Is there just one Zero?

Question

Once i was discussing Zero with my kids, i picked up a pen and asked what it is they said 1 pen, then i kept the pen and showed them empty hand and asked how about now, they said Zero pen, then I picked up notebook, did the same, 1 notebook, Zero notebook, pencil etc. Then one of the kids remarked that Zero of Everything looks exactly same!!!! So is it that there is SEPARATE one of each thing(different from others) BUT JUST ONE Zero for EVERYTHING. So is it that Zero as an object, an entity or anything, exists as one single copy. Any Equation any place any time if wrote this symbol >> "0", that symbol is pointing to just one same single instance of Zero.

I don't know whether it makes sense to ask the question that way, but I found the idea intriguing...

[EDIT]

Just to give you some context, I am studying Mind as much as I am studying any particular Subject, so when I ask such questions it's like WHOLE UNIVERSE. Since I also look at Mind so it becomes a question of "IN THE WHOLE UNIVERSE". So if you have an Apple, universe knows about it, if you don't have a Apple, Universe doesn't know that there is, was or can be a thing called Apple. There is no system or context smaller than the WHOLE UNIVERSE.

Answer 1

In group theory, the "zero" is better referred to as the identity element, usually within what are additive groups.

This is not the same in every group.

For matrices, it is the n × n zero-matrix.

For functions from ℝ → ℝ, it is the function mapping every number to zero.

For the permutation group, it is the operation mapping every element to its own position (some conceptions of zero would exclude this as a zero, and merely consider it an identity element, since it is not in relation to an additive group):

For vectors in ℝⁿ, it is the zero-vector of dimension n.

• Note that saying there are "no notebooks, no pencils, and no pens" can be translated to the zero-vector of dimension 3, where each dimension is the count of a different kind of thing. This could be extended to cover the entire space of countable things.
• Zero notebooks would be [0, b, c], for any values of b and c.
• Zero pencils would be [a, 0, c], for any values of a and c.
• Zero pens would be [a, b, 0], for any values of a and b.
• So a hand with "zero pens" does not necessarily look the same as a hand with "zero notebooks." The first might have a notebook in it; the second might have a pen in it.
• But the actual "zero" in this group is the full zero-vector.

While authors will often denote these various zeros in different groups using the same symbol "0" or "∅", they refer to different conceptual objects in each group.

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^{Your thinking as presented in the question and some of your follow-up comments somewhat conflates the concept of zero with the concept of nothing (what some other answers identify with the empty set). This may in part just be terminological, since you may not have had these terms at the ready. In any case, since you used the word "zero," I have presented an answer focusing on its varied forms in different groups. Other answers are also correct where they say there is only one empty set: {}.}

Answer 2

Try holding only a book in your hand, and asking them how many pens you're holding.

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Zero doesn't exist at all. Neither does one, two, three, etc.

These are just concepts we often (but not necessarily) ascribe to collections of things that actually exist in the world.

* Well, mathematical realism says these concepts do actually themselves exist, but even there, the concepts are still separate from what we ascribe them to.

When you're saying there is 1 pen, there's a pen that actually exists in the world, which you're referring to. When you say there are 0 pens, that's referencing an absence of things (pens) in the world. If there are 0 pens, there may 0 books, or there may more than 0 books. So they're kind of "the same" in the sense that they both refer to an absence of things, but they're kind of different in the sense that they're referring to an absence of different things.

As for the concepts themselves: There is only 1 concept* for zero, but there's also only 1 concept for one, and 1 concept for two, etc. When you say there's one pen, and you say there's one book, that's the same concept of one being used there, even though it's being ascribed to different collections of things.

* At least on the most simple level. One might debate whether an integer one is different from a real-valued one, and whether ones in different groups are the same, and there's also set theory.

For set theory, as other answers mention, you could have many different sets of 1 element. And there's only 1 set with 0 elements. This is because sets themselves don't have a type. It's not "a pen set" or "a book set". A set containing 0 elements would "contain" 0 pens and 0 books and 0 everything else (which is not to say there's a "0 books" element inside it, but rather that it doesn't contain any book elements). A set containing 1 element would also contain 0 books and 0 everything else.

Somewhat related: the history of the number zero, and how some societies didn't even have a symbol or placeholder for zero, and some opposed using it as a number.

Answer 3

There are many related issues:

• the symbol 0: different in different languages/notations,
• collections of objects with the same number of objects: how collections of NO objects at all can differ? and finally,
• the number ZERO: what is it?

Note: if we accept to speak of "the number zero" we already assume that there is one individual named "zero"; compare with "the US president".

"there is SEPARATE one of each thing(different from others) BUT JUST ONE Zero for EVERYTHING".

Not exactly. All collections of NO objects are the same: there is no way to see any difference, AND the number of objects in collections of no objects at all is ZERO (the same for all).

Collections of ONE objects are different because there are differente objects in each one, but they all have the same number of objects, that is ONE.

Thus also the number ONE is unique.

See Frege on numbers:

Frege argues that numbers are objects and number statements assert something about a concept. Frege defines numbers as "logical objects" related to concepts [their extension]: 'The number of F's' is defined as the extension of the concept 'G is a concept that is equinumerous to F'. Frege defines 0 as the extension of the concept 'being non self-identical'. So, the number of this concept is the extension of the concept of all concepts that have no objects falling under them.

In this way, zero is exactly the number of all collections having no elements.

And see also Number Concepts. An Interdisciplinary Inquiry.

Answer 4

This is a corollary of the fact that there exists exactly one empty set: {}. Therefore any intensional formula that defines a condition that no object satisfies, such as the three examples you gave, {x in the classroom : x is a pen}, {x in the classroom : x is a notebook}, {x in the classroom : x is a pencil}, all evaluate to the empty set: {}.

{x in the classroom : x is a pen} = {x in the classroom : x is a notebook} = {x in the classroom : x is a pencil} = {}.

Answer 5

For each natural number n different from zero there are many sets with n elements. But for n = 0 there is a unique set with zero elements, the empty set.

Aside: It would be interesting to know at which age children capture the idea of the empty set. Did Piaget investigate this question? What about the age of your children?

Answer 6

There are three zeros in 1000, but they mean different things: zero units, zero tens and zero hundreds.

Answer 7

Your question: "Is there only one 'zero'?" First, we need to clarify: what is the role or purpose of the 'zero' in question?

Zero as a representation of "nothingness": According to von Neumann's assignment, zero corresponds to the empty set ∅. By the ZFC axioms, this is, of course, unique.

Zero as the starting point of natural numbers: Here, zero is a specially designated element within the Peano system. According to its axioms, zero must be unique. However, if one does not adhere to some or all of Peano's axioms, it is possible for multiple distinct "zeros" to exist.

Zero as the identity element of an operation (commonly referred to as 'zero' when the operation is written as addition): According to the axioms for any identity element e, where e+a=a+e=a, it follows that 0=0+0'=0'. This clearly ensures its uniqueness.

Answer 8

123456789012345678901234567890

Answer 9

There is exactly one 'zero' in the same sense that there is exactly one 'one' or one 'two'. Numbers are linguistic quantifiers that call for a reference object. The reason people think that 'zero pencils' looks like 'zero notebooks' is that the referent isn't visible. But you wouldn't say that having zero girlfriends is the same as having zero lattes — at least, I assume you understand the difference between the things referenced — so there's really no problem here.

Answer 10

Well, if you add a unit to it they are different. Zero pens versus zero notebooks. For example, zero meters is physically different from zero seconds.

Answer 11

Egyptians used a zero line level in construction, with measurements above and below.

But zero in it's full form, is about positional notation, like the decimal system. Roman numerals get increasingly clunky as you get to bigger numbers, whereas positional notation is scalable. People were very uneasy about zero for a long time, and Roman numerals were still being used in accounting until the 18th C because of this.

A case has been made that zero in Arabic numerals, actually from India, emerged from Buddhist culture, where contemplating emptiness was seen as positive, and encouraged. Check out the Radio 4 episode of In Our Time on Indian Mathematics, where these scholars of the issues and history discuss this: George Gheverghese Joseph, Honorary Reader in Mathematics Education at Manchester University; Colva Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews; Dennis Almeida, Lecturer in Mathematics Education at Exeter University and the Open University.

People uncomfortable with algebra until at least as late as Newton's time, causing him to publish his proofs in the form of geometric descriptions, which were considered 'more real' and reliable. Infinity is still a cause of unease and philosophical disputes, despite it's widespread use and efficacy in defining modern mathematics.

In short, there is zero on the numberline, and zero in positional notation, and these are different, though they are reconciled.

Answer 12

I am reminded of Wittgenstein, because ambiguities like these where all the words seem different but seem to reflect no underlying difference in reality are really his jam in Philosophical Investigations.

Specifically, I'll bring up an excerpt from remark 33:

And what does 'pointing to the shape', 'pointing to the colour' consist in? Point to a piece of paper.---And now point to its shape---now to its colour---now to its number (that sounds queer).---How did you do it?---You will say that you 'meant' a different thing each time you pointed. And if I ask how that is done, you will say you concentrated your attention on the colour, the shape, etc. But I ask again: how is that done?

A few remarks later, he suggests an explanation, that there is a different spiritual or mental or intellectual activity attached to focusing on zero pens or zero books or zero apples. After all, zero food makes me hungry and zero water makes me thirsty!

Of course, Lowri is already correct on the current mathematical practice, but I think that it's a mistake to accept mathematical practice as orthodox philosophy too soon.

Answer 13

In addition to the many answers pointing out the large difference between:

• 0 = "no instances of any thing", and
• 0 = "no instances of this thing"

I'd just like to add that while the former has exactly one instance, the latter has an infinite number of instances! Consider the options you have for how many pens you can show and ask how many notebooks there are.

EDIT:

On chatting about this with a friend, perhaps a better way of visualising this a set of axes with the origin at the center.

All axes share this origin, and 0 along any axis will be the same point. On the other hand, you can have 0 for any specific axis but non-0 for any other value. All of these are valid 0s of that axis, and form some kind of surface or area across the other axes.

What your examples are doing is confusing each separate axis with the multi-dimensional space.

Answer 14

...they said Zero pen, then I picked up notebook, did the same, 1 notebook, Zero notebook, pencil etc. Then one of the kids remarked that Zero of Everything looks exactly same!!!! So is it that there is SEPARATE one of each thing(different from others) BUT JUST ONE Zero for EVERYTHING.

No, the children are focused on the concrete object (pen, notebook etc.) in front of them, and are confusing that with the actual number (a quantitative relationship to a unit), which exists as a concept.

Both the single pen and the single notebook represent the same number, a quantitative relationship of 1:1 with a unit, even though they are different physical objects.

(The unit in this case being the generic/abstract "instance of an object".)

"Nothing" always looks the same, and always points to the same quantitative relationship of zero.

You could show them two different pens, of the same brand: are these two pens, shown one-at-a-time, a separate type of "one"?

No, they both "point" to the same "one", and so on.

Answer 15

What Property Characterizes Zero?

i) Being an Additive Identity

x + 0 = x

Then we can, and do have many different Additive Identities.

The Zero Vector is different from the Zero Natural Number, which is Different from the Zero Dedekind Cut etc....

However, it's not clear that the property of being an additive identity is intrinsic to the object, and not how we define addition in the structure we are working with.

ii) Zero is a notion of emptyness.

Then there is only one set which has no elements.

This is the notion you used when describing The Set of No Pens etc...

iii) Zero up to Isomorphism ( 0 As Minimal Element)

If We have an Isomorphism f:(X,R) ≅ (N,<) - N is Set of Natural Numbers

Then, for some element x in X, f(x) = 0

We then have that x is Zero in X.

Basically, we have thet x is the R-Minimal Element in X

Remark: ii) is a somewhat like a special case of iii) where we consider a set that is ∈-minimal in the Universe. i.e. There is no set which is an element of the emptyset.

Note that no matter what, whether we use being R-minimal for some Order R, or being an Additive Identity- we use other concepts to have that something has a property appropriate for being 0.

Answer 16

Is there just one Zero?

Yes. Suppose x+y = x and x+z = x where x, y and z are real numbers.

Solving for y and z in both equations, we obtain:

y = x-x = 0

z = x-x = 0

By substitution, we have:

y = z = 0

There is only one value, namely 0, that when added to a number gives you the same number.

Answer 17

Zero is defined as an additive identity element of a group. In context of numbers, it is such number x, which satisfies the equation x+a=a, for any number a.

Can there be two such numbers, x1 and x2 that satisfy this equation? Let's see.

x1+x2=x2

but also

x2+x1=x1.

So, either x1=x2 or addition is non-commutative. Since in all number systems and vector spaces addition is commutative, zero is unique.

Answer 18

Your example is a bit deceiving and putting you on a wrong path because not only is there no e.g. pen in your hand, there is nothing at all in your hand. And, indeed: Of nothing at all there is only one ;-). Of no x though, there are as many as there are x!

In order to illustrate this take a car: There is a big difference between zero gas or zero windshield wiper fluid. Or if you take a set of crayons: There is a big difference between zero red crayons in the set and, say, zero blue ones, especially around Christmas ;-).

You could perfectly well illustrate this by holding a pen, a notebook, a pencil etc. together in your hand and then taking away just one of them, say, the pen. The hand holds zero pens! Very different from holding zero notebooks (but pens).

In this respect, the zero is not at all different from one or any other number: If we count things, we must say which things.

If you were the Dalai Lama you could order the opposite of a pizza without toppings: A pizza with everything. While the absence of everything is nothing, the presence of everything is the universe. The two concepts complement each other.