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I understant why it has to exist, but how can zero exist, if zero is nothing, then nothing is something wich means that zero cant exist, ive seen similar questions but i still dont get it, help

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    zero isn't nothing. At worst zero is a symbol that represents nothing. Just like the word "void" isn't empty or the word "circle" isn't round.
    – armand
    Oct 16, 2023 at 11:35
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    See Zero. It is a number, i.e. a symbol and a concept: symbols and concepts exists also if not always as "physical" objects. Oct 16, 2023 at 11:53
  • Does this answer your question? How is zero different from nothing? Oct 16, 2023 at 12:53
  • The same way adjective "red" exists without being red, and names of fictional characters exist without the characters themselves. Zero is not nothing, it is a numerical label for it.
    – Conifold
    Oct 16, 2023 at 13:20
  • Nothing means "no thing". Zero counts "no things". Nothing and Zero are synonyms, that express "no things to count or counting zero things".
    – user71091
    Jan 24 at 7:08

4 Answers 4

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Here's a simple way through your dialectical illusion:

  1. If zero were nothing, zero wouldn't exist.
  2. Zero exists.
  3. Therefore, zero isn't nothing, it's something.

Similarly, but more odd-sounding:

  1. If Nonexistence didn't exist, not anything wouldn't exist.
  2. Some things don't exist.
  3. Therefore, Nonexistence exists.

You are, in Wittgenstein's sense, in need of conceptual therapy, though, regarding that word "exists." Your sense of it is eliciting abstract pain, in your mind. Why not define it, or redefine it, so that your pain goes away?


Commonly, zero is taken for an interpolant of an intrinsically empty set, i.e. an abstract "container" with nothing in it. Furthermore, one might speak of ⧄, a contingently empty set, one that can sustain eventual elements, which "until then" contains only the absence of its possible elements.

Platonically, this will not work: the Forms self-participate, so the Form of Nothingness would be an exemplar of nothingness, i.e. would be nothing, and then everything would still exist. So see again my remark about conceptual therapy!


Addendum: a structuralist option

Mathematical structuralism can be taken for a realist, or at least demi-realist, position in the philosophy of mathematics. A perspicuous depiction of the difference might be: the "normal" realist takes the referents of mathematical names to be objects with properties, e.g. 2 is a "substance" with attributes such as "is the square root of 4, is prime, is the natural successor of 1," and so on. The structuralist can take structures for higher-order objects, can adopt a Humean bundle theory about objects therefore sans a substantive attribution of subsistence(!) to these objects, though. So the structuralist might say that zero doesn't exist on its own, but is rather a slot in the linear lattice of ℕ. (ℕ, then, might be the higher-order object; but ignore this complication for the time being.) Now, if all such slots are in some sense empty "on their own," it might not do to push the equivalence "zero = the empty slot," but one might want to say, "Zero = the slot that is necessarily empty," while one = the slot that necessarily has only one filling-in if it has any, two is the one with necessarily only two fillings-in if any, etc.

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  • Re your sentence: "Your sense of it is eliciting abstract pain, in your mind." I think this is true for most concepts. We need to hold them at arm's length.
    – Scott Rowe
    Oct 16, 2023 at 13:09
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This is a problem mainly if you are a "realist about mathematical entities," meaning that you believe that mathematical entities such as 1 and 0 have their own real existence. Some people do believe this, whereas others see math just as a system of notation, in which case the use of "0" to mean nothing is no more problematic than the use of the word "nothing" to denote nothing.

It would also be possible for you to be a realist about 1 but not about 0. You might conceptualize 0 as a privation, meaning just as "cold" is only the absence of heat, and "darkness" is only the absence of light then 0 is just the absence of quantity. You can find a good discussion of this here: https://philarchive.org/archive/KOOAAR

There are people who are realists about 0 but no generally accepted consensus about what that means.

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As far as mathematics is concerned, zero isn't defined as "nothing".

Definition 1

Zero is an additive identity element in some set, where addition is a defined operation on. We denote the identity element with the symbol 0, if there is only one such element.

What this means is that for any x in our set. x + 0 = x

Does this imply 0 is nothing? No! 0 is clearly an element in our set, what we are saying here is that the addition operator treats the zero element as something which doesn't "change" elements.

It really says more about how we defined addition, than anything about 0.

By the way, 1 is defined similarly, but for multiplication. We call 1 the multiplicative identity.

0 need not even be a number, it could be a function,matrix, or any mathematical object that we define some operator that we call addition for.

Ex. Let {T,F} be our set and let + be defined to mimic an OR logic Gate

So,

T + T = T

T + F = T

F + F = F

F + T = T

As for every element in the Set, x + F = x

we call F our additive identity, and can Now denote F as the symbol 0

In this case, 0 doesn't mean nothing, it means false, or F

Note: There is a layer or complexity where we denote an element as a right or left identity if x + y = x, is true but y + x ≠ x and vice versa, This added layer of terminology I feel isn't that important but I add here for completeness.

Definition 2

Zero is the set y, such that

y = {x | x ≠ x}

" Zero is the Set which contains every set, not equal to themselves- i.e Zero contains no elements"

Here, zero is't "nothing" zero is defined to be a set.

As an analogy, we could think of zero as a folder on a computer with no files inside. Or as an empty container.

In my experience, zero is never defined to be "nothing" in mathematics, thinking of zero as nothing is a strictly nonmathematical conception of 0.

There is some Platonic Notions of 1, where 1 represents Unity, and 1 is the form which contains everything- "All is One". Similarly 0, Platonically can be interpreted as "Nothing" which is usually conceived of as the negation of 1, Funny enough this does correspond to 0,1 representing false/true as seen in Discrete Math.

As we can view "Reality/Oneness" as all that is true, and "Nothingness" as its negation, i.e. falsehood.

Now, in this conception, how does nothingness exist, if it is not part of the "Reality/Oneness". ?

Perhaps we can view it as the absense of Oneness, similar to how a shadow "exists" as an absence of light.

Considering that most of us walk around with self-identity. Nothingness, in this framework is the shadow on the wall we are all percieving as "Reality".

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Zero is the absence of a measurable quantity. This is not equivalent to nothing.

If my coffee cup has zero coffee in it, then my cup does not contain nothing. It has air in it. But it contains no measurable quantity of coffee. Furthermore, in experimental science, zero has units: 0 inches of precipitation, 0 red balls in the bin, 0 newtons of force, etc.

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