[Halc]

How would you distinguish between zero and non-existent?

My sister is nonexistent.
I have zero sisters.

For me, existence is to be measured, so I can measure the count of my sisters, and it comes up zero. But the first wording is a reference to a specific person, and that reference does not correspond to anything, hence is nonexistent.
That seems to be the best distinction I can come up with.

[Bill S]

I have zero money. Money is non-existent on Mars.

In my current situation, money is non-existent.  As far as I am aware, there is zero money on Mars.

You can make it work both ways, if you really want to.

[Bill S]

My sister is nonexistent.
I have zero sisters.

For me, existence is to be measured, so I can measure the count of my sisters, and it comes up zero. But the first wording is a reference to a specific person, and that reference does not correspond to anything, hence is nonexistent.
That seems to be the best distinction I can come up with.

I see your distinction, but it is purely lexical semantics.  If we were discussing philosophy, it would be fine.
 
Consider the definition of non-existent at:

https://www.google.com/search?q=opposite+of+nonexistent&oq=opposite+of+nonexisent&aqs=chrome.1.69i57j0.14448j1j1&sourceid=chrome&ie=UTF-8

not existing or not real or present

Your sister is non-existent, therefore, she is not real.
You have zero sisters, therefore, you do not have a sister who can be said to exist.
Both examples fall under this definition.

[alancalverd]

The money example is robust.

Having zero money implies that the concept of money has some meaning in context and that I could have a nonzero amount - the amount in my account  is a member of an extant and meaningful set - the sums of money owned by people on Earth.

The set of sums of money owned by people on Mars is non-existent and meaningless. There being no people there, the concept of money (i.e. the common property of the members of that set) is undefined, so the set does not exist. 

Neatly, this leads to a definition of money: the set of tokens that people exchange for goods and services. It covers every form of monetary exchange but it entirely reliant on the presence and consensual activity of humans.

[Halc]

My sister is nonexistent.
I have zero sisters.

For me, existence is to be measured, so I can measure the count of my sisters, and it comes up zero. But the first wording is a reference to a specific person, and that reference does not correspond to anything, hence is nonexistent.
That seems to be the best distinction I can come up with.

I see your distinction, but it is purely lexical semantics.  If we were discussing philosophy, it would be fine.
 
Consider the definition of non-existent at:

https://www.google.com/search?q=opposite+of+nonexistent&oq=opposite+of+nonexisent&aqs=chrome.1.69i57j0.14448j1j1&sourceid=chrome&ie=UTF-8

not existing or not real or present

Your sister is non-existent, therefore, she is not real.
You have zero sisters, therefore, you do not have a sister who can be said to exist.
Both examples fall under this definition.

Totally agree.  That is consistent with my usage of the words.

[Bill S]

The money example is robust.

One of the things I like about this thread is that it keeps introducing different perspectives.

What you say about the concept of money, and the introduction of sets is instructive. 

this leads to a definition of money

Interesting stuff, but still throws no light on any difference there might by between zero and non-existent in terms of physical objects.

Sometimes I’m glad I’m not a mathematician!  Life is complicated enough, as it is.

[Bill S]

That’s another thing I like about this thread; it’s possible for two posters to “Totally agree” when they seem to be disagreeing.  Perhaps we need a good definition of “agree”. 

[Halc]

I'm saying you're using the words correctly in your post. The words mean different things, which is easily illustrated by trying to swap them.

My sister is nonexistent.  My sister is zero.
I have zero sisters.  I have nonexistent sisters.

In both cases, the left side conveys the intended meaning and the right side doesn't, if it make sense at all.
Prisoner zero has escaped!

[Bill S]

My sister is nonexistent.  My sister is zero.
I have zero sisters.  I have nonexistent sisters.

That's why I didn't try to swap your statements.  However, doing so does highlight one point that is very relevant to the OP.
Saying there could ever have been "nothing" is not dissimilar from saying " I have nonexistent sisters".

[alancalverd]

Interesting stuff, but still throws no light on any difference there might by between zero and non-existent in terms of physical objects.

I have zero apples. Unicorns are non-existent.

Or to be really, really pedantic: There are zero Indian elephants in my garden (because I've eaten all the apples). Martian elephants are non-existent.

This has important forensic consequences. There might be a lingering smell of elephant in a garden, or even a huge pile of sh1t, that would lead Poirot to unmask the murderer as a rogue mahout, and the defence "I was working the native elephants on Mars" is no alibi. 

[Bill S]

I have zero apples. Unicorns are non-existent.

I believe you, and have no problem with this distinction, in this context.

It still leaves me with the ongoing questions:

If something is infinitely small, does it, physically, exist?

Does the concept of "infinitely small" have any meaning? 

I know we can wander off into definitions of “exist” and various other terms; but am I the only one who sees a fundamental problem here?

[Halc]

If something is infinitely small, does it, physically, exist?

Does the concept of "infinitely small" have any meaning?

We've been over this.  It means zero size.
The center of gravity of Earth exists (is real and present), and lacks a nonzero dimension.

[Bill S]

The center of gravity of Earth exists (is real and present), and lacks a nonzero dimension.

Of course it is, but, but it exists only in that it is differentiated as part of the Earth.  It has no independent existence.

How big is its “nonzero dimension”? 

[Halc]

The center of gravity of Earth exists (is real and present), and lacks a nonzero dimension.

Of course it is, but, but it exists only in that it is differentiated as part of the Earth.  It has no independent existence.

All correct. The angular momentum of Earth similarly exists (and hasn't a location), and similarly does not exist independent of Earth.

How big is its “nonzero dimension”?

I said it lacks one.  It is a point in space, or a worldline in spacetime.  The latter has a length of perhaps 12-13 billion light years.

[Bill S]

I said it lacks one.

Sorry, the question should have been: how big is a zero dimension?

It is a point in space,

So, a zero dimensional entity has zero dimensions, and that is a point, which is also zero dimensional.  A trace of tautology, here, don’t you think?

or a worldline in spacetime.  The latter has a length of perhaps 12-13 billion light years.

How can it be zero dimensional if it has length?    Wouldn't that be one dimensional?

[Halc]

or a worldline in spacetime.  The latter has a length of perhaps 12-13 billion light years.

How can it be zero dimensional if it has length?    Wouldn't that be one dimensional?

A worldline (of a spatial point like the center of gravity of Earth) is a line in spacetime, yes.  It is one dimensional in that it has no width or anything, but it is 4 dimensional in that it isn't a straight line and thus follows 4 nonzero coordinates in its path.  A wiggly line on a planar graph is one dimensional (has no area), but still requires coordinates in two dimensions to describe.

Technically, the center of gravity of Earth moves around at faster than light speed, and thus its worldline is sort of broken instead of a nice continuous line.  This is because the material that constitutes Earth is not fixed.

[alancalverd]

Does the concept of "infinitely small" have any meaning? 

It's a poor phrase, be cause logically it means "Bigger than you can imagine small" "Infinitesimal" is the mathematical term you are looking for. 

am I the only one who sees a fundamental problem here?

Apparently! Surely everyone else can distinguish between the local absence of stuff that could be present at a point, and the universal non-presence of stuff that couldn't. James and John are not at school today. James has a toothache. Thanks to effective contraception, John does not exist. Massive difference! The taxpayer has to buy desks for kids who happen to have a day off, but not for those who were never born.

[Bill S]

The center of gravity of Earth exists (is real and present), and lacks a nonzero dimension.

It “lacks a nonzero dimension”.  Sounds like a good explanation, but what does it mean?  If it lacks a nonzero dimension, then it must have “a zero dimension”.  Therefore, it occupies no space.  It is, in fact, the conceptual point at which the total mass of the Earth must be considered as residing, in order to work with Newtonian gravity.  I have no problem with that.

I said it lacks one.  It is a point in space, or a worldline in spacetime. 

Just for clarity; a point in space lacks a non-zero dimension, so it is zero-dimensional; whereas a worldline lacks a non-zero dimension, but has, as you explain in #255, either one or four dimensions.  (?)

[Halc]

The center of gravity of Earth exists (is real and present), and lacks a nonzero dimension.

It “lacks a nonzero dimension”.  Sounds like a good explanation, but what does it mean?  If it lacks a nonzero dimension, then it must have “a zero dimension”.

Yes, it does. All three of its spatial dimensions are zero.  It has zero length, width, and height.

Therefore, it occupies no space.

Correct.

It is, in fact, the conceptual point at which the total mass of the Earth must be considered as residing, in order to work with Newtonian gravity.  I have no problem with that.

Newtonian gravity doesn't require all mass to be a point.  He proved that the mathematics can be simplified to that case if the object is spherically symmetrical, which Earth almost is. But Newtonian gravity works fine for irregular objects. It just needs a longer pencil to work out.

Just for clarity; a point in space lacks a non-zero dimension, so it is zero-dimensional; whereas a worldline lacks a non-zero dimension, but has, as you explain in #255, either one or four dimensions.  (?)

The worldline of a point is still a mathematical (not straight) line, which has no thickness in any spatial dimension.  Earth is not a point, so it has a worldline that is some 11000km thick in any of the three dimensions, at least in a frame near that of Earth.  In other frames, some of those three values might be significantly less.

[alancalverd]

Lacking a nonzero dimension means having zero spatial (or spatiotemporal, or indeed any of the 256 dimensions that pay the rent for string theorists) extent in any direction. What it doesn't mean is "having zero dimensionality", because a point is defined by the confluence of vectors.

(x,y,z) is a point in 3D. Any or all of the coordinates can be zero, depending on where you assign the origin of the coordinates, and conventionally they all originate at (0,0,0). (1,2,3) obviously has dimensionality because we get to or from it by travelling through three dimensions. And if we started there, we could get to (0,0,0) without becoming spaghettified or losing any information, so (0,0,0) is a point with dimensionality but no spatial extent.