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.
Mathematical Concept removal, subsequent sharing and separation
Introduction:
The notion of a point, that which has no parts or no extent, is basic in math. The ancient Greeks thought about points, but what if they were not entirely correct?
They asked, what if two points were placed next to each other? They thought that this would be one point and stopped there.-Aristotle : Physics- “neither can two points be contiguous with one another”
But what if something like the contiguousness of two points could be possible. Since something of no extent combined with something of no extent would still have no extent- but there could be two items of no extent there! The two objects could be hidden as one!
One can regard the overlapping shadow diagram below:
Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:
Now as seen in the overlap, two different shadows can take up the space of one location, we can regard these as e and p. (e is another possible entity of no extent. P represents a point) and not points, p and p combining to a single point . If I take away one light, one shadow still remains. The table is like an underlying space.
Since in mathematics we wish to have this concept of no parts or no extent, so too can we have this further notion. The further notion is that two items could be together as one, if they both had no extent.
But they could not both be points as we know them now, as they would just become one point as Aristotle thought. Aristotle thought that two points could not be contiguous since there would have to be no line in between them and we would have no extent.-Aristotle : Physics- “as there are an infinite number of points between two selected points”. The idea is that the two points would combine into one. So there might be another way of thinking about entities with no extent.
Suppose I have a combination of a point and another, different entity of no extent, call it e. To have this combination appear I have to be able to remove a point and replace it with this combination.
So a point could also be thought of as different from the way we think of it now. It could have another level along with it.
It would have to exist with a place of places. Since I am taking a point out there has to be an underlying level where the point can be. This has to coexist with the location level.
This shows the way to make the new entity, e, different from a point.
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If I take out a point, it leaves the place of places of the point, somewhere where a point could be, another level to location. A location of location. This has to coexist with the usual location, so it is another underlying dimension of location.
If we then have a dimension of new places of the original place of the new entity, the new entity can move in this dimension. The new entity has a location but can also have new locations of its original location. This makes it different from a point which we can say still has its location only. Then I can switch this new combination with a point and have a new geometry.
P is now with a fixed location of its original location making it different from what I originally called p, but similar enough to leave it labeled p. The idea has not changed, only another level added in. We can think of it as taking p out to reveal a new level, then putting p back in but keeping the new level.
So the new entity e can be made different from p by allowing it to move in this new dimension of location of original locations. So that e not only has location but also can have new locations of its original location.
The two items would be in some way different as we want two there, but in some way the same as these are at the same place. So these two are equivalent but not identical.
It would be like being next to each other, as we have two here, but unlike being next to each other since these both are in one location. The two items would have to be different enough so that they would not combine.
The two points would have to touch in their entirety.
Yet what if we didn’t have two points, but two different geometric entities of no extent?
Then we have to take out the idea that a point is the only item of no extent. We switch the point with this new idea of a point and an e together.
Suppose this is the case then. Then the two new items must be different so that they don’t combine into one.
Because we want to keep the notion of an item of no extent we have to accept that two different items of no extent can be together since they can take the space of one point. We can remove the concept of a point as the only item of no extent.
Then remove the point as the idea of the only item of no extent.
P is fixed as usual. E is different from p in that it can move, unchanging in an extent of p’s to share with different p’s, since it has a location already.
Unlike a p moving in an extent of p’s which is constantly changing with changing position, since p has position only.
E has a new location of its original location away from its original position
Also we can have an extent of e’s to go along with the p’s so can have a sharing like e(1)*p(2)*e(2).
E has multiple new locations of its original location where it could be. Each p has one location only.
E can be different from p as e has the extent of p it can move in. Before we only had p alone so a moving p could only be in coincidence with a p in the plane of p’s. E can move unchanged in a plane of p’s or in a combined plane of e’s and p’s.
Now, we have in the same location two different entities. How do we number these?
It is not the number 2, as we know it, since the two entities are in the same position. Since we have concept shared a point so too can we concept share the concept of a number. So I can take out the number one and label the entities 1(1) and 1(2). The next level of numbers is the new number of different numbers. This can be 2 instead of 1. So I may have the new numbers 1(1) and 1(2) representing the new entities. The two new numbers are sharing the concept of the number 1.
There could be at least two different geometric entities of no parts. For then these two could be together and still be two, hidden as one, since this would still have no extent. We can remove a point, in so doing replace it with this sharing and discover how the two new items are different.
If we want the notion of an object of no extent we could have other objects of no extent as well, since two could be hidden as one. The two entities would have to be different enough so that they would not combine.
Because there is an entity of no extent, a point, and we base geometry on this, there might be other entities of no extent. We could have a non-combined entity of no extent and have another geometry, which would fit and extend the geometry which is already there. This is because we are still keeping to the items to no extent as a basis. We would just need to make the other items different.
The two items could be two different items of no parts, if the two items were different entities. That way we could not conclude that the combination was not a point or the other entity but we would have to leave it as these two, together, since now we have two entities of no extent.
E’s and p’s don’t combine. E’s share the concept of no extent with p’s. E’s are also shared with different e’s. So e’s have to share with other e’s so they can all be sharing a single location. P’s are still at the level of places and e’s can move, this is what makes e’s different from p’s.
The new space is a space of possible new locations of existing locations a next level to location. Locations can have new locations.
This is possible since locations are zero-dimensional and could fit into different locations of locations, which would again be zero-dimensional.
We can keep both of these ideas if two points together are defined to be one point, yet e(the new entity) and p placed together do not combine.
Since we want to keep this concept of no extent, or no parts it must also be possible to have two together and still have two, as long as the two parts are different enough so that they do not combine into one. That is, since we want to keep this notion of an entity with no extent and since two could be together and still be two if they were different entities, or different individual entities.
Yet let’s keep the math we already have just add a new item to the concept of “no extent”. That way the usual idea of two points together being one, still is valid, just create another item for the other case. In the original case the two items lead to an item of no extent which we say is the same as the original two. In the new case there are two items together which together have no extent, but they do not combine, being different entities.
Each e can be considered an origin and can move along the extent of e which is defined as being with the extent of p.
Yet we want this concept of point removal and sharing (the point is removed and the two new items are sharing the notion or concept of an object of no extent) to join up with the knowledge that is already in existence.
In doing this we have to add to and change the mathematics which already exists. This turns out not to destroy but heal mathematics, as this can be seen as something which is basic, but is missing. The math from this leads to a way to solve many old problems which are easy to state but previously hard to solve, such as:
The knot problem:https://calctutor.ca/category/mathematics/knots-mathematics/the-knottedness-and-chirality-of-the-trefoil/
The twin prime conjecture:https://calctutor.ca/category/mathematics/the-twin-prime-conjecture/
The Goldbach conjecture:https://calctutor.ca/category/mathematics/goldbachs-conjecture/
Fermat’s last theorem;https://calctutor.ca/category/mathematics/a-clearer-and-simpler-demonstration-of-fermats-last-theorem-wiles-theorem/
The Poincare conjecture:https://calctutor.ca/category/mathematics/the-poincare-conjecture/
The Riemann Hypothesis:https://calctutor.ca/category/mathematics/the-riemann-hypothesis/
The Collatz conjecture:https://calctutor.ca/category/mathematics/the-collatz-conjecture/
In fact this idea can complete mathematics:
The completion of mathematics:https://calctutor.ca/category/mathematics/the-completion-of-mathematics/
Definition and description of e:
When two points come together or start together, it seems as if there could be two there as 0+0=0.(two items of no extent could still combine to no extent). Only that points have location only, and having them together means we have one location so we have to conclude that there is one point there.
What if an object of no extent could be expressed as two different items of no extent, starting together. Since an item of no extent together with another different item of no extent would still have no extent, yet there could still be two items here and not one. The other item of no extent would have to have something more than location (it would have to be different from a point).
Yet since it is logical that two items of no extent could be together and still be two items, if they were two different entities, this must be mathematically possible, somehow. It would work if the two items were not both geometric points, in the sense that they were not the same type of entity. For points are defined to have location only and here we are at the same location.
Call this new indivisible item “e”. Then this together with a point can be called e*p. We can separate e and p (not divide as this is a separation not a division) if we move the original e along an extent made up of p’s or e’s. The e extent is different from the p extent in that it is a series of positions of the original e position. That is it is the next level of position.
Then we have to have an object of no extent which is somehow different from the usual conception. I am thinking of putting two items of zero size together. One can think of putting a dot on top or beside a dot which is there already. One cannot visualize two items of no extent being together, but one can also not visualize an item of no extent as well.
Yet they are an e( new entity) and a p at the same place, if we have a preexisting space made up of ordinary points (p’s).
In terms of symbols, one may write pop=p as the idea that for an ordinary point two points starting together or being brought in coincidence form again an ordinary point. O is the idea of coincidence.
Then e would be something different. E then cannot be in coincidence with p, if we redefine coincidence as the starting together or coming together of two or more points. so we invent a way for the entities to be together and call it sharing, E*p is e sharing with p . Sharing is a way for two items to come together or be defined together and not combine. This should be possible as seen in the analogy of the overlapping shadows.
This must exist somehow in mathematics, since we must have the notion of something with no parts or no extent and we should be somehow able to put two objects together and still have two. So we have p and e as different entities which do not combine.
Then we can see that e can be different from p, as in an extended space of e’s, I could move an e to another new place of the original place of e, along the e continuum. The e continuum can coexist with the p continuum.
So e is another type of point. It is able to be with another entity which has no extent and not be equal to that entity. P doesn’t share with p, as this is already seen as coincidence. Since points have location only this is the same location and so the same point.
Looking at e, we can’t have e(1)oe(1)=e(1) as then e=p. But possible is e(1)*e(2), if we have an extent of e outwards from p(1). e(2) is some e on the e extent outwards from p(1). This is what makes e different from p, with p we only have pop=p. That means we can have a shifting movement of one e along another continuum of e’s and p’s. Then e’s can have a new location of the original location of an e which makes e’s different from p’s, which have location and the underlying space only.
e(1)*p(1) is e and p sharing a location. We can separate a copy of e(1) away, since 0-0=0 and we have two extents. Think of a jigsaw puzzle of a landscape being taken apart. There might also be an extent of p, along with the e, In which case we can have e(1)*e(2)*p(2).
This is how e is different from p.
To be complete…=p(1)op(1)op(1)op(1) =p(1)op(1)op(1)=p(1)op(1)=p(1), and also… e(1)*e(2)*e(3)*e(4) is not equal to e(1)*e(2)*e(3) is not equal to e(1)*e(2) is not equal to a single e .Also e(1)*e(2) is possible.. That is, e or multiple e’s can travel along an extent of e’s and p’s.
These would not be points, then, as points have location only. Two moving points could come together to form one point. In Physics or mathematics if we have a plane of points(fixed) there can be a point moving in this plane which takes up the locations of these planar points as it moves, and then we have pop=p as it moves.
But also e’s can move as well and we could have two e’s come together to form e*e when the initial location is moved to the extent of e’s.
Two little nothings still add up to nothing, yet there can be two little nothings there. The two items would just be hidden as one. Let’s hide!-they say. Who can see us? The two items would not be points, for points have a singularity to them. That is, when I combine two moving points, in the usual geometry, it leads to one point, but this doesn’t have to be so with another possible entity.
These two items could be multiple. This is how we can hide! In order to do this we would have to take out the usual concept of a point as being the only entity with no extent and replace it with these new conceptions as other entities which can have no extent. There are these little entities that don’t combine to form a point. This is because a point is already there as the only idea of no extent. We need some room!
If you accept the notion of an entity with no extent you have to accept this new possibility as well. If you want a point, you have to have us too! The new entities say! The notion of no extent leads naturally to concept sharing! The concept of no extent is an example of a concept.
To remove the concept of a point as the only item of no extent, we need to have a concept space, consisting of a concept of multiplicity. This must exist because I must have this further concept of a doubled item of no extent somewhere.
Extension to other concepts:
Furthermore, all math concepts are point-like in that they are exact, universal ideas which have no existence in physical reality. They are mental constructions which are not available to the senses. They are in thought only.
We can replace e and p, as two items sharing the concept of no extent with two items sharing the concept of a number as well. Or the concept of a set, or group, etc.
Math concepts are universal so that they don’t change from person to person or over time or space. A number, point, function or group, etc. is unchanging, eternal, unalterable.
So also they all can be multiple as the basic ideas of math are all based on geometry and numbers. Functions map inputs to outputs based on formulas which are algebraic expressions of variables(numbers) to points on a graph. Groups are collections of the rotations or flips of geometric objects. Elements of groups are exact in the end numerical or geometric. So numbers, sets, groups, functions, etc. can all have concept spaces!
The idea is to extend the concept using the idea of the concept and the extender “of”. So for example; location of location, number of numbers.
The idea is to go backwards into the idea.
Look at the example of location. For a new location of an initial location we need an extended location, that is there must be an extended location, somewhere to put one of the entities, the extended entity.
Then the extended entity has the location of an initial location, which makes it different from a point, having only location.
The concept sharing of a number:
We can start with a line of p’s with two lines of e’s sharing locations. All in a plane of e’s as shown below:
The numbers 1(3), 1(1) and 1(2) can be associated as shown, an e is removed and replaced with an e*e and we can associate the new numbers 1(1) and 1(2), concept sharing with the number 1(3). The number of numbers is 3 and not 1. We can then unfold the two lines of e’s to make an axis where we can have the numbers 1(1) and 1(2). The two numbers 1(1) and 1(2) are sharing the concept of 1. We have the usual number line on the midline of the new plane.
With numbers, they represent a position, an amount or a label.
Think about position and consider a race where 2 runners are tied for 10th place. We can say the runners are tied for tenth place giving the number 10 to both runners. But suppose instead of a foot race, two points are together in a race along the number line. Then in 10th place two points could be together as has been seen. Then two number 10’s can be given one to each point. These numbers could go with the points as they are mapped into the new plane which has been mentioned.
A circle which is the 10th circle to be formed could contain the two e’s with the two number 10’s. Then 10(1) and 10(2) could also show an amount of points, 2. As well these could be the labels we are giving to these points, or the position in a number line, 10.
If sets are made of points, the sets could have concept spaces. If sets were made of numbers, the numbers could be associated with points and we could have concept spaces. Similarly if groups are made of numbers or diagrams which are made of points they too could have lower concept spaces.
In the equation x^2+y^2=1 the two points (0,1) and (0,-1) can be mapped to the point (0,0).
This can be notated ….p(1)*p(2)*p(3)=p(1)*p(2)= p(0).
Where * is the idea of different points coming together in the usual plane.
When mapping to points there is a many to many map, a many to one map or a one to one map. Yet there is another possibility with e’s. There can be, for example, a two to two map. Where two separated e’s are in the new plane and combine to form two e’s at a single location. I say two e’s at that location now because they are defined in two different places in the new plane. Therefore there are two here as in the case of the overlapping shadows.
Since it is possible to have two entities together and still be two entities, this other case must exist somewhere. The entities must be zero-dimensional but not be points.
Yet the overlapping shadows show us that there could be a constant number of entities. This is because the shadows have somewhere to be cast onto. Rather than a notion of no extent alone, which could or could not be multiple, the surface makes it possible to show a finite number of entities of no extent.
Then postulate a place of places where ‘e” the entity of no extent which is like a point in that it has no extent but unlike a point being different in the following way:
e(0) is not equal to e(1)*e(2) is not equal to e(1)*e(2)*e(3) where again * is the idea of movement but this time in the new plane.
Suppose I represent a location by (0,0). What if we can have a case of (0,0)(1) not equal to (0,0)(2)?. This is possible if (0,0)(1) and (0,0)(2) were somehow different. Since something with no extent can be “added” to something with no extent and the result is something with no extent, there could be two items here. (see the overlapping shadow diagram) We would have to somehow make the two “points” different and only two “points”.
Since two items (“points”) of no extent could appear to be a single item, and we could conceivably fix this at 2 or three, or as many as we choose, this other plane also must exist.
The usual idea of points can be notated …p(1)*p(2)*p(3)=p(1)*p(2)=p(0).This is the idea of two or more points coming together to form one point. This is all happening in the usual plane.
The other case can be notated e(0)=e(0) not equal to e(1)*e(2)=e(1)*e(2) not equal to e(1)*e(2)*e(3)=e(1)*e(2)*e(3). So e is not equal to p because it exists in placement space and has this other feature which is different from the way p behaves. In placement space we have 1 or 2 or 3 e’s together and also capable of being separated to different placements.
Then usually the idea of points can be notated pxp=p or pxpxp=p…etc. Where x is the idea of coming together or separating apart and p is a point. But what if there were another entity of no extent, call it e such that exe=exe, e is not equal to p so that exe is not equal to p and also exe=is not equal to e as that would be the same as pxp=p. We can call these entity equations.
It seems like exe are two identical entities of no extent and it should result in e. But consider that to have exe=p, I have to take out pxp=p that is the idea that a point is the only item of no extent.
This means I need a space of places coexisting with the original places, so that I can take out the place pxp=p and replace it with the new places, exe=exe. Then we give up the notion of a fixed plane of points, instead we can have three planes, coexisting with one another.
The most basic new plane is in a sense at a lower level than the usual plane. This is a plane of places of new places .Any e in the usual plane can move off in any direction into this new plane, leaving its partner behind. Most basically, the entire plane can move, as shown above.
That means exe are not 2e’s at the same place, as is usually thought of as place but two e’s at the same place in new places. A new level to place. Now we have more room. Since they are in this sense not in the same place, they don’t combine. Briefly we can write this exe=exe (sharing).
Take out the concept of place and put in this new concept of place. The only way it can be different is if the places don’t combine to form a single place but stay separate while being together. (sharing as in the overlapping teacup shadows)
Then we can separate the two e’s, but the only way this can be different from the usual idea of separation in points, They are the same in that they share a place and a place of new and original places.
The place has been removed so we have an underlying dimension where places have other places of original places. Like a jigsaw puzzle of a landscape being taken apart. In this way two e’s are sharing a place and a place of new places.
Then this leads to a new extent, a line with two distances one being this new zero and the other being the usual concept of distance, extended.
Wait, the little entities say, how can this help anything mathematical? We can just scatter anywhere! Yes but you could also stay together as a line!
Then this leads to a new extent, a line with two distances one being this new zero and the other being the usual concept of distance, extended.
This is a new dimension. Each e of the extent is different as any other e, yet they originally shared. This is just a new dimension in length. We can notate any two e’s as e(1,m) and e(1,n).
This extent may be considered as negative distance as we need to shrink it to get back to the new zero and then take this out and replace it with pxp=p to get back to the usual zero. Since for e(1,m) and e(1,n) the place is the same, any point that is bound to e(1,m) is also bound to e(1,n). Just not to both at the same time. We may have a closed loop of e’s which can move off and the shape could be altered if we have different distances associated with each e.
We can set a mathematical system with exe=p or choose three e’s so that exexe=p or the number of e’s could be variable.
This must fit into our current structure of mathematics as I am not adding any new notion in, merely clarifying the concept of a point as having no extent, then adding in the necessary new entities. The notion of no extent is the same. We already have this notion of a point as being pxp=p, we have to extend this.
Additionally, there is also the case exr=exr where e and r are two different types of entities as well. This can be for future work.
So we have the idea that a point is an entity with no extent, and also another notion that it could be exe=p but how do these fit together?
It must be that we have replaced the usual idea of a point as being pxp=p with this new idea of a point as being exe=exe.. This is the space, the space of doubled e’s. It can be modeled after the usual idea of extent, yet the distances are negative.
Then this also means I can separate exe=exe in the new space and move in a space between two of the same e, like so, the displacements from the 2e’s are shown.
Then we can have the idea of a multiple point or two tangent points.
With the tangent point we measure the diameters from the point of tangency outward. These can be separated as usual with the usual distance appearing between them. The exe=exe points can be separated as well, with the new space appearing between them.This is the space of places of new places. This is the negative space.
So we must have a plane or a space in which the ordinary places e or p, take on other places.
A picture of this would look like below if we have only two e’s at the origin and I move one e off up and to the right.: The notation is (()) are places of new places and () are places.
This is a movement of one piece of a doubled origin, a single e.
Not only the origin but each identified exe=p of the new space can act as its own centre, The two e’s can move away from each other.
We could have a closed loop of these points all moving together as shown in the diagram above. As well, this loop could be knotted, if instead of a plane we consider a three dimensional space..
Then this is also the entry into Concept Sharing as math concepts such as number, set, group, ect. Can all be thought of as point-like. That is to say they are all ideas which could have multiple expressions. They can all have sharings.
They are all exact and have no physical reality, they are just ideas.
Since they can all be multiple, there must exist lower concept spaces.
The Concept Sharing of a number:
A number is an amount, as in a counting number, or a position on a number line, or a label.
It is point-like in that it has no existence in physical reality, it is a mathematical object, not a physical object. Therefore we can make a correspondence between the idea of sharing in points and an idea of sharing in numbers.
So that means the concept of a number can be extended downwards so that we have a number of original numbers space and this number of sharing numbers after we take out the original number. So, for example, with the number 1; we have a number of numbers space, let the number of original numbers be 2, instead of 1. Take out the number 1, then we can have 1(1) and 1(2) sharing.
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.
