Most parents have some experience with teaching their children the art of
counting. Nobody can deny that there exist large individual differences
between one child or another. Personally, I find children who are not
exhibiting so much "intelligence" from
themselves the more interesting. Maybe because they mimick accurately the
difficulties a stupid machine would have with counting the elements in a set.
(I mean: how would a computer or a robot perform on cardinal numbers ?) But,
in order not to embarrass somebody, let me give that "dumb" child of ours the
fictive name: Han (: my own). When teaching our little Han how to count,
we started with a few apples in a dish.
And we asked him: how many apples are there in that dish ?
Though Han knew some of the counting words (ordinals): one, two, three, four,
five,
..., he couldn't apply them to that set of apples (cardinals). This remained
so for a while. Observing him, however, it became clear that he always made
the same mistake. One way or another, he wasn't able to count each apple only
once. So he attached more than one counting word to each apple.
And he kept running in circles, until he was bored. "Ten, mama !".
Obviously, the most important idea to be grasped here is that the discrete
elements in a set have to be distinguished from each other. If our
little Han doesn't effectively understand how, or why, to do this, then he
will go on, until he becomes bored or - if you are lucky - until his limited
set of ordinal numbers becomes exhausted. As a rule, this will happen rather
quickly. Indeed, while looking
into that dish, all apples are looking the same. If the child is not allowed
to mark each of the apples, with a dirty wet finger for example, then he has
no means to distinguish one apple from another and he will count some of them
more than once. The keyword here is marking. But, instead of the dirty
wet finger, there is another, neater solution that can be thought of.
After a couple of fruitless attempts, we did the following. Han was forced to
take an apple out of the dish. And at the
same time say a counting word.
Then the miracle happened: he stopped counting after the last apple was taken
out. The Halting Problem was solved !
So, with every act of the kind, let it shout the next ordinal number. Then,
instead of the endless loops in the past, suddenly a stopping criterion
seems to be present. And, as soon as the dish is empty, there is no reason to count any further.
Photographs
Examine now Han's counting process in detail. Taking apples out of the dish
does actually destroy the set you want to determine the cardinality of.
Even more general, it is impossible to count a set without
disturbing it, one way or another. Even if you count apples by just taking a
good look, you have to shoot photons at them, since you can't see them without
light.
Apart from that, you have to remember which one has been enumerated
already. So you have to put marks upon the apples, at least in your mind.
But as soon as an element is marked, then it isn't the "same" element anymore.
The elements have been changed while counting them. Or, which is the
same: they belong to another set now.
All this should'nt be surprising, since counting is a physical process
called measurement, in its simplest form. According to quantum mechanics (and
common experience confirms it) any form of measurement implies a kind of
disturbance.
We conclude that, in the process of counting, at least two sets are
involved:
- the set of apples which is still to be counted ("future");
- the set of apples which has already been counted ("past").
The "future" set is destroyed, while the "past" set is created. (Re: What is
time ?)
A child fails to comprehend how to assign a number to the elements in a set.
I went through Han's school-books and, after all, I wasn't much surprised.
According to the teaching methods in those books, the children learn the order
of the counting words. Sure. And they learn when two sets have an equal number
of elements. But perhaps a more difficult problem is: how to assign a counting
word to the number of elements in a set. Mathematically speaking: what is the
relationship between ordinals and cardinals? In a sense that can be made clear
to a little child.
Because the teaching methods in our schools are related to modern mathematics
("New Math") we shouldn't be surprised that there are no adequate tools.
A theory like that of Cardinals cannot be used for deriving a suitable method.
I'm pretty sure that the above practice does not belong to standard equipment
for teaching numbers: because the dynamics of it is contradictory to the
statics of Modern Mathematics, with those sets that eternally "exist" and never
change ...