Does zero exist?

Question

I suggest that zero is a cognitive crutch, a way for humans to grasp the concept of absence, but not an objective existence in itself. By using zero, we're essentially creating a placeholder for something that's inherently difficult to comprehend – the absence of quantity or being.

This would resonate with the idea that mathematics is a human construct, designed to help us make sense of the world. In this context, zero serves as a tool to facilitate our understanding, rather than an inherent aspect of reality.

This also raises questions about the limits of language and cognition. If we rely on zero as a crutch, what does that reveal about our capacity to understand the world without quantification? Can we truly grasp the concept of absence without resorting to a symbol or concept like zero?

Answer 1

You can make a case for zero being a simple idea, rather than the esoteric creature you take it to be. If I have two eggs and eat them both, I no longer have any eggs- that is a concept a child could grasp. As for the limits of language and cognition, I can congnize the fact that I no longer have any eggs without summoning to mind any words or symbols at all.

Does zero exist? That is simply a matter of wordplay. Zero does not exist in the sense that a physical object does. It does exist as a word, a symbol, a mathematical concept and a common idea.

Answer 2

I suggest that zero is a cognitive crutch, a way for humans to grasp the concept of absence, but not an objective existence in itself.

The same could be said for any number. They're all abstract concepts that allow us to think about mathematical operations. The number three is how we conceptualize the number of X characters in this string:

XYXYX

By the same logic, the number zero is how we conceptualize the number of X characters in this string:

YYY

There's little reason why its existence as a concept should be any less legitimate than any other number. When using it in addition, subtraction, or multiplication, it behaves just like other numbers. There's a problem with using it as the divisor in division, so we just say that this isn't allowed, we don't go so far as saying that it can't exist.

If zero is a crutch, what about π? You can't put π apples in a bushel or cut a cake into π pieces.

Answer 3

In ancient times, before the invention of the Hindu-Arabic numeral system, zero was not treated as a number. The Romans used the familiar Roman numerals, with one as the smallest number, and the Greeks and others mostly did the same.

Now if Atticus and Belarius are merchants, and Atticus borrows a hundred denarii from Belarius and then later repays the debt, how much does he owe? The obvious answer is zero. The ancients would sometimes write a symbol in their ledgers to indicate a zero balance, but they didn't regard this symbol as representing a number, just a placeholder for where a number would go.

But this is rather limiting. What if you want to add up all the balances in your ledger to see whether you are solvent? If zero is not a number you cannot add with it, so you have to just ignore all the zeros, but that seems rather arbitrary. My spreadsheet program expects cells to contain numbers when I'm adding a column, and it would be rather awkward if I had to treat zeros differently from other numbers.

Using a zero numeral is an essential feature of the Hindu-Arabic numeral system. 302 is three hundreds, plus zero tens, plus two units. Without treating zero as a number, we cannot simply account for how large numbers are constructed from smaller ones in this way. This numeral system is immensely superior to the old Roman numerals for many purposes. Try doing long division using paper and pencil with Roman numerals. I suppose it would be possible to claim that '0' is a numeral but it is not the name of any number, but again this seems rather arbitrary.

All that said, it is a matter of convention as to whether you choose to treat zero as a number. I know a few mathematicians who don't. Typically, they also do not regard the empty set as a set. Those two often go together. From a pragmatic point of view, it is just convenient to treat zero as a number and the empty set as a set. Doing otherwise makes things messy by requiring extra conditions to handle the edge cases.

Answer 4

Elementary number theory of mathematics is based on the idea of successively amplifying the range of numbers for calculation:

• We start with the numbers 1, 2, 3, … and addition.
• Then we introduce subtraction and the number zero to enable 3-3 = 0, 4-4 = 0, …
• The next step is to introduce negative numbers to calculate 3-5 = -2, e.g., in order to represent debts.
• Eventually Gauss et al. introduced complex numbers like “i” to take the root from (-1).

Insofar I agree with you assessment of mathematics as a human construct.

History shows the usefulness of introducing numbers and calculation. Introducing the number zero is similar to introducing other numbers like the negative, or the rational, or the irrational, or the real, or the complex numbers, or the quaternions in order to allow certain calculations.

These calculations help to understand and to explain reality by quantification and facilitate the science of mathematical physics.

Answer 5

I will say this, we know that infinity or ∞, is just but a representation of ultimate continuity and therefore is not a number, similarly 0 is just a representation of neutrality along the number line i.e., non-positive and non-negative; one might even say a good representation of a center for the number line. You will be surprised that pretty much every other number is also not "real" - in the materialistic sense. You can shift the value of 0 to 0.1 by increment and do the same to shift the value of 1 to 0.2 and do this until infinity and still have the same mathematical properties (some) hold true especially in number theory and their proofs (esp. when dealing with rings). You can even as far go as not use any number to "do/perform" math.

For your question:

If we rely on zero as a crutch, what does that reveal about our capacity to understand the world without quantification?

Take a look at this excerpt from Wikipedia, it shows us that the crutches of quantification are not necessary but makes everyday interaction and formulation much easier in that humans and our brains like simplicity and heuristics:

Mathematical fictionalism was brought to fame in 1980 when Hartry Field published Science Without Numbers which rejected and in fact reversed Quine's indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as a body of truths talking about independently existing entities, Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real. He did this by giving a complete axiomatization of Newtonian mechanics with no reference to numbers or functions at all. He started with the "betweenness" of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields. Hilbert's geometry is mathematical, because it talks about abstract points, but in Field's theory, these points are the concrete points of physical space, so no special mathematical objects at all are needed.

As for your question:

Can we truly grasp the concept of absence without resorting to a symbol or concept like zero?

I don't know if you consider a group of letters to be a symbol but in Computer Science, terms like null or None or NaN is used to indicate absence in the case of data.

Answer 6

Does zero exist?

As always, it depends on what you use the word "zero" to mean.

If we mean the mathematical concept, then it certainly exists, otherwise I don't know how mathematicians could even understand each other when they talk about zero.

If we use the word "zero" to mean a particular number of things, as presumably mathematicians do, and in these very enlightened times as most people seem to do as well, then it is a matter of conventions between willing adults. Whenever we think that there is no elephant in the room, we say that the number of elephants in the room is zero. This seems to make perfect sense, not only to mathematicians and philosophers, but to most people as well.

The impression that zero is problematic may come from the idea that it seems that we cannot count things when there is no thing there to count. This is true! Yet, whenever there are not too many things to count, we don't actually need to count things, and zero thing cannot be too many. If we see four cookies, we somehow immediately think that there are four cookies. When there is no cookies left, it is the convention that we may say either that there is no cookie in the box, or equivalently that there is zero cookie in the box.

If this is a crutch, then it is an inordinately convenient crutch, as it is probably one of the smartest innovations ever introduced in the way we conceive of numbers, and accordingly one which has been adopted in all cultures. Even the most conservative of zealots think nothing of using the zero.

This would resonate with the idea that mathematics is a human construct, designed to help us make sense of the world. In this context, zero serves as a tool to facilitate our understanding, rather than an inherent aspect of reality.

Exactly! The key word is "understanding", which really just means that the concept of zero helps us make sense of the world. No, there is no zero out there, but there is no 143,566,847 either and nobody thinks there is.

I suspect that this question is motivated by the idea that mathematics somehow contains some implicit claim that zero exists as an entity out there in the world! I don't think this is the case at all. Mathematicians presumably all believe that there is a concept of zero, but this is a true belief as far as I can tell, and that there is a concept of zero does not imply that there is a zero out there in the physical universe, in between quarks and leptons somehow. To say that zero lepton is a quark just means that no lepton is a quark.

You can relax, zero has no metaphysical significance.

This also raises questions about the limits of language and cognition. If we rely on zero as a crutch, what does that reveal about our capacity to understand the world without quantification?

Nothing further from the truth! The concept of zero on the contrary shows how versatile language is and how intuitive most humans are. It is simply a marvel that we should be able to discuss Black Holes, the Holy Trinity, quantitative easing and what not as if it was just the sort of things we do every morning when we brush our teeth! Just consider how incredibly far our nearest cousins in the animal kingdom are of achieving the same sort of things.

Can we truly grasp the concept of absence without resorting to a symbol or concept like zero?

Yes, of course. The notion that the things that counts the most for us may disappear and therefore be absent is probably one of the most important humans have to understand if they are to feel at ease in this world. Zero is just a name and we seldom use it outside actual calculations.

It is perhaps interesting to signal that mathematicians themselves see 0 as standing appart from natural numbers. Perhaps to fit with the historical genesis of numbers, the set ℕ is sometimes said to be the union of ℕ*, the historical (or perhaps actual) set of natural numbers, i.e. {1, 2, 3, ...}, and {0}, making 0 itself not a natural number.

Answer 7

That depends on what you mean by exist. Annoying sounding response, I know. But I wouldn't say that the absence of anything is difficult to understand, only that it is in principle impossible to understand. Why? Because there is nothing there to be understood! The concept of nothing finds itself in a similar place, as nothing is not a thing, and the concept is a mere placeholder to talk about absence. I wouldn't go so far as to call it a crutch, because, as I said, it isn't some defect or limitation of our minds that prevents us from comprehending nothing, but rather that there simply is nothing there to be comprehended in the first place.

In that sense, we can probably understand "zero" in similar terms, a kind of syntactic placeholder that communicates something to us, but not through its content, as it has no content save to function as a placeholder.

You ask: what does this tell us about our capacity to understand reality? Well, I won't agree that mathematics is not about reality. Any sensible constructivism is, in my mind, a matter of methodology, not metaphysics (these various "logics" that are used are not logic per se, but formal instruments that presume some kind of ur-logic by which we judge such formal instruments). That we have such signs as zero or nothing does nothing to diminish our ability to know the real, unless you make the mistake of reifying nothingness. This won't obliterate your capacity to know reality as such, but it will lead you into error when reasoning with such a reification. Zero, like nothing, while a placeholder, is not strictly intentional, as it is not about anything, it has no signification in the real, save a state of affairs in which something is lacking. It is a logical or syntactic instrument. It is very much a matter of negation, and so, in that sense, nothing and zero are not special. They're just specific negations; nothing is not something, and zero is not some number or amount. They're negations of particular or existential quantification; "no X is Y" is but the negation of "some X is Y".