The Number Sense by Stanislas Dehaene, 拢25,
ISBN 01951100483, July in Britain
LET鈥橲 not beat about the bush. Buy this book. You want more? Don鈥檛 trust me,
eh? Oh, very well . . .
I remember our elder son, at about 18 months old, watching cars go past his
buggy. 鈥淐ar. Nudder one car. Nudder one car. Lots of cars!鈥� Not bad at that
age鈥攂ut even today, some adults get little further.
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At the other extreme, mathematician Andrew Wiles proves Fermat鈥檚 Last
Theorem, which has baffled the best minds for three and a half centuries. Talent
for mathematics is unevenly distributed across the human race. Those that have
it make giant strides in comparison with those that don鈥檛. And our numerical
ability grows with age鈥攖hat 18-month-old is now a professional computer
programmer.
Is number sense innate or learnt? A bit of both? How do our brains do maths,
anyway? And where did the ability come from? Stanislas Dehaene, a mathematician
who became a neuroscientist, is uniquely qualified to answer such questions, and
The Number Sense is a delight. While reading it, I stole a colleague鈥檚
technique of turning down the top corner of pages that say something really
interesting, and turning up the bottom corner of pages that say something really
dumb. The book gets dog-eared, but you can find the bits you need to refer to
when you鈥檙e writing a review. I abandoned the technique halfway through: I鈥檇
turned down nearly every page, and turned up just one.
A few high notes. Our arithmetical abilities have precursors in many animals.
Around 1958, psychologist Otto Koehler showed that rats can learn to press
levers a chosen number of times. But however much they practice, they don鈥檛
achieve total accuracy. Their number sense is analogue, not digital. Pigeons can
recognise a small number of objects, even when the objects are distributed in
different spatial patterns鈥攑acked close, or strung out in a long line,
say.
But according to the educational theories of Jean Piaget, which dominated
educational thought from the 1940s to the 1980s, human children younger than
five are inferior to pigeons. Show them five marbles in a row, and four in a
shorter row. Ask which has the more marbles. They pick the row with five. Now
show them a short row of five marbles and a longer row of four marbles, and ask
the same question. They pick the row with four. 鈥淎ha!鈥� said Piaget, 鈥淐hildren of
that age cannot conserve number.鈥�
Only recently has the fallacy in his methodology come to light. The kids,
unable to believe that an adult would ask such a stupid question, decide that
鈥渕ore鈥� must now mean 鈥渓onger鈥�. If you play the same game with two-year-olds,
using sweets instead of marbles, they unerringly swipe the row with five,
whether it is long or short. Babies can distinguish two from three a few days
after birth. But this infant arithmetic sense does not endure much beyond the
age of three: maybe as far as four years old, but certainly no further. Adult
chimps do a little better.
Adult humans do better still, with technique, but their innate number sense
reflects the early limitations. In 1886, James Cattel briefly showed adult
humans cards bearing several black dots: they could enumerate the dots with
complete accuracy only when there were three or fewer.
In 1908, Bertrand Bourdon measured the time taken by an adult to name given
numbers of dots. It is roughly constant for numbers up to three, and it takes an
extra 200 to 300 milliseconds to identify each dot beyond three. The smaller the
difference between the number of dots, the longer it takes adults to select the
larger from two groups.
In Western adults, there seems to be a strong sense of a 鈥渘umber line鈥� which
moves in space from left to right. It鈥檚 not the conventional number line of
school mathematics, though, of course, it may have been influenced by that. It
is wobbly, and compressed, and some numbers, such as 10 and 20, stand out while
76 is just some fuzzy bit between 50 and 100. The most likely cause of this
directional bias鈥攚hich does not occur, for instance, in Iranians鈥攊s
the Western habit of writing from left to right.
Many people also think of numbers as having colours, with some degree of
consistency between individuals in different cultures. This association of
numbers with other attributes affects our mathematical thinking.
We must ask whether numbers are a human invention or part and parcel of the
Universe. I remember a discussion topic with a former head of department in
which he gestured around the common room and said: 鈥淚t is a physical fact that
there are three chairs in this room鈥濃�攊mplying that numbers are an aspect
of reality. I wasn鈥檛 totally convinced. 鈥淚 don鈥檛 think the Universe knows that
they鈥檙e chairs,鈥� I said. 鈥淚t doesn鈥檛 classify them as similar objects. In fact,
it treats them as independent things. All it does is move atoms around.鈥�
Dehaene reckons that our sense of number involves a rather arbitrary and
human act of clumping things into convenient packages. We talk of one deck of
cards rather than 52 cards, for instance. Survey the common words for numbers,
and you find strange discrepancies between the result and the frequency with
which particular numbers of similar objects occur in our environment.
It was, perhaps, no coincidence that my superior chose a small number, three,
and culturally familiar objects such as chairs. He could have made his point
equally well鈥攐r equally badly鈥攂y telling me there were 4388 objects
in the room that combined the features 鈥済reen鈥�, 鈥渟maller than a wastebasket鈥�,
鈥渉arder than a carpet鈥�, and 鈥渁t least 50 per cent of whose molecules have
molecular weight greater than 10 000鈥�. The Universe would 鈥渒now鈥� this just as
readily as it 鈥渒new鈥� about the three chairs鈥攊f that meant anything. But we
飞辞耻濒诲苍鈥檛.
There鈥檚 more, much more, in The Number Sense. There鈥檚 Einstein鈥檚
brain鈥攑hysiologically unremarkable, except for an excess of glial cells in
the angular gyrus, within the inferior parietal lobule. Did Einstein鈥檚
extraparietal cells make him a better mathematician? Or did he grow more
parietal cells because he was thinking about mathematics? And it鈥檚 worth
remarking鈥攂ecause Dehaene doesn鈥檛鈥攖hat in Einstein鈥檚 own estimation
he wasn鈥檛 much good at maths, so we may be looking at the wrong brain.
I am struck by how intelligent the discussion is, and how Dehaene avoids
leaping to simplistic conclusions. For example, he offers a revealing antidote
to the statistical rubbish in the book The Bell Curve, and deals with
Camilla Benbow鈥檚 work on sex differences in SAT scores (American Standard
Aptitude Tests) for talented seventh-graders. He demolishes the naive uses of
PET scans to decide which bit of the brain is doing what (Positron Emission
Tomography scans measure blood flow to areas, missing fine detail, and the
mathematical devil lies in the details.)
He gives a brilliant discussion of the sense in which a brain is not a
computer and a neuron is not a digital switch, all wrapped up in the observation
that our brains use analogue representations of digital concepts to model an
analogue world. As neuroscientist Randy Gallistel said of the animal that learnt
to push levers: 鈥淚nstead of using numbers to represent magnitude, the rat uses
magnitude to represent number.鈥� We do the same, and nowadays increasingly use
calculators instead of our heads.
My only criticism of The Number Sense is that it is too true to its
title: it looks at how the mind handles simple arithmetical concepts. The
discussions of advanced mathematical thought near the end are more philosophy
than neurology.
But that鈥檚 not a reason for you not to buy the book. It鈥檚 a reason for the
author to write a sequel. Soon, please.