Does the concept of a quantity we call ‘five’ – a non-changing idea of a quantity greater than four but fewer than six – exist independently of any one person thinking about it, a concept that has been stable over time, across culture, independent of language? In other words, do numbers fall into the category of concepts that meet the definition of metaphysics: first principles that do not change over time, a universal notion that can be applied to particulars, and the teleological doctrine of causation that provides us with evidence of design or purpose in nature? Numbers have often been used as an example of a metaphysical ‘objective reality’ existing beyond or externally to ‘subjective experiences’ of this reality. But is this evidence actually true?
In this fascinating article we look at how our brains conceptualize quantity by use of numbers.
To start off with a bang, it appears that numbers are probably no more than about 10,000 years old! How do we know this?
When numbers are spread out evenly on a ruler, the scale is called linear. When numbers get closer as they get larger, the scale is called logarithmic. And it turns out the logarithmic approach is not exclusive to Amazonian Indians – we are all born conceiving numbers this way. In 2004, Robert Siegler and Julie Booth at Carnegie Mellon University in Pennsylvania presented a similar version of the number-line experiment to a group of kindergarten pupils (average age: 5.8 years), first-graders (6.9) and second-graders (7.8). The results showed in slow motion how familiarity with counting moulds our intuitions. The kindergarten pupil, with no formal maths education, maps out numbers logarithmically. By the first year at school, when the pupils are being introduced to number words and symbols, the curve is straightening. And by the second year at school, the numbers are at last evenly laid out along the line. There is a simple explanation. The logarithmic scale also takes account of perspective. For example, if we see a tree 100 metres away and another 100 metres behind it, the second 100 metres looks shorter. Our understanding of the passing of time tends to be logarithmic. We often feel that time passes faster the older we get. Yet it works in the other direction too: yesterday seems a lot longer than the whole of last week.
Our deep-seated logarithmic instinct surfaces most clearly when it comes to thinking about very large numbers. For example, we can all understand the difference between one and 10. It is unlikely we would confuse one pint of beer and 10 pints of beer. Yet what about the difference between a billion gallons of water and 10 billion gallons of water? Even though the difference is enormous, we tend to see both quantities as quite similar – very large amounts of water. Likewise, the terms millionaire and billionaire are thrown around almost as synonyms – as if there is not so much difference between being very rich and very, very rich. If our brains can represent numbers only approximately, then how were we able to “invent” numbers in the first place?.
“The ‘exact number sense’ is a [uniquely] human property that probably stems from our ability to represent number very precisely with symbols,” concluded Nieder. Which reinforces the point that numbers are a cultural artefact, a man-made construct, rather than something we acquire innately.
And the oldest anthropological evidence for precise symbols for numbers only goes back about 10,000 years!