The question of the relevance of natural language to counting and calculating capacities has been raised in recent years in connection with two similar research projects.
In a well-known study published in 2004, the counting abilities of the Piraha people of the Amazon were examined. Their language lacks number words (other than for one and two). The researchers suggested that language (rather than other societal or environmental factors) was the crucial factor in explaining the poor counting abilities of members of this tribe.
Though not everyone was convinced by the researchers' claims, more recent research on several adults in Nicaragua who were born deaf and never learned Spanish or a formal sign language provided some slightly more convincing evidence of the importance of language for counting ability.
Elizabeth Spaepen (of the University of Chicago) and her colleagues conducted experiments involving, for example, the experimenter knocking her fist against the subject's fist a number of times and asking the subject to respond with the same number of knocks. (Iteration, note, rather than objects.)
'So if I were to knock four times on their fist,' commented Dr Spaepen, 'they might knock my fist five times.' *
The earlier research on the Piraha involved similar tests and similar results, but there was nothing to say that language was the crucial factor. A stronger case for language being the key factor can be made on the basis of the more recent research, as the Nicaraguans, unlike the Piraha, were living in a culture rich in counting systems.
Daniel Casasanto (of the Max Planck Institute for Psycholinguistics) points out that the human brain is good at approximating, e.g. distinguishing between ten and twenty objects, but needs a counting system to distinguish between ten and eleven, say.
'What language does,' he explains, 'is give you a means of linking up our small, exact number abilities with our large, approximate number abilities.'
As I see it, language provides for individuals, societies and cultures a kind of bridge to sophisticated forms of counting and calculation. Number words (in conjunction with other aids like fingers) facilitate simple forms of counting and these form a basis for more advanced techniques incorporating symbols and calculating devices.
Though number words are an intrinsic part of language, counting systems by and large are not. And - significantly - the more sophisticated the counting and calculating systems are, the less dependent they are on natural language.
So I don't see any necessary or intrinsic link between natural language and counting systems.
Historically, it may well be that only societies with number words went on to develop sophisticated counting systems and mathematics generally. And it may well be that, for most human children, learning number words is a prerequisite for learning to count and do basic arithmetic.
But this does not mean that arithmetic is in any fundamental way dependent on natural language.
Even in terms of human psychology, the link between language and calculating ability is pretty tenuous.
Think of autistic savants, for example. Are there not many instances of individuals who lack the ability to use and process language and yet whose brains display advanced calculating abilities?
* Wittgenstein would have had a field day with this!