### Central ideas

The main idea is that two ‘points’ can exist together and still be two ‘points’. Points are as they are usually thought of as that which has location only. ‘Points’ have location but also extended location in a new plane as described below.

On philosophical grounds it should be possible to create an object of two ‘points’ together where we have one point before, since points and ‘points’ both have no extent, this should be possible. (two ‘points’ together should have no extent also, but we could have two items here). Please see the teacup shadow diagram in the post at the bottom. This is not an abstract or imaginary extension of what’s real.

I write my mathematics in plain language to reach the widest audience possible. These ideas do not have to be expressed at the university level, but could be understood much earlier.

We have the concept of the point being an exact location in the plane. Two new ‘points’ can be created together if they are different from the usual idea of points. One may say why two? Why not more? But we can state at the beginning that we have only two. In some other mathematics we may have more.

We can think of two skew lines in three-dimensional space moving together until they intersect. Usually the intersection is thought of as a common point or location. We can also use the phrase ‘two of the same point’. Since if we connect the two lines to form a figure eight, we can go around once and we go over the common point twice.

Suppose we want a different result than a common location. From the philosophy we know we must be able to do this. For this we need to take away the possibility that ‘two of the same point’ leads to one point. In order to have another possibility I have to be able to replace the existing possibility. To do this we need to remove and replace the common point, so we can put something else there. I am not removing a point from a given set of points (the plane), but removing the point itself from the subset of points that make up points of interest from the plane.

This means we need to create a new level of ‘locations’ of a location- where a location is a ‘point’ of interest from a subset of the plane- and ‘locations’ are a new plane of locations which takes over from the first plane where a ‘point’ of interest could be. To take away the concept of the common point I need to add a concept of a location of a point of interest having a location itself.

Then the common point can be removed from the larger set (the subset of points that make up points of interest from the plane can be regarded as a set of points which is contained by the plane of ‘locations’) and replaced and we can create two ‘points’ which do not combine to form a common point-even though they are together. They are two ‘points’ on this new plane. We still have two of the ‘same’ ‘point’. They are the ‘same’ since they have the same location. They are different since they can be moved to any two different ‘locations’ of location.

Needing to be able to take this one point away creates the need and existence of this new level of location. I need to take away the concept of no extent being a point and replace it with a new concept of no extent. This can be an object which has no length, breadth or height but does have parts to it. In this special case the number of parts is two.

I can’t take a ‘point’ away without a new level. I need this space to be able to put two ‘points’ in. (another possibility) If this new plane isn’t here then the two points are coincident and we have just one point. This new plane of ‘locations of locations of points of interest from the plane’ is considered fixed, for now.

These new objects (‘points’) I call ‘parts’ (as in pieces of the whole but in this case the whole is the same size as the parts). Parts are different from points in that I have two items there and not one. They can be labelled the same and are mobile in this new plane. Each doubled location can have two ‘locations’ of location. We can imagine a whole plane of these ‘doublets’. This is the beginning of the work on knots.

We may imagine the empty space in a three-dimensional container. We can extend the point concept stated above to spaces as I can have a set of points be the three-dimensional space. This empty space can be moved to take up the same “space of spaces” as the still spaces, if we move two empty spaces together. Then two empty spaces take up the same “space of spaces” left. . The spaces may be full of something such as the string which makes up a knot. Then the two full spaces can still be at one “space of spaces”. Two items are not together at the same space.

This may be the reality we live in. I am working on some physics in another section of this website. It may be for the far off future, but if we can move space itself then we may be able to travel to one of the exoplanets. Since the speed of light is limited by space but we may not be limited by this speed if we move space itself.

The new level can be what I call placement space. It is a space where a location may have other ‘locations of a location’. Or a small piece of space can have another “space of spaces”. Basically a location or section of space is mobile. At the beginning it is stated how many locations are at any given placement. For the work on knots this is two, and the resulting entity I call a “doublet”. Then two different locations may share the same placement (location of locations) and I call this a “sharing”. A special type of sharing may exist in the manipulation of diagrams which I call a “joining”.

I use the knottedness and chirality or achirality of knots to demonstrate the usefulness of these ideas.