When does zero exist?

ex: if i say, "I dont have a hat" it seems more fair to say -1(hat) rather than 0(hat) ,because 0(hat)=0 would mean (hat)=0 and -1(hat)=(-hat) implying there is a hat in existence. ..How does zero work when we empirically measure the world?

sorry if this is unclear, im trying to word this as best I can

***this question got closed by 5 people, but I am unclear as to why it was closed. If you closed the question can you clarify in the comments why it got closed, or edit the question? I am semi-new to the stack exchange forum

  • zeroes just like infinity, probable do not exist in the "real" world. There is not a single observed entity that has a no properties, by the very definition, if it did, it would be unobservable. The closest things to "zero" as we get is empty space, but quantum indeterminacy prohibits completely zero energy in space. So far, nobody has observed 0, nor infinity, nor a perfect circle. They are mathematical points and real phenomenon are asymptotes to these points.
    – Weezy
    Commented Feb 25, 2020 at 10:25
  • 3
    ZERO is a number; it exists "where" numbers are. The issue is the well-known philosophical problem regarding Abstract Objects. Commented Feb 25, 2020 at 10:37
  • 1
    To say that we have "two apples" is not less mysterious than saying that we have "zero apples". Consider your bank account: an amount of zero cash is not less real and significant than an amount of 100 USD. Commented Feb 25, 2020 at 10:42
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    In the words of the great philosopher J.P. McCartney, "Baby you can drive my car." But only if you understand "empty" so you know when to fill up the tank.
    – puppetsock
    Commented Feb 25, 2020 at 14:31
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    I voted to close because your use of scalar multiplication like -1(hat)=-hat is neither conventional nor clear. Your question is unclear, because in philosophy, the nature of "exist" is contentious among ontologists. You seem to confuse negative numbers with being used as nothing, when in practice, they denote something. Ex: I owe you five dollars. -$5 in accounting doesn't mean that I don't owe you anything, rather it means that you don't have them in your possession and they are due. Empirically, if something doesn't exist, we indicate it by 0. Zero, the number, exists when we count nothing
    – J D
    Commented Feb 28, 2020 at 19:50

3 Answers 3


Lovely question!

First, "when does anything exist"? I know this is only math, but it helps to remind ourselves that everything that exists exists because it occupies some space for a time. I call these the modes of existence because it doesn't literally have to be physical space, in the sense that 0 occupies very small space from the physical universe but it can mean a lot of things, interpreted many different ways.

Mathematical ideas exist as part of our communication with each other, specifically a very narrow and formal part of it.

0 doesn't just belong to this small group, it has its place (it exists) in different places where it's not mathematical (like in roulette where you can get 00 even, certainly not as a result of counting hats).

But when it's about math, it's an integer and the easiest to imagine it is as a relationship between two natural numbers.

Natural numbers are the results of some counting operation. Counting hats for example. Sometimes you want to compare natural numbers, in those cases you can have an approximate result (smaller, bigger, equal) or an exact result which would be the integer result of the said comparison (also called subtraction). If the two natural numbers were the same, you get 0.

So to answer your specific question at the end, in empirical measurements you get zero when you subtract one measurement from another. If your tools already have 0 on them, they already compare two measurements. Usually the difference from the starting point or default state.

  • Abstract objects do not occupy space or time, yet they exist. plato.stanford.edu/entries/abstract-objects
    – Dcleve
    Commented Feb 25, 2020 at 17:37
  • @Dcleve I think that is saying the same thing I am saying but in a way that's more oriented to the classical western way of thinking which I am trying to avoid. I believe that it's unreasonable to say that abstract things don't occupy space, it's just that the space they occupy manifests indirectly in our thoughts, mental processes, formalisms, writings, laws. The point is not really that our thoughts occupy space, but that in fact space is completely misunderstood, which then leads to a lot of paradoxes in physics for example.
    – Ashnur
    Commented Feb 25, 2020 at 20:57
  • "I believe in direct realism" and "in fact space is completely misunderstood", are contradictory. If we are all confused about space, then we are absolutely not directly apprehending its reality and nature.
    – Dcleve
    Commented Feb 25, 2020 at 21:29
  • @Ashnur the last paragraph really explains it for me in a way that makes sense! thumbs up
    – Noah
    Commented Feb 26, 2020 at 7:33

If you don't have a hat, then you've got zero hats.

ex: if i say, "I dont have a hat" it seems more fair to say -1(hat) rather than 0(hat) ,because 0(hat)=0 would mean (hat)=0 and -1(hat)=(-hat) implying their is a hat in existence.

You seem to be making a math mistake here:

0(hat)=0 would mean (hat)=0

0x=0 doesn't imply that x=0.

If you divide both sides by 0, we get

  • x = 0/0

, which is a classic indeterminate form. This means that we can't precisely determine a specific value for x given only that x=0/0.

For example:

  • If x=0, then 0*x=0*0=0, which works.

  • If x=1, then 0*x=0*1=0, which works.

  • If x=2, then 0*x=0*2=0, which works.

Point being that 0*[hat]=0 doesn't mean that [hat]=0.

  • 1
    To note it, I realize that I casually hid a reduction of 0/0 to 1 in the above. In that specific context, 0/0 does equal 1. I'm hoping that anyone able to see it will also understand it, as it seems liable to confuse.
    – Nat
    Commented Feb 25, 2020 at 11:19
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    Another way to look at it: 0 is the unique finite quantity such that, when you add it to a number, that number doesn't change. If you have no hats, and get 5 hats, you now have 5 hats. Therefore, you started with 0 hats.
    – J.G.
    Commented Feb 25, 2020 at 17:38
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    Plus, it's not clear what kind of algebra there should be on 'hat type of things' and numbers. Do they commute? Like 5 hats = hats 5 ... probably not. Should it distribute over the numbers, like 3 hats + 2 hats = 5 hats? probably yes. How does the additive inverse and the additive identity element interact with the 'hat type of things'? If you lay out those rules, then go ahead and do with them as you please.
    – Marlo
    Commented Feb 25, 2020 at 23:01
  • this makes sense, I need to play with the math a little more. My initial reaction to x=0 (in the which works example) would be that x=0 would be an impossible claim because this would mean their are no hats. this made a lot of sense how you explained it, I think i was mistaken
    – Noah
    Commented Feb 26, 2020 at 7:28

Zero exists in mathematical space. This is the well known concept of math realism: https://plato.stanford.edu/entries/platonism-mathematics/. The vast majority of mathematicians accept math realism, and the plurality of philosophers do as well: https://philpapers.org/surveys/results.pl

One encounters these sorts of virtual spaces repeatedly in theoretical physics, and in any math solution in which one does a transform of variables.

Note that we do not directly experience the physical world, its existence is inferred from the success of the assumption of its existence. This is indirect inferential realism, and it is how empirical science operates: https://en.wikipedia.org/wiki/Direct_and_indirect_realism

Quine, in 2 Dogmas of Empiricism, argued that the same inferential process, applied tot he success and utility of mathematics, leads to the inference of the reality of math, and of other abstract objects. https://www.theologie.uzh.ch/dam/jcr:ffffffff-fbd6-1538-0000-000070cf64bc/Quine51.pdf Popper applied the same method to awareness as well, and inferred the reality of three worlds: those of matter, consciousness, and abstractions. https://tannerlectures.utah.edu/_documents/a-to-z/p/popper80.pdf

So, zero exists in world 3.

  • arguments in this answer for platonism based first on - begging the question (you know about 0, so zero has to be real) - appeal to authority - red herring (even if we don't experience the real world directly, it's irrelevant to the question at hand if zero exists or not) - wikipedia is not a good source for deciding on how science works (I believe in direct realism, I find the evidence for it overwhelming), 'the accepted view' is not as accepted as wiki says - reducing the modes of existence to 3 is shortsighted. Popper is great, but not everything is gospel...
    – Ashnur
    Commented Feb 25, 2020 at 21:10
  • @Ashnur -- every part of your comment is incorrect. There is no "question begging" in my reply, and my reply does not start its reasoning by assuming zero exists. Appeal to authority is not a fallacy, it is how we arrive at all knowledge. the fallacy is to cite inappropriate authority. Citing the considered opinion of the experts in a field is the GOLD STANDARD of providing a reference to knowledge! Wikipedia is a weak reference, but it is often accurate, and provides a good simple digest of a question. If you reject it, you are obligated to show it wrong, rather than just declaring it so.
    – Dcleve
    Commented Feb 27, 2020 at 15:22
  • @Ashnur - Meanwhile direct realism is as falsified a view as possible. It was refuted classically, just by referencing illusions and delusions, plus the Ship of Theseus problem refuted the central idea of "identity" in our psychogical desire to believe direct realism. But physicists chemists shattered it, with 19th century science of energy, inertia, elements, waves, fields, etc. Then germ theory, genetics, QM, and relativity buried the ashes. No scientist accepts it -- the world is NOT directly perceived. Modern neurology re-demonstrates this every day. See "Incognito" for a good summary
    – Dcleve
    Commented Feb 27, 2020 at 15:33

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