Plane of new numbers/Fermat’s last theorem
Introduction:
In this article I describe concept sharing and take a look at Fermat’s Last Theorem. 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.
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? Since an item of no extent placed together with another similar item of no extent would still have no extent, yet there could be two items here. The two items would just be hidden as one. Why does there only have to be one entity which has no extent?
These two items could share the concept of a point (concept sharing). In order to do this we would have to take out the usual concept of a point and replace it with this new conception.
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. A place of places. Another level to 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.
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).
Further, all math concepts are point-like in that they are exact ideas which have no existence in physical reality. 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 further defined in a new plane as separate.
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.
What if a”point” could be expressed as two items of no extent which were not points? Since an item of no extent placed together with another similar item of no extent would still have no extent, yet there could be two items here. Why does there only have to be one entity which has no extent?
These two items could share the concept of a point. In order to do this we would have to take out the usual concept of a point and replace it with this new conception.
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.
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.
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, 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)
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.
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 New Plane of Numbers:
We can build another set of numbers by creating a number plane rather than a number line.
Start with place(2) space (places of places). This can be begun by considering two points which have an infinite uncountable number of parts. They can be extended outward perpendicular to each other. Then let this be the beginning. The places of places space so created contains points which are equivalent but not identical. They are equivalent in that they all start defined together at some place of place of places but not identical in that they all have a different place of place. Then let us place an arbitrary number of points(2)(these are the points which can move in the more primitive underlying place of places space) placed at the same origin. These can consist of a countably infinite number of points.
Now we can build a new number line by moving two points(2) out to some unit length of 1 unit. Do this for both axes. So that in each direction we have a distance of 1 unit marked, now also I have 1 point(2) at 1 and mark it with 1 This is a location (0,0) moved from the origin, so represents some distance. If I want to I can move these numbers in the place(2)a axis as they are associated with points(2).
In the first quadrant we can make a number (1(1) by moving a part at 1 on one axis and the other part at 1 on the other axis to combine at the place(2) (1,1). To do this we have to move two parts out to each axis, but this is okay because we can have an arbitrary number of parts there.
Now we can make (2(2) by the same process. Move two parts from the origin out 2 units and move one part up 2. units from the x axis and one part from the y axis to come together at the number (2(2).
We can go on making a midline of new numbers (1(1), (2(2), (3(3),…
Right now we are only interested in the numbers in this number plane that makes sense to us. That is, the numbers on the midline. 1 and 3 could also share but there would be no higher number here at the first level.
Similar to negative numbers, these could have some meaning, but I won’t explore this here.
We can have operations on these numbers such as addition (1(1) +(1(1)= (2(2) = 2*(1(1)- if we combine the new number system with the system of numbers point(2) numbers-and multiplication (1(1)* (1(‘)= (1(1) or (1(1) *(2(2) = (2(2)= 2*(1(1) as well.
But here is something interesting, since we can combine the two point(2) number systems (1(1) =sqrt(2), if I consider the midline as continuous and the axis to be discreet values of natural number on the axis and multiples of sqrt(2). Now we have to make this space different from that of which we are already aware. Let any distance be in the new dimension and let it be negative, as we are moving away from zero, but not in the first dimension.
Then since (1(1)^n=(1(1) then sqrt(2) =sqrt(2)*sqrt(2) =sqrt(2)*sqrt(2)*sqrt(2)… And we have an equivalence of these numbers. This is also seen intuitively as I am moving parts out to the different place(2)’s and these parts are equivalent for a certain set of numbers in this space.
Fermat’s Last Theorem:
Now I want to take a look at the general statement of Fermat’s Last Theorem, seeing if I can find a way to show its truth. It is said that it is unlikely that Fermat had a proof but my ideas could have been known at his time( since in a sense they are basic), so maybe with this new understanding he worked out what I did.
Many people have tried to prove the theorem in the past with elementary methods that would have been available at Fermat’s time. But here I am introducing something new and powerful which didn’t exist before, but might have been known and then subsequently lost.
Now consider that I might have three numbers a, b, c with c related to a and b. I am thinking specifically of a+b=c or a^2+b^2=c^2 or a^3+b^3=c^3…..
Let these be extended to the second form of numbers so that I have (a^n, a^n), (b^n, b^n), (c^n, c^n) or a^n(1(1), b^n(1(1), c^n(1(1)=a^n(1(1)+b^n(1(1), n is some natural number all these (1(1)-type numbers are on the midline. Here are two examples:
So for example with n=3, 1^3+2^3=(9^(1/3))^3 but is 9^(1/3) an integer?
b(1(1) is between c(1(1) and a(1(1). C’ is the largest number.
I can move the midline so it is congruent with the horizontal axis, as shown previously.
Then since (1(1)^n=(1(1) and that means sqrt(2)=sqrt(2)*sqrt(2)..
Then a and b are integers, c^n(1(1)=a^n(1(1)+b^n(1(1) and we assume c is an integer too.
If I take the nth root of (c^(1/n))^n I must have an integer times (1(1) on the midline.
I am left with c^(1/n)*(1(1). but (1(1)=(1(1)^q. This expression has to be a multiple of sqrt(2).
Does c^(1/n)*(1(1)^q=m*sqrt(2) for m a natural number or m a multiple of sqrt(2)? or c^(1/n)*sqrt(2)^q=m*sqrt(2). M can be a natural number to give the multiples of the sqrt(2) or m can be a multiple of the sqrt(2) to give the natural numbers that might occur as we expand sqrt(2)^q.
So an equation we can come up with is m=c^(1/n)*(2)^(1/2)(q-1). I wish the horizontal axis to only consist of integer values or multiples of sqrt(2) of places but any value of places of places.
Since a^n and b^n are at least 1 each (n=1) and a and b are at least 1, then c is at least 2. so m is somewhere between 2 and c.
But c and 2 are not points but points(2) which can move(at the other level). Let c and 2 and vary on the places(2) axis. These are fixed at their values in place(1) so the first equation in place(1) still holds.
These places of places (place(2)’s)can be numbered. Let them have numbers associated with them which mirror the number system given to the first level.
Before the moving numbers move they are associated with numbers on the place(2) number system. Since these associated numbers in the place(2) system have the same relative values, the equation holds for them. We can have a sharing of two concepts at two different levels. The equation still holds true for these sharings as well as the individual numbers are not moving at the beginning. Then as the places move, the placements change at different rates so that the equation still holds in the place(2) number system.
Then let c and 2 move together on the horizontal place(2) axis so that we have them all together at m. We can move at two different rates as the underlying placements change in such a way to always keep the equation true in placement space as well as place space. As in algebra when we give a letter to a number, here we are giving a letter ‘p’ to a specific number we come to which is fixed at m while the places move. In this number system it is not the numbers but the number of the numbers which count.
Then p = (p)^(1/n)*(p)^(1/2)*(q-1). We are not interested in the number but the number of the numbers. This general number is a number so it obeys the exponent rule. So looking at the exponents then, we can ask is there any n and q for which 1=(1/n)+(1/2)*(q-1)?. In the cubic case, i.e. let n=3. For which the answer is no. For the squared case 1=1/2+1/2 (n=2, q=2) and the linear case 1=1+0 (n=1, q=1). For any case of n>2, this does not work.
So for the extended number system we are restricted when we examine (c(c)^n, (a(a)^n and (b(b)^n given they are linked by Fermat’s last Theorem . This number system was found by starting with the co-ordinate axis and forming a midline. I started this way to give a point of reference.
But suppose we look at it as the midline going forward to the usual number system. The concept sharing of a number is a more basic understanding of numbers. With (4(4) for example the sum of the underlying points leads to single item, what we know of already as a point. The so too the numbers which represent it follow. So that (4(4) creates 4. It makes sense that this result is a more basic feature of numbers.
Once the discovery is made for n, we can see how this is true in the usual number system. The underlying point and number systems can be considered to be previously hidden from us.