New APPS

A few weeks ago I had a post on different ways of counting infinities; the main point was that two of the basic principles that hold for counting finite collections cannot be both transferred over to the case of measuring infinite collections. Now, as a matter of fact I am equally (if not more) interested in the question of counting finite collections at the most basic level, both from the point of view of the foundations of mathematics (‘but what are numbers?’) and from the point of view of how numerical cognition emerges in humans. In fact, to me, these two questions are deeply related.

In a lecture I’ve given a couple of times to non-academic, non-philosophical audiences (so-called ‘outreach lectures’) called ‘What are numbers for people who do not count?’, my starting point is the classic Dedekindian question, ‘What are numbers?’ But instead of going metaphysical, I examine people’s actual counting habits (including among cultures that have very few number words). The idea is that Benacerraf’s (1973) challenge of how we can have epistemic access to these elusive entities, numbers, should be addressed in an empirically informed way, including data from developmental psychology and from anthropological studies (among others). There is a sense in which all there is to explain is the socially enforced practice of counting, which then gives rise to basic arithmetic (from there on, to the rest of mathematics). And here again, Wittgenstein was on the right track with the following observation in the Remarks on the Foundations of Mathematics:

This is how our children learn sums; for one makes them put down three beans and then another three beans and then count what is there. If the result at one time were 5, at another 7 (say because, as we should now say, one sometimes got added, and one sometimes vanished of itself), then the first thing we said would be that beans were no good for teaching sums. But if the same thing happened with sticks, fingers, lines and most other things, that would be the end of all sums.

“But shouldn’t we then still have 2 + 2 = 4?” – This sentence would have become unusable. (RFM, § 37)

Working out an account of numerical cognition as a social practice has been on my to-do list for a while, but somehow I never got around to it. Well, as it turns out, now I probably don’t need to do it anymore: at the highly enjoyable conference  'Deductive mathematical cognition and philosophy' earlier this week in Bristol, our very own Helen de Cruz gave a fantastic lecture which covered pretty much all I would have liked to say on the topic and much more, with her characteristic command of the empirical literature combined with impressive philosophical sophistication. She correctly pointed out that much of the literature on numerical cognition as well as the philosophical discussions on the topic rely too much on the problematic idea of the child as the ‘lone scientist’ who explores the world on her own. This is a picture of learning that goes back at least to Rousseau, and reaches its pinnacle in 20^th century psychology and education with Jean Piaget and Maria Montessori. This passage from Rousseau’s Emile quoted by Helen summarizes the idea quite aptly:

Make your pupil attentive to natural phenomena, and you will soon make him curious; but, in order to nourish his curiosity, never be in haste to satisfy it. Ask questions that are within his comprehension, and leave him to resolve them. Let him know nothing because you have told it to him, but because he has comprehended it himself; he is not to learn science, but to discover it. If you ever substitute in his mind authority for reason, he will no longer reason; he will be but the sport of others’ opinions. (Emphasis added)

The picture of the child as the lone scientist is starting to crumble down (about time!), and is convincingly criticized for example in Paul Harris’ recent book Trusting what you’re told – How children learn from others (here is a previous post of mine discussing Harris' work). With respect to learning the natural numbers and developing numerical cognition in particular, Helen argued that (as with learning in general) testimony and imitation play a phenomenal role, and this social component so far has been mostly neglected. Indeed, learning number words and their ‘proper’ sequence is in first instance for the child no different from learning other rhymes and chants, with no particular meanings attached to each of the words. Clearly, this is a thoroughly social practice, and what motivates the child to learn the sequence is the drive to imitate, please and share experiences with caregivers. (Indeed, it is now known that humans have a tendency towards overimitation that chimps, for example, do not display.)

Naturally, being able to recite number words in the right sequence does not suffice for the emergence of what we refer to as number cognition: the child must also learn that each of these words is mapped into a particular quantity of discrete objects in a collection. And how does the child learn this? As everyone who is or has been the caregiver of a young child probably recalls, this is done by continuous, tedious repetition of another routine, that of ‘counting’ — often consisting in bringing the child’s hand to each of the objects in a collection and counting with them: ‘one, two, three, four… how many toys do we have here?’ True enough, this routine usually does not go beyond 10 (except for the more anxious caregivers perhaps), and yet at some point the child is able to make the ‘inductive step’ of concluding that the procedure could be repeated for an arbitrarily large (finite) collection of objects. (When she was about 5, my older daughter once said to me: "Mummy, there is no such thing as the uncountable; you can always keep counting!")

This remains a fascinating cognitive step (often explained in terms of 'bootstrapping'), but it clearly presupposes the social reinforcement of learning number words, and then learning to associate them with specific quantities – the counting routine properly speaking. In cultures where counting is not a socially pervasive practice (the more extreme cases being the cultures with no words for numbers beyond 3, such as the Pirahã and the Munduruku in the Amazon), predictably, numerical cognition does not evolve in the same way. This is not to say that there is no numerical cognition in these cultures; but it does not follow the pattern of ‘our’ sequence of the natural numbers, as famously studied by S. Dehaene and colleagues.

Let me conclude with the concluding points from Helen’s own talk: children grow up in a world seeded with artifacts, counting songs, and other features that help them learn; they are taught by benevolent communicators who point out salient features of cardinality (exactness, discreteness). Any explanation of numerical cognition must not focus exclusively on what goes on in the child’s mind, but crucially also on what happens in a broader teaching, social context. The picture of the child developing numerical cognition on her own, by interacting with the objects in the world around her as a ‘lone scientist’, is simply dead wrong. (Helen, I can’t wait for you to write up this paper!)