlpiris in Australia, for example, have words for only 1, 2, "some," and "a lot." The people in such tribes manage without the need for counting. For instance, they may identify each sheep or cow in their flocks individually, so they would know if one is missing, or they may use the correspondence principle, whereby the herdsman uses a bag of pebbles, with each pebble representing a sheep.

Unlike language, counting systems are not universal. They possibly arose from religious or spiritual ritual encompassing concepts such as duality, pattern, and sequence. The historian Abraham Seidenberg, who made extensive studies in the 1960s, speculates that counting was invented in a few, or even only one, civilized centers, for ritual purposes and from there spread to other cultures. There is some evidence to support this idea. There are many superstitions and taboos regarding numbers. Many peoples believe counting brings misfortune or angers the gods. For example, some African mothers will not count their children, for fear this would bring them to the notice of evil spirits. There appears to be no practical origin for such taboos, so it seems unlikely that they arose spontaneously in so many different places, supporting the view that counting arose in relatively few places and then dispersed to other tribes and cultures.

Counting begins with body parts, especially fingers. There are different methods of finger counting. Some cultures start counting one on the thumb, some on the little finger. Some start with the right hand and others the left. Certain societies extend the fingers from a closed hand while others fold them into the palm from an open hand. The distribution of these different methods adds weight to the argument that counting developed in a few places and then spread to other cultures.

Counting existed for thousands of years before it progressed into mathematics in just a few civilizations. Several developments had to have occurred before the field of mathematics could be established. First, a symbolic representation of numbers, or notation, was required. This began with a one-to-one correspondence, such as one pebble for each sheep, or one knot in a piece of string for each sack of corn. Bones as old as 30,000 years old, marked with notches, which appear to have served as a tally, have been found in the ruins of ancient civilizations. Individual symbols for particular numbers developed much later. More sophisticated number representation also required a place value system, a system of numerical notation in which the same symbol can have different meanings, depending on its position in the number—that is, 1 can mean 1 or 10 or 100, depending on its position. This also necessitated the use of a base number (in our number notation, the base is 10), with successive places representing successive powers of the base. Zero became necessary as a "place holder." Leaving a space in a column can be ambiguous, but inserting a zero makes the representation clear. While base numbers and forms of notation developed widely, very few societies developed a place value system and the concept of zero. These innovations, along with sophisticated language skills, gave us true mathematics.

Mathematics and Language

Archaeological evidence indicates that language predates numeracy. Our complex language skills have allowed us to develop symbolic number systems. This, in turn, has made possible abstraction in mathematics.

Examples of the close relationship between language and math are not hard to find. Consider learning the multiplication tables. For most people, these are learned, not as abstract patterns of number but through language, almost like a poem, and it is not unusual for people to recite the table in order to find the answer to a mathematical problem.

Asian people, especially the Chinese, are often particularly good at math because of their language. The system of words used for numbers in Chinese is far more clear and logical than in Indo-European languages. There are Chinese words for the numbers 1-9 ( yi, er, san, si, wu, liu, qi, ba, jiu ), plus multipliers 10 ( shi ), 100 ( bai ), and 10,000 ( wan ). There are no special words for the numbers 11-19 or multiples of ten (20, 30, and so on). Thus 10 is shi yi (ten one). The number 35 is san shi wu (three ten five). Note that the words relate more closely to the symbols, making them easier to understand. Kevin Miller and his colleagues found that four-yearold Chinese children, on average, could count to 40, while American children of the same age could count only to 15. This delay is due to the difficulties with the "teens" numbers, which do not follow the same pattern as other numbers. While American children do eventually catch up, by that time, Chinese children would have had several years more experience handling larger numbers than their American counterparts.

The conciseness of the Chinese number words is also an advantage. Our memory span for a list of numbers relates directly to the length of the words used for those numbers. Native Chinese speakers can routinely remember strings of nine or 10 numbers, whereas native English speakers can only manage six or seven.

Fluent bilingual people can often think in their second language—they can understand and answer a question without translating it into their first language. However, anecdotal evidence suggests that this ability breaks down when the question is an arithmetic problem. In such cases, the person solves the problem in his or her native language, translates it, and then gives the answer in the second language.

Mathematics and the Brain

Stanislas Dehaene and his colleagues have recently shown that there is a significant difference in brain activity in subjects performing exact and approximate arithmetic and that the use of language appears to be important. Exact arithmetic involves parts of the brain normally used in word association, and multilingual subjects have been found to perform better on exact arithmetic problems if the problems are posed in their native language.

Approximate arithmetic, where the subject is presented with a problem and a set of possible answers and is asked for the solution closest to the correct answer, is language independent, and creates activity in the parietal lobes, an area usually associated with visuo-spatial processing. This non-language-dependent ability may be the same as the number sense observed in babies and animals. Evidence in support of this idea comes from the British cognitive scientist Brian Butterworth, who has identified a small area of the left parietal lobe, which he calls the "number module," believed to be the location of our number sense.

It seems clear that we have an intuitive sense of quantity for small numbers and an ability to manipulate them to a limited extent. The fact that we share this with many other species has been used to support the hypothesis that mathematical ability is innate, but these experiments suggest that "number sense," while necessary for higher mathematical reasoning, is not sufficient proof of innate mathematical ability.

Using positron-emission tomography (PET) and magnetic resonance imaging (MRI) studies, Dehaene, in his book entitled The Number Sense, states that when we perform complex tasks that are not part of our evolutionary behavioral repertoire, many different, specialized areas of the brain become activated. It would be interesting to analyze the brain activity of a skilled mathematician performing complex calculations to see how the brain operates when faced with a highly abstract problem.

Conclusion

Humans have certain innate abilities such as number sense, pattern identification, and spatial awareness that contribute to mathematical reasoning. We can see that mathematical ability is built on these fundamental units, but we must learn how to organize and use them. The development of symbolic notation and the ability for abstraction were essential for the emergence of mathematics. Thus, mathematics itself is not innate but rather a cultural acquisition. In fact, we might say that mathematics is an emergent property of the human mind and culture combined, in as much as the immense intellectual system that is mathematics today is far greater than the sum of the properties that form it.

—ANNE K. JAMIESON

Further Reading

Barrow, John D. Pi in the Sky: Counting, Thinking and Being. New York: Little Brown, 1993.

Butterworth, Brian. What Counts: How Every Brain Is Hardwired for Math. New York: Free Press, 1999.

Cardoso, Silvia Helena. "Why Was Einstein aGenius?" Brain and Mind (Oct.-Dec. 2000). <http://www.epub.org.br/cm/n11/mente/eisntein/einstein.html> .

Dehaene, Stanilaus. The Number Sense: How The Mind Creates Mathematics. New York: Oxford University Press, 1997.

——, et al. "Sources of Mathematical Thinking: Behavioural and Brain-Imaging Evidence." Science 284 (May 7, 1999): 970-74.

Dennett, Daniel. Consciousness Explained. NewYork: Little Brown, 1993.

Devlin, Keith J. The Math Gene: How Mathematical Thinking Evolved & Why Numbers Are Like Gossip. New York: Basic Books, 2000.

Hersch, Reuben. What Is Mathematics, Really? London: Random House, 1997.

Lackoff, George, and Rafael E. Nunez. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books, 2001.

Motluk, Alison. "True Grit." New Scientist no.2193 (July 3, 1999): 46-48.

Pinker, Steve. The Language Instinct. New York:William Morrow, 1994.

Stewart, Ian. "Think Maths." New Scientist no.2058 (November 30, 1996): 38-42.

KEY TERMS

ABSTRACTION:

The transformation of a concrete object into an intangible idea.

EMERGENT PROPERTIES:

Properties of a complex system that are greater than the sum of the properties of the parts of that system.

EMPIRICISM:

View that all knowledge is acquired through the senses; generally opposed to the view that any knowledge is innate.

INFERIOR PARIETAL CORTEX:

Region of the brain thought to be responsible for mathematical ability.

NEUROSCIENCE:

Science of the study of the brain and its functions.

PLACE VALUE SYSTEM:

A system of numerical notation, where the value of a symbol varies depending upon its position in the number.

SUBITIZATION:

The innate ability to comprehend the size of small numbers (1-3) without having to count. The "knowledge" of numbers.