For human beings, no-thing is an abstract concept, meaning it has no-thing to do with our real experience of life, and in effect no-thing has to be imagined as a total blank, say the paper that something is written on, but of course this paper is a whole thing and it is only our focus on the writing that makes it a blank, no-thing, background.
No-thing is this background to focus, and once, space, the heavens, were seen as a black background of no-thingness against which the stars appeared as things. Nowadays this no-thingness is thought to be filled with fields, sequential influences and almost-things, and the no-thing is in doubt. At the other end of the scale we have atoms, quantum particles, strings and the something that they appear from, but if we focus on the strings and ignore the fields etc. that bring them into our imaginary view of the sub-ataomic world, we see the background something that describes them as no-thing again.

However, if I have an apple and someone steals it, takes it away, then I have no apple, no-thing.
If I recover my apple I have a unified thing and if I cut it into sections it is a divided thing, and maths is based on this unified principle even as it ignores its own reality. This is the division of unity into things, or the many things, like a lot of apples, that create a unified concept.
What is lost in maths today is the concept of a unified background, the unity that things appear from within or the unity that is being enumerated as things, and the sooner maths re-invents itself into a concept of unity and sees its no-thing for what it is, the better for everyone.

No-thing exists in my human experience when something is taken away, but what I experience before this event is a unified concept that can be divided into things or the things that can represent another unified concept.
O as part of the numerical symbol for ten, a hundred and so on only describes a decimal form of mathematical unity, and the modern decimal system is based on this unified concept of ten things.
O shows that the integers in a column have been unified as the 1 in the next column, it represents a unity of the ten in this previous column, and so on with 100 and 1000 etc., but unfortunately, mathematics ignores its own begginings and limits its focus to the abstract background that it prefers. This says that no-thing exists in the unified column and this ability to ignore reality, the paper that maths is written on or the human being that first divided things in a numbered or quantified way, is forgotten today.

Reintroduce the concept of unity as the background that maths is built on and which it uses all the time, and maths could make sense to everyone, but of course abstract thinkers will probably choose to rely on the unwritten rule of preference that created the zero. Their abstract way of thinking depends on it. They will take away our humanly unified reality and leave us with an abstracted no-thing again.

Can I have my unified life back please.

zac, it’s not quite a philosophy of numbers course as much as high school mathematics with a bit more than the students bargained for. But they enjoy the opportunities to think outside of the basic curriculum diet.

You’re right about the degree to which you can remove pieces of paper and still describe it as a piece of paper compared to removing pieces or parts of a chair, but I still think that the idea has some merit, albeit tentative. If you asked for a piece of paper and I gave you a fragment of paper measuring 1 mm by 1 mm then you would think I was crazy. It might contain the same matierial as paper but it would not function as paper. The phrase “one piece of paper” refers to a usable and practical measure of paper. Admittedly the point at which it is no longer describable as a piece of paper is somewhat subjective, but you would never describe it as half a piece of paper unless you were comparing it with, say, a piece of A4 paper which had been torn in half. In which case you are comparing its size, not really describing it as half a piece of paper in terms of its function. Simialrly with the chair, if I continue to remove parts of it then at some point it ceases to be describable as a chair. Even if I took a chainsaw to through the centre of it then you might look at one “half” and say, “That’s half a chair.” But you would only be saying that in the sense that you recognise it as one half of a complete chair. Technically what you’re looking at is no longer a chair. It does not function as a chair any longer.

But I recognise that even here with all this philosophising I am skating on very thin ice, and I wouldn’t be foolish enough to defend my points with any authority. I simply don’t possess it.

The bottom line in the discussion of “Is 0 a natural number?” is that there is sufficient confusion to invalidate it as a universally acceptable phrase. You might have noticed at Project Euler, where our problems often venture into the realm of Number Theory, that we are careful to define the set of whole numbers not including zero, as the set of positive integers. However, at one of my other websites: http://mathschallenge.net, I do use the phrase “natural number” without stating if it including zero or not, but the nature of the problem would exclude it as a possibility. For example, “Given that n is a natural number, when is n^4 + 4 prime?” It doesn’t matter whether or not you include zero, it will not affect the solution.

Thanks for the input about conventions in your part of England. It’s interesting that these things are not even necessarily standard across one whole country, let alone universally.

I wrote about your Project Euler here:
http://www.squarecirclez.com/blog/project-euler/1558

Back to the discussion…

Arguably it is a matter of opinion on one level, but I’m with John Foster on this. The term “natural” strongly suggests a sense of intuition. Hence Euler, et al., having such a bias.

I must say that I am unaware of England making such a distinction between 0 being included among the natural numbers. I teach mathematics in England and on the journey into the world of rational versus irrational numbers my older students take a brief tour into the world of “natural” numbers; we discuss the abstract and philosophical notions and implications of non-integers, negatives, and zero. For example, you cannot have half a piece of paper or half a chair. You can remove pieces, but it remains what it is until it is no longer what it was. (I hope that makes sense.) In other words, fractions exist to define relative comparisons or measures, whereas natural numbers define the actual quantity of (usable) items.

So in the same way we talk about how unnatural the concept of zero actually is. It is quite natural to talk about three books or one calculator, but it makes no sense to talk about zero anythings. If zero were natural then an infinite number of them would occupy some space. The room where I am typing this comment right now contains zero elephants with one written on its back, zero elephants with two written on its back, and so on. There are an infinite number of zero elphants with N written on its back and yet there is space for me to be here. There is nothing natural about zero! (c;

Anyway, I edited it and I think I have what you originally intended (including the corrected typo).

@Esquio: Thanks for your input. Often the simplest explanation is the best!

I consider a whole number as a value attached to counting the number of the same things I can see. If I see no apples, then I see 0 apples.

I my explanation 0 is not a natural number.

We already have that ℤ₊ = {x ∈ ℤ | x ≥ 1} = {1,2,3,…}

Thus if we want to represent the set {x ∈ ℤ | x ≥ 0} = {0,1,2,3,…} we’ll have to write ℤ₊ ∪ {0}, which is cumbersome to write all the time. Especially if you are referring to that set often. Therefore, I find it more practical to define the set of the natural numbers to include 0, i.e. N = ℤ₊ ∪ {0}.

If I’m writing something in a course I’m taking, I’ll use whatever convention the textbook uses, but otherwise I’ll say that 0 ∈ N.

The mere FACT that this is post #30 should be enough for the readers of this post to conclude that zero being a natural number IS a matter of opinion.

I realize that certain people have very strong opinions one way or another. It seems that they want to influence others with those opinions. I have my own opinions but I will not try to influence anyone with them. There are books that include zero and books that do not include it. There are teachers that include zero and teachers that do not include it.

A hypothetical situation:

I happen to give a student my opinion. That student does his math work based on my opinion. The teacher teaches based on a different opinion. Where do you think that would leave the student?

You may notice in my previous post (#26) that I never stated that zero was or wasn’t included. Instead, my answer was that it is a matter of opinion. It will continue to be a matter of opinion until it is proven well enough (one way or the other) to become fact.

My suggestion: Refer to your own teacher/textbook or use the titles you’re sure about (counting numbers & whole numbers). If the question was, “Is 0 a whole number or counting number;” then I doubt there would be any disagreements and think this thread would be much shorter.