What Are Numbers, Really? A Cerebral Basis For Number Sense
In a recent book as well as in a heated discussion at the EDGE forum, the mathematician Reuben Hersh has asked "What is mathematics, really?". This is an age-old issue that was already discussed in Ancient Greece and that puzzled Einstein twenty-three centuries later. I personally doubt that philosophical inquiry alone will ever provide a satisfactory answer (we don't even seem to be able to agree on what the question actually means!). However, if we want to use a scientific approach, we can address more focused questions such as where specific mathematical objects like sets, numbers, or functions come from, who invented them, to what purpose they were originally put to use, their historical evolution, how are they acquired by children, and so on. In this way, we can start to define the nature of mathematics in a much more concrete way that is open to scientific investigation using historical research, psychology, or even neuroscience.
This is precisely what a small group of cognitive neuropsychologists in various countries and myself have been seeking to do in a very simple area of mathematics, perhaps the most basic of all : the domain of the natural integers 1, 2, 3, 4, etc. Our results, which are now based on literally hundreds of experiments, are quite surprising: Our brain seems to be equipped from birth with a number sense. Elementary arithmetic appears to be a basic, biologically determined ability inherent in our species (and not just our own ? since we share it with many animals). Furthermore it has a specific cerebral substrate, a set of neuronal networks that are similarly localized in all of us and that hold knowledge of numbers and their relations. In brief, perceiving numbers in our surroundings is as basic to us as echolocation is to bats or birdsong is to songbirds.
It is clear that this theory has important, immediate consequences for the nature of mathematics. Obviously, the amazing level of mathematical development that we have now reached is a uniquely human achievement, specific to our language-gifted species, and largely dependent on cultural accumulation. But the claim is that basic concepts that are at the foundation of mathematics, such as numbers, sets, space, distance, and so on arise from the very architecture of our brain.
In this sense, numbers are like colors. You know that there are no colors in the physical world. Light comes in various wavelengths, but wavelength is not what we call color (a banana still looks yellow under different lighting conditions, where the wavelengths it reflects are completely changed). Color is an attribute created by the V4 area of our brain. This area computes the relative amount of light at various wavelengths across our retina, and uses it to compute the reflectance of objects (how they reflect the incoming light) in various spectral bands. This is what we call color, but it is purely a subjective quality constructed by the brain. It is, nonetheless, very useful for recognizing objects in the external world, because their color tends to remain constant across different lighting conditions, and that's presumably why the color perception ability of the brain has evolved in the way it has.
My claim is that number is very much like color. Because we live in a world full of discrete and movable objects, it is very useful for us to be able to extract number. This can help us to track predators or to select the best foraging grounds, to mention only very obvious examples. This is why evolution has endowed our brains and those of many animal species with simple numerical mechanisms. In animals, these mechanisms are very limited, as we shall see below: they are approximate, their representation becomes coarser for increasingly large numbers, and they involve only the simplest arithmetic operations (addition and subtraction). We, humans, have also had the remarkable good fortune to develop abilities for language and for symbolic notation. This has enabled us to develop exact mental representations for large numbers, as well as algorithms for precise calculations. I believe that mathematics, or at least arithmetic and number theory, is a pyramid of increasingly more abstract mental constructions based solely on (1) our ability for symbolic notation, and (2) our nonverbal ability to represent and understand numerical quantities.
So much for the philosophy now, but what is the actual evidence for these claims? Psychologists are beginning to realize that much of our mental life rests on the operation of dedicated, biologically-determined mental modules that are specifically attuned to restricted domains of knowledge, and that have been laid down in our brains by evolution (cf. Steve Pinker's How the Mind Works). For instance, we seem to have domain-specific knowledge of animals, food, people, faces, emotions, and many other things. In each case - and number is no exception -, psychologists demonstrate the existence of a domain-specific system of knowledge using the following four arguments:
one should prove that possessing prior knowledge of the domain confers an evolutionary advantage. In the case of elementary arithmetic, this is quite obvious.
there should be precursors of the ability in other animal species. Thus, some animals should be shown to have rudimentary arithmetic abilities. There should be systematic parallels between their abilities and those that are found in humans.
the ability should emerge spontaneously in young children or even infants, independently of other abilities such as language. It should not be acquired by slow, domain-general mechanisms of learning.
the ability should be shown to have a distinct neural substrate. My book The Number Sense is dedicated to proving these four points, as well as to exploring their consequences for education and for the philosophy of mathematics. In fact, solid experimental evidence supports the above claims, making the number domain one of the areas in which the demonstration of a biologically determined, domain-specific system of knowledge is the strongest. Here, I can only provide a few examples of experiments.
Animals have elementary numerical abilities. Rats, pigeons, parrots, dolphins, and of course primates can discriminate visual patterns or auditory sequences based on number alone (every other physical parameter being carefully controlled). For instance, rats can learn to press one lever for two events and another for four events, regardless of their nature, duration and spacing and whether they are auditory or visual. Animals also have elementary addition and subtraction abilities. These basic abilities are found in the wild, and not just in laboratory-trained animals. Years of training, however, are needed if one wants to inculcate number symbols into chimpanzees. Thus, approximate manipulations of numerosity are within the normal repertoire of many species, but exact symbolic manipulation of numbers isn't ? it is a specifically human ability or at least one which reaches its full-blown development in humans alone.
There are systematic parallels between humans and animals. Animals' numerical behavior becomes increasingly imprecise for increasingly large numerals (number size effect). The same is true for humans, even when manipulating Arabic numerals: we are systematically slower to compute, say, 4+5 than 2+3. Animals also have difficulties discriminating two close quantities such as 7 and 8. We too: when comparing Arabic digits, it takes us longer to decide that 9 is larger than 8 than to make the same decision for 9 Vs 2 (and we make more errors, too).
Preverbal human infants have elementary numerical abilities, too. These are very similar to those of animals: infants can discriminate two patterns based solely on their number, and they can make simple additions and subtractions. For instance, at 5 months of age, when one object is hidden behind a screen, and then another is added, infants expect to see two objects when the screen drops. We know this because careful measurements of their looking times show that they look longer when, a trick makes a different number of objects appear. Greater looking time indicates that they are surprised when they see impossible events such as 1+1=1, 1+1=3, or 2-1=2. [Please, even if you are skeptical, don't dismiss these data with the back of your hand, as I was dismayed to discover Martin Gardner was doing in a recent review of my book for The Los Angeles Times. Sure enough, "measuring and averaging such times is not easy", but it is now done under very tightly controlled conditions, with double-blind video tape scoring. I urge you to read the original reports, for instance Wynn, 1992, Nature, vol. 348, pp. 749-750 ? you'll be amazed at the level of detail and experimental control that is brought to such experiments.]
Like animals and adults, infants are especially precise with small numbers, but they can also compute more approximately with larger numbers. In passing, note that these experiments, which are very reproducible, invalidate Piaget's notion that infants start out in life without any knowledge of numerical invariance. In my book, I show why Piaget's famous conservation experiments are biased and fail to tell us about the genuine arithmetical competence of young children.
Brain lesions can impair number sense. My colleagues and I have seen many patients at the hospital that have suffered cerebral lesions and, as a consequence, have become unable to process numbers. Some of these deficits are peripheral and concern the ability to identify words or digits or to produce them aloud. Others, however, indicate a genuine loss of number sense. Lesions to the left inferior parietal lobe can result in a patient remaining able to read and write Arabic numerals to dictation while failing to understand them. One of our patients couldn't do 3 minus 1, or decide which number fell between 2 and 4! He didn't have any problem telling us what month fell between February and April, however, or what day what just before Wednesday. Hence the deficit was completely confined to numbers. The lesion site that yields such a number-sense deficit is highly reproducible in all cultures throughout the world.
Brain imaging during number processing tasks reveals a highly specific activation of the inferior parietal lobe, the very same region that, when lesioned, causes numerical deficits. We have now seen this activation using most of the imaging methods currently available. PET scanning and fMRI pinpoint it anatomically to the left and right intraparietal sulci. Electrical recordings also tell us that this region is active during operations such as multiplication or comparison, and that it activates about 200 ms following the presentation of a digit on a screen. There are even recordings of single neurons in the human parietal lobe (in the very special case of patients with intractable epilepsy) that show specific increases in activity during calculation.
The fact that we have such a biologically determined representation of number in our brain has many important consequences that I have tried to address in the book. The most crucial one is, of course, the issue of how mathematical education modifies this representation, and why some children develop a talent for arithmetic and mathematics while others (many of us!) remain innumerate. Assuming that we all start out in life with an approximate representation of number, one that is precise only for small numbers and that is not sufficient to distinguish 7 from 8, how do we ever move beyond that "animal" stage? I think that the acquisition of a language for numbers is crucial, and it is at that stage that cultural and educational differences appear. For instance, Chinese children have an edge in learning to count, simply because their number syntax is so much simpler. Whereas we say "seventeen, eighteen, nineteen, twenty, twenty-one, etc..", they say much more simply: "ten-seven, ten-eight, ten-nine, two-tens, two-tens-one, etc."; hence they have to learn fewer words and a simpler syntax. Evidence indicates that the greater simplicity of their number words speeds up learning the counting sequence by about one year! But, I hasten to say, so does better organization in Asian classrooms, as shown by UCLA psychologist Jim Stigler. As children move on to higher mathematics, there is considerable evidence that moving beyond approximation to learn exact calculation is very difficult for children and quite taxing even for the adult brain, and that strategies and education have a crucial impact.
Why, for instance, do we experience so much difficulty in remembering our multiplication tables? Probably because our brain never evolved to learn multiplication facts in the first place, so we have to tinker with brain circuits that are ill-adapted for this purpose (our associative memory causes us to confuse eight times three with eight times four as well as will eight plus three). Sadly enough, innumeracy may be our normal human condition, and it takes us considerable effort to become numerate. Indeed, a lot can be explained about the failure of some children at school, and about the extraordinary success of some idiot savants in calculation, by appealing to differences in the amount of investment and in the affective state which they are in when they learn mathematics. Having reviewed much of the evidence for innate differences in mathematical abilities, including gender differences, I don't believe that much of our individual differences in math are the result of innate differences in "talent". Education is the key, and positive affect is the engine behind success in math.
The existence of mathematical prodigies might seem to go against this view. Their performance seems so otherworldly that they seem to have a different brain from our own. Not so, I claim ? or at the very least, not so at the beginning of their lives: they start in life with the same endowment as the rest of us, a basic number sense, an intuition about numerical relations. Whatever is different in their adult brains is the result of successful education, strategies, and memorization. Indeed, all of their feats, from root extraction to multidigit multiplication, can be explained by simple tricks that any human brain can learn, if one were willing to make the effort.
Here is one example: the famous anecdote about Ramanujan and Hardy's taxi number. The prodigious Indian mathematician Ramanujan was slowly dying of tuberculosis when his colleague Hardy came to visit him and, not knowing what to say, made the following reflection: "The taxi that I hired to come here bore the number 1729. It seemed a rather dull number". "Oh no, Hardy", Ramanujan replied, "it is a captivating one. It is the smallest number that can be expressed in two different ways as a sum of two cubes."
At first sight, the instantaneous realization of this fact on a hospital bed seems incredible, too amazingly bright to be humanly possible. But in fact a minute of reflection suggests a simple way in which the Indian mathematician could have recognized this fact. Having worked for decades with numbers, Ramanujan evidently had memorized scores of facts, including the following list of cubes:
1x1x1 = 1
2x2x2 = 8
3x3x3 = 27
4x4x4 = 64
5x5x5 = 125
6x6x6 = 216
7x7x7 = 343
8x8x8 = 512
9x9x9 = 729
10x10x10 = 1000
11x11x11 = 1331
12x12x12 = 1728
Now if you look at this list you see that (a) 1728 is a cube; (b) 1728 is one unit off 1729, and 1 is also a cube; (c) 729 is also a cube; and (d) 1000 is also a cube. Hence, it is absolutely OBVIOUS to someone with Ramanujan's training that 1729 is the sum of two cubes in two different ways, naming 1728+1 and 1000+729. Finding out that it is the smallest such number is more tricky, but can be done by trial and error. Eventually, the magic of this anecdote totally dissolves when one learns that Ramanujan had written this computation in his notebooks as an adolescent , and hence did not compute this on the spur of the moment in his hospital bed: he already knew it!
Would it be farfetched to suggest that we could all match Ramanujan's feat with sufficient training? Perhaps that suggestion would seem less absurd if you consider that any high school student, even one that is not considered particularly bright, knows at least as much about mathematics as the most advanced mathematical scholars of the Middle Ages. We all start out in life with very similar brains, all endowed with an elementary number sense which has some innate structure, but also a degree of plasticity that allows it to be shaped by culture.
Back to the philosophy of mathematics, then. What are numbers, really? If we grant that we are all born with a rudimentary number sense that is engraved in the very architecture of our brains by evolution, then clearly numbers should be viewed as a construction of our brains. However, contrary to many social constructs such as art and religion, number and arithmetic are not arbitrary mental constructions. Rather, they are tightly adapted to the external world. Whence this adaptation? The puzzle about the adequacy of our mathematical constructions for the external world loses some of its mystery when one considers two facts.
First, the basic elements on which our mathematical constructions are based, such as numbers, sets, space, and so on, have been rooted in the architecture of our brains by a long evolutionary process. Evolution has incorporated in our minds/brains structures that are essential to survival and hence to veridical perception of the external world. At the scale we live in, number is essential because we live in a world made of movable, denumerable objects. Things might have been very different if we lived in a purely fluid world, or at an atomic scale ? and hence I concur with a few other mathematicians such as Henri Poincare, Max Delbruck, or Reuben Hersh in thinking that other life forms could have had mathematics very different from our own.
Second, our mathematics has seen another evolution, a much faster one: a cultural evolution. Mathematical objects have been generated at will in the minds of mathematicians of the past thirty centuries (this is what we call "pure mathematics"). But then they have been selected for their usefulness in solving real world problems, for instance in physics. Hence, many of our current mathematical tools are well adapted to the outside world, precisely because they were selected as a function of this fit.
Many mathematicians are Platonists. They think that the Universe is made of mathematical stuff, and that the job of mathematicians is merely to discover it. I strongly deny this point of view. This does not mean, however, that I am a "social constructivist", as Martin Gardner would like to call me. I agree with Gardner, and against many social constructivists, that mathematical constructions transcend specific human cultures. In my view, however, this is because all human cultures have the same brain architecture that "resonates" to the same mathematical tunes. The value of Pi, thank God, does not change with culture ! (cf. the Sokal affair). Furthermore, I am in no way denying that the external world provides a lot of structure, which gets incorporated into our mathematics. I only object to calling the structure of the Universe "mathematical ". We develop mathematical models of the world, but these are only models, and they are never fully adequate. Planets do not move in ellipses - elliptic trajectories are a good, but far from perfect approximation. Matter is not made of atoms, electrons, or quarks - all these are good models (indeed, very good ones), but ones that are bound to require revision some day. A lot of conceptual difficulties could be clarified if mathematicians and theoretical physicists paid more attention to the basic distinction between model and reality, a concept familiar to biologists.