I am seeing five students so far in the Fall. One online and the rest in person at the Thornhill Community Centre Library. I have one grade 9, one grade 10, one grade 11 and two grade 12 Math students.
I am also working on three articles for three different mathematics educator’s magazines. One for OAME in Ontario, one for Vector, based in B.C and one for the AMTNYS based in New York. These are based on some of the mathematics I created shown at the left under the category: Mathematics.
I have five summer students whom I am reviewing the previous grade and teaching ahead the next grade level for. One student is online and the rest are in person at the Thornhill Community Centre Library. I am giving homework to 3 of the five students.
Also I am working on an article for Vector Magazine, based in B.C. The article uses some of the math I created, which is shown at left under the category: Mathematics.
It seems the axiomatic foundation of Mathematics is made of concepts and thus subject to concept sharing. Then the axioms must have other levels. Then there is no irreducible axiomatic system.
A better foundation of Mathematics is seen when we look at the truths that concept sharing allows. Since we must have concept sharing we need to look at the Mathematics that comes from this.
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.
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. This can be notated (1(1)(1(2)). We can show that there are really 2, 1’s there by calling it (1(1) like so:
(1)(x2) = (1<….(1) =(1(1).
The left 1 is really together with the right 1, yet I need to work with both of these so both of these are shown.
The twin prime conjecture:
There is a lower level of prime numbers using the concept sharing as it applies to numbers. For examples: (1(1), (2(2), (3(3), (5(5),…I drop the (1), (2) notation for clarity and brevity.
Yet there is now a plane of numbers as shown below. With entries such as (3(5), (5(7), (5(11). These are partially shared prime numbers. We can make squares as shown, these are the primes and composites as seen with the extra added dimension. One is centered at (4(4). We connect the prime shared numbers.
There is a split between the natural numbers and the primes at (3(3).
We might also make copies of these prime’s associated with these squares. P=exe can be repeated indefinitely. Yet there is a countable infinity of these possible. At the first of these new squares, there can be an indefinite amount of copies. Yet we can balance this infinity by stretching the copies out to the other infinity that is available to us.
There is an infinite number of naturals (composites considered) and an infinite number of primes (larger squares considered)
The composites and primes can’t be anywhere else in the plane but on the midline. Since we need to recover the ordinary composites and ordinary prime’s, there can only be one copy of these. The rest of the copies are moved into the infinity that is available. Then in this way we are able to recover the composite and prime numbers. Take away all the shared points except for one at each step. Then (1(1),(2(2),(3(3),(4,4),(5,5),(6,6),(7(7) can become 1,2,3,4,5,6,7…
Since I can make an indefinite amount of copies, the same gap must be possible infinitely as we go further up. Therefore the twin prime conjecture is seen, as well as we see other gaps repeating.
Starting with the diagram of the Culprit knot, I am trying to find a way of showing that I can form the unknot without increasing the crossing number.
I do so by placing the knot in placement space. The the locations are free to move in a space of locations of locations. We keep the locations connected in the knot as they were originally connected.
Specifically, they can move around in a loop along the path of the knot.
Once we decompose two crossings into a joining (alpha-beta’s), we can double up the knot diagram again. One diagram is still and the other I can move the locations around again along the path of the new knot diagram. Then I can decompose again, etc. This means I can move one alpha-beta past another.
I then decompose completely and look for another way to put the knot back together. This new diagram is obtainable from the original diagram by the usual Reidmeister moves.
Hide and seek is a fun game. But suppose instead of children, numbers would like to hide.
Suppose you are the number 1. Where could you hide? Numbers are so obvious. They go 1,2,3,4…etc. How could 1 hide?
But suppose I have two number 1’s. I could hide both together, by putting them together so that the two of them could not be seen, they just look like 1 number 1.
LIke so 1…..>1(x2)<……1.
Now wait you say, there aren’t two number 1’s!
But why not? We only have to take the usual 1 that’s there out of the way and two number 1’s can be where that 1 was.
To do this we need another level of numbers like a one story house has a basement. Then I could remove the number 1, like removing the first story, and still have the basement. Then build a two story house on top of the basement that’s left over. For some clarity on this see the notes below.
Now since two number 1’ s are playing hide and seek, how do we tell that it is not the case that we have the usual case of 1 number 1?
We can show that there are really 2, 1’s there by calling it (1(1) like so:
(1)(x2) = (1<….(1) =(1(1).
The left 1 is really together with the right 1, yet I need to work with both of these so both of these are shown.
So not only would 1 like to play this game but the other numbers would like to join in. 2,3,4,5,6,…etc would also like to hide.
So we can have (2(2), (3(3), (4(4)…
Then also we can have the other games that numbers like to play like addition (+) and multiplication (x).
So (1(1)+(1(1)=(2(2) and (2(2)x(3(3)=(6(6) for examples.
These numbers can all be on a line. The hidden number line:
But now 1 says I would like to hide with 2. Since part of 2 is 1 I could hide with part of 2. Then this could be labeled (1(2) and it would be equal to (2(1). We just have a smaller number hiding with a larger number. 1 and 2 are in the exact same place.
But where are these hidden numbers on the hidden number line?
The answer is that these numbers are on a number plane. A plane of numbers that looks like the diagram below:
On the midline we have the usual hidden numbers.
Then we can ask, how can these hidden numbers play the usual games of numbers? (addition and multiplication)
Let’s investigate!
(1(2) + (1(2) =(2(4)
But also,
(1(2) + (2(1)=(3(3). So it appears that in order for (1(2) to play (2(4)=(3(3). But there’s nothing to say this isn’t okay, since there are no rules for these partially hidden numbers-yet!
Goldbach’s Conjecture:
On June 7th, 1742 Christian Goldbach wrote a letter to Leonhard Euler In which Goldbach guessed or “Conjectured” that every even number is the sum of two prime numbers. So for example: 16=3+13.
Let’s look at the guess in terms of the hidden numbers!
We can think about new types of numbers (1(1)(1(2)), (2(1)(2(2)), (3(1)(3(2)),..These are lower numbers than the naturals, the idea of concept sharing being deeper than the usual idea of concepts. The notation is meant to show that the numbers are together in the number space, fitting into one another or hidden as one. I show a revelation of one number “peeking out” from concept sharing with another number. Shown more clearly here: Like so (a)x2….(a<……(a) leading to (a(a). For clarity and brevity I drop the (1),(2) extra designation from now on.
There are other numbers such as (2(3), (3(4), (3(5),… We can think of this as partial sharing. For (2(3) think of three dots colored red. We can have two blue dots overlapping with two red dots forming two purple dots. I say that to give the idea, numbers are not colored dots. Numbers are the exact concept of amount or position only. So that (2(3) can be thought of as two sharing with three and (3(2) can be thought of as three sharing with two. In the first case 1 is left over and in the second case 1 is an extra number. These are the same concept though so (2(3)=(3(2).
Consider the even shared numbers (4(4), (6(6),…
We can break them down into a sum of two prime shared pairs such as (4(4)=(2(2)+(2(2) if we define the other types of shared numbers with dissimilar basic numbers to fill in the rest of the possibilities of a number plane. They can form products too.
Some numbers have more than one decomposition. Such as (16(16)=(13(3)+(3(13) and (16(16)=(11(5)+(5(11).
Now think of the even shared numbers as being created from the primitives. I designate them (4(4) … but these numbers are not created from the usual numbers on the number lines, as these are lower numbers, these lower numbers; (4(4)…. must come from somewhere else. These even shared numbers must go on indefinitely, as to lead to the numbers 1,2,3,…at the higher level . Then the natural numbers are all at a higher level from these numbers.
For multiplication, in the usual number system we have prime numbers. We can look for the lower level of prime numbers. Suppose I build a lower level of primes from the usual prime numbers by concept sharing two prime numbers. For example (7(3)=(3(7).
Then starting at (4(4), we can ask, how is (4(4) created? (4(4)=(2(2)(2(2). (4(4) is an even shared number so there is a division by (2(2) possible. This can be the definition of an even shared number. When I divide both sides by (2(2) I have a shared number on both sides.
Then also (6(6)=(2(2)(3(3). Also (6(6) =(2(2)(5(1) also as (5(1)+(5(1)=(5(1)+(1(5)=(6(6).
This is a factorization but notice 2 is multiplied by 1, so we don’t allow it. As this is not a prime factorization.
But we wish a prime factorization. For example (2(2) is not equal to (2(1)(1(2) as this is the shared number counterpart to not allowing a prime to be expressed as a product of 1 and itself. (this is a decomposition but not a prime factorization).
So in the shared number system we need a new definition of prime factorization. Let’s look at some more examples.
(6(6)=(2(2)(4(2) but (4(2)=(4(1)(1(2)=(2(1)(2(1)(1(2) which is does not work as I can break it down further into (6(6)=(2(2)(2(1)(2(1)(1(2)= (2(1)(2(1)(1(2) +(1(2)(1(2)(1(2)=(4(2)+(1(8)=(5(10). 5+10=15 not 12.
Also (8(8) is not equal to (2(2)(4(4) as (2(2)(4(4)=(2(2)(4(2)(1(2)=(2(2)(2(2)(2(1)(1(2). This can be further broken down to (2(2)(2(1)(1(2)+(2(2)(1(2)(1(2)=(4(4)+(2(8)=(6(12). This is not allowed. We find (8(8)=(5(3)+(3(5)=(2(2)*(3(5)=(2(2)*(5(3)=(10(6), and 10+6=16=8+8.
It turns out we do not allow for example, (12(12)=(2(2)(11(1) as this cannot appear as a prime factorization.
It seems it’s okay for 1 to appear as a factor more than once, for it means the other part of the shared number is composite. It just can’t appear once with a single prime number.
And so on. Each even shared number must be a multiple of (2(2) and another prime shared number.
Every even shared number can be divided by (2(2). The other factor must be a prime shared number as we need this to work in the shared number system. For example (12(12)=(2(2)(6(6). (6(6) must separate into two prime numbers, the left number moving up by one and the right number moving down by one. Otherwise (12(12) can’t appear (we assumed it was there). It must exist since I need to have 12. I can now take (12(12) out and replace it by 12. So we must have a decomposition with (2(2)(a(b) with a, b both being ordinary prime numbers.
You see as we now have (4(4), (6(6), (8(8),…. we can now go on to create the usual natural numbers. Divide (4(4) by (2(2) to get (2(2) and divide (2(2) again by (2(2) to get (1(1). The we can go forward by dividing (6(6) by (2(2) to get (3(3) and (8(8) by (2(2) to get (4(4) etc. Then all this sequence can lead to the natural numbers. I can replace (1(1) by 1, (2(2) by 2, (3(3) by 3 etc.
This also demonstrates the equality of every primitive decomposition as well (which the lower level was hinting at) since for any even shared number there may be more than one decomposition. We go back to form (16(16) and then can separate once again to find another decomposition. Here we find two, (2(2)(13(3) and (2(2)(11(5)
Then if we look at one half of these binary decompositions, that is, for example, look at 16(16)=(13(3)+(3(13) and see 16 =13+3 we can see Goldbach’s conjecture is true. It was just a part of a deeper understanding of numbers.
Notes:
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.
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.
For Concept Sharing we needed to start with sharing numbers. The idea is to form sharing concepts. For this we need these new numbers, to be specific so that we can start with certain mathematical systems.
In mathematics, concepts are mental constructions. They are ideas like shadows with boundaries. They can be thought of like points. The foundation of mathematics is based on concepts. Here then we need to find a new, extended foundation.
We can concept share a point since two items of no extent will still have no extent, but there could be two items here.
So similarly numbers, as we have seen, points, sets, groups, ect. Can all concept share.
In order to do this we must remove the concept which is present initially and replace it with 2 or more sharing concepts. Since it is possible to share concepts there must exist a more underlying concept space.
The two sharing concepts must be different in some way which we can specify based on the nature of the concept itself.
Then an infinite level concept space can form as I can continue into the next level of the concept and so on. One may utilize as many levels as is necessary.
Additionally there can be a finite number, an infinite countable number of an infinite uncountable number of sharing concepts, as this is the understanding of numbers.
Abstract: Here I introduce concept sharing. In uncovering extended space, I show a new way of understanding knots.
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.
The Concept Sharing of a Number:
Numbers are exact concepts. In the above case, we can think of them as the number of shadows at the center. They have exact boundaries and some way of showing we have two there or three there, ect.
Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two hidden as one” as well. (concept sharing) if the mathematical objects represented by the numbers had the same boundaries, like the shadows at the center.
Other than the further darkness of the overlapping shadows, we cannot see or imagine that there are two separate shadows there. Two or more numbers can be hidden as one since natural numbers represent exact positions. Similarly with two numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.
Then to this end let us create another number dimension, a dimension of number of numbers. This must already exist since it should be possible to put two or more numbers together at a beginning. Let the usual case be that the number of numbers is only 1. But now let us expand into the next dimension and allow the number of numbers to be 2.
So for example with the number 1, let us take away the original number 1 (since we have another underlying dimension, we can do this) and replace it with two new numbers 1’(1) and1’(2). These are together like the two shadows but do not form one number.
The two numbers together can be notated ((1’(1)((1’(2)). 1’(1) is “peeking out” from behind 1’(2). Shown by the use of a half parentheses. Seen more clearly here: (a(a).
Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects, also concept sharing, hidden as one.
The objects are concept sharing as well and are somehow different from each other. We give the two hidden objects two new numbers 1’(1) and 1’(2).
In the case of mathematical objects there is no external way of telling how many objects there are, previously it was assumed it was only one. We can state how many we wish at the onset thus fixing a certain mathematical system. Then we need the concept sharing of a number to indicate how many objects we wish to be there.
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 created 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 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 interest of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a subset of sharing points co-existing with a plane of places of places.
One of the new points can be fixed, while the other one is capable of “shifting” away in this new dimension of place. In this way these two can be different. Then all of the sharings in the new plane can become new origins-one point being fixed while the other point is capable of shifting away.
