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Article 2037 of comp.ai.philosophy:
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>From: rctthdb@dutrun2.tudelft.nl (Han de Bruijn)
Newsgroups: sci.philosophy.tech,comp.ai.philosophy
Subject: Re: Existence
Summary: How about CHANGE?
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Date: 11 Dec 91 10:42:29 GMT
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Organization: Computing Centre of the Technical University of Delft The Netherlands.
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In article <1991Dec8.181146.2226@arizona.edu> Bill Skaggs:
> [ ... ] Number, defined in this way, gives rise to an equivalence relation
> between groups of things (the relation of "having the same number").
> In short, the psychological concept '7' is the property that all groups of 7
> things have in common.

Yes, this is _one_ way of looking at it. But I had quite different experiences
while raising one of my own children.

Disclaimer: this article appeared previously in "sci.philosophy.tech" under the
name "Teachability of Cardinals (WAS: New Math)".

When little Martijn was four years old, his parents attempted to learn him how
to count objects. Like: Martijn, how many apples are there in that basket?
  ____________      Though Martijn knew some of the counting words (ordinals):
 /      O     \     one, two, three, four, ..., he could'nt apply them to that
|  O         O |    set of apples (cardinals). This remained so for a while.
|        O     |    Observing him, however, it became clear that he allways
|  O      O    |    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. (The latter happened very quickly.) "I've counted
10 apples, mama!"

After a couple of fruitless attempts, we did the following.
Martijn was forced to take an apple OUT OF the basket, and at the same time say
a counting word. Then the miracle happened: he STOPPED counting after the last
apple was taken out. Halting problem solved!

  ____________             Examine now Martijn's counting process in detail.
 /            \            Taking apples out of the basket actually DESTROYS
|  O           |    O      the set you want to determine the cardinality of.
|              |  O        || Even more general, it is IMPOSSIBLE to count a
|  O      O    |   O       || set _without_ disturbing it, one way or another.
 \____________/            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 them, at least in your mind.
But as soon as an element is marked, then it isn't the "same" element anymore.
You have _changed_ the elements while counting them.

All this should'nt be surprising, since counting is a _physical_ process called
measurement, in its simplest form. According to quantum mechanics (and ordinary
experience confirms that) any form of measurement implies kind of disturbance.

We conclude that, in the process of counting, at least TWO sets are involved:
1. the set of apples which is still to be counted ("future");
2. 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 Martijn'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 much 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: the _dynamics_ of it is contradictory to the statics of
Modern Mathematics, with sets that eternally "exist" and never change ...

Last question: how would a computer or a robot perform on cardinal numbers?
-
* Han de Bruijn; Applications&Graphics | "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354   | would be NO idleness in  * Oil
* 2600 AJ  Delft; The Netherlands.     | Mathematics" (HdB).      * for
* E-mail: Han.deBruijn@RC.TUDelft.NL --| Fax: +31 15 78 37 87 ----* Blood


