Does zero have a 'value'

[Dubium]

Does zero have a mathematical value? Just something we were discussing at work today. I said I'd ask. It sounds really simple but when you think about it... maybe not.

 

[scribble]

I'm no mathematician, so let me add to your question: does 1 have a value? How about 2?

 

[crimresearch]

Does zero have a mathematical value? Just something we were discussing at work today. I said I'd ask. It sounds really simple but when you think about it... maybe not.

It is a place holder between 1 and -1, so it has a value in the sense that all such integers do.

It can be plugged into operations and equations with predictable results, the same as any other value.

And it can certainly be treated as a value in the same manner:

'Hey! I wanted a pony for Christmas, and all I got was zero!'

or

'Hey! I wanted 4 ponies for Christmas, an all I got was 3!'

 

[Soapy Sam]

Three ponies ?
Or just three ?

Is "value" the same as "quantity"?

If not, does zero have a quantity?

When I was a kid in primary school, we did not use the word "zero". We said "nothing"- as in "You write ten by putting a one in the tens column and a nothing in the units column."

The term "zero" appeared in high school. We were told not to say "nothing".

I presume there was some sort of psychology involved.

 

[drkitten]

Three ponies ?
Or just three ?

A distinction without a difference.

The rules of English (like many other languages), permit cardinal numbers to be used as units of item measure (as in "three apples," or "seven brides for seven brothers"), and to be omitted when the items counted are clear from context. ("I have only one sister, but my cousin Gregory has three.")

"Zero" is an cardinal number like any other. And like any other cardinal number, it has a value.

Of course, cardinal numbers can be used in other contexts than item measure. They can be used as direct measurements ("that board is two meters long, while the longer one is three,") or as scalar units ("The temperature last night was got down to five [degrees].") Again, it's clear that the number zero has a value -- just as "five" describes a particular degree of coldness, so does "zero."

Of course, there are lots of other things that can be done with numbers besides simple measurement -- but these examples should be sufficient.

 

[Soapy Sam]

I think you miss my meaning.

Crimresearch said
'Hey! I wanted 4 ponies for Christmas, an all I got was 3!'

In a normal context of conversational English, that would usually mean "3 ponies", as you say, but in the context of a question about mathematics, might mean just "three" -ie the numerical concept.

"Three ponies" is a quantity,- a physical group of ponies.
"Three" is not a quantity. It's a value, an abstraction.

I see a fundamental difference here.

Zero is an oddity. I agree it has a value, but is that value also a quantity?

If so, how would we distinguish between "Zero garages" and "Zero invisible unicorns"?

 

[drkitten]

If so, how would we distinguish between "Zero garages" and "Zero invisible unicorns"?

The same way we distinguish between "three garages" and "three invisible unicorns" -- by context.

"Hey, my invisible unicorn detector is messed up. He is detecting three invisible unicorns, while I'm detecting zero!" obviously does not refer to garages.

 

[crimresearch]

I think you miss my meaning.

Crimresearch said
'Hey! I wanted 4 ponies for Christmas, an all I got was 3!'

In a normal context of conversational English, that would usually mean "3 ponies", as you say, but in the context of a question about mathematics, might mean just "three" -ie the numerical concept.

"Three ponies" is a quantity,- a physical group of ponies.
"Three" is not a quantity. It's a value, an abstraction.

I see a fundamental difference here.

Zero is an oddity. I agree it has a value, but is that value also a quantity?

If so, how would we distinguish between "Zero garages" and "Zero invisible unicorns"?

In both of my examples, the speaker has received a thing that they valued at a lower level than they expected. One has received 3, and one has received 0.

And a lower value is still a value.

 

[Achán hiNidráne]

This conversation reminds me of a scene for Stargate SG1:

Daniel Jackson and Maj. Carter are trying to rig a space mine to attack a G'ould ship. To do so they have to open a combination lock conceived by one of the many human cultures that the G'ould snatched off Earth in ancient times.

Their first attempt is a failure and the mine is counting down for detonation. Carter thinks the reason that they got it wrong is that they didn't figure zero into the combination. Daniel points out to her that most ancient earth cultures didn't even understand the concept of zero, so how could it? Carter replies that a culture that became technically advanced enough to create such a device would have to eventually come up with the notion of zero.

"Trust me, it's a math thing," she says.

 

[69dodge]

Originally posted by Soapy Sam
I think you miss my meaning.

Crimresearch said
'Hey! I wanted 4 ponies for Christmas, an all I got was 3!'

In a normal context of conversational English, that would usually mean "3 ponies", as you say, but in the context of a question about mathematics, might mean just "three" -ie the numerical concept.

"Three ponies" is a quantity,- a physical group of ponies.
"Three" is not a quantity. It's a value, an abstraction.

I see a fundamental difference here.

Zero is an oddity. I agree it has a value, but is that value also a quantity?

If so, how would we distinguish between "Zero garages" and "Zero invisible unicorns"?

If there's a garage in front of you, but no unicorns, then "zero garages" doesn't apply but "zero unicorns" does. So there's a difference.

Of course, if there are neither garages nor unicorns present, both phrases apply. There's no reason more than one description can't apply to the same situation.

 

[Paul C. Anagnostopoulos]

This is why I don't understand the old claim that such-and-such a civilization had no "zero." Of course they had "zero." They had to be able to say "I had a horse yesterday, but now I have none."

Perhaps what they didn't have was a place-value system with a digit for zero.

~~ Paul

 

[drkitten]

This is why I don't understand the old claim that such-and-such a civilization had no "zero." Of course they had "zero." They had to be able to say "I had a horse yesterday, but now I have none."

Perhaps what they didn't have was a place-value system with a digit for zero.

"None" is not "zero." More specifically, "zero" is a number, but "none" is not (and the difference is that you can work with numbers mathematically, but cannot do the same with simple words).

 

[Dubium]

Hmm, some interesting views. I was thinking something along the lines of:

My wallet is empty, my financial reserves do not exist. They therefore have a value of $0, which can also be stated as my wallet has no value at all in it.
The chair has no value, which equally states the chairs value is zero. How can "no value" also equal "value equals zero" giving a thing that is defined as valueless, a value at all?

Maybe zero doesn't actually exist per se, maybe there is no exact middle point between 1 and -1. I think this is actually the case because an entity that could go into such realms would encounter a decreasing positive value, and immediately on transition would begin to encounter negatives. There would be no "rest stop" in the middle.

Or another concept...there is matter, and anti-matter. There is no middle matter, simply one side or another. Black or white. No greys...

So zero is a flexible concept, which has no value and to use it in context where it is perscribed a value is technically incorrect. But it does make life easier...

 

[Wudang]

The chair has no value, which equally states the chairs value is zero. How can "no value" also equal "value equals zero" giving a thing that is defined as valueless, a value at all?

Ah no, just because two sentences can refer to the same concept does not necessarily mean that 2 equivalent phrases from those sentences are identical outwith the current domain of discourse.
"no value" in this case means that the concept of value applies and is 0. I think you're violating Russells theory of types in that you're equating a measurable property of an instance of an object with an abstraction.

 

[Paul C. Anagnostopoulos]

Drkitten said:
"None" is not "zero." More specifically, "zero" is a number, but "none" is not (and the difference is that you can work with numbers mathematically, but cannot do the same with simple words).

In current mathematical parlance, true. But if that backward civilization I was talking about begins to keep logs or ledgers of their horse trading, they are going to come up with a symbol that means "I don't have any horses." If they can do crude math with that symbol, along with the symbols for other values, then I'd say they had the concept of zero, no? They just didn't have the modern concept of zero.

This is not to say that their arithmetic is slick, but you can still do arithmetic with roman numerals and "none."

This little article equates "discovering zero" with place notation:

http://www.jimloy.com/math/zero0.htm

~~ Paul

 

[drkitten]

In current mathematical parlance, true. But if that backward civilization I was talking about begins to keep logs or ledgers of their horse trading, they are going to come up with a symbol that means "I don't have any horses."

Not necessarily. If they don't have any horses, they don't need to symbolize it (or to symbolize it in any way that would be usable in "crude math"). For example, one of the standard "backwards civilization" method of counting things involves tally marks on sticks. If I have three horses, I make three scratches on a stick, one per horse. If I have fifty, I make fifty scratches, and so forth.
I can then give you the marked stick as a representation of the number of horses I have.

So let's say that I have no horses. This system permits me to represent "zero horses" by using a stick with no marks on it. But it just as equally permits me not to represent the number zero, simply by not using a stick at all in that case. This is a clear example where the concept "none" (no stick) is different from the representation for "zero" (an unmarred stick). I actually consider it to be more likely that if I were to hand said primitive an unmarred stick as a representation of how many horses I have, he wouldn't know what the hell I was on about.

 

[69dodge]

Originally posted by Dubium
The chair has no value, which equally states the chairs value is zero. How can "no value" also equal "value equals zero" giving a thing that is defined as valueless, a value at all?

I don't understand. What's the difference between saying, "the chair has no value" and saying "the chair's value is 0"? Either way, you aren't willing to pay any money to get it, as you would if its value were positive, nor are you willing to pay any money to get rid of it, as you would if its value were negative.

Maybe zero doesn't actually exist per se, maybe there is no exact middle point between 1 and -1. I think this is actually the case because an entity that could go into such realms would encounter a decreasing positive value, and immediately on transition would begin to encounter negatives. There would be no "rest stop" in the middle.

Why should 0 be special in this regard? Why not also say, for example, that as a number decreases from slightly above 7 to slightly below 7, it never hits 7 exactly?

I don't think this line of reasoning is useful. It makes much more sense to say that there is a number exactly halfway between -1 and 1, namely 0, just as there is a number exactly halfway between 6 and 8, namely 7.

So zero is a flexible concept, which has no value and to use it in context where it is perscribed a value is technically incorrect. But it does make life easier...

I'm not sure I agree with that, but I do agree that context is important. Can you tell us more about how your discussion at work started? Was it just an abstract discussion about numbers, or was there some practical situation that gave rise to it?

 

[gnome]

I've gone to the moon a number of times.

Zero's a number

 

[gnome]

Something can have a value, and it's not the same question as asking if something has a positive value. The only reason the question seems confusing is that in common terminology we often mean the latter when we say the former. If you define the question more carefully the answer is obvious.

 

[Santa666]

Not necessarily. If they don't have any horses, they don't need to symbolize it (or to symbolize it in any way that would be usable in "crude math"). For example, one of the standard "backwards civilization" method of counting things involves tally marks on sticks. If I have three horses, I make three scratches on a stick, one per horse. If I have fifty, I make fifty scratches, and so forth.
I can then give you the marked stick as a representation of the number of horses I have.

So let's say that I have no horses. This system permits me to represent "zero horses" by using a stick with no marks on it. But it just as equally permits me not to represent the number zero, simply by not using a stick at all in that case. This is a clear example where the concept "none" (no stick) is different from the representation for "zero" (an unmarred stick). I actually consider it to be more likely that if I were to hand said primitive an unmarred stick as a representation of how many horses I have, he wouldn't know what the hell I was on about.

I would have to disagree here. You still end up with a representation of zero = none. Using the primitive example to maintain the flow. If I were to approach said primitive and ask (assuming for a moment I can communicate with him using his medium of communication) how I know how many horses each person has, how might he answer that? I would surmise, by showing me the sticks of others with tic marks to demonstrate the number of horses each person owns. Now if I ask, what if you do not have any horses? This question in and of itself is going to cause him to demonstrate how they represent having zero or no horses. He is not going to look at me funny because I asked what if you do not have horses, as if such a concept is beyond him. He is simply going to demonstrate how he represents no (zero) horses, likely by the means you mentioned, a stick with no tic marks.

The very concept of counting will invariably lead to the most basic concepts of zero. At its most fundamental level, zero = none, and this basic function of zero is what a primitive people would start with even they do not have a name for such a concept.


Santa