Monday, 28 September 2015

Easy as 1, 2, 3 ? -- Wittgenstein on counting

By Catarina Dutilh Novaes
(Cross-posted at NewAPPS)

“A B C
It's easy as, 1 2 3
As simple as, do re mi
A B C, 1 2 3
Baby, you and me girl”

45 years ago, Michael Jackson and his troupe of brothers famously claimed that counting is easy peasy. But how easy is it really? (We’ll leave aside the matter of the simplicity of A B C and do re mi for present purposes!)

Counting and basic arithmetic operations are often viewed as paradigmatic cases of ‘easy’ mental operations. It might seem that we are all ‘born’ with the innate ability for basic arithmetic, given that we all seem to engage in the practice of counting effortlessly. However, as anyone who has cared for very young children knows, teaching a child how to count is typically a process requiring relentless training. The child may well know how to recite the order of numbers (‘one, two, three…’), but from that to associating each of them to specific quantities is a big step. Even when they start getting the hang of it, they typically do well with small quantities (say, up to 3), but things get mixed up when it comes to counting more items. For example, they need to resist the urge to point at the same item more than once in the counting process, something that is in no way straightforward!

The later Wittgenstein was acutely aware of how much training is involved in mastering the practice of counting and basic arithmetic operations. (Recall that he was a schoolteacher for many years in the 1920s!) Indeed, counting and adding objects can be described as a specific and rather peculiar language game which must be learned by training, and which raises all kinds of philosophical questions pertaining to what it is exactly that we are doing when we count things. Perhaps my favorite passage in the whole of the Remarks on the Foundations of Mathematics is #37 in part I:

Put two apples on a bare table, see that no one comes near them and nothing shakes the table; now put another two apples on the table; now count the apples that are there. You have made an experiment; the result of the counting is probably 4. (We should present the result like this: when, in such-and-such circumstances, one puts first 2 apples and then another 2 on a table, mostly none disappear and none get added.) And analogous experiments can be carried out, with the same result, with all kinds of solid bodies.---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.

What Wittgenstein seems to be saying here, among other things, is that the practice of counting is useful only insofar as certain conditions are in place, in particular the discreteness (so that they can be counted) and perdurance of objects, which ensures that collections of objects have a stable numerical value when no external interference occurs. If objects would regularly fuse and multiply, or appear and disappear spontaneously, then counting and doing sums would not be a very meaningful, useful (usable!) practice at all. (This also means that the practice of counting is not a purely arbitrary construction; its fruitfulness depends crucially on how things happen to be in the world.)

In fact, the stability of objects in this sense may be a necessary condition for the meaningfulness of the practice of counting, but it still does not seem to be a sufficient condition. Recent anthropological work by Pica and others argues that some tribes in the Amazon speak languages that do not have words for counting beyond 2 or 3, as beyond that counting words are not used in any consistent way. In terms of the metaphysical properties of objects, members of these cultures live the same world as ours, and yet for various reasons the need for the practice of counting seems not to have emerged in these cultures. Indeed, when given tests with simple arithmetical operations with exact quantities, members of this culture perform ‘poorly’ for our standards; this seems like a clear win for Wittgenstein on counting. (Notice that they perform just fine when it comes to tasks involving approximations.)

Recent developmental work on numerical cognition also seems to lend support to the Wittgensteinian view of counting as training. While there is still much debate on the extent to which infants and very young children have some ‘innate’ sense of quantities, quite some results suggest that mastery of the practice of counting in exactly the way we do it – that is, based on the concept of a linear progression of the natural numbers, where the difference between 1 and 2 is ‘the same’ as the difference between 9 and 10 – requires extensive training, and becomes fully established only once children have been suitably ‘indoctrinated’, typically by formal schooling.

The upshot is thus that, pace the Jackson brothers, there is nothing easy about 1, 2, 3 – something that the later Wittgenstein already knew all too well, and which has been recently corroborated by anthropological as well as developmental research on numerical cognition. To be sure, there are still competing theories on numerical cognition among psychologists and cognitive scientists, and some of them are closer to the Wittgensteinian conception of counting as a language game requiring extensive training than others. But many of Wittgenstein’s insights have been borne out by recent research, which means that they provide an empirically-motivated starting point for a fully-fledged philosophical account of counting and basic arithmetic.

(This is a piece written for Qualia, the student-run magazine at the Faculty of Philosophy in Groningen.) 


  1. In this connection I very much like Wittgenstein's discussion of counting raindrops. He says, "It is like saying of falling raindrops, `Our vision is so inadequate that we cannot say how many raindrops we saw, though surely we did see a specific number.' The fact is that it makes no sense to talk of the number of drops we saw."

    1. Exactly! And this is because raindrops do not have the required metaphysical properties to be something one can count: they merge, they multiply, and they don't have proper individuating conditions.

    2. Not sure if that's quite right, see the quote from Wittgenstein in the page of the book linked below for instance. I've only found this blog post and that excerpt as sources on this particular phrasing of the quote so who's to say.

    3. Needs a source! Where did you find this?

  2. Having a child in the process of becoming numerate/literate, I've been struck by the correlation (as I see it) between the necessity of teaching a child how to count objects without duplication (usually by teaching them to count in an ordered fashion, e.g., back and forth in lines) and the necessity of teaching a child to string letters in a similarly ordered fashion. Gwen loves taking dictation -- she'll name me a word and I'll spell it for her to write down, but she for the most part lacks a sense of writing them in a line, left to right. Writing a line has been developing; she's pretty good at that. But often she'll write left to right until she reaches the edge of the page, and then start the next line right to left!

    The idea that the sequential ordering of things is relevant to the information that they carry seems to be the (for her, at least) the primary barrier between not reading and reading. She knows all her letters, she knows all the sounds they make, she knows that words can be turned into letters and written down, but the step from turning written letters back into sounds hasn't happened yet, in part because I don't think she's grasped the importance of doing it in a linear fashion, in a particular order.

  3. I can't believe I found this just now! I've recently been thinking heavily about what counting really is, and found it disturbing that a statement such as "2+2 = 4" in the everyday sense (outside of some kind of formal arithmetic) is really not as "innate" or precisely meaningful as I thought it would be. It seems that it is a kind of encapsulating "name" for a learned process that I can't really describe. Maybe I'm confusing myself over nothing, I'm not so sure. Perhaps you have some sort of insight/references on this?

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