

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 ageold issue that was already discussed in Ancient Greece and that puzzled Einstein twentythree 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 languagegifted 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.


