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.

[EDIT]

Thanks everyone for answers. The biggest learning I have from this is that for some reason this question is very interesting, I never knew that!!! I read the answers, I don't understand fully most of them, however I am still 100% sure there's just one because I look at it not not in purely Mathematical sense but rather more as Absence, the way I have seen "Absence" there cannot be two of it because they look exactly same. Zero, the concept and even the Symbol (0 << I like it!!! It's COOL) is the bestest and most popular "Definition" we have got so I used the name Zero...

[LATEST EDIT] As I was thinking about Zero after reading the answers, I feel there is a "defined" Zero and there is an Undefined Zero, meaning not bounded by any context. This Undefined Zero is Absence, of anything and everything. Now by Definition Zero has MEANING with any thing, it doesn't need a "thing" for it to have meaning, but 1 can never mean anything without a "thing" attached to it. So since Zero is not attached to a "thing", anything, JUST ONE OF IT covers absence of ALL THINGS.

Hence there is JUST ONE ZERO FOR EVERYTHING....

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.

---

^{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.

---

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

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 5

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

Answer 6

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 7

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 8

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 9

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 10

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 11

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 12

In formal set theory, say ZF, there is a unique empty set. However, note that for every finite number, there are many different sets with that number of elememts. This makes the empty set as being unique as the odd one out. Indeed, we can conceptualise a set as a bag and there are many different bags with nothimg inside it.

Thus its philosophically acceptable that we say that there are many empty sets but they are all isomorphic. I don't know of any formal set theory that takes this tack.

Another way to think about this is through the theory of rings. The integers form a ring. Here zero is the identity. There are many more rings other than the integers and they all have identities. This there are as many zeroes as there are rings.

Answer 13

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 14

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 15

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 16

Sets are extensional: two sets are the same iff they have the same elements. Hence any two empty sets are the same. But two non-empty sets might have the same cardinality without being the same: this is a subtler notion of 'identity', that of isomorphism.

Cardinals are the most elementary form of numbers, and indeed they arise from 'decategorifying' sets, i.e. cardinality is the finest property of sets invariant under isomorphism (meaning: two sets are isomorphism iff they have the same cardinality).

Thus for sets, it is true there are many of the same non-zero cardinality. Numbers, however, are by definition unique.

Answer 17

...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 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.

Answer 19

Unless you are a Platonist, there are zero zeros. There are similarly zero ones, and zero twos. "one" is used colloquially as a noun, and is referenced as a noun in Mathematics, which does not overly concern itself with whether the abstractions in which it deals are "real". However, "one" is not a thing you can reach out and touch - it is an adjective. Saying that "one cat" is a "different 'one'" than "one dog" makes as much sense as saying "fat cat" is a "different 'fat'" than "fat dog". It's not a "different 'fat'", because there's not a physical "fat" to begin with. The thing that differs is the subject - the noun. The cat is different from the dog. In this sense "one cat" is different from "one dog", but not because of a difference in "oneness". In this sense also, "zero cats" is different than "zero dogs", because cats are different from dogs - if you walked into a pet store and said "I would like to buy a cat" and the associate responded "Sorry, but we currently have zero dogs." you would not find this very informative, because "zero" is just an adjective for the actual subject of the conversation, which is either "cats" or "dogs".

Answer 20

It depends on what you mean by "Zero". If you take a technical, formal language like mathematics, then you can have many "zeros". For instance, when a quadratic equation crosses the abscissa of the Cartesian plane, one has two distinct "zeros" (a,0) and (b,0) which one could write 0_a and 0_b, so yes, you can have two zeros.

If you are using your Magic Wand of Letter Capitalization to turn 'zero' into 'Zero' like Harry Potter, and make a symbol that represents 'nothing' into 'Nothing', then as a scientist, I'd simply suggest that you are committing the reification fallacy if you're trying claim Zero exists. From WP:

Reification (also known as concretism, hypostatization, or the fallacy of misplaced concreteness) is a fallacy of ambiguity, when an abstraction (abstract belief or hypothetical construct) is treated as if it were a concrete real event or physical entity.

'Zero' is simply an abstraction that varies from context to context, and while we can then generalize across many uses of 'zero' to claim there is a concept 'Zero', that concept is not a physical thing with an identity that refers to something in the real world. Of course, that elicits disagreement from superstitious neo-Platonists who think because human brains tend to have the same experiences and prefer to use the same symbols to represent them to foster communication, it somehow proves that zero's are real! So, you will find many mathematical realists (SEP) who will help you ponder your navel with you. But, your question is a pseudo-problem. As 'Zero' is just a class of abstractions to deal with the lack of something (varying what with context), you'll find no zero, other than the physical medium used to represent the idea, has anything that comes to close to physical existence, and therefore it's nothing more than another confusion of de dicto and de re. The map, after all, is not the territory.

Answer 21

I wonder: If you show an empty hand to a child, asking "what's in it?", then would it ever say something like "zero pens", or would it say "nothing"?

I mean: "Zero" has no type in the sense that it has all types, and thus one zero can cover them (the types) all.

Or in the other sense you could state "I have no money" as "I have zero nickles plus zero dimes plus zero dollars plus zero euros plus ..." - You get the idea?

(This answer is similar to https://philosophy.stackexchange.com/a/119809/58307)

Answer 22

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 23

The number zero has a long and intriguing history, in many parts of the ancient world. The way zero was thought about and written depended on the geographical region, culture, religion, and time period, and for quite some time it was never designated a formal symbol. Even the ancients Greeks, the first-known mathematicians, were at first unsure if zero was a number or not. In some cultures an empty space was often left in writing to indicate the number or digit zero, conveying the idea of zero representing "nothing", the "absence of anything". But does "nothing" exist? If nothing exists, does that means everything doesn't exist? It only makes sense to say that the concept of "nothing" exists, in our minds, an absence of some property in some context. A glass that is called "empty" by a human wouldn't be called "empty" by say, a bacterium living in that glass.

When talking about our physical world, we often use the language of physics, which is itself built upon the language of mathematics. But unlike in most mathematics, numbers in physics always have units (like meters, seconds, notebooks, etc.) attached to them at the end of the day. This "attaching" of units can be thought of as multiplying the number with its unit. So in your example, "0 notebooks" can be thought of as "0 * notebook". Using this way, adding would work by distributing the units out, like this:

(2 * notebook) + (3 * notebook) = (2 + 3) * notebook = 5 * notebook.

We obviously wouldn't write it all out like that in a real setting, I'm just writing what happens behind the scenes to show why what we do works. But multiplying anything with 0 is 0 right, so it would be still right to say that 0 * notebook = 0 * pen = 0, right? Yes, but although by itself it would be mathematically the same, it would be semantically different, and be generally less clear what's going on. It would not make sense to write, for instance:

(2 * notebook) + (0 * pen)

or

(2 * notebook) + 0

since numbers with different units cannot be combined with addition. It wouldn't make much sense for one measurement to have a combined units like this, since we want units to be able to be factored out of the entire value. In pure mathematics, zero by itself is a completely fine number and is even often specifically sought after, or the number zero can be thought of as a set that doesn’t contain anything, which also has its own uses. But in our physical world, we prefer having explicit units to know the wider context of what we're talking about, what we're counting relative to that zero, and it lets us combine it with a larger system. "Zero" of what?

The "zero" we talk about in our physical world, where we usually only talk about non-negative integers or real numbers, is only a specific case of a general "zero" in the general area of math called abstract algebra, where a "zero element", or an "identity element" for addition, is any element of our set that does not change any element in the set when combined with it. A zero element of a set, if there is one, is typically unique, including the 0 we all know and love. At least, as unique as every other number is. So is that really still unique? Is any number actually special then? Or is it just a result of our cognitive bias as humans, the inventors of these numbers?

Even though numbers can be said to be "not real", they are real concepts that represent real things about our world, even if we can't physically see or touch them. We often find applications for things in pure mathematics much after the fact, but that's part of what makes it beautiful. It's why we have fractions, negative numbers, imaginary numbers, and beyond. And you'd certainly feel a real sensation if you saw a negative value in your bank account.