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.)

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."

ReplyDeleteExactly! 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.

DeleteHaving 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!

ReplyDeleteThe 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.

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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