"One, two three, four. My mathematics finishes here." Those are the words of
Signora Gaddi, an alert, 59-year-old Italian woman whose puzzling impairment has
helped neuroscientists understand how the brain does arithmetic. Signora
Gaddi suffered a stroke that damaged the left parietal lobe of her brain. Since
then, she has become largely hopeless with arithmetic. She cannot read, write,
compare or calculate with any numbers other than one, two, three and four.
Even with numbers below four, she is definitely not performing normally. For
instance, when shown two wooden blocks, she has to laboriously count on her
fingers in order to establish their numerosity. Because Signora Gaddi performs
normally on many other tests that do not involve numbers, her affliction can be
described as a selective loss of arithmetic.

The detailed study of Signora Gaddi is just one of many fascinating pieces of
evidence gathered by Brian Butterworth in his effort to illuminate the relations
between brain, mind and mathematics in his book "The Mathematical
Brain".
The title itself is something of a misnomer, for one would search this volume in
vain for investigations of the cognitive bases of higher mathematics, or even of
simple geometry, algebra or topology. The book focuses on a single
mathematical object, but one that is rightly seen by Butterworth as a
fundamental cornerstone of the mathematical edifice: the concept of number.

Butterworth's central hypothesis is that our brain is "born to count". Our genes
contain instructions that specify how to build a number module, a set of neural
circuits specialized for processing numbers. Those circuits, which are associated in
part with the left inferior parietal lobe, make us sensitive to numerosities in our
environment and allow us to understand and to manipulate numbers mentally.
Loss of those circuits, as in Signora Gaddi's case, results in a selective inability to
grasp the meaning of numbers. The number module is not unique to humans:
behavioural experiments reveal that many animals can also attend to
numerosity. What makes the human numerical ability unique, however, is that
it can be extended through the invention and spreading of cultural tools, such as
number symbols and arithmetic algorithms.

In recent years, the cognitive neuroscience of numeracy, or 'numerical
cognition', has emerged as an important area where the interaction between
brain architecture and human culture can be studied empirically. The hypothesis
of a modular architecture underlying number processing has been fruitful in
many areas of research, from developmental psychology to brain imaging,
animal behaviour or behavioural genetics. Several previous reviews of these
findings are available, some aimed at specialists (for example, "The Nature and
Origins of Mathematical Skills" by J. I. D. Campbell; Elsevier, 1992), others at a
wider audience (for example, "The Number Sense" by S. Dehaene; Oxford
University Press, 1997). "The Mathematical Brain" falls into the second category:
it is a skilful overview of the area for the non-specialist, with remarkable depth
and breadth in many cases, but with occasional oversights that may frustrate the
expert.

Butterworth's review of prehistory is particularly original and commendable. He
convincingly pulls together little-known evidence from cave-paintings and
bone-carvings to suggest that the dawn of arithmetic in stone-age populations
dates back at least as far as 30,000 years. More puzzling, however, is the almost
complete omission of brain-imaging evidence in the discussion of the neural
bases of the number module. Although the modern tools of positron emission
tomography, functional magnetic resonance imaging and electro- and
magneto-encephalography have been applied only recently to mathematical
cognition, a review of the available evidence would have been welcome, especially
since it confirms the presence of numerical circuits in a localized brain region:
the left inferior parietal region.

Specialists will be delighted, however, by Girelli and Butterworth's latest
evidence on developmental dyscalculia, some of which is published here for the
first time. If there is a genetic plan for a number module, then one might
expect to find an occasional child who is born without it, either due to a genetic
defect or to pre- or perinatal cerebral damage. Butterworth claims to have
identified one such patient, Charles, who is "born blind to numerosities".
Although Charles is now a very bright adult, with a university degree in
psychology, he has experienced profound, lifelong difficulties in mathematics,
to the point of still having to count on his fingers in order to solve single-digit
addition problems.

Chronometric tests reveal at least two major impairments.
First, Charles cannot "subitize": he cannot decide how many items are
presented on a computer screen, even if there are only two or three, unless
he painstakingly counts them one by one. Second, he has an abnormal
intuition of number size, which is reflected in an inverse distance effect
in a number-comparison task: whereas we normally take less time to decide
which of two numbers is larger as the distance between them gets larger,
Charles takes more time for more distant numbers, presumably because he is
using a very indirect counting strategy.

Charles has not been subjected to
brain imaging, but another case of developmental dyscalculia, recently
scanned with the novel technology of magnetic resonance spectroscopy, shows
a small, isolated area of damage exactly where number circuits are
postulated to lie -- the left inferior parietal cortex.

The finding that early focal brain damage can have such a permanent and
restricted effect on mathematical competence is perhaps the best evidence to
date in favour of the number-module hypothesis. Such evidence imposes
strong limits on brain plasticity and clearly speaks against purely
constructivist theories that view mathematical competence as the result of
a general learning device.

In the end, I suspect that Butterworth's hypothesis of a direct link between
genes, number circuits and higher mathematical competence may be too
simple. Still, the cogent arguments of "The Mathematical Brain" should be
required reading for anyone interested in the modularity of higher
cognitive functions.

Stanislas Dehaene
is at Unité INSERM 334,

Service Hospitalier Frédéric
Joliot,

4 Place du Général
Leclerc,

91401 Orsay cedex, France.

© Nature Macmillan Publishers
Ltd 1999