# Relative Mathematics

#### topsquark

Forum Staff
When A is used as value and multiplied by zero as space then the sum is A
When 0 is used as value and multiplied by A as space the sum is 0
I'm going to echo what someone else asked: What do you mean by "sum" here?

-Dan

#### Conway

Dan

Valid question, thank you

all symbols on the left of the equal sign are composites of numbers. That is a space and a value separately.

all symbols on the right of the equal sign (the sum) are a full fledged number containing both value and space not separated.

In the op, I stated that it is that a value is placed into a space there by generating a number.....for multiplication....and division....slightly more to it then that, we can get there latter if you wish.

#### topsquark

Forum Staff
Dan

Valid question, thank you

all symbols on the left of the equal sign are composites of numbers. That is a space and a value separately.

all symbols on the right of the equal sign (the sum) are a full fledged number containing both value and space not separated.

In the op, I stated that it is that a value is placed into a space there by generating a number.....for multiplication....and division....slightly more to it then that, we can get there latter if you wish.
Okay, you are apparently using symbols and terms in unusual ways. If I may, please let me suggest some notation.
Definition: Let the ordered pair (v,s) be defined in the following way: $$\displaystyle ( ~, ~ ) : V \times S \to Q: \forall v, ~s \in V, ~S \mapsto (v,s) \in Q$$
where V, S, and Q are "value," "space," and "quantity" spaces.

Then we have some defining properties (A, 0) = A, (0, A) = 0, etc. It's still not great, as A in Q space still looks like A in V space, but I think it's a bit better. (Perhaps even $$\displaystyle (A, 0) = A_Q$$?)

Just a thought.

-Dan

#### Conway

Dan

I would love to use notation. I am very weak in such matters (set theory). I however have done some research on the symbols used in order to better understand you at this point. I hope you will bare with me as we continue along these lines. Please note I will post nothing of "certainty" until you have helped me understand. As far as the notation you used....can you explain more about why you chose Q. What is it that you mean by "quantity space" as opposed to "space". I will await more from you on this matter before I "judge" the notations that you gave. Until then I pose the following notations.

Where V = value, S = space, A = any number in the Set

let the ordered pair (V,S) be described as follows

(V + S) → A: ∀A, S ∈ V, S→(V,S) ∈ A, V ∈ S, V→(V,S) ∈ A
∀A ≠ 0: (V,S) = (A,A)
∀A = 0: (V,S) = (0,1)

Or alternatively

(V,S) ∈ ∀A
∀A ≠ 0 : (V,S) = (A,A)
∀A = 0 : (V,S) = (0,1)

?????What do you think????

#### topsquark

Forum Staff
Dan

I would love to use notation. I am very weak in such matters (set theory). I however have done some research on the symbols used in order to better understand you at this point. I hope you will bare with me as we continue along these lines. Please note I will post nothing of "certainty" until you have helped me understand. As far as the notation you used....can you explain more about why you chose Q. What is it that you mean by "quantity space" as opposed to "space". I will await more from you on this matter before I "judge" the notations that you gave. Until then I pose the following notations.

Where V = value, S = space, A = any number in the Set

let the ordered pair (V,S) be described as follows

(V + S) → A: ∀A, S ∈ V, S→(V,S) ∈ A, V ∈ S, V→(V,S) ∈ A
∀A ≠ 0: (V,S) = (A,A)
∀A = 0: (V,S) = (0,1)

Or alternatively

(V,S) ∈ ∀A
∀A ≠ 0 : (V,S) = (A,A)
∀A = 0 : (V,S) = (0,1)

?????What do you think????
If I am understanding you what you want is a notation to make a difference between values (V) and space (S). I chose the ordered pair notation to avoid the "+" notation: You are adding value and space and they are not the same kind of thing. So the addition is potentially screwing up your meanings. I mean, how can you add 3 + apples? One is a (numeric) value and the other is your space. But we can easily say that we have (3, apple). Do you see what I mean?

I admit that I am still not quite in order with your concepts. I was writing the (3, apple) = 3 apples. I know that's not precisely what you are trying to say but am I right in saying that (3 + apples) lives in neither V nor S? I was calling that a "quantity," hence it belongs to a new set Q . I was also trying to use a convention where the elements of V (value) would be in lower case letters to distinguish it from the set V. So $$\displaystyle v \in V$$. It makes element vs. set a bit easier to read.

As far as your definitions are concerned:
1. (V,S) ∈ ∀A. Are you trying to say that all elements in both V and S are drawn from a set A? Or is A a specific member of V and S?

2. ∀A ≠ 0 : (V,S) = (A,A). Again with the all values question. Are you saying that (V, S) = (A, A) for any members of V and S?

3. ∀A = 0 : (V,S) = (0,1). What are 0 and 1 representing here?

-Dan

1 person

#### Conway

Dan

I will listen carefully to you, so as to learn. I understand what you are saying regarding (3 + apples). As long as we are agreed that 3 is not the numeric representation, but rather the space. So too apples is not the numeric representation but the value. When value is (+) to a space then a "numeric" value is created. This is why I have not heavily invested in notation. As neither V nor S (as I use them are sets.) I will switch to (v,s) as you suggested. I understand and agree that distinguishing elements from a set is necessary. I would like to say here...no matter the SET, all elements there in contain a (v,s). That is value and space are property's of all elements in any given set. I use A as (any number in any set). As to your numerical points

1. Yes (v and s) are drawn, or elements of A (any number in any set).
2. Yes (v, s) are members of all elements of any set.
3. (0,1) is the equivalent to (v,s) for the element 0 that belongs to a set.

Allow for the following

1(as a number) is composed of (v,s) = (1v,1s) = (1,1). The "numbers" in the parenthesis are not numerical.
(_) = one space, no value
(1) = one value, no space
(1) = one value added to one space = 1 (a number)

2(v) X 3(s)

(2(as value) is placed into 3(as space))

(_,_,_) three as space only
(1,1) two as value only, alternatively (2)

(2+2+2) value of two placed into three spaces

note the (+) comes from the axiom of multiplication when value is "placed" into space.

so then with 1 and 0

0 as value only = (0)
0 as space only = (_)

1 as value only = (1)
1 as space only = (_)

The space of zero is equivalent to the space of one, the difference is in 1 having a single defined value, where as 0 has a single undefined value.

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#### chiro

MHF Helper
Hey Conway.

When you combine different objects you need some sort of rule to combine them.

The two main structures in mathematics are numbers and sets and you use arithmetic to combine numbers [and synthetic structures like vectors] and you use set intersections and unions for sets.

Can you express the algebra [i.e. way of combining them] with one or both of these structures? [This is as general as it really gets for current mathematics]

1 person

#### Archie

One reason that you get a bad reaction from your suggestion is that what you are saying is muddled from the start.

Conway said:
It is the inherent nature of all things thatthey are a compilation of two different and distinct things. It is axiomaticthat these two things are space and value.
It may be axiomatic to you, but it isn't an axiom that the majority can by into without some justification which you fail to give. On the other hand, if you are just setting up a (new) number system, you should say so without the psuedo-philosophical mumbo-jumbo. Just state what the system consists of: ordered pairs (a,b) with operations that do whatever it is they do.

#### Conway

Chiro

The combining of value and space are done so with the axioms of multiplication and division. As given in the op.

Multiplication = The value given is placed equally into the spaces given, then all values are added in all spaces.
Division = The value given is subtracted equally into all spaces given. Then all values and all spaces are subtracted but one.(inverse of multiplication)

Archie

I am not setting up a new number system. I have only....

Redefined numbers (within the CURRENT system). I have done so using axioms. You either agree or you don't. I would love to discuss the validity of these axioms, but that would require more of that "mumbo-jumbo". With the redefining of what a number is...a single field axiom can be written to allow for division by zero. Allow for multiplication by zero with sums other than zero. Allow for varying amounts of zero.

Thank Again Archie for you time.

#### Archie

I am not setting up a new number system. I have only....

Redefined numbers (within the CURRENT system).
That is creating a new system. Numbers in are defined in the current system and the way they are defined is nothing like what you are writing.

The problem is that you say what you are doing is pure maths (i.e. set up some rules and follow them) which is fine, but then you try to say that it is the existing system of maths. That simply can't be because the current system is different to what you are proposing.

On the other hand, you talk about the validity of your axioms. Axioms don't have to be valid, they just are. But if you are claiming that your system is a better model for reality than the existing standard system, you have to justify your axioms.

Pure maths: just state your axioms in a mathematically precise way without grandiose claims about relevance to the real world and start proving stuff.
Applied maths/physics: justify why your axioms are a better model for the real world than the current system. Then proceed as for pure maths.

The existing mathematical system has no need of an ability do divide by zero. In fact, the inability to divide by zero is integral to the proper functioning of the system.

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