-1

I'm a novice in math, just learning the basics as of the moment and I'm naturally out of me depths when it comes to the real rigorous and technical stuff. I would like some help understanding some concepts in the subject (vide infra)

  1. Nothing (Zero is sometimes said to nothing)
  2. Zero (the solution of an equation like $m - m = n$, what is $n$? Take away $m$ from $m$, what are you left with? Nothing.)
  3. The empty set, {}, $\emptyset$, $\phi$ or the null set (the set that contains nothing)

All three ideas seem to be about nothing and given how little I know, I'm unable to tell them apart in any meaningful way, but legends say that a mathematician/philosopher can.

A Neutrino Boy
  • 73
Agent Smith
  • 951
  • 4
    Zero is 'a number' and 'is' a genuine element of 'many' sets. "Nothing" is not mathematically defined, except maybe for the interpretation of the logical symbol $\not\exists$. The empty set is just a set. We assert its existence, and you'll see later on that that's very convenient. For now probably just don't worry about this, your opinion of 'zero' will change as you study more maths – FShrike Dec 27 '22 at 10:30
  • 2
    "Nothing" is not a standard word in mathematics, so that it does not correspond to a specific definition. "Zero" is the number you know well and, more generally, the identity element of an additive operation (i.e. the "quantity that adds nothing"). The empty set can be seen as an empty box : nothing exists inside the box, but the box itself (as a container) exists. – Abezhiko Dec 27 '22 at 10:33
  • 1
    @Abezhiko Nothing is what's in the empty set. You can say a lot about Nothing, because statements can be vacuously true. For example I can say Nothing is alive and dead at the same time. It makes perfect sense because I'm talking about a member of the empty set. So we tacitly use nothing when proving sets to be empty or the non-existence of some mathematical object. – CyclotomicField Dec 27 '22 at 10:50
  • 1
    Whether "nothing" exists is a very philosophical question. In mathematics , we do not speak of "nothing" , we can have , for example , a situation where no (real) number satisfies a particular equation. I would not say that the empty set contains "nothing". It is just the (unique!) set that has no elements. Zero has many meanings including a digit in every base $b$-system , where $b\ge 2$ is an integer. – Peter Dec 27 '22 at 10:52
  • 1
    @Peter even if Nothing doesn't exist, which I claim it doesn't, it's still useful to think of the empty set as containing nothing. Proving a set is empty allows you to prove the non-existence of certain objects and that ends up being a proof about nothing. It's also easier to see why vacuously true statements exist when you realize you're talking about nothing. – CyclotomicField Dec 27 '22 at 10:55
  • 1
    To further muddy the waters, is $2$ a point on the number line, or a point on the real axis in the complex number plane, or a positive integer one uses in counting (and thus not a point having some geometric location), or a certain constant function (with domain all reals, positive reals, all complex numbers, $\ldots),$ or only a symbol for $1+1$ (with no separate existence of its own), or only a symbol for the least conceptualizable multiplicity (with no separate existence of its own), or $\ldots$ – Dave L. Renfro Dec 27 '22 at 11:07
  • Thanks to all for their most helpful comments. Here's what I discovered due in no small part to all your contributions. $n(\phi) = 0$ [All of the 3 ideas are contained in that single mathematical statement. The empty set contains nothing and the number of elements it has is 0. Can someone one tell me how to deepen this understanding if it is one, if I am on the right track. – Agent Smith Dec 27 '22 at 14:12

1 Answers1

1

Nothing is the most fundamental concept of the three. It typically relates to non-existence or absence of something. However you can actually say a great deal about Nothing as vacuously true statements are always true. Python code uses the None keyword as a token to represent this concept but mathematically it's not useful to do so since we don't have to run it on hardware.

Now the next construction is the empty set. Sets are characterized by their members, which is to say which distinct objects exist within the set and I can recover those elements from the set. However, if no elements in the set exist, then it has Nothing in it. I capitalize to draw attention to the fact we're still using Nothing like a noun here but it reads correctly in natural language as well. Note that it's just acting as the antithesis of existence here.

So the empty set exists. It has the little braces after all $\{\}$ and that's not Nothing. So on the set level we have existence, but on the element level we have non-existence, i.e. Nothing. This works exactly the same way a box does. What's in the box? Could be something, could be Nothing.

Finally the natural numbers are used for counting. I prefer including $0$ as a natural number for this reason and as a number it exists, so it's not Nothing. A simple case would be a set of oranges and (disjoint) set of apples. I can count the oranges and apples then add them together to get the total.

So what if I go on an apple binge and now I only have the empty set. Well, to count those I'd add $0$ to the total. In fact, $0$ is the only natural number such that $x+0=x$ for all $x$. In modern algebra we typically define $0$ this way as the identity element in some system since it generalizes well to more complicated objects. For example the $0$ vector plays a major role in linear algebra.

It's this property that makes $0$ and the only way to count Nothing. In fact, the empty set is an identity element with respect to set union, which is to say $\emptyset \cup X =X$ for all sets $X$. So they do have that commonality among them.

So in short, nothing is Nothing which in turn is a state of non-existence. The empty set exists but contains Nothing. The amount of Nothing we have is $0$. The distinction is subtle but they all clearly have their own role.

CyclotomicField
  • 11,018
  • 1
  • 12
  • 29