Humans have become the dominant species on this Earth, and that is partially because humans are numerate and and are able to do calculations. But are humans the only life form that has a sense of numbers? Why are some people better with numbers than others? Does it require a particular kind of brain to become a (good) mathematician? This book is not about mathematics, but it gives some partial answers to the previous questions. Nieder gives an extensive account of what has been learned from experiments about how the human (and animal) brain deals with numbers.
To be able to give some answers, it requires first to define exactly what is meant by the concept number. So a first part of the book is required to define cardinal numbers as objective properties of a set. Moreover there is an intrinsic ordering (ordinal numbers) and numbers can be represented in many different ways: visually by dots (of the same or different size) or different objects or by symbols, or by sounds or by tactile input. All these require different brain activity.
In a second part, numerous experiments are reported to illustrate that in view of the Darwinian theory there must have been a common ancestor in the evolutionary tree. These experiments show indeed that insects, fishes, birds, and humans have some notion of (small) numbers since some instinctive concept has been observed in all living creatures. The particular reptiles of the test were an exception to this general rule, but not all the different kind of reptiles were tested. Of course, not every life form had the same ability to discriminate between quantities. Nevertheless, there must be an evolutionary advantage for survival to have an approximate idea of quantities. This instinctive knowledge is also observed in human babies. Of all the experiments resulted the well known Weber's law of psychology. It says that the change in stimulus (e.g., the number of dots presented) is noticed when it is above a certain percentage and Weber's student Fechner added to this that the perception of that change in stimulus is logarithmic: $dp=k\ln (S/S_0)$ where $p$ stands for perception and $S$ for stimulus.
To locate the numerical brain activity, Nieder describes the structure of the human brain in part three. All kinds of experiments enable us to locate the parts of the brain that are active when the subject is exposed to a number and even the activity of neurons can be measured. These experiments confirm that there is some innate number instinct.
Homo sapiens differs from other animals by its ability to deal with numbers in symbolic form. This is the subject of part four. Here Nieder explains the origin and history of our notation and symbolic representation of numbers, and how we can learn animals to connect numbers to symbols. These symbolic representations are essential when one wants to do calculations that go beyond the small numbers. Where in the brain does calculation take place? Surprisingly, again there is some notion of approximate addition and subtraction of small numbers present in animals and even in babies. Professional mathematicians do not have a different brain and even mathematical prodigies do not have a brain that differs physiologically as has been found in postmortem determination. It may also come as a surprise that symbolic number representation is not necessarily connected with the way we process language. Another surprise is the strange connection between numbers and space. It seems like we have mentally an innate idea of a horizontal number line with numbers growing from left to right, which corresponds to our order of reading text.
Part five deals with how children evolve from saying one, two, three,... as a sequence of meaningless words to consciously associating these words with abstract numbers and how they learn to calculate. Some people suffer from dyslexia, others from dyscalculia, and it has been observed that dyscalculia affects life in a way that is worse than being illiterate. It has been investigated if that may have genetic causes, but genes seem to be only partially responsible if a person has difficulty to calculate.
The last part is about how the brain behaves when the number concept deviates from empirical reality. For example the concept of zero. It is a major step to accepted it as a number. A number can be visualized by showing a number of objects or dots on a screen. It is however difficult to represent "nothing", but nevertheless there is some sense of zero present in babies as it is illustrated with experiments. Zero is an important step towards an abstraction that leaves experimental reality. It opens the gate to negative numbers which can only be represented symbolically. Imagine five dots disappearing behind a screen, then seeing two dots leaving. When the screen comes down, one is expecting to see three dots. However when two dots are hiding and five emerge, there is no visual expectation about what to see when the screen comes down. A minus three can only be imagined symbolically.
This brief summary of the content illustrates what one has to expect from this book. There is no mathematical model of the brain. There are actually not really mathematics, but there is an extensive, yet very accessible, description of what happens in the brain when we catch some idea of quantities and numbers from sensory perception, how it is stored and what happens when we calculate. That is at the very lowest and basic level that can hardly be called mathematics, but it is a first and an essential ingredient to start doing mathematics. Nieder describes numerous experiments on animals and humans that reveal some surprising facts. He is not only citing the latest generally accepted results, but he also describes the historical evolution, illustrating how more and better insight was obtained. Whether it comes to the mathematical concepts of numbers, or the history of the symbolic notation, or the biological structure of the brain, he takes the time to explain all the necessary concepts so that any layman can follow the details of the experiments and the conclusions. Thus the reader will not learn how to become a better mathematician, a computer scientist will not learn how to make models for artificial intelligence, and educators will not learn how to teach mathematics properly. It is however an intriguing story about the magnificent engine of the human brain which allows us to deal with numbers. Numbers are an important, yet only a small part of the overall human intelligence. We know already many, sometimes surprising, things, about our brain, but basically it is still a big mystery and extremely hard to be modelled by any artificial (so called intelligent) software.