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Abstract: Here I explain and use concept sharing. I take a look at Goldbach’s conjecture.

Overlapping shadows:

An interesting notion can come about from something observed in nature, the overlapping of two shadows. Consider a cup of tea placed on a table and lit with two lights from above, one from the left and another from the right, as illustrated below:

Three areas of shadows are formed. The one in the center is the overlapping of two shadows. 

Consider two points intersecting from the areas of the overlapping shadows. We may make an analogy of two points being placed together with the overlapping of the shadows at the centre. Think of items you see around you, some touching other items, some like a book on a table, by the force of gravity. Others like books placed together on a shelf. Here is where two points of the different items, one from each item, can be seen as being together.

Now consider a point on one of the shadows which doesn’t intersect another shadow. This too has no extent. But as we’ve seen, something of no extent( two points placed together) can be expressed as the combination of two items which also have no extent( two separate points). Can this one point be seen as multiple as well as singular? Think of a hand of bananas. Each banana is connected to the others at the top of the hand. Do we have one banana or could there be many? Points could be multiple.

Duplication also exists in nature, we can think of cell division.

Seeing this duplication with the shadows is possible. Consider two teacups and four lights. Creating sets of shadows as shown above.

The shaded area is where two shadows from teacup 1 and two shadows from teacup 2 intersect. Labeling points with “p”, I can associate p(1)(1) and p(1)(2) two duplicates and p(2)(1) and p(2)(2) the two other duplicates with this area. Then the original points p(1) and p(2) can be groups of these two. Then we may think p(1) and p(2) as two points which may be separated as in the analogy of the items touching each other but not p(1)(1) and p(1)(2) or p(2)(1) and p(2)(2). They cannot be separated in the usual sense.

In the case of a mathematical point, a duplication would have to be located in the same position.Since points are defined as having position only, a duplicate would have to take the same position. But I already have a concept of no extent without duplicates. If two duplicates are together it should lead to a single point. The usual case of a point is singular. So this is not a duplication in the usual sense.

Duplication exists in the form of concept sharing. The idea that concepts can be multiple. This duplicates the concept. Yet since the duplicate is identical we must remove the initial concept so there is room for this new concept.

The case of a point being only singular could be removed. Then I have room for this concept sharing idea. The concept sharing of point.

we must be able to then remove the concept which was there originally and replace it with the shared concept. It must be possible for shared concepts to exist somehow, as we can imagine it.. 

A point and its partner must be multiple so must take up the space of one point only. Yet we already have this notion of no extent without extra points.

All other possibilities for the number of shared concepts are removed.

There must be a “container” of concepts to take the concept out of. A concept of concept space. Here the two shared concepts can be seen as different. That is, I can separate them. I have another dimension of the concept, in concept space. So for points or places I have “places of places”

So I need the concept of a “concept container”

It is the same concept as the concept in question, yet it contains the concept in question. Meaning that it exists at a lower level so that the concept in question could be removed by two or more sharing concepts (equivalent to the concept in question). 

We cannot have both the initial concept  and sharing concepts at the same time. We need to remove the initial concept which means we need the lower level. Since we can always have the same concept as the concept in question and another dimension of the concept is possible since I can have the sharing of two or more of the initial concept and the concept can be extended using the extender “of” as in concept of concept. You can think of this as a nesting of concepts.

We can imagine that concepts can be multiple. Since it must be somehow possible to have sharing concepts, this lower level must also exist.

Akin to the overlapping shadows, concepts are not concrete-as they are abstract ideas, two or more of the same concept can co-exist. They share and fit into the concept which was there originally.

A symbol may represent two different concepts, such as “8” representing an amount, a measurement or a label.

In the case of points or places, the lower concept level is a place of places or location of locations. The two shared points may move off to two different places of places and we can then see how they can be made different (they can be defined at two different places of places).

When they separate, they do not leave a void, there must be a place of places. They may be separated as a place can take on a new place of places. See the diagram below. I am using a jagged line to show the opening into the new place of places space. This can be made two dimensional, leading to a new plane.

The concept sharing of a number:

Numbers are exact concepts. In the above case, we can think of them as the number of overlapping shadows at the center. They have exact boundaries and some way of showing we have two overlapping shadows there or three there, etc.

Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two hidden as one” as well. (concept sharing) if the mathematical objects represented by the numbers had the same boundaries, like the shadows at the center. 

Other than the further darkness of the overlapping shadows, we cannot see or imagine that there are two separate shadows there. Similarly with two numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Then to this end let us create another number dimension, a dimension of number of numbers. Let the usual case be that the number of numbers is only 1. But now let us expand into the next dimension and allow the number of numbers to be 2.

So for example with the number 1, let us take away the original number 1 (since we have another underlying dimension, we can do this) and replace it with two new numbers 1(1) and1(2). These are together like the two shadows but do not form one number. The number then has a one-dimensionality to it as it is at least possible to separate it linearly.

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects hidden as one. Or you can think of the two number 1’s as fitting into each other.

Since numbers are exact we can consider the objects to be like points. Since something with no extent placed onto something with no extent would still have no extent but there could be two objects here. Also we could think of a combined single object of no extent( at the start) created form two other objects of no extent.

When we are numbering two mathematical objects which fit into each other, the objects are somehow different from each other and we give the two hidden objects two new numbers 1(1) and 1(2). 

In the case of mathematical objects there is no external way of telling how many objects there are, previously it was assumed it was only one. We can state how many we wish at the onset thus fixing a certain mathematical system. Then we need the concept sharing of a number to indicate how many objects we wish to be there. 

Goldbach’s conjecture:

We can think about new types of numbers (1(1)(1(2)), (2(1)(2(2)), (3(1)(3(2)),..These are lower numbers than the naturals, the idea of concept sharing being deeper than the usual idea of concepts. The notation is meant to show that the numbers are together in the number space, fitting into one another or hidden as one. I show a revelation of one number “peeking out” from concept sharing with another number. Shown more clearly here: Like so (a<……(a) leading to (a(a). For clarity and brevity I drop the (1),(2) designation from now on.

There are other numbers such as (2(3), (3(4), (3(5),… We can think of this as partial sharing. For (2(3) think of three dots colored red. We can have two blue dots overlapping with two red dots forming two purple dots. I say that to give the idea, numbers are not colored dots. Numbers are the exact concept of amount or position only. So that (2(3) can be thought of as two sharing with three and (3(2) can be thought of as three sharing with two. In the first case 1 is left over and in the second case 1 is an extra number. These are the same concept though so (2(3)=(3(2).

Consider the even shared numbers (2(2), (4(4), (6(6),…

We can break them down into a sum of two prime shared pairs such as (4(4)=(1(3)+(3(1) if we define the other types of shared numbers with dissimilar basic numbers to fill in the rest of the possibilities. They can form products too.

Some numbers have more than one decomposition. Such as (16(16)=(13(3)+(3(13) and (16(16)=(11(5)+(5(11).

Now think of the even shared numbers as being created from the primitives. I designate them (4(4) … but these numbers are not created from the usual numbers on the number lines, as these are lower numbers, these lower numbers; (4(4)…. must come from somewhere else. These even shared numbers must go on indefinitely, as to lead to the numbers 4,6,…at the higher level . Then the natural numbers are all at a higher level from these numbers.

For multiplication, in the usual number system we have prime numbers. We can look for the lower level of prime numbers. Suppose I build a lower level of primes from the usual prime numbers by concept sharing two prime numbers. For example (7(3)=(3(7).

But (10(10) = (7(3)+(3(7) =(2(2)(7(3). (7(3) can’t be equal to (7(1)(1(3) as I can change this to (7(1)(3(1)=(21(1). The other part, (3(7) can be changed to (1(21). I can add to obtain (10(10)=(22(22), a contradiction as 10 does not equal 22. This is outside the sharing domain. So (3(1) and (1(7) can’t be factors. So I don’t factorize (a,b) further where a and b are prime.

Then starting at (4(4), we can ask, how is (4(4) created? (4(4)=(2(2)(2(2). (4(4) is an even shared number so there is a division by (2(2) possible. This can be the definition of an even shared number. When I divide both sides by (2(2) I have a shared number on both sides, these are equal but not necessarily identical. For (4(4)=(3(1)+(1(3)=(3(1)+(3(1) =(2(2)(3(1) also. In this case we can think of the shared number (2(2) separating in the number space to form the number (3(1)=(1(3).

Then also (6(6)=(2(2)(3(3). Also (6(6) =(5(1)+(1(5) =(5(1)+(5(1) =(2(2)(5(1) also.

Also (8(8) is not equal to (2(2)(4(4) as (2(2)(4(4)=(2(2)(4(2)(1(2)=(2(2)(2(4)(1(2)=(4(16). This is outside what we could consider the sharing domain as 8+8=16 but 4+16=20. We find (8(8)=(5(3)+(3(5)=(2(2)*(3(5)=(2(2)*(5(3)=(10(6)

And so on. Each even shared number must be a multiple of (2(2) and another prime shared number. We allow 1 shared with another prime number to be a prime factor. This is not the same as saying 1 is a prime number.

Imagine an extended number line. For example, since (5(5) can separate, lets look for a way (10(10) can be created. (10(10)=(a(b) +(b(a)=(2(2)(a(b). I am taking apart (5(5) so as to create (10(10). Two (5(5)’s are separated and then recombined. ex. (7(3)+(3(7). (10(10) can be created from (a(b) +(b(a) but this is also (a(b)+(a(b) =(2(2)(a(b) so this must work too. As shown in the case above with (8(8), a,b must both be prime. There must be one such factorization.

Every even shared number can be divided by (2(2). The other factor must be a prime shared number as we need this to work in the shared number system. For example (12(12)=(2(2)(6(6). (6(6) must separate into two prime numbers otherwise (12(12) can’t appear (we assumed it was there). It must exist since I need to have 12. I can now take (12(12) out and replace it by 12. So we must have a decomposition with (2(2)(a(b) with a, b both being ordinary prime numbers, or one of the numbers being 1.

This demonstrates the equality of every primitive decomposition as well (which the lower level was hinting at) since for any even shared number there may be more than one decomposition. We go back to form (16(16) and then can separate once again to find another decomposition. Here we find two, (2(2)(13(3) and (2(2)(11(5)

Then if we look at one half of these binary decompositions, that is, for example, look at 16(16)=(13(3)+(3(13) and see 16 =13+3 we can see Goldbach’s conjecture is true. It was just a part of a deeper understanding of numbers.