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:
