Interesting, Margarita!
The logical fallacies you brought up reminded me of a book I bought in London last week,
Nothing, edited by Jeremy Webb. It contains a number of articles from New Scientist, and three of them deal with the number zero. Zero is extremely useful in math, but it is also dangerous. It is against the law of mathematics to divide anything by zero, because the answer will be infinity. But you are also not allowed to multiply zero by zero. Ian Stewart explained:
0 divided by 0 should mean ´whatever number gives 0 when multiplied by 0´. But since this is true whatever number you use to divide 0 by, unless you're very careful, you can fall into many logical traps - the simplest such being the ´proof´ that 1=2 because bot equal o when divided by 0.
By the way, I think I found a mix-up here - the quote should have said that both equal 0 when
multiplied by 0. Anyway, that's kind of fascinating.
In another text in the book, also written by Ian Stewart, we learn what ´
1´ really is. What is it really? ´1´can't be ´1 apple´ or ´1 shoe´or something like this, because ´1´ isn't a physical thing. So what is it really?
Ian Stewart answered that you have to use sets, because a number is always a part of a set. If you want to count the number of days in a week, you may compare the number of days in the week with the number of dwarfs of Snow White. You may say that Doc is the first of the dwarfs, and Monday is the first of the weekdays, so Doc is day one, or Monday. Grumpy, number two, is Tuesday, and Happy (3) is Wednesday, Sleepy (4) is Thursday, Bashful (5) is Friday, Sneezy (6) is Saturday and Dopey (7) is Sunday. By this logic, there are Dopey days in the week. (Imagine - TGIB (Thank God It's Bashful. Or, Germany beat Brazil by Dopey Doc in the World Cup semifinal.)
But this won't do, because we can't really count apples, shoes or Snow White's dwarfs. And there is no way to describe the final score of the final between Germany and Argentina that way: How do you say 1-0 in Snow White dwarfese?
Ian Stewart wrote:
Zero is a number, the basis of our entire number system. So it ought to count the members of a set. Which set? Well, it has to be a set with no members. These aren't hard to think of: ´the set of all honest bankers´, perhaps, or ´the set of all mice weighing 20 tonnes´. There is also a mathematical set with no members: the empty set. It is unique, because all empty sets have exactly the same number of members: none. Its symbol, introduced in 1939 by a group of mathematicians that went by the pseudonym Nicholas Bourbaki, is θ. So theory needs θ for the same reason that arithmetic needs 0: things are a lot simpler if you include it. In fact, we can define number 0 as the empty set.
What about the number 1? Intuitively, we need a set with exactly 1 member. Something unique. Well, the empty set is unique. So we define 1 to be the set whose only member is the empty set: in symbols, {θ}. This is not the same as the empty set, because it has one member, whereas the empty set has none. Agreed, that member happens to be the empty set, but there is one of it. Think of a set as a paper bag containing its members. The empty set is an empty paper bag. The set whose only member is the empty set is a paper bag containing an empty paper bag. Which is different: it's got a bag in it.
The key step is to define the number 2. We need a uniquely defined set with two members. So why not use the only two sets we've defined so far: θ and {θ}? We therefore define 2 to be the set {θ, {θ}}.
Now a pattern emerges. Define 3 as {0,1,2}, a set of three members, all of them already defined...
Interesting! That's what I thought, anyway. And talk about logic (and logical fallacies).
Ann
