Patterns in language yield patterns in thought. Extensive research has now demonstrated that differences between languages can yield differences, often subtle ones, in the cognitive habits of their speakers. This finding, commonly referred to as linguistic relativity, has now been supported by dozens of studies on topics like spatial awareness, the perceptions of time, and the categorisation of colours. For instance, “where” the future and past “are” depends on the language you speak. Similarly, the manner in which you recall and discriminate colours is affected in sublte ways by the basic colour term inventory of your native language. Our tour of the numberless worlds ultimately led to the conclusion that numeric language also yields difference in how people think. Number words, present in the vast majority of the world’s languages (though not all of them), certainly influence quantitative cognition. Only those people who are familiar with number words and counting can exactly differentiate most quantities. The presence of numbers in a language does not just subtly influence how we think about certain quantities, then; it also opens up a door to the world of arithmetic and mathematics. The first step through that door is the realisation that quantities, regardless of size, can be precisely differentiated. But how exactly do numbers first open this door? And what happens after we walk through it?
The findings from numberless worlds suggests plainly that we need numbers to really “get” quantities in ways that are uniquely human, but, this raises a paradox. If we need numbers to appreciate most quantities precisely, how did we get numbers in the first place? How could we ever name the amounts in particular sets of items, if we could not recognise the amount?
Given the apparent intractability of this paradox, some have concluded that humans must be innately predisposed to acquire number concepts. But, if we are predisposed to recognise different set sizes as separate abstract entities, then what is the limit to this predisposition? Are we naturally predisposed, for example, to eventually realise that 1,023 is not 1,024? This seems fairly implausible. Framed differently, nativist views on numbers just delay the point at which we reach the paradox.
James Hurford noted that number words are names for the “non-linguistic entities denoted by numbers.” That is, the number words label conceptual entities. In a related vein, Karenleigh Overmann recently suggested that “quantity concepts must surely precede their lexical labels, or there would be nothing to name… A method of invention cannot presuppose that which it invents.” This latter stance is understandable, but it arguably trivialises the extensive evidence, according to which, words for quantities beyond three do not simply label pre-existing concepts, because these concepts do not exist for most people until they actually learn numbers.
In my view, this is the key to resolving the paradox: words for quantities beyond three make concrete the precise numerical abstractions that are only occasionally and inconsistently made by some people. Some of these people may eventually invent numbers, but if they do not, their fleeting abstractions are not transferred to others. The naming of such ephemeral realisations is what eventually enables people to consistently show the ability to make a simple but powerful realisation, the realisation that sets of quantities greater than three can be identified precisely. This simple realisation has led, in all likelihood more times than could be documented, to the invention of symbols for such larger quantities. These symbols are chiefly verbal in nature, judging from the fact that the overwhelming majority of the world’s cultures have words for such quantities though most cultures traditionally lack written numerals or elaborate tally systems. Some people invented number words to concretise the potentially transient recognition of the existence of exact higher quantities.
Does this mean that number words simply serve as labels for the concepts? Not really. The truth seems a bit more nuanced than the forced dichotomous choice assumed by the paradox. Number words are not simply labels, yet they do describe conceptual realisations that some people make some times. The term ‘label’ implies that the words simply denote concepts that we all think about: concepts all humans are born ready to appreciate (at least eventually), regardless of their cultural environment. But clearly not all humans have such concepts at the ready even as adults, and likely most people would never make the relevant realisations that can be described via numbers. Just as clearly, though, some people have made those realisations, even if inconsistently. In those real historical cases in which people managed to describe that realisation with a word, they invented numbers. The concept they named was subsequently recognised by other members of their culture through the adoption of the relevant word(s). Number words are conceptual tools that get passed around with ease, tools most people want to borrow.
Caleb Everett, Numbers and the Making of us: Counting and the Course of Human Cultures