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
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?
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. 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.
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 e’s 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, the surface makes it possible to show a finite number of entities of no extent.
Then postulate a place of places where ‘e” the entity of no extent which is like a point in that it has no extent but unlike a point being different in the following way:
e(0) is not equal to e(1)*e(2) is not equal to e(1)*e(2)*e(3) where again * is the idea of movement but this time in the new plane.
Suppose I represent a location by (0,0). What if we can have a case of (0,0)(1) not equal to (0,0)(2)?. This is possible if (0,0)(1) and (0,0)(2) were somehow different. Since something with no extent can be “added” to something with no extent and the result is something with no extent, there could be two items here. (see the overlapping shadow diagram) We would have to somehow make the two “points” different and only two “points”.
So I say “added”, let us postulate another level of places. That is, an underlying plane where places of the usual plane may exist in other “placements” of places. Where a placement is not a place but a lower level of place. The same notion as place, yet let places be capable of shifting off into this new plane of placements. Then we no longer have a fixed plane of places, yet the placements could be fixed.
Since two items (“points”) of no extent could appear to be a single item, and we could conceivably fix this at 2 or three, or as many as we choose, this other plane also must exist.
The usual idea of points can be notated ….p(1)*p(2)*p(3)=p(1)*p(2)=p(0).This is the idea of two or more points coming together to form one point. This is all happening in the usual plane.
The other case can be notated e(0) not equal to e(1)*e(2) not equal to e(1)*e(2)*e(3). So e is not equal to p because it exists in placement space and has this other feature which is different from the way p behaves. In placement space we have 1 or 2 or 3 e’s together and also capable of being separated to different placements.
Then usually the idea of points can be notated pxp=p or pxpxp=p…etc. Where x is the idea of coming together and p is a point. But what if there were another entity of no extent, call it e such that exe=exe, e is not equal to p so that exe is not equal to p and also exe=is not equal to e as that would be the same as pxp=p. We can call these entity equations.
It seems like exe are two identical entities of no extent and it should result in e. But consider that to have exe=e, 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 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 or choose three e’s so that we have exexe 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 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.
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. 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.
The first time I encounter a new square type, 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 countable number of copies present. Then I can move these copies into the other countable infinity I have present. The next such square is similar so that I do the same with this. Then this leaves a finite number of squares as we climb higher. This leaves a finite number of exe copies present at any junction. 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 primes and after that only composite numbers. The new definition of prime in the plane, is to have a prime sharing at every corner.
The second square is the new definition of a prime in the plane, with a prime shared number at every corner. There must be an infinite countable number of copies here as well. 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.
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.
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?
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. 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.
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.
Suppose I represent a location by (0,0). What if we can have a case of (0,0)(1) not equal to (0,0)(2)?. This is possible if (0,0)(1) and (0,0)(2) were somehow different. Since something with no extent can be “added” to something with no extent and the result is something with no extent, there could be two items here. (see the overlapping shadow diagram) We would have to somehow make the two “points” different and only two “points”.
So I say “added”, let us postulate another level of places. That is, an underlying plane where places of the usual plane may exist in other “placements” of places. Where a placement is not a place but a lower level of place. The same notion as place, yet let places be capable of shifting off into this new plane of placements. Then we no longer have a fixed plane of places, yet the placements could be fixed.
Since two items (“points”) of no extent could appear to be a single item, and we could conceivably fix this at 2 or three, or as many as we choose, this other plane also must exist.
The usual idea of points can be notated ….p(1)*p(2)*p(3)=p(1)*p(2)=p(0). This is the idea of two or more points coming together to form one point. This is all happening in the usual plane.
The other case can be notated e(0) not equal to e(1)*e(2) not equal to e(1)*e(2)*e(3). So e is not equal to p because it exists in placement space and has this other feature which is different from the way p behaves. In placement space we have 1 or 2 or 3 e’s together and also capable of being separated to different placements.
Then usually the idea of points can be notated pxp=p or pxpxp=p…etc. Where x is the idea of coming together and p is a point. But what if there were another entity of no extent, call it e such that exe=exe, e is not equal to p so that exe is not equal to p and also exe=is not equal to e as that would be the same as pxp=p. We can call these entity equations.
It seems like exe are two identical entities of no extent and it should result in e. But consider that to have exe, 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 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 or choose three e’s so that we have exexe 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 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 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.
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.
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.
Suppose I represent a location by (0,0). What if we can have a case of (0,0)(1) not equal to (0,0)(2)?. This is possible if (0,0)(1) and (0,0)(2) were somehow different. Since something with no extent can be “added” to something with no extent and the result is something with no extent, there could be two items here. (see the overlapping shadow diagram) We would have to somehow make the two “points” different and only two “points”.
So I say “added”, let us postulate another level of places. That is, an underlying plane where places of the usual plane may exist in other “placements” of places. Where a placement is not a place but a lower level of place. The same notion as place, yet let places be capable of shifting off into this new plane of placements. Then we no longer have a fixed plane of places, yet the placements could be fixed.
Since two items (“points”) of no extent could appear to be a single item, and we could conceivably fix this at 2 or three, or as many as we choose, this other plane also must exist.
The usual idea of points can be notated ….p(1)*p(2)*p(3)=p(1)*p(2)=p(0).This is the idea of one point being mapped to two or more points or 2 or more points coming together to form one point. This is all happening in the usual plane.
The other case can be notated e(0)=e(0) not equal to e(1)*e(2)=e(1)*e(2) not equal to e(1)*e(2)*e(3)=e(1)*e(2)*e(3). So e is not equal to p because it exists in placement space and has this other feature which is different from the way p behaves. In placement space we have 1 or 2 or 3 e’s together and also capable of being separated to different placements.
Then usually the idea of points can be notated pxp=p or pxpxp=p…etc. Where x is the idea of coming together and p is a point. But what if there were another entity of no extent, call it e such that exe=exe, e is not equal to p so that exe is not equal to p and also exe=is not equal to e as that would be the same as pxp=p. We can call these entity equations.
It seems like exe are two identical entities of no extent and it should result in e. But consider that to have exe=p, I have to take out pxp=p. This is not as simple as taking out a point out of a given subset of points of the plane as I have to be able to put something back in that is truly different.
This means I need a space of places coexisting with the original places, so that I can take out the place pxp=p and replace it with the new places, exe=exe. Then we give up the notion of a fixed plane of points, instead we can have three planes, coexisting with one another.
The most basic new plane is in a sense at a lower level than the usual plane. This is a plane of places of new places .Any e in the usual plane can move off in any direction into this new plane, leaving its partner behind. Most basically, the entire plane can move, as shown above.
That means exe are not 2e’s at the same place, as is usually thought of as place but two e’s at the same place in new places. A new level to place. Now we have more room. Since they are in this sense not in the same place, they don’t combine. Briefly we can write this exe=exe (sharing).
Take out the concept of place and put in this new concept of place. The only way it can be different is if the places don’t combine to form a single place but stay separate while being together. (sharing as in the overlapping teacup shadows)
Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:
Now as seen in the overlap, two different points of the shadows can take up the space of one point. This is analogous to two e’s sharing.
Then we can separate the two e’s, but the only way this can be different from the usual idea of separation in points, They are the same in that they share a place and a place of new and original places.
The place has been removed so we have an underlying dimension where places have other places of original places. Like a jigsaw puzzle of a landscape being taken apart. In this way two e’s are sharing a place and a place of new places.
Then this leads to a new extent, a line with two distances one being this new zero and the other being the usual concept of distance, extended.
This is a new dimension. Each e of the extent is different as any other e, yet they originally shared. This is just a new dimension in length. We can notate any two e’s as e(1,m) and e(1,n).
This extent may be considered as negative distance as we need to shrink it to get back to the new zero and then take this out and replace it with pxp=p to get back to the usual zero. Since for e(1,m) and e(1,n) the place is the same, any point that is bound to e(1,m) is also bound to e(1,n). Just not to both at the same time. We may have a closed loop of e’s which can move off and the shape could be altered if we have different distances associated with each e.
We can set a mathematical system with exe=p or choose three e’s so that exexe=p or the number of e’s could be variable.
This must fit into our current structure of mathematics as I am not adding any new notion in, merely clarifying the concept of a point as having no extent, then adding in the necessary new entities. The notion of no extent is the same. We already have this notion of a point as being pxp=p, we have to extend this.
Additionally, there is also the case exr=exr where e and r are two different types of entities as well. This can be for future work.
So we have the idea that a point is an entity with no extent, and also another notion that it could be exe=p but how do these fit together?
It must be that we have replaced the usual idea of a point as being pxp=p with this new idea of a point as being exe=exe.. This means there is another level to space. Since I’ve taken out the usual notion of a point, I must have taken it out from somewhere. This is the space, places of new and original places. It can be modeled after the usual idea of extent, yet the distances are negative.
Then this also means I can separate exe=exe in the new space and move in a space between two of the same e, like so, the displacements from the 2e’s are shown.
Then we can have the idea of a multiple point or two tangent points.
With the tangent point we measure the diameters from the point of tangency outward. These can be separated as usual with the usual distance appearing between them. The exe=exe points can be separated as well, with the new space appearing between them.This is the space of places of new places. This is the negative space.
So we must have a plane or a space in which the ordinary places e or p, take on other places.
A picture of this would look like below if we have only two e’s at the origin and I move one e off up and to the right.: The notation is (()) are places of new places and () are places.
This is a movement of one piece of a doubled origin, a single e.
Not only the origin but each identified exe=p of the new space can act as its own centre, The two e’s can move away from each other.
We could have a closed loop of these points all moving together as shown in the diagram above. As well, this loop could be knotted, if instead of a plane we consider a three dimensional space..
Then this is also the entry into Concept Sharing as math concepts such as number, set, group, ect. Can all be thought of as point-like. That is to say they are all ideas which could have multiple expressions. They can all have sharings.
They are all exact and have no physical reality, they are just ideas.
Since they can all be multiple, there must exist lower concept spaces.
Introduction:
In this article I describe concept sharing and take a look at the Riemann Hypothesis. While it’s a common belief that math is cumulative, so that for example, to do calculus you need to know how to do the math at the lower grades, it might be that there is some basic math missing from our understanding of the overall mathematical structure. Here I present a different concept which subsets existing mathematics and has many applications.
It seems to me there may be an easier way to express the zeta function: Z(s)=1/1^s+1/2^s+1/3^s…..using the ideas of concept sharing as it applies to a new geometry.
The concept sharing of a number:
A number is an amount, as in a counting number, or a position on a number line, or a label.
It is point-like in that it has no existence in physical reality, it is a mathematical object, not a physical object. Therefore we can make a correspondence between the idea of sharing in points and an idea of sharing in numbers.
So that means the concept of a number can be extended downwards so that we have a number of original numbers space and this number of sharing numbers after we take out the original number. So, for example, with the number 1; we have a number of numbers space, let the number of original numbers be 2, instead of 1. Take out the number 1, then we can have 1(1) and 1(2) sharing.
The Riemann Hypothesis:
One may imagine a type of grid with the first square being 1, the next being 1/2^2 the next being 1/3^2… if we use s=2 as an example. See pictures in the notes below. The higher numbers of s can be seen by increasing the dimension. Yet there is always a plane possible with any dimension equal to or higher than 2. For we are just standing on it. We can always project downward to a plane. For dimension 1 there is a line and dimension zero an infinite point at zero.
Since with concept sharing geometry there comes a place of places, in which places can vary, we may vary the distance as we choose to always make the zeta function defined. The zeta function can be continued into the extended geometry. Then there is no longer a need for analytic continuation. I can always make the grid into a 1 by 1.
So we can create a grid specific to the Zeta function defined in placement space.
Then we have that there are two types of number involved. A real part and an imaginary part. This is to make the Zeta function equal to zero.
I think this can be seen more primitively as a numbers which lead to a square with a positive area and numbers which lead to a square with negative area ie. the negative distance is -i. These can be sharing space.
We can concept share two different numbers in the following way: (-1(-1)*(-i(-i) where * is a concept sharing of a concept sharing= ((-1(-i))((-1(-i)). But -1 and -i have to be different. Let -i be the negative distance and -1 be the other, real distance. Then let this be how the square comes about. We have to expand the zero-dimensionality of the concept sharing. Let -i and -l be numbers at the next level of numbers. That is they are no longer point-like but line like. We can start with a point consisting of an infinite uncountable number of sharing parts and expand it outwards into a line.
The 2nd next to last image shows how there are trivial zeros at -2,-4,-6… and how the zeta function could equal -1/12 when s=-1. We are adding an infinite series to get a finite sum. This comes about as we have a addition of positive and negative area. This works for the plane as we can have i and i^2=-1.
If we look at the next to last image, there are three possible cases. One where s=2, one where s=1/2 and some where s=-2,-4,-6,…These all lead to a plane where we can also share with i, in the last two cases so that we might cause the series to converge.
If we look at the last image, this is showing how we can have the complex numbers 1/1^(a+bi), 1/2^(a+bi), 1/3^(a+bi),…on the bottom of the grid and also on the side of the grid. We can give up the idea of negative areas and look to cancel the lengths, thinking of the complex numbers as vectors. Then instead of the square areas, count the diagonals in the squares as lines which could rotate at different origins. To find the diagonal lengths multiply the numbers by sqrt(2)/2. Add two of these to find the diagonal lengths. This is in the case of the plane. In three dimensions multiply by sqrt(3)/3.
Then we have for example 1/sqrt(2)x1/2^(1/2+bi). This is seen as the line with a rotation from the imaginary part as 2^(1/2+bi)=2^(1/2)x2^(bi)=2^(1/2)xe^(ln2(bi)=2^(1/2)x(cos(ln2(b)+isin(ln2(b)). All these rotations of lines and all these other dimensions can lead to a result of zero as the possible rotations can cancel the vectors.
August 25, 2006
As the new school year is approaching, one thing is certain: Many students will struggle with their courses this year. As always, one of the most prominent areas of difficulty will be Mathematics.
It is not very difficult to understand why math is considered by most Canadian high-school and college students to be such a difficult subject. Math is a discipline that requires a basic foundation and a natural transition from core knowledge to more advanced concepts. However, the reality of North American education is that the transition rarely takes place naturally.
It is often the case that throughout elementary school and early High school, math is taught in a very disorganized fashion. While arithmetic gets more than its fair share of attention, intermediate core concepts are often glossed over by everyone except the very astute students. Only the dedicated students fill in the big education gaps in their spare time. What happens to the less dedicated students?
For other students, the true math difficulties begin in grades 11 and 12. At those levels, math transforms into a serious subject and cracks in the knowledge foundation begin to emerge. Those concepts that are natural extensions of what is considered to be basic mathematical abilities become difficult to grasp for many students.
As a result, many struggling students turn to a private math tutor for additional help, but this is often not a sure-fire path to better understanding. Most tutors have the ability to help students with their immediate areas of difficulty. However, only the more experienced tutors are able to detect true deficiencies in the students’ knowledge and fill in the gaps before concentrating on more complicated topics.
For those looking for a math tutor, it is very important to consider the tutor’s knowledge, experience, and approach, and not just the hourly rate. With math tutoring, like with anything else, you will often get exactly what you pay for.
In order to get the most out of tutoring, it is vitally important to establish specific short and long-term learning objectives early on. A good tutor will be able to use this information effectively in creating a structured learning progression, rather than concentrating on the irrelevant concepts.