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Vol.
XXVIII No. 36
September
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The
Number Sense:
Small
Numbers Have Large Impact
Prabir
Purkayastha
A
FAIRLY technical paper in Science Express, August 19, on number sense
amongst an obscure tribe in the Amazons numbering only about 200 has ignited a
huge controversy. Briefly, Peter Gordon, a researcher found that the Pirahas,
the tribe in question do not have any words for numbers above three and also
lack in simple mathematical skills that are found even in small children. Set
the task of matching objects or registering taps on the floor larger than three,
they performed very poorly. From this, the conclusion that unless we have words
for numbers, we are unable to count beyond three. This reinforces a hypothesis
known as the Whorf hypothesis (or more correctly the Shapir Whorf hypothesis),
which dates back more than 50 years that language limits thought: language
affects the way we perceive reality. In other words, nature is dissected by the
mind along the lines laid down by language
PIRAHA
LANGUAGE
Before
we plunge into the issue of how the human mind perceives mathematics, we need to
add a note of caution regarding the Pirahas and their language. The Piraha
language has been studied by only two researchers, who have spent more than 20
years with them, a husband and wife couple Dan and Karen Everett. Peter
Gordons interaction was also through Dan and Karen. Dan disagrees with Peter
Gordons findings and argues that instead of language being a limit on thought
here, it is the cultural impact on language that is important for the Piraha.
For linguistics, the challenge is even larger than for mathematics. If Dan is
right, Piraha language has many more deficiencies than only lacking words for
higher numbers; it has then serious implications for foundations of linguistics
including Chomskis universal grammar. While there is no reason to believe
that all the results quoted by Peter and Dan are not correct, a group of 200
speakers and two linguists are too small a sample size to come to such startling
conclusions.
What
are numbers is not the preoccupation of cognitive psychologists alone.
Philosophers and mathematicians have spent centuries debating what numbers are.
It is a shift in the last twenty years that neuroscience and cognitive
psychology have been brought into such questions. And there is little doubt that
they have opened avenues of enquiry after the philosophical examination of what
numbers are had reached a mathematical dead-end.
ABSTRACT
OBJECTS
If
we look at the numbers as mathematicians and philosophers have looked, the
numbers are either abstract objects that exist independent of us or are
intuitively grasped from the real world. In the first scheme, the mathematical
objects including numbers exist as objective ideas as Plato would have put it
and any mathematical knowledge is a discovery of the properties of this Platonic
mathematical universe. For intuitionists, all mathematics is strictly to be
constructed from the natural numbers. The natural numbers do not need an
explanation as they are intuitively grasped. In other words, while
Neo-Platonists believe that all mathematics is discovery, the intuitionists
believe it to be human construction, apart from natural numbers, which are the
only given.
Much
of the mathematical developments would be thrown out in the intuitionists
scheme of things, as they do not accept any development, which cannot be
constructed. As a major part of todays mathematics rests on logical devices
such as reductio ad absurdum, (proving a result by showing its opposite leads to
a contradiction), the intuitionist agenda was rejected by the mathematical
community in general. But the neo-Platonists suffered a body blow when it was
found that it was impossible to put the Platonist universe on a firm logical
footing. In spite of best efforts of a Bertrand Russell, Frege and others, the
foundations of mathematics remained mired in contradiction. Hilbert proposed
a solution, which makes mathematical truth independent of the external world.
His agenda was to show that if we have a set of axioms (Euclids axioms in
geometry for example), the rest can be derived logically out such axioms. His
project was to show that all the rest could be derived out of the axioms by a
set of steps, which could even be programmed into a machine. This is the formal
system of mathematics; mathematics is derived using logic from a set of
independent axioms. Hilberts problem rose when Kurt Godel showed that such a
scheme would necessarily be incomplete: even if mathematics gave up the
meaning of mathematics, it cannot reach completeness. Hilberts project was
doomed from the start.
ALTERNATE
APPROACH
It
is here that an alternate approach to numbers and numbers are the building
block of mathematics emerged. The question of what are numbers was
reformulated as how does the mind know numbers? And in the last two decades very
many interesting results have arisen from this enquiry, which belongs to
neuropsychology or cognitive psychology.
The
most important one is that a simple number sense of being to identify up to
three is common between humans and the animal kingdom.
Birds and animals can count easily up to three and even can do elementary
additions and subtractions. For instance if set of hunters enters into a hut and
leaves one by one, when will the bird or the animal emerge from hiding? If it
involves three hunters, the prey can subtract from this number as the hunters
leave one by one and emerge only when the hut is empty. But they start running
into problems higher up we go. The more the hunters, more their confusion. Even
for humans, experiments show that numbers between 1 and 3 are recognised much
faster and with much fewer errors than larger numbers.
RESULT
OF
From
this, cognitive psychologists have theorised that the number sense humans
have is a result of evolution. We have two kinds of number of number sense,
one for the numbers from 1 to 3, the other which is less fine grained and refers
to many without being exact. The ability to quantify numbers exactly above three
is the result of mathematical abstraction. But the ability these two types of
numerical sense a fine grained one up to three and a coarse grained one for more
is hardwired into the brain. Cognitive neuropsychologist, Stanislas Dehaene, the
author of The Number Sense and others have given detailed accounts of how
the mind is able to build on this innate mathematical hardwiring of the brain to
develop further mathematical notions.
Dehaenes
and such work have serious implications for the Neo-Platonists. If the human
mind constructs mathematical truths, and it can construct them in more than one
way. This spells the end of the Platonic immutable mathematical universe. But
the human mind cannot construct them in infinite number of ways, as social
constructivists would argue. George Lakoff, in his commentary on Dehaene in
Edge, an Internet forum, states, since it is embodied, that is, based on the
shared characteristics of human brains and bodies as well as the shared aspects
of our physical and interpersonal environments. As Dehaene said, pi is not an
arbitrary social construction that could have been constructed in some other
way.
SMALL
NUMBERS VS
LARGE NUMBERS
So
we come back to what does Peter Gordons results with the Piraha show? It does
confirm that the way we look at numbers the fine-grained structure of
small numbers up to three differs from that of larger numbers. Without
linguistic support for larger numbers, the human mind may not make the
transition to exact large numbers which otherwise the brain perceives in a
coarse grained way. The human mind may differentiate between 20 and 50 but not
between 20 and 21 unless the numeric construct is supported by the words for 20
and 21.
This
has serious implications for those who believe that the human brain is capable
of intuitively grasping all natural numbers and language helps only to refine
it. It shows the innate abilities of the brain are more limited and the
magnificent structure of mathematics that we have built up is neither a
discovery nor a reflection of an inner reality of the mind. Nor is it a
formal exercise devoid of any meaning. It is shared cultural construct built on
an innate number sense of the brain. In this it is very much like language,
which is also a shared cultural construct but based on an intrinsic hardwiring
of the brain. Noam Chomskis greatest achievement, whom the readers of Peoples
Democracy know otherwise as a radical activist and intellectual, is creating
almost single handed the discipline of modern linguistics. And it is this
insight that divorces modern linguistics from its predecessors. Not that
therefore we should try writing poetry in algebra (though the algebraists may
believe they are). But it is
nevertheless a special language with its specific characteristics. That it helps
to describe the natural world so well is no surprise, as its constructs are
created from this real world.
