FOM: One thing and another (Barbie hath said)

[Dean Buckner]

I asked for arguments, and got arguments.  Apologies to William Tait if I
missed any of his points.

Some thoughts ...

(a) Tennant points to young children needing to count twice (first the
steps, then the stiles).  Is that because they simply can't count more than
one lot of things at a time?  If I put 5 peaches in a pail, then I put the
five stones in too.  But if I count 5 peaches, do I also count the five
stones?  There's an intentionality built into "count".  Young children may
notice the stiles, but maybe they can't count them.  Is noticing the same as
counting?

(b)  More damaging is the "nursery rhyme" objection.  There is a point
before which they just recite the numbers as they go up the stairs, and it
seems like an accident, depending on the speed they recite, that they
"count" at all.  But at some point they get the intentionality right, they
match stairs to numerals.  Question, at this point do they also grasp that
there is something common to all similarly-numbered sets?

(c) Aristotle wrote, we presume, in ancient Greek, but I can still say (in
English) what he said.  Similarly I can grasp that there are two things,
(e.g. if I'm an ancient Greek)  without understanding the meaning of the
English word "two".  Whether I can grasp this without understandig any
language is I concede a different question.

Another related point.  For young children often grammar gets in the way of
maths.  Lizzie (7) can understand 10-7 = 3.  but she still has a problem
with "the number that remains when three is subtracted from ten".

(d) Tait & a number of people (offlist) were rightly suspicious on the move
from being able grasp that there are two things, to a general conception of
number.  I concede this.  I just think the children who grasp this, grasp
something .  Maybe nothing important.

But, then, what is it to grasp that there are two hundred and eighteen
things?  A study of Chinese children found they were much quicker at
grasping the numbers 11 ("ten-one"), 12 ("ten-two")and so on.

(e)  If we do accept (as all respondents actually did) that if there is one
thing and another thing, then there are two things, couldn't we devise a
theory of number without any of the psychologising that Tait so despises?
Or the Platonising that I despise?

Let's define:

    there is one thing and another thing =df there are 2 things
    there is one thing and another thing and another thing =df there are 3
things

Then if we are allowed to substitute identically-meaning expressions, you
can show that if there are two things and another thing, then there are 3
things.  This without having to assume anything exists!

It's much harder to prove that if there are two things and another thing,
then there is a thing and two other things, but will leave as an exercise.

(f) On Piaget (Hazen raised this) I think beware.  A number of recent
experiments have raised questions about the idea of "stages" & other
Piagetiana.

I am assembling list of recent (psychology) literature for anyone
interested.  I am as sceptical as Tait about this sort of stuff, but
nonetheless watch with interest.





Dean Buckner
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