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Concept Sharing is a term used in education. But here I am giving it another definition.

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. These are “little nothings”.

What if something of no extent could be expressed as two items of no extent, placed together. Since an item of no extent placed together with another similar item of no extent would still have no extent, yet there could still be two items here and not one.

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 points, it leads to one point, but this doesn’t have to be so with another possible entity. See the teacup shadow diagram, below. Why does there only have to be one entity which has no extent?

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.

These two items could share the concept of a point (concept sharing). Let’s share the concept of nothing, the two little nothings say! This is how we can hide! In order to do this we would have to take out the usual concept of a point and replace it with these new conceptions. There are these little entities which do not combine and this combination. This is because a point is already there as the idea of no extent. We need some room! Kick out what’s already there!

What does it mean to take out the concept of a point? It is not the same as removing a point from a given subset of points of the plane. We wish to open up a new possibility, so we have to reject the usual concept of a point and accept this new possibility. 

Yet to reject the concept of a point and replace it, it means we have to define a space where a point can be truly taken out of. We need something like another plane! Places of original places. Let’s let the points have another level to them, where they can move off into another plane, coexisting with what is already there. That way it is possible to take them out, otherwise, how can we remove them? I can’t take them out of nothing!

What if a point had another type of item of no extent that it normally lived in. It is so tiny that it only needs a tiny house to house it!-we have already seen it is necessary to have other types of entities with no extent. This would necessitate another kind of plane of these items co-existing with the usual plane of points. A place where the original place can be taken out of. Another level to places, coexisting with existing places. That is, a concept space. 

A place where the original place can be taken out of. Another level to places, coexisting with existing places. That is, a concept space. 

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!

To remove the concept of a point, we need to have a concept space, consisting of different nested levels of the concept. This must exist because I must have this further concept of a doubled item of no extent somewhere and I need to take out the concept of a single item of no extent to have it appear by itself (unconfused). Also we must take out any other possibility of any number of items of no extent, except 2.

Furthermore, all math concepts are point-like in that they are exact ideas which have no existence in physical reality. They are mental constructions which are not available to the senses. They are in thought only. So number, set, group, function, etc. can all have concept spaces.

It seems like the two items should be the same. But not if they can be also further defined in a new plane as separate( in the case of points).

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 points at a single location.I say two points at that location now because they are defined in two different places of original 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”.

So I say “added”, let us postulate another level of places. That is, an underlying plane where places of the usual plane may exist in other “placements” of places. Where a placement is not a place but a lower level of place. The same notion as place, yet let places be capable of shifting off into this new plane of placements. Then we no longer have a fixed plane of places, yet the placements could be fixed.

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 one point being mapped to two or more points or 2 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 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. This is not as simple as taking out a point out of a given subset of points of the plane as I have to be able to put something back in that is truly different.

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.

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 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=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.

Introduction: 

In this article I describe concept sharing and take a look at the Riemann Hypothesis. While it’s a common belief that math is cumulative, so that for example, to do calculus you need to know how to do the math at the lower grades, it might be that there is some basic math missing from our understanding of the overall mathematical structure. Here I present a different concept which subsets existing mathematics and has many applications.

It seems to me there may be an easier way to express the zeta function: Z(s)=1/1^s+1/2^s+1/3^s…..using the ideas of concept sharing as it applies to a new geometry.

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.

The Riemann Hypothesis:

One may imagine a type of grid with the first square being 1, the next being 1/2^2 the next being 1/3^2… if we use s=2 as an example. See pictures in the notes below. The higher numbers of s can be seen by increasing the dimension. Yet there is always a plane possible with any dimension equal to or higher than 2. For we are just standing on it. We can always project downward to a plane. For dimension 1 there is a line and dimension zero an infinite point at zero.

Since with concept sharing geometry there comes a place of places, in which places can vary, we may vary the distance as we choose to always make the zeta function defined. The zeta function can be continued into the extended geometry. Then there is no longer a need for analytic continuation. I can always make the grid into a 1 by 1.

So we can create a grid specific to the Zeta function defined in placement space.

Then we have that there are two types of number involved. A real part and an imaginary part. This is to make the Zeta function equal to zero.

I think this can be seen more primitively as a numbers which lead to a square with a positive area and numbers which lead to a square with negative area ie. the negative distance is -i. These can be sharing space.

We can concept share two different numbers in the following way: (-1(-1)*(-i(-i) where * is a concept sharing of a concept sharing= ((-1(-i))((-1(-i)). But -1 and -i have to be different. Let -i be the negative distance and -1 be the other, real distance. Then let this be how the square comes about. We have to expand the zero-dimensionality of the concept sharing. Let -i and -l be numbers at the next level of numbers. That is they are no longer point-like but line like. We can start with a point consisting of an infinite uncountable number of sharing parts and expand it outwards into a line.

The 2nd next to last image shows how there are trivial zeros at -2,-4,-6… and how the zeta function could equal -1/12 when s=-1. We are adding an infinite series to get a finite sum. This comes about as we have a addition of positive and negative area. This works for the plane as we can have i and i^2=-1.

If we look at the next to last image, there are three possible cases. One where s=2, one where s=1/2 and some where s=-2,-4,-6,…These all lead to a plane where we can also share with i, in the last two cases so that we might cause the series to converge.

If we look at the last image, this is showing how we can have the complex numbers 1/1^(a+bi), 1/2^(a+bi), 1/3^(a+bi),…on the bottom of the grid and also on the side of the grid. We can give up the idea of negative areas and look to cancel the lengths, thinking of the complex numbers as vectors. Then instead of the square areas, count the diagonals in the squares as lines which could rotate at different origins. To find the diagonal lengths multiply the numbers by sqrt(2)/2. Add two of these to find the diagonal lengths. This is in the case of the plane. In three dimensions multiply by sqrt(3)/3.

Then we have for example 1/sqrt(2)x1/2^(1/2+bi). This is seen as the line with a rotation from the imaginary part as 2^(1/2+bi)=2^(1/2)x2^(bi)=2^(1/2)xe^(ln2(bi)=2^(1/2)x(cos(ln2(b)+isin(ln2(b)). All these rotations of lines and all these other dimensions can lead to a result of zero as the possible rotations can cancel the vectors.

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).

Thanks to Lou Kauffman for sending me a picture of the Culprit knot.

Thanks to Lou Kauffman for showing me swing moves.

Izzy’s giving a look at the Culprit knot.