Subject: Re: What is a number?
Date: Sun, 07 Mar 2000 13:21:37 -0500
From: Alex Bogomolny
If I may ask you: What is a star? What would you say. I am pretty sure that you'd be cornered very easily. A falling star? The one on the USA banner? Star performer? Cut out of paper? Abstract symbol?
So numbers are also different. There are numbers that have nothing to do with quantities. It's not quite circular to start with natural numbers just because every one has some idea about them. The problem is to abstract from the properties of natural numbers something that may become the a common attribute of all possible "numbers". Quantity does not fullfill this requirement.
The best one can do is concentrate on algebraic operations defined for the natural numbers. You'll get groups, rings, fields, etc. To the best of my knowledge the word "number" seldom plays a significant role in any of these theories. Thus, it appears that, as far as the current usage goes, search for the meaning of the word "number" is rather misplaced. You say "I use the word number, therefore, there must be some meaning to it." I say "Who cares? I shurely do not. Neither any body of my acquaintance."
To sum up, there is no definition of "number" as such but only a few terms like "natural number", "rational number", "complex number", etc. (The latter, in particular, has nothing to do with quantities.) That's all - no philosophy involved.
If you want this to be otherwise, you should define the word number somehow before trying to grasp its meaning. Personally, I do not care whether you or anybody else goes for it.
Now regarding zero. If you say that 0 is not quite a number, I'll reply that, in my view, you are wrong. Furthermore, from the algebraic perspective - which is the only unifying aspect of all various "number" I can think of - 0 is not only a number, but the most important of them all. Without 0 there would never be all "other numbers" - anything that goes beyond pure counting.
But it is an exceptional number. Have you never heard there are exceptions sometimes? Very often you can turn right on red. But sometimes you can't.
How do you define division? a/b = c is the same as a = bc, right? But this does not make sense for 0. For, if b = 0, then you should claim a = 0*c, for any a and c, which is, of course, not true. Multiplication properties of 0 are different from those of any other number. Why its division properties must be the same as everyone's?
All the best,