The Number Sense - How the Mind creates Mathematics by Stanislas Dehaene, Allen Lane The Penguin Press, UK, 1998, £20.00, ISBN 0-713-99170-4
How old are numbers? Some people claim that they are at least as old as our
universe. Unless we consider the natural numbers as ``god-given'' as Leopold
Kronecker did, the question where they originate from and how we represent
them in our brain should have by now arisen.
The Number Sense from Stanislas Dehaene, a mathematician who became a
neuroscientist, asks exactly this question. The Number Sense is also one
of very few books to look at the psychology of mathematics.
The human understanding of mathematics has developed enormously in the last
five thousand years. However, the bigger part of this evolution happened
through Dawkins' Memes and not through biological evolution; our brain is
pretty much the same as the brain of our ancestors hundreds of years ago.
Dehaene guides the reader through the whole spectrum where numbers can appear.
He is not only interested in the number sense in adult humans, but also in small
children and even in animals.
In 1904, a horse named ``Clever Hans'' stirred up some excitement by its
apparent ability to solve arithmetical problems. A committee of experts,
which included the German psychologist Carl Stumpf, certified Hans' abilities.
However shortly after, the psychologist Oskar Pfungst conducted systematic
experiments in which he discovered, that the arithmetical ability was in fact
the horse's ability to sense tensions in the auditory.
This case brought a lot of discredit to the idea of number sense in animals
and the research stopped for many years. It was not until the 1970s,
when Thompson discovered ``number-detecting'' neurons in the cat's brain.
This number sense is far from accurate for numbers beyond three, in fact
it is some sort of fuzzy-counting.
To get a picture of what Dehaene means by fuzzy-counting, let us look at one of the examples he used in his book. It is a paragraph of ``Alice in Wonderland'' by Lewis Carroll, who is often cited by Dehaene. It is interesting to examine the thoughts Lewis Carroll had about number sense.
"Can you do Addition?" the White Queen asked. "What's one and one and one and one and one and one and one and one and one and one?"
"I don't know", said Alice. "I lost count."
"She can't do Addition," the Red Queen interrupted.
With this in mind, we can find the fuzzy-counting not only in animals but also
in ourselves. When looking at an arrangement of objects, our number guess
becomes the more inaccurate the more objects there are.
Therefore, some sort of rough number sense may be innate, whereas having an
exact number sense or even to understand non-natural numbers like 
is a
distinguishable ability that can only be learned by adult humans.
Jean Piaget, the founder of constructivism, claims that numbers in children
have to be constructed and are not represented in the genes.
Dehaene blames him for underestimating a child's power of abstraction.
Babies at the age of only a few days are able to distinguish from two
to three sounds or toys.
This again strengthens the idea of a separation of the innate fuzzy number
sense and the one we use for arithmetics.
You have probably noticed that you can grasp the number of some objects without
explicitly counting them when their number is one, two or three. Starting with
four objects, you will probably arrange them in groups and count.
One theory supposes that this has to do with the geometrical arrangement of the
objects and that the immediate counting is already done in our visual system.
Dehaene suggests that it also had a large effect on how we decided to
invent written numbers. The most obvious is the Roman number system, where
1=I, 2=II, 3=III, but 4=IV. This jump between three and four can also
be seen in many other number systems like the Arabic or Chinese.
A difference in cultures however can be seen in remembering digits.
An English speaking adult can keep up to seven digits in memory by hearing or
reading them once; this is often considered as a fixed maximum since all
studies have been conducted on English speakers. When we do the same tests
with Chinese speaking adults, the number of remembered digits goes up to ten.
The answer to this difference is that the Chinese number words only consist
of one syllable each. Therefore, more digits fit into their short term
memory.
Having an associative memory, we often involuntarily come up with 6*8=68
instead of 6*8=48. I have to disagree with Dehaene that our associative
memory is a pure disadvantage for mathematics. New mathematical proofs and
inventions only exist because of some humans using their associative memory
and therefore their creativity. Still, associating a tiger and a lion and
running away from both of them has been much more important than creating
mathematics for the last few thousand years.
An interesting question is in which parts of the brain arithmetics can be
localized. There is no easy answer. Numbers can be perceived in many different
ways, and it is not the same whether hearing the number seven , seeing
seven objects, reading 7 or the word seven . Dehaene presents some
examples of tests with lesion patients. Some of these people can't pronounce
a number when reading it, but can tell if one is bigger than the other.
Some numbers can only be pronounced using a different route. For example:
``504'', ``...was my first car...Peugeot...504!''.
A study from Dehaene and Cohen from 1995 (which can be found in The
Number Sense ) suggests that the inferior parietal cortex is involved in
quantity representation. However, many other parts of the brain are involved
as well. Dehaene also suggests that number representation,
spatial mapping, and the mapping of the ten fingers are positioned in
neighbouring regions in the brain. This could explain why some people do not
imagine the number line as a straight equidistant line, but imagine it as
a curve in space.
The Number Sense is an informative book and very interesting to read.
It gives us many new ideas, which, unfortunately, it is impossible to
cover in complete depth in a book of this size.
Verena Vanessa Hafner is studying Computer Science and Artificial Intelligence at the University of Sussex, UK
