Peano Postulates

[olmari]

Originally Posted by R.D. Silverman

Zero equals Zero.

"Nothing" is not a mathematical concept. You are applying the common
English meaning of the word "nothing" in a context where it is not meaningful.

0 is a number. It is not "nothing"

Which value isn't positive or negative. Yes, it is numerical, you mark it with 0. You are right, I am not math-person. Instead I try to look things by common sense. How do you buy 0 pieces of apples?

And I am not saying that 0 isn't number and that, I wanted to tell why 0 isn't positive nor negative.

[ewmayer]

Originally Posted by olmari

How do you buy 0 pieces of apples?

Quite cheaply, one would hope.

[Wacky]

Originally Posted by olmari

You are right, I am not math-person. Instead I try to look things by common sense

But this is the fallacy. Math is not "common sense". It is a rigorous set of rules that result in a system that is self-consistent. If you change any of the rules, you either get a system that is not self-consistent (and therefore of no interest) OR you get a different mathematical system.

For example, two non-coincident parallel lines never meet is a "common sense" statement. It is also true in certain (Euclidian) geometries. However, you can create another (non-Euclidian) geometry where they do meet. In this geometry, many other things are not what you might expect. But, since the system is self-consistent, it too is a mathematical system. And you can derive some very interesting results therein.

But the point that Dr. Silverman was making is that BY DEFINITION, zero is well defined and required to have certain properties.

[olmari]

Originally Posted by Wacky

But this is the fallacy. Math is not "common sense". It is a rigorous set of rules that result in a system that is self-consistent. If you change any of the rules, you either get a system that is not self-consistent (and therefore of no interest) OR you get a different mathematical system.

For example, two non-coincident parallel lines never meet is a "common sense" statement. It is also true in certain (Euclidian) geometries. However, you can create another (non-Euclidian) geometry where they do meet. In this geometry, many other things are not what you might expect. But, since the system is self-consistent, it too is a mathematical system. And you can derive some very interesting results therein.

But the point that Dr. Silverman was making is that BY DEFINITION, zero is well defined and required to have certain properties.

And again you are correct, as I said earlier. I am now strictly saying reasons why zero has or has not positive nor negative value.

[Wacky]

Originally Posted by olmari

And again you are correct, as I said earlier. I am now strictly saying reasons why zero has or has not positive nor negative value.

No, your statements may attempt to make a "common sense" interpretation.
But the ONLY reason is ...
because IT IS DEFINED that way.

[olmari]

Originally Posted by Wacky

No, your statements may attempt to make a "common sense" interpretation.
But the ONLY reason is ...
because IT IS DEFINED that way.

Which leads to fact that someone needed define it that way first

[Numbers]

olmari,
I’m afraid that your comment defeats your own argument. By pointing out that zero was invented by being defined exactly as Wacky and Dr. Silverman explained, you have shown that it is clearly not synonymous with “nothing” which, as a concept, already existed. If “nothing” served its purpose as zero, there would have been no need to invent something else.

[olmari]

Originally Posted by Numbers

olmari,
I’m afraid that your comment defeats your own argument. By pointing out that zero was invented by being defined exactly as Wacky and Dr. Silverman explained, you have shown that it is clearly not synonymous with “nothing” which, as a concept, already existed. If “nothing” served its purpose as zero, there would have been no need to invent something else.

Well, gotta mark that nothing in some way. And why we all stick with that same argument? As I only tried to understand WHY it isn't neither positive or negative. It IS defined that way, but REASON is because it is essentially way of marking that you don't have anything, positive nor negative, and you need to still tell that you don't have it...

[Numbers]

Zero is not “nothing”.

In your post you have used the idea of having nothing, and the idea of not having anything as though they were the same. But since “nothing” is not the same as “no thing”, there is a clear difference between having nothing and not having anything. And yet you use these different philosophical concepts as though they were both exactly equal to zero. When in fact neither of them is even remotely close to zero because they come from different languages.

Think of maths as being a language. Zero is a word in that language. Nothing is not a word in the language of maths. So that I can write a mathematical sentence that includes the word 0, but I cannot write a mathematical sentence that includes the word “nothing” or even the concept of “nothing”. I can even define a function that operates on a set that includes the number 0. But the closest I can get to “nothing” in maths is the empty set, and that doesn’t even have a zero in it, and if it did it would no longer be the empty set.

Zero is not “nothing”.

So whilst you might feel the need to “mark that nothing in some way” you cannot do it in the language of maths.

“but REASON is because it is essentially way of marking that you don't have anything” Wrong. When you have zero, that is exactly what you have, zero. And that is very different to not having anything, because you have zero and that is something.

Zero is not “nothing”.

[ewmayer]

Originally Posted by olmari

As I only tried to understand WHY it isn't neither positive or negative.

Look, it's very simple: positive is *defined* as "greater than zero," negative as "less than zero." Given any 2 real numbers x and y, upon comparing them there are precisely three mutually exclusive possibilities: x > y, x = y, x < y. It is axiomatic that any number equals itself. So, since zero itself is neither greater than nor less than itself, it is neither positive nor negative. Do we really need dozens of postings about this kind of thing?

If you want to quibble endlessly about semantics or how many angels can dance on the head of a pin, I suggest you make a career in politics or theology, respectively. In the meantime, stop wasting everyone else's bandwidth and time.

[maxal]

The two definitions of natural numbers have the following nature:

1) Natural numbers are those used for counting things: one, two, three, ...

2) Natural numbers are cardinalities of finite sets. They are zero (the cardinality of empty set), one, two, ... This definition was sold by Burbaki.

Which definition is more popular actually depends on the country: e.g. the first is used in Russia, while the second is used in France. But now mathematicians seem to avoid using the concept of "natural number" (unless when it is precisely defined) in favor of "positive" and "non-negative" integers.