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Concept Sharing and Wile’s Theorem

Abstract: Here I introduce concept sharing. In uncovering extended space, I show a demonstration of Wile’s Theorem.

As an entry into Concept Sharing let’s start with the concept of a point. In math this is the notion of an entity with no extent, or in Cartesian geometry the notion of something with position only.

We have the familiar idea of two items just touching or resting upon one another as we see in everyday life. For example a book resting on a table, or two books packed tightly together, on a shelf.

Then the point of contact can be separated into two points, one for each item. Mathematically a single point is replaced by two distinct points, with a small gap, then this gap can be increased..

What if a point could be expressed as two items of no extent which were not points? Why does there only have to be one entity which has no extent? These would not exist in space as we know it, but in another space.

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=p, e is not equal to p as then exe=e 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. This means I need a space of places which contain the original places, so that I can take out the place p=pxp and replace it with the new places, exe=p.

That means exe are not 2e’s at the same place but 2e’s at the same place of original places, a lower level to place. Since they are not in the same place, as there is no place there, 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)

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 points of the shadows can take up the space of one point. This is analogous to two e’s sharing.

Then we can separate the two e’s, but the only way this can be different from the usual idea of separation in points, is as if both entities are the same and different (not just different as in points). In the case of points, there exists this gap from one common point to the case of two distinct points.

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 they are the same place yet they are at different 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.

This is a new dimension. Each e of the extent is both the same and different as any other e. 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 by adding positive distance (usual distance) 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.

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=p where e and r are two different types of entities as well. E is not equal to p and r is not equal to p, additionally e is not equal to r. 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=p.. This means there is another level to space. Since I’ve taken out the usual notion of a point, I must have taken it out from somewhere. This is the space, places of new and original places. It can be modeled after the usual idea of extent, yet the distances are negative.

Then this also means I can separate exe=p 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, this is the idea of pxp=p. There is a small gap before we reach p, from two distinct points. The exe=p 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:

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 well, this loop could be knotted.

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.

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 space can be thought of as akin to a jigsaw puzzle being taken apart over an underlying space.. 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. 

Wile’s Theorem:

Now the statement of Wile’s theorem is that the sum of two squares can equal a square, but the sum of two cubes or any higher power cannot equal a single cube, a fourth power or higher (more widely known as Fermat’s Last Theorem).

It seems then that it should be possible to demonstrate this with geometry. One of these new geometries mentioned above is a possible way of demonstrating this.

Let’s start by considering a line of places of places defined in a plane of places of places of places and a line segment which can consist of two or more superimposed lengths of places (two or more lengths).

At the start we can only have two types of points, fixed or mobile. Let the places of places be the fixed points, then since we can move off into two directions we must have 2 line segments with one 1 point each one line moving left and one line moving to the right. It can have two integer lengths (or multiple lengths), yet a single length of lengths which can vary. Since length is not the same in the new geometry.

It’s length of lengths might be one unit, but its lengths can be two, three or four units, for example. It’s lengths can only be multiples of the length of lengths and the length of lengths can vary.

Then let one line segment, consisting of two different sets of places and place of places be decomposed (simplified) in the space of places of places along the line of places of places. It has a length of length.

We can only move out in two directions along this line. It is seen that it is only possible to have two different places of places at the beginning. The places of places are mobile, and they can only move out left or right. So we double the mobile points and weight each one point, since I want to form the sum of two lines.

Suppose we map these two lengths of lengths co-linearly, inside the original by shrinking each line. Then this is the demonstration that a+b=c is at least possible for some cases of a, b and c. a, b and c being some lengths. Since the sum of two lengths of lengths is also a length of length as well. 

Then this at least makes it possible that a+b could equal c. b may be too small or too big and not equal c, but there may be a case when a+b could equal c. Now the intention is to move up in dimension.

Now we can move to the next dimension by rotating the line of places of places out of the line and into a plane. When perpendicular we have a square, the side length of which is again two possible integers. Let there be a set of two squares making up the initial square, I can only have fixed or mobile points. Then since I can move off into four different compass directions n,e,s,w. This one mobile square must be made of two squares and must be rated at ½ points each.

Since I must move the copied squares out into an area of places of places it must be following the parallel lines which are places of places. I can move out four possible squares. 

This indicates that I am moving the sum of two squares out to become four squares, which means the points of the squares are weighted ½ each. Then map these squares and move them all into the original square. See the diagrams below.

If we use the same pattern as in the case of one dimension this is the demonstration that a^2+b^2=c^2 is at least possible for some values of a, b and c , since the summed squares can add to a square in some cases. I can start with one square and add area around that square, which adds up to a square to try and form a final square.

In three dimensions and higher this is not possible to do. In three dimensions I create six cubes instead of the required eight. Each of the six cubes can be weighted ⅓ but we cannot form an added cube, since I need 8 cubes to do this. See the sketch below.

In a fourth dimension I would also not have the required number of hypercubes and so on. This shows a geometric proof of Wile’s theorem (Fermat’s conjecture).