You break things down and explain them well. You really know your stuff!-S. Meshmeyer
How did you do that! (when shown the solution to a problem) I could never do that!-M. Wismer
My teacher is always away. You are always here for me. –Amanda L.
I’m glad you’re here. This is the hardest math I have ever seen. –Jeff K.
Hi, I’m Rob. I can be reached at 647-218-1407 or robburchett1@gmail.com. I have 4 online students at the moment and anticipate more as the semester continues.
I have picked up a new laptop, a new writing pad and am using the Zoom platform for meetings. The new whiteboard feature with Zoom allows the student to write to me, I can have up to 12 boards. The student and I can work on problems together, in real time, this is a good way for them to learn. As well I can make recordings of our sessions available to review later. This is an important feature which we don’t have with in-person meetings.
I apologize for the construction going on at my website and hope to have it back to normal shortly. You can find all the necessary information about my tutoring by going to the category: Tutoring, in the column on the left hand side.
There you can find my degree from the University of Toronto and my Education Certificate. Also my police records checks and my letters of recommendation.
I usually meet a student for the first time for a half-hour session, just to see how things go and charge $12.50.
My usual fee is $25/hr. for online meetings. I can be reached at 647-218-1407 or robburchett1@gmail.com. I look forward to hearing from you soon.
I continue to work on my own original math and self publish. Also I am looking to publish in a magazine or journal. Some of this work can be seen under the category: Mathematics at the left.
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 original 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.
Then this can be a new basis of Mathematics
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. Yet we would need another level to the number 2, as these two items are in the same place of places. And so another level to places, somewhere to put these two new entities.
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 moving points, in the usual geometry, 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 only 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 (a new entity with no extent) 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 that don’t combine to form a point and a new co-existing dimension where these can be. 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 co-existing 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 unless there is something underlying! Since two items of no extent can exist together and still be two items of no extent, it must be possible to have this. Since a single item of no extent already exists, we must be able to take this out to accommodate that which must also exist!
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.
A place where the original place can be taken out of. Another level to places, coexisting with existing places. That is, a concept space. The idea is to have an infinite number of descending levels as shown below: For we can keep going lower seeing as the new level of place may also be subjected to concept sharing!
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 somewhere. 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!
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. Yet we can now describe this situation better by taking out the number 10 and replacing by it by (10(10), (10)(x2) = (10<….(10) =(10(10). here the 10th position is shared, similar to the concept sharing of a point, this is the concept sharing of a number. We need this idea to come together with the idea of the concept sharing of a point for two e’s to share the concept of a point (that which has no extent). Here then is the extension of the number 2.
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.
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, similar to what we have seen in points, 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 position of the number 1. 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). We remove the original 1 or 2 or 3.. and replace by (1(1), (2(2), (3(3) etc. as shown below:
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. These numbers can move because there exists an underlying plane as mentioned above. The numbers are associated with moving e’s. We have 1 dot or 2 dots or 3 dots as shown below:
The twin prime conjecture:
There is a lower level of prime numbers using the concept sharing as it applies to numbers. For examples: (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 (1(2), (2(3), (3(2). 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.
There is a split between the natural numbers and the primes at (3(3). After this, larger squares with partially shared prime numbers at the corners appear. These are built from the smaller squares.
We might also make copies of these prime’s associated with these squares. exe can be repeated indefinitely. Yet there is an uncountable infinity of these possible. Think of the first square, this is the model we use to build the structures. There must be an uncountable number of these available at the beginning. We can think of “folding-up” 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.
The first, primary square is used in repetition, to build the entire structure.
It must be repeated indefinitely, further along two diagonals as well as filling in space in the larger squares.
Since the larger square built of four, ends, it must also be repeated indefinitely.
It ends, yet the structure must somehow be repeated as the lines which make up the edges of the square can be multiple.
The first time I encounter a new square type, that is a larger square, there is no rule for how many copies exe=exe I have present. The only way I can have it exist is if there are an infinite uncountable number of copies present. Then I can move these copies into the other uncountable infinity I have present. I do this because it is necessary to have only one copy of the shared numbers present in order to recover the usual numbers.
The next such larger square is similar so that I do the same with this. This larger square is at a different, “lower”, level than the smaller square, above it.
Then this leaves one copy of each square as we climb higher. Then I can recover the usual numbers by replacing exe with pxp=p ie. (1(1) with 1.
The first small square I encounter, I spread out three prime squares as shown above and after that only composite numbers. The new definition of a prime square in the plane, is to have a prime sharing at every corner. Here is seen what happens when we have a prime gap of 4.
The second square is the new construction of what comes before a prime in the plane, with a prime shared number at every corner. There must be an infinite uncountable number of copies here as well, when you consider the folded up case. I have to be able to spread this out too, so I have an infinite number of twin primes, primes that differ by 2, and an infinite number that differ by 4, etc.
As there is an infinite number of naturals (composites considered), these go inward to build the larger prime squares 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 out into the infinity that is available, that is, the rest of the number plane. Then in this way we are able to recover the composite and prime numbers. 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. I can make squares of any size by accessing the next lower levels. All levels must exist, one after the other, to create the structure which leads back to all the prime and natural numbers.
I can fold up, like a two-dimesional accordion, all of the squares with partially shared prime numbers at the corners. Also the smallest squares with the composites sharing the corners can be also folded up.
Then we obtain the squares of length 1,2,4,6,8 as infinitely uncountable. The squares of lengths 3,5,7 come along as well. We must be able to access all the different levels. All folded up the squares look like below:
All the squares of side length 1 fold up into the first unit square, all the squares of side length 2 fold up into the second unit square, etc. The notion is that the squares with an uncountable number of copies is more basic than the squares all spread out. This is given at the very beginning. When folded up the squares are on different levels, when spread out they are on the same level.
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 one number 1.
LIke so 1…..>1(x2)<……1.
Now wait you say, there aren’t two number 1’s! But there are if we have two number lines, perpendicular to each other, as shown. Then the usual number line can be thought of as the diagonal line and we are able to remove the usual 1 that’s there and replace with 2, new 1’s.
Why do we want to have two number 1’s? We can extend numbers into a number plane and see an underlying level of numbers. The symmetry of these numbers can clear up old problems, since we are looking at an underlying level to numbers.
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.
The two number 1’s sharing is an example of Concept Sharing. 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, since they are not defined to be somewhere else where the two items could be separated. That is, when I combine two moving points, with the usual plane, it leads to one point, but this doesn’t have to be so with another possible entity. See the teacup shadow diagram, below. If we have another underlying plane there can be another 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 (a new entity with no extent) 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 mathematically we would have to take out the usual concept of a point and replace it with these new conceptions. There are these little entities that don’t combine to form a point, this new non-combination of two entities.
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, a new dimension, where the usual plane is, yet co-existing with the usual plane! A new dimension of places where original places can be. 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 unless there is something underlying! This must exist because I have to have the double point somewhere (we must keep the notion of no extent, yet also be able to extend it) and in order to have it I have to remove a point from somewhere!
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 dimension to places, coexisting with existing places.
A place where the original place can be taken out of. Another level to places, coexisting with existing places. That is, a concept space. The idea is to have an infinite number of descending levels as shown below: For we can keep going lower seeing as the new level of place may also be subjected to concept sharing!
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 somewhere. 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!
This, then is how the numbers move, I create a new plane, “lower” than the usual plane. See below:
The locations move off into the other dimension, coexisting with the plane that is originally there. The numbers come along with the moving points. We can have one point at 1, two points at 2, three points at 3, etc.
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). But what are the numbers representing? The number of dots. As shown below:
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. The number of numbers is two.
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 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 higher 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 by multiplication, similar to the way composite numbers are created from primes.
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. That is I do not choose 4 and 4 and put them together. I create (4(4) from more primitive numbers and then to get to 4, I have to replace (4(4). 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 have a lower level of primes by concept sharing two prime numbers. For example (7(3)=(3(7).
Prime numbers are numbers which have 2 factors, 1 and itself. 1 is not a prime number as it has only one factor, 1. It can be expressed as 1×1 or 1x1x1…etc. Any prime factorization of a composite number is unique and so does not include 1 otherwise any composite number could have any number of factors, we could just extend it indefinitely by extending 1.
If we look at (3(7) we see it can be further broken down into (3(1)(1(7). Numbers like these two can be considered the shared counterpart to primes. Yet to have it, it is not a prime factorization. Yet, (2(2)(11(1) could be a prime factorization, since I have a single 2 and 1 but I also have two prime factors. Here then 1 is considered to be a prime number. Non-prime factorization just opens the shared number up and allows for non-finite representation. Yet, so does prime factorization!
So a prime in this system might be represented as (1(p) or (p(1) where p is a prime in the higher system. Then (p(q) where p,q are primes would be a different kind of number, we could call it a binary prime shared number.
If I do include this decomposition then (3(7)=(1(3)(1(7) =(1(21) and then if I have (10(10)=(2(2)(7(3)=(7(3)+(3(7)=(2(2)(21(1)=(42(2)=(22(22). So (10(10)=(22(22). This may lead to interesting mathematics, but if we allow it it means that the representation of a shared number is not finite. We need a finite representation otherwise the sum of the shared numbers is not conserved.
We can make a further restriction to only allow shared number decompositions where the sum of the two component numbers are equal over all possible decompositions. These means the sum of original numbers is conserved. Since we are sharing numbers, this makes sense that the amount we are sharing one way or another can vary but the sum of the numbers must remain the same.
This works out with a factorization where one of the factors is (2(2) as we can switch the two numbers in the other factor. This makes the sum the same. If we allow the sum to be different, there is no finite representation.
Here is an example: (2(2)=(3(1)
If we move by one up or down from (3(3), for example we can obtain (2(4) and ask is (2(4) okay? But (2(4)=(2(1)(1(4) (since 4 is composite this is an allowed decomposition)=(2(1)(1(2)(1(2)=(1(2)(1(2)(1(2)=(1(8) and 1+8=9 and not 6. We seek a prime decomposition which maintains addition of shared numbers.
Then for (16(16) we get (16(16)=(2(2)(p(q) and (13(3)=(11(5). No other binary numbers work as I can further decompose them and by switching lead to two different sums. Also I can’t decompose (13(3) and (11(5) further as this will not be prime factorization.
Think of (12(12)=(2(2)(6(6). So if we are at (6(6) I can move it to (7(5) and this works out because (7(5) has no prime factors of either 7 or 5. This is the way it will work. I must have for example (16(16)=(2(2)(11(5) the second factor has to be where both numbers have to be primes. This is then the definition of these shared even numbers, greater than or equal to (4(4), since they are conserving the sum of the numbers.
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, one is considered a prime number.
In general (2n(2n)=(2(2)(p(q).
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) (4 is composite so it is okay to express the decomposition this way as the shared product is that of a composite)=(2(1)(2(1)(1(2) which 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.
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.
We also can’t have multiple factors like (12(12)=(2(2)(2(2)(3(3) because I can have then (12(12)=(2(2)(6(2)(1(3) and then (12(12)=(2(2)(2(6)(1(3)=(2(2)(2(18) and it won’t add up. Having multiple factors again opens the shared number up for a non-finite representation. There must be some creation of (12(12) which has only finite representation. It must exist since I need to have 12. There must be a stable way of defining it. You see, if we allow (12(12)=(4(36) I can further extend (4(36) indefinitely.
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. The picture below, would now be in reverse. We would replace (1(1) by 1 and (2(2) by 2, etc.
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).
Also, since we include 1 as a prime in some cases, it can be seen that there must always be at least two of these different representations of any even shared number greater than (4(4). So if we have one of them being (p(1) another one must be (r(s) where p,r,s are primes. For example with (10(10) we have (5(5) and (7(3). Since once I form the number 10, I need at least two of these to feedback to the other, original number line. See the picture below. We might also have more than two representations, In which case there are extra added dimensions.
Then there must be at least two of these decompositions for each even shared number. We might have one (p(1) but then I have another one (r(s). 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.
General Math Concept 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.
