the Poincare conjecture
I suggest that the Poincare conjecture is a direct result of a new understanding of the basic structure of space itself.
In the concept sharing of a point, we can see that space has this new leveled structure.
Abstract: Here I introduce concept sharing. In uncovering extended space, I develop new ways of understanding Poincare’s conjecture.
Overlapping shadows:
An interesting notion can come about from something observed in nature, the overlapping of 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.
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).
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 shadows at the center. They have exact boundaries and some way of showing we have two there or three there, ect.
They are ideas or concepts so that they are further like the shadows in that there is no real substance to them.
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.
Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects hidden as one.
The objects are somehow different from each other. 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.
A new plane:
Points are also exact concepts. In the Euclidean plane they are places, with the notion of no extent, in the plane. We should be able to place two together using two new numbers 0’(1) and 0’(2) identifying that we have two points. (0 is indicating an origin)
An object of no extent placed together with another object of no extent, would still have no extent- but there could be two objects here, under another mathematical system.
The two points 0’(1) and 0’(2) can be different by first uncovering a new place dimension, a place of places. This must already exist because there must be some way to have two points exist together and still be two points.
In a similar way as we uncovered the new number dimension (the number of numbers) we can uncover the new place dimension.
Take the original point out (we can do this since we have a new underlying dimension of place, a place of places) and replace it with the two new points. This can be done for the whole plane of points.
That is, there is nothing special about the origin, so each point of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a whole plane of doubled points co-existing with a plane of places of places.
One of the new points can be fixed, while the other one is capable of “shifting” away in this new dimension of place. In this way these two can be different.
Then all of the sharings in the new plane can become new origins-one point being fixed while the other point is capable of shifting away.
So also that any loop may be considered to be a point, because the point can have an infinite number of parts, and any point considered to be a loop. Therefore any space which does not contain “holes” or twists in any dimension will be necessarily simply connected.
