**What happens when you can't count past four?**

by Brian Butterworth

**The Guardian: 21st October, 2004.**

Brian Butterworth asks whether you can do maths without words for big
numbers.

"Some Americans I have spoken with (who were otherwise of quick and
rational parts enough) could not, as we do, by any means count to
1,000; nor had any distinct idea of that number," wrote the English
philosopher John Locke in 1690.

He was referring to the Tououpinambos, a tribe from the Brazilian
jungle, whose language lacked names for numbers above five. Locke's
point was that number names "conduce to well-reckoning" by enabling
us to keep in mind distinct numbers, and can be helpful in learning
to count and to calculate, but they are not necessary for the
possession of numerical ideas.

Two recent studies of Amazonian Indians reported in the journal
"Science", take a crucially different view. These studies, far
from maintaining that number words are convenient, propose they
are actually necessary.

The theory that language shapes thought is sometimes called the
Whorf hypothesis, after the anthropologist Benjamin Lee Whorf.
Berinmo, a stone-age tribe in New Guinea, does not put a
linguistic boundary between blue and green but does have a
boundary between "nol" and "wor" within what we would call
green. Research by Jules Davidoff, Ian Davies and Debi
Roberson showed that the Berinmo categorise blue and green
together but "nol" and "wor" separately, whereas we do the
opposite - see blue and green as separate colours, but
"nol" and "wor" as variants of the same colour.

So if categorising objects by colour can be shaped by
colour vocabulary, why shouldn't categorising the number
of objects? The idea advanced in the two studies of Amazonian
Indian tribes support a strong Whorfian view that number
vocabulary is necessary for categorising the world numerically.
The idea being tested in the Amazon is that humans, and many
other species, are born with two "core" systems of number that
do not depend on language at all. The first is a small number
system related to the fact that we can recognise the exact
number of objects up to three or four without counting. We
use a second system to deal with numbers larger than four,
but it only works with approximations. To get the ideas of
larger numbers, of exactly five, exactly six, and so on, you
need to be able to count, and to count, you need the counting words.

The Pirahã, a tribe of 150 people who live by the banks of a
remote tributary of the Amazon, studied by Columbia linguist
Peter Gordon, have words for one and two, and for few and many.
That's all. Even the words for one and two are not used
consistently. So the question is, do they have the idea of
exact numbers above three?

Not having much of number vocabulary, and no numeral symbols,
such as one, two, three, their arithmetical skills could not
be tested in the way we would test even five-year-olds in
Britain. Instead, Gordon used a matching task. He would lay
out up to eight objects in front of him on a table, and the
Pirahã participant's task was to place the same number of
objects in order on the table. Even when the objects were
placed in a line, accuracy dropped off dramatically after
three objects.

The Mundurukú, another remote tribe, studied by a French team
led by Pierre Pica and Stanislas Dehaene, only have words for
numbers up to five. Pica and colleagues showed that the
Mundurukú could compare large sets of dots and add them
together approximately. However, when it came to exact
subtraction, they were much worse.

Mundurukú participants saw on a computer screen dots dropping
into a bucket, with some dots falling through the bottom. They
had to calculate exactly how many were left. The answer was always
zero, one or two, and they had to select the correct answer. They
were quite good, but not perfect, when the initial numbers dots going
in and falling out were five or fewer, the limit of their vocabulary,
but many of them were doing little better than guessing when the numbers
were more than five, even though the answers were always zero, one or
two. Pica and colleagues concluded that "language plays a special role
in the emergence of exact arithmetic during child development".

Tribal societies in the Amazon differ in many ways from a numerate
society like ours. The Pirahã are essentially hunter-gatherers who
rarely trade, and the Mundurukú also have little need for counting
in their everyday lives. It is therefore very difficult to tell
whether it is only the difference in the number vocabularies that
hold the key to their unusual performance on exact number tasks.
It could be lack of practice at using the ideas of number themselves,
in counting or calculating. Pica and colleagues seem to recognise
this, since even in the range of their vocabulary, the Mundurukú
are approximate - "ebadipdip" is typically used for four, but also
used for three, five and six. The words alone are not enough, they
conclude. The number names need to be used to do counting, and some
conception of what it is to count must co-exist with the vocabulary.

So maybe Locke was right. Counting can exist without number names, but
is greatly helped by them.

Brian Butterworth is the author of the Mathematical Brain, and is at
the Institute of Cognitive Neuroscience at UCL.