# Numerus “Numerans-numeratus”

#### SlipEternal

MHF Helper
I think your question is very valid.

If I am comparing these two expressions...

n/(_) : n/(_)

Yes, I agree, there is no way to say which is zero and which is the number 1.

However the op shows that it can be expressed as...

n/(0_) and n/(1_)

also as:

n/(0) and n/1

Where:
n/0 = n/0_ = n/(_)

Where:
n/1 = n/1_ = n/(_)

This then shows exactly when (_) is the dimension of zero, as opposed to the dimension of 1.

So technically when we divide in this system by zero, or any number...it is not a numerical quantity it is a dimensional quantity, that is "placed into".

So if I divide a cookie or zero cookie into the dimensional space of zero, or the dimensional space of any number....that cookie is still there.

So I may only say the cookie divided into the "space" of zero, or the cookie divided into the "space" of 1. "Both of which are the same"

So in any and all semantic statements, I must clarify which "thing" is the cookie, or numerical value, and which "thing" is the space, or dimensional unit.

I claim that zero has space...because of many reasons...

I also claim zero is the absences of a given numerical quantity...(as is currently accepted)...but it is also a quantity of dimension.

"Nothingness" does not exist...so if zero is not "nothingness" and it is also the absence of a numerical quantity...what then is it...the only remaining option is a dimensional unit quantity.

Also the equation 1 + (-1) = 0....proves the space of zero...and that it is equivalent to the space of 1.

We never have a "number" unless a numerical quantity and dimensional unit quantity are "together". This is so in the math and so in the semantics.

I hope I have understood your questions. I hope I have answered them.

Thanks!
You use the equation 1+(-1)=0 as a proof of "the space of zero", but I am not following that proof. I thought zero has space because you defined it to. What about the equation 1+(-1)=0 would bring me to the conclusion that 0 has space? What is the space of $2+(-3)$?

• 1 person

#### AndyDora

You use the equation 1+(-1)=0 as a proof of "the space of zero", but I am not following that proof. I thought zero has space because you defined it to. What about the equation 1+(-1)=0 would bring me to the conclusion that 0 has space? What is the space of $2+(-3)$?
To answer your question, I beg of you to allow me the use of semantics again.

The equation tells me to start on the numerical quantity of the number 1 (on a number line). The operator tells me to move one "space" to the left. That "space" is zero. Therefore zero is space. If it were not so...the sum yielded would be the numerical quantity of the number (-1).

As given in the op...(2 + (-3) = (-1)...addition exists without change.

I start on the numerical quantity of the number 2 (on a number line), then I move three "spaces" to the left. Which is the "space" and "numerical quantity" of the number (-1).

Even in this last example...we must count zero as space...or never arrive at the sum...or the wrong sum...

#### SlipEternal

MHF Helper
To answer your question, I beg of you to allow me the use of semantics again.

The equation tells me to start on the numerical quantity of the number 1 (on a number line). The operator tells me to move one "space" to the left. That "space" is zero. Therefore zero is space. If it were not so...the sum yielded would be the numerical quantity of the number (-1).

As given in the op...(2 + (-3) = (-1)...addition exists without change.

I start on the numerical quantity of the number 2 (on a number line), then I move three "spaces" to the left. Which is the "space" and "numerical quantity" of the number (-1).

Even in this last example...we must count zero as space...or never arrive at the sum...or the wrong sum...
I am getting more and more confused. To me, it is seemingly arbitrary how you decide how much "space" zero has. Why did you arrive at a space of 1? Why not the same space as 2? What makes the space of 1 correct? Why does 2 not have the same space as 1? It does not take 2 space to "move past" it on the number line. So, what makes the space of 2 equal to _,_? And why would it be bad if zero had no space? It still has a numerical value, and so addition and subtraction that yield 0 still will make sure not to miss it on the number line. You start at two and go three numerical units to the left. Why would you need space for the number zero at all?

Let's consider measure theory. Measure theory takes careful precautions to ensure that a single number (or point) does not take up any measurable space. If each individual number did take up space, then integration and much of advanced statistics would break. So, conceptually, I am having a hard time with the notion that you want to specifically give space to each number. How would this work with advanced mathematics? Is all analysis and topology wrong? Or do numbers not have space when you are studying measure theory, analysis, topology, advanced statistics, etc.?

Edit: It is not necessarily a bad thing if you decide that all of these branches of mathematics are wrong so long as you can recreate them using your number system. But, the fact that they all break and it is unclear if they can be recreated would give any mathematician heart palpitations at the thought of a number system that did not directly correct some major underlying problem within mathematics. For instance, if your number system lead to a measure theory that would prevent odd occurrences like the Banach-Tarksi paradox, it might be embraced. Think about it from a purely psychological standpoint. If your number system were to ever be adopted, it would require enough people to agree with it that they start using it for other branches of mathematics. But, if it does not work with other branches of mathematics and they have a stable system that has worked for thousands of years, what motivation could they possibly have to even give your system a chance?

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#### AndyDora

I am getting more and more confused. To me, it is seemingly arbitrary how you decide how much "space" zero has. Why did you arrive at a space of 1? Why not the same space as 2? What makes the space of 1 correct? Why does 2 not have the same space as 1? It does not take 2 space to "move past" it on the number line. So, what makes the space of 2 equal to _,_? And why would it be bad if zero had no space? It still has a numerical value, and so addition and subtraction that yield 0 still will make sure not to miss it on the number line. You start at two and go three numerical units to the left. Why would you need space for the number zero at all?

Let's consider measure theory. Measure theory takes careful precautions to ensure that a single number (or point) does not take up any measurable space. If each individual number did take up space, then integration and much of advanced statistics would break. So, conceptually, I am having a hard time with the notion that you want to specifically give space to each number. How would this work with advanced mathematics? Is all analysis and topology wrong? Or do numbers not have space when you are studying measure theory, analysis, topology, advanced statistics, etc.?

Edit: It is not necessarily a bad thing if you decide that all of these branches of mathematics are wrong so long as you can recreate them using your number system. But, the fact that they all break and it is unclear if they can be recreated would give any mathematician heart palpitations at the thought of a number system that did not directly correct some major underlying problem within mathematics. For instance, if your number system lead to a measure theory that would prevent odd occurrences like the Banach-Tarksi paradox, it might be embraced. Think about it from a purely psychological standpoint. If your number system were to ever be adopted, it would require enough people to agree with it that they start using it for other branches of mathematics. But, if it does not work with other branches of mathematics and they have a stable system that has worked for thousands of years, what motivation could they possibly have to even give your system a chance?
You scold me well, in the same manner as you have before. I can not offer what is considered "sufficient" reason to mathematicians. Yet...that is. But I am working on this. I can offer the reason that it provides solutions for questions Siri can not answer...as suggested.

Any time I use the term "space" or dimensional quantity unit....it's length and width remains abstract and undeclared. This is from the op.

I never claim zero has more than "a single" space. It is arbitrary...just as you say.(nor can you EVER define its' length and width).

But if you do NOT allow zero to have a single "space"...then you can not divide by zero, or multiple by it with products other than 0.

Therefore...at no time is there a violation in topology and measure theory...as they require length and width to "not" exist (within each number)...not the actual dimension itself. This I know is a philosophical point. But it is the exact thing you are asking me regarding the use of 0 with only "1" space.

It is that zero has a single quantity of dimension...it is not that it has more than that.
It is that (1) has a single dimension...of which a single numerical quantity inhabits.
It is that (2) has two single dimensions...each of which a single numerical quantity inhabits.

The only difference between 1 and 0....is that 1 has a numerical value assigned to a dimensional unit, or a single space.
Zero...just has not had a numerical value assigned to the "single space". That is why zero has a "single" space....as opposed to "two" etc...

I will investigate this paradox you suggest...very likely it is over me, and will take me much time...

Again...so far...I can only say this is good for a unification of semantics and mathematics.

I can only.... at this time.... say...it is now possible to divide by zero...semantically...without breaking the rules of mathematics.

#### SlipEternal

MHF Helper
That's a fair assessment

• 1 person

#### AndyDora

Ok regarding Measure Theory:

https://en.wikipedia.org/wiki/Measure_(mathematics)

From this article we can see that only zero, must remain without a length and width. It says nothing of dimension. Zero remains without a length and width.

Regarding Topology and set theory:

https://en.wikipedia.org/wiki/Zero-dimensional_space

This is a link referencing both:

At this point I must say...for this to continue in higher branches of mathematics a slight re-wording is required. But nothing else. We must say or allow that a "dimension" may exist abstractly...without a length and width. Further work of course is needed here.

So perhaps I was hasty to dismiss you on these two points Slip...but...one point is easily taken care of...the other should be also. More work to come.

https://en.wikipedia.org/wiki/Dimension

A bit late....but from this we can see...all I really need to do...is claim all "MY" spaces or... all dimensional unit quantities... are abstract....

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#### Archie

So consider a black whole...the mathematics of which yields divsion by zero.
Division by zero in physical theories is useful. It tells you when the theory breaks down and no longer describes reality. It tells you that refinements to the theory are necessary.

#### AndyDora

Division by zero in physical theories is useful. It tells you when the theory breaks down and no longer describes reality. It tells you that refinements to the theory are necessary.
yes...OR...it is just possible that there is a certain portion of these theories... that only fail...because the math fails to describe the action of division by zero. Yes...this is an opinion...but I am working diligently to prove it. I have to some degree done this regarding semantics. And in my opinion in other areas as well. Just not meaningfully as of yet.