The Number Sense: Small Numbers Have Large Impact

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 Gordon’s interaction was also through Dan and Karen. Dan disagrees with Peter Gordon’s 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 Chomski’s 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 intuitionist’s scheme of things, as they do not accept any development, which cannot be constructed. As a major part of today’s 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 (Euclid’s 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. Hilbert’s 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. Hilbert’s 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 EVOLUTION

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.

Dehaene’s 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 Gordon’s 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 Chomski’s 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.