British science writer Philip Ball, author of How to Grow a Human, offers an even-handed account of a controversy on the origin of our ability to understand numbers (numeracy). Numeracy is the beginning of mathematics, the most abstract of all human pursuits. It isn’t possible to get very far in mathematics without some ability to abstract.

Ball cites as an example the difference between 152 and 153. Many life forms, competing for a pile of food items, can distinguish between 2 and 3. But distinguishing between 152 and 153 clearly requires abstraction. It’s the same principle as the chiliagon, a geometric figure like a triangle except that it has 1000 sides. A triangle can be envisioned concretely. A chiliagon can only be envisioned abstractly.

The controversy he covers in his recent essay at Aeon, “How natural is numeracy?”, is whether the ability to work with numbers in an abstract way is cultural or biological:

In support of culture, he cites cognitive scientist Rafael Núñez of the University of California at San Diego and neuroscientist Wim Fias of the University of Ghent in Belgium.

‘The idea of an inherited number sense as the unique building block of complex mathematical skill has had an unusual attraction,’ said the neuroscientist Wim Fias … ‘It fits the general enthusiasm and hope to expect solutions from biological explanations,’ in particular, by coupling ‘the mystery of human mind and behaviour with the promises offered by genetic research.’ But Fias agrees with Núñez that the available evidence – neuroscientific, cognitive, anthropological – just doesn’t support the idea.

Philip Ball, “How natural is numeracy?” at Aeon

In support of biology, he cites neurobiologist Andreas Nieder of the University of Tübingen in Germany:

Nieder remains convinced that ‘our faculty for symbolic number, no matter how much more elaborate than the non-symbolic capacity of animals, is part of our biological heritage’. He feels that Núñez’s assertion that numbers themselves are cultural inventions ‘is beyond the reach of experimental investigation, and therefore irrelevant from a scientific point of view’. And he believes that a neurobiological foundation of numerosity is needed to understand why some people suffer from dyscalculia – an inability of the brain to deal with numbers. ‘Only with a neurobiological basis of the number faculty can we hope to find educational and medical therapies’ for such cases, he said.

Philip Ball, “How natural is numeracy?” at Aeon

A critic might retort, “Of course it’s just biology! Lots of animals can count!”

Well yes, they can, but for them it is not an abstraction. It seems always related to small, practical quantities:

In the late 1980s, researchers showed that chimpanzees could add up the number of chocolates in two food bowls (up to five pieces of chocolate in each bowl), compare it with the sum of two other food bowls, and correctly choose the larger of the two sums 90 percent of the time.

Joseph Castro, “Can Animals Count?” at LiveScience (December 3, 2017)

The ability to distinguish more from fewer chocolates probably has a biological explanation. But chimpanzee chocolate fans are not interested in the chiliagon. We need a different account of that interest. Evolution won’t help because we don’t know when or even how language emerged.

But now here’s the conundrum: If we say that number sense is cultural, what sustains it? Do numbers really exist or does human culture simply make them up? It turns out that that is a philosophical question:

Yet whether numbers really exist independently of humans ‘is not a scientific debate, but a philosophical, theological or ideological one’, said Núñez. ‘The claim that, say, five is a prime number independently of humans is not scientifically testable. Such facts are matters of beliefs or faith, and we can have conversations and debates about them but we cannot do science with them.’

Philip Ball, “How natural is numeracy?” at Aeon

And yet we can’t “do science” without an understanding of numbers either…

Humans have developed a large number of different counting systems: binary (used in computing), quinary (based on five), decimal (based on 10), duodecimal (based on 12), vigesimal (based on 20) and many others.

What all these systems have in common is that they are abstractions; they are gateways to an abstract world that our minds inhabit.

Keep going and you will end up with Chaitin’s unknowable number. And learn how to manipulate infinity in math. And ponder whether we can add new numbers to math.

There is no current science explanation for how humans come to inhabit such an abstract world. The answer can’t lie in biology because, given our biological similarity to chimpanzees, chimps should be able to do it too. But they are stuck with the “pile of chocolates” stakes. If our ability to abstract lies in culture, well, what’s the cultural key to the abstract world?

Mathematics may well be an argument for dualism, the idea that the universe is intrinsically dual. It is both concrete and abstract, depending. Both the Chimp Chocolate Stakes and Chaitin’s Unknowable Number.

You may also wish to read: Gregory Chaitin on the great mathematicians, East and West: Himself a “game-changer” in mathematics, Chaitin muses on what made the great thinkers stand out.

and

Yes, there really is a war on math in our schools. Pundits differ as to the causes but here are some facts parents should know.