The Riemann hypothesis
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?
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 p=pxp. 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 p=pxp 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)
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, 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.
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.
