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Mathematical Concept removal and sharing

Introduction:

The notion of a point, that which has no parts or no extent, is basic in math. The ancient Greeks thought about points, but what if they were not entirely correct?

They asked, what if two points were placed next to each other? They thought that this would be one point and stopped there.-Aristotle : Physics- “neither can two points be contiguous with one another”

But what if this could be regarded as two points? Since something of no extent placed next to something of no extent would have no extent- but there could be two items of no extent there! The two points could be hidden as one!

Yet, as Aristotle thought the two indivisables of no parts would overlap for the two items can be considered together, side by side or to be on top of one another. This is possible with items of no extent as we can consider them either to be on top of each other or side by side. If two dots are placed side by side, they can be considered on top of one another if we look at them from the side. Also if two dots are placed on top of one another we can look at them as being side by side from the top.

Since we want to keep this concept of no extent, or no parts it must also be possible to have two together and still have two, as long as the two parts are different entities so that they do not combine into one. That is, there must be another possible entity with no extent!

One can regard the overlapping shadow diagram below:

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, we can regard these as e( e is another possible entity of no extent)  and p and not points, p and p. If I take away one light, one shadow still remains. 

This concept cannot be visualized as one would be trying to visualize two different items in the same place, but it is still logical, nonetheless. Even the concept of no extent cannot be fully visualized as well. Only usually we represent a point with a dot, but this is not really a visual of an object with no extent.

Also, consider that we have to take a point out in order to have this e*p present. That is e, the new entity sharing position with the point, p. That means we need another level to place, this must exist since we have to have e*p somehow.

Then this is how e can be made different from p, as in an extent of e and p which can come about, we can move e through this e extent, it then has a new location of original location, making it different from p. That is, e is the next level of location.

Yet we want this concept of point removal and sharing (the point is removed and the two points are sharing the notion or concept of an object of no extent) to join up with the knowledge that is already in existence.

In doing this we have to add to and change the mathematics which already exists. This turns out not to destroy but heal mathematics, as this can be seen as something which is basic, but is missing. The math from this leads to a way to solve many old problems which are easy to state but previously hard to solve, such as:

The knot problem:https://calctutor.ca/category/mathematics/knots-mathematics/the-knottedness-and-chirality-of-the-trefoil/

The twin prime conjecture:https://calctutor.ca/category/mathematics/the-twin-prime-conjecture/

The Goldbach conjecture:https://calctutor.ca/category/mathematics/goldbachs-conjecture/

Fermat’s last theorem;https://calctutor.ca/category/mathematics/a-clearer-and-simpler-demonstration-of-fermats-last-theorem-wiles-theorem/

The Poincare conjecture:https://calctutor.ca/category/mathematics/the-poincare-conjecture/

The Riemann Hypothesis:https://calctutor.ca/category/mathematics/the-riemann-hypothesis/

The Collatz conjecture:https://calctutor.ca/category/mathematics/the-collatz-conjecture/

In fact this idea can complete mathematics:

The completion of mathematics:https://calctutor.ca/category/mathematics/the-completion-of-mathematics/

Definition and description of e:

When two points come together or start together, it seems as if there could be two there as 0+0=0.(two items of no extent could still combine to no extent). Only that points have location only, and having them together means we have one location so we have to conclude that there is one point there.

What if an object of no extent could be expressed as two different items of no extent, starting together. Since an item of no extent together with another different item of no extent would still have no extent, yet there could still be two items here and not one. The other item of no extent would have to have something more than location (it would have to be different from a point).

Yet since it is logical that two items of no extent could be together and still be two items, this must be mathematically possible, somehow. It would work if the two items were not both points, in the sense that they were not the same type of entity. For points are defined to have location only and here we are at the same location.

Call this new indivisible item “e”. Then this together with a point can be called e*p. We can separate e and p (not divide as this is a separation not a division) if we move the original e along an extent made up of e’s. This extent of e’s can be together with an extent of p’s. The e extent is different from the p extent in that it is a series of positions of the original e position. That is it is the next level of position.

Then we have to have an object of no extent which is somehow different from the usual conception. I am thinking of putting two items of zero size together. One can think of putting a dot on top or beside a dot which is there already. One cannot visualize two items of no extent being together, but one can also not visualize an item of no extent as well.

Yet they are an e( new entity) and a point (p) at the same place, if we have a preexisting space made up of ordinary points (p’s). 

In terms of symbols, one may write pop=p as the idea that for an ordinary point two points starting together or being brought in coincidence form again an ordinary point. O is the idea of coincidence.

Then e would be something different. E then cannot be in coincidence with p, if we redefine coincidence as the starting together or coming together of two or more points. so we invent a way for the entities to be together and call it sharing, E*p is e sharing with p . Sharing is a way for two items to come together or be defined together and not combine. This should be possible as seen in the analogy of the overlapping shadows. This must exist somehow in mathematics, since we must have the notion of something with no parts or no extent and we can put two together and still have two. So we have p and e as different entities which are not identical. 

Additionally we must remove p for p*e( p and e together) to exist otherwise there is confusion. If we start with a space with P’s everywhere we have to remove a p to put in an e*p. So there must be another level to space. A container of points and e’s. This would then be a new place of original places, another level to place. Then we can see that e can be different from p, as in an extended space of e’s, I could move an e to another new place of the original place of e, along the e continuum. The e continuum can coexist with the p continuum.

So e is another type of point. It is able to be with another entity which has no extent and not be equal to that entity. P doesn’t share with p, as this is already seen as coincidence. Since points have location only this is the same location and so the same point.

Looking at e, e(1)oe(1)=e(1). But also possible is e(1)(e(2)), if we have an extent of e outwards from p(1). e(2) is some e on the e extent outwards from p(1). This is what makes e different from p, with p we only have pop=p. That is an e can act as a type of container for other e’s, as two different e’s do not combine similar to a p and an e. That means we can have a shifting movement of one e along another continuum of e’s. Then e’s can have a new location of the original location of an e which makes them different from p’s, which have location only.

e(1)*p(1) is e and p sharing a location. We can separate a copy of e(1) away, since 0-0=0. Think of a jigsaw puzzle of a landscape being taken apart. e(1)*p(1) =e(1)oe(1)*p(1). Let e(0) be a bridge into the other dimension. Then e(0)(e(1))oe(1)*p(1)=(e(0)*e(1))o*p(1) and p(1) falls away leaving e(0)(e(1)). Then we have e(2)(e(1)) with e(1) moving to an e(2). There might also be an extent of p, along with the e, In which case we can have e(1)*e(2)*p(2).

The e can be multiple, also e can move along another extent of e’s. This is how e is different from p.

To be complete…=p(1)op(1)op(1)op(1) =p(1)op(1)op(1)=p(1)op(1)=p(1), and also….=e(1)oe(1)oe(1)oe(1)=e(1)oe(1)oe(1)=e(1)oe(1)=e(1) and also.. e(1)*e(2)*e(3)*e(4) is not equal to e(1)*e(2)*e(3) is not equal to e(1)*e(2) is not equal to a single e .Also e(0)(e(1)*e(2)*…)). That is, e or multiple e’s can travel along an extent of e’s.

These would not be points, then, as points have location only. Two moving points could come together to form one point. In Physics or mathematics if we have a plane of points(fixed) there can be a point moving in this plane which takes up the locations of these planar points as it moves, and then we have pop=p as it moves.

But also e’s can move as well and we could have two e’s come together to form e*e when the initial location is moved to the extent of e’s.

A point is the notion of that which has no extent or location only, labelled, p.

But since the idea of no extent is to have no dimension.

Two ‘items’ of no extent could exist together and still have no extent. The sum of any number of these new items of no extent is also no extent. The two will be hidden as one. They are hidden as one, at the ordinary point in the pre-existing space. 

It would be two of these new locations at a regular location. By regular location, we mean an ordinary point of the pre-existing space. 

Since points have location only, and not this multiplicity, this could not be two points, but would have to be two of something else, call them ‘parts’.

Represent them by the letter e.

e has no extent but can also be multiple. E’s have the position to them given by the pre-existing points of the pre-existing space.

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. 

These two items could be multiple. This is how we can hide! In order to do this we would have to take out the usual concept of a point as being the only entity with no extent and replace it with these new conceptions as other entities which can have no extent. There are these little entities that don’t combine to form a point. This is because a point is already there as the only idea of no extent. We need some room!

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! The concept of no extent is an example of a concept.

To remove the concept of a point as the only item of no extent, we need to have a concept space, consisting of a concept of multiplicity. 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 as the only item of no extent to have it appear somewhere. Also we must take out any other possibility of any number of items, except 2.

Extension to other concepts:

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.

We can replace e and p, as two items sharing the concept of no extent with two items sharing the concept of a number as well. Or the concept of a set, or group, etc.

Math concepts are exact so that they don’t change from person to person or over time or space. A number, point, function or group, etc. is unchanging, eternal, unalterable.

So also they all can be multiple as the basic ideas of math are all based on geometry and numbers. Functions map inputs to outputs based on formulas which are algebraic expressions of variables(numbers) to points on a graph. Groups are collections of the rotations or flips of geometric objects. Elements of groups are exact in the end numerical or geometric. So numbers, sets, groups, functions, etc. can all have concept spaces!

The idea is to extend the concept using the idea of the concept and the extender “of”. So for example; location of location, number of numbers.

The idea is to go backwards into the idea.

Look at the example of location. For a new location of an initial location we need an extended location, that is there must be an extended location, somewhere to put one of the entities, the extended entity.

Then the extended entity has the location of an initial location, which makes it different from a point, having only location.

The concept sharing of a number:

We can start with a line of p’s with two lines of e’s sharing locations. All in a plane of e’s as shown below:

The numbers 1, 1(1) and 1(2) can be associated as shown, an e is removed and replaced with an e*e and we can associate the new numbers 1(1) and 1(2), concept sharing with the number 1. The number of numbers is 2 and not 1. We can then unfold the two lines of e’s to make an axis where we can have the numbers 1(1) and 1(2). The two numbers 1(1) and 1(2) are sharing the concept of 1, with 1 on the number line. We have the usual number line on the midline of the new plane.

With the Collatz conjecture, we can think of a set of partial sharing numbers, The first ordinary number being an input and the second ordinary number being an output. Then the output of one partially sharing number is the input of the next.

So for example we can create a set such as (10(5), (5(16), (16(8), (8(4), (4(2), (2(1). The rules (if even divide by two, if odd multiply by 3 and add 1) are applied to the input ordinary number to create the output ordinary number.

These can be thought of as existing in the plane of shared numbers as described in the link. We can say that order matters so that (10(5) is not equal to (5(10). Starting with 10 dots and sharing 5 is not considered the same as starting with 5 dots and adding 10.

Then consider that we can show that the distance between each partially shared number is zero, so that they are all equal.

We can create a combined number (10(5(16(8(4(2(1) if we bring back coincidence of the ordinary numbers and then allow sharing of the one number so that all ordinary numbers appear once and are shared here.

But then we can see (10(5)=(5(16), as well as all the other equalities, as all these ordinary numbers are sharing together.

The other thing to realize is that these partially sharing numbers are rules and the rules are partially sharing numbers. So that all rules must be represented by partially sharing numbers. Which means all ordinary starting numbers with associated ordinary output numbers are represented and are all equal. Since One set leads to the loop (4(2), (2(1), (1(4), they all do.

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

Mathematical Concept removal and sharing

Introduction:

The notion of a point, that which has no parts or no extent, is basic in math. The ancient Greeks thought about points, but what if they were not entirely correct?

They asked, what if two points were placed next to each other? They thought that this would be one point and stopped there.-Aristotle : Physics- “neither can two points be contiguous with one another”

But what if this could be regarded as two points? Since something of no extent placed next to something of no extent would have no extent- but there could be two items of no extent there! The two points could be hidden as one!

Yet, as Aristotle thought the two indivisables of no parts would overlap for the two items can be considered together, side by side or to be on top of one another. This is possible with items of no extent as we can consider them either to be on top of each other or side by side. If two dots are placed side by side, they can be considered on top of one another if we look at them from the side. Also if two dots are placed on top of one another we can look at them as being side by side from the top.

Since we want to keep this concept of no extent, or no parts it must also be possible to have two together and still have two, as long as the two parts are different entities so that they do not combine into one. That is, there must be another possible entity with no extent!

One can regard the overlapping shadow diagram below:

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, we can regard these as e( e is another possible entity of no extent)  and p and not points, p and p. If I take away one light, one shadow still remains. 

This concept cannot be visualized as one would be trying to visualize two different items in the same place, but it is still logical, nonetheless. Even the concept of no extent cannot be fully visualized as well. Only usually we represent a point with a dot, but this is not really a visual of an object with no extent.

Also, consider that we have to take a point out in order to have this e*p present. That is e, the new entity sharing position with the point, p. That means we need another level to place, this must exist since we have to have e*p somehow.

Then this is how e can be made different from p, as in an extent of e and p which can come about, we can move e through this e extent, it then has a new location of original location, making it different from p. That is, e is the next level of location.

Yet we want this concept of point removal and sharing (the point is removed and the two points are sharing the notion or concept of an object of no extent) to join up with the knowledge that is already in existence.

In doing this we have to add to and change the mathematics which already exists. This turns out not to destroy but heal mathematics, as this can be seen as something which is basic, but is missing. The math from this leads to a way to solve many old problems which are easy to state but previously hard to solve, such as:

The knot problem:https://calctutor.ca/category/mathematics/knots-mathematics/the-knottedness-and-chirality-of-the-trefoil/

The twin prime conjecture:https://calctutor.ca/category/mathematics/the-twin-prime-conjecture/

The Goldbach conjecture:https://calctutor.ca/category/mathematics/goldbachs-conjecture/

Fermat’s last theorem;https://calctutor.ca/category/mathematics/a-clearer-and-simpler-demonstration-of-fermats-last-theorem-wiles-theorem/

The Poincare conjecture:https://calctutor.ca/category/mathematics/the-poincare-conjecture/

The Riemann Hypothesis:https://calctutor.ca/category/mathematics/the-riemann-hypothesis/

The Collatz conjecture:https://calctutor.ca/category/mathematics/the-collatz-conjecture/

In fact this idea can complete mathematics:

The completion of mathematics:https://calctutor.ca/category/mathematics/the-completion-of-mathematics/

Definition and description of e:

When two points come together or start together, it seems as if there could be two there as 0+0=0.(two items of no extent could still combine to no extent). Only that points have location only, and having them together means we have one location so we have to conclude that there is one point there.

What if an object of no extent could be expressed as two different items of no extent, starting together. Since an item of no extent together with another different item of no extent would still have no extent, yet there could still be two items here and not one. The other item of no extent would have to have something more than location (it would have to be different from a point).

Yet since it is logical that two items of no extent could be together and still be two items, this must be mathematically possible, somehow. It would work if the two items were not both points, in the sense that they were not the same type of entity. For points are defined to have location only and here we are at the same location.

Call this new indivisible item “e”. Then this together with a point can be called e*p. We can separate e and p (not divide as this is a separation not a division) if we move the original e along an extent made up of e’s. This extent of e’s can be together with an extent of p’s. The e extent is different from the p extent in that it is a series of positions of the original e position. That is it is the next level of position.

Then we have to have an object of no extent which is somehow different from the usual conception. I am thinking of putting two items of zero size together. One can think of putting a dot on top or beside a dot which is there already. One cannot visualize two items of no extent being together, but one can also not visualize an item of no extent as well.

Yet they are an e( new entity) and a point (p) at the same place, if we have a preexisting space made up of ordinary points (p’s). 

In terms of symbols, one may write pop=p as the idea that for an ordinary point two points starting together or being brought in coincidence form again an ordinary point. O is the idea of coincidence.

Then e would be something different. E then cannot be in coincidence with p, if we redefine coincidence as the starting together or coming together of two or more points. so we invent a way for the entities to be together and call it sharing, E*p is e sharing with p . Sharing is a way for two items to come together or be defined together and not combine. This should be possible as seen in the analogy of the overlapping shadows. This must exist somehow in mathematics, since we must have the notion of something with no parts or no extent and we can put two together and still have two. So we have p and e as different entities which are not identical. 

Additionally we must remove p for p*e( p and e together) to exist otherwise there is confusion. If we start with a space with P’s everywhere we have to remove a p to put in an e*p. So there must be another level to space. A container of points and e’s. This would then be a new place of original places, another level to place. Then we can see that e can be different from p, as in an extended space of e’s, I could move an e to another new place of the original place of e, along the e continuum. The e continuum can coexist with the p continuum.

So e is another type of point. It is able to be with another entity which has no extent and not be equal to that entity. P doesn’t share with p, as this is already seen as coincidence. Since points have location only this is the same location and so the same point.

Looking at e, e(1)oe(1)=e(1). But also possible is e(1)(e(2)), if we have an extent of e outwards from p(1). e(2) is some e on the e extent outwards from p(1). This is what makes e different from p, with p we only have pop=p. That is an e can act as a type of container for other e’s, as two different e’s do not combine similar to a p and an e. That means we can have a shifting movement of one e along another continuum of e’s. Then e’s can have a new location of the original location of an e which makes them different from p’s, which have location only.

e(1)*p(1) is e and p sharing a location. We can separate a copy of e(1) away, since 0-0=0. Think of a jigsaw puzzle of a landscape being taken apart. e(1)*p(1) =e(1)oe(1)*p(1). Let e(0) be a bridge into the other dimension. Then e(0)(e(1))oe(1)*p(1)=(e(0)*e(1))o*p(1) and p(1) falls away leaving e(0)(e(1)). Then we have e(2)(e(1)) with e(1) moving to an e(2). There might also be an extent of p, along with the e, In which case we can have e(1)*e(2)*p(2).

The e can be multiple, also e can move along another extent of e’s. This is how e is different from p.

To be complete…=p(1)op(1)op(1)op(1) =p(1)op(1)op(1)=p(1)op(1)=p(1), and also….=e(1)oe(1)oe(1)oe(1)=e(1)oe(1)oe(1)=e(1)oe(1)=e(1) and also.. e(1)*e(2)*e(3)*e(4) is not equal to e(1)*e(2)*e(3) is not equal to e(1)*e(2) is not equal to a single e .Also e(0)(e(1)*e(2)*…)). That is, e or multiple e’s can travel along an extent of e’s.

These would not be points, then, as points have location only. Two moving points could come together to form one point. In Physics or mathematics if we have a plane of points(fixed) there can be a point moving in this plane which takes up the locations of these planar points as it moves, and then we have pop=p as it moves.

But also e’s can move as well and we could have two e’s come together to form e*e when the initial location is moved to the extent of e’s.

A point is the notion of that which has no extent or location only, labelled, p.

But since the idea of no extent is to have no dimension.

Two ‘items’ of no extent could exist together and still have no extent. The sum of any number of these new items of no extent is also no extent. The two will be hidden as one. They are hidden as one, at the ordinary point in the pre-existing space. 

It would be two of these new locations at a regular location. By regular location, we mean an ordinary point of the pre-existing space. 

Since points have location only, and not this multiplicity, this could not be two points, but would have to be two of something else, call them ‘parts’.

Represent them by the letter e.

e has no extent but can also be multiple. E’s have the position to them given by the pre-existing points of the pre-existing space.

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. 

These two items could be multiple. This is how we can hide! In order to do this we would have to take out the usual concept of a point as being the only entity with no extent and replace it with these new conceptions as other entities which can have no extent. There are these little entities that don’t combine to form a point. This is because a point is already there as the only idea of no extent. We need some room!

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! The concept of no extent is an example of a concept.

To remove the concept of a point as the only item of no extent, we need to have a concept space, consisting of a concept of multiplicity. 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 as the only item of no extent to have it appear somewhere. Also we must take out any other possibility of any number of items, except 2.

Extension to other concepts:

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.

We can replace e and p, as two items sharing the concept of no extent with two items sharing the concept of a number as well. Or the concept of a set, or group, etc.

Math concepts are exact so that they don’t change from person to person or over time or space. A number, point, function or group, etc. is unchanging, eternal, unalterable.

So also they all can be multiple as the basic ideas of math are all based on geometry and numbers. Functions map inputs to outputs based on formulas which are algebraic expressions of variables(numbers) to points on a graph. Groups are collections of the rotations or flips of geometric objects. Elements of groups are exact in the end numerical or geometric. So numbers, sets, groups, functions, etc. can all have concept spaces!

The idea is to extend the concept using the idea of the concept and the extender “of”. So for example; location of location, number of numbers.

The idea is to go backwards into the idea.

Look at the example of location. For a new location of an initial location we need an extended location, that is there must be an extended location, somewhere to put one of the entities, the extended entity.

Then the extended entity has the location of an initial location, which makes it different from a point, having only location.

The concept sharing of a number:

We can start with a line of p’s with two lines of e’s sharing locations. All in a plane of e’s as shown below:

The numbers 1, 1(1) and 1(2) can be associated as shown, an e is removed and replaced with an e*e and we can associate the new numbers 1(1) and 1(2), concept sharing with the number 1. The number of numbers is 2 and not 1. We can then unfold the two lines of e’s to make an axis where we can have the numbers 1(1) and 1(2). The two numbers 1(1) and 1(2) are sharing the concept of 1, with 1 on the number line. We have the usual number line on the midline of the new plane.

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 as the only item of 10th position 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 , 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 squares. exe can be repeated indefinitely.  Yet there is an uncountable infinity of these possible, also a countable infinity within this. 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 the diagonal as well as filling in space in the larger squares.

Since the larger square exists 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. This means that the countable infinity must also exist.

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 a space where we can have doubled locations. See other papers on knots in my series. The the e’s are free to move in a space where we have an extent of e’s. 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.

We have two number lines, perpendicular to each other, as shown. There is a way of moving the two 1’s so that they can be placed together and not be one 1, this is shown later, below.

We would need another level to the number 2, as these two items here are in the same place. These two items just “look like” one item. We cannot imagine two items there, but our logic tells us this could be so. Numbers are in a sense zero-dimensional.

Since we can have this, there is no way of knowing how many are hidden here ahead of time. Simply state ahead of time how many we want.

We need to take away the number 2 and replace it with a label like (1(1)(1(2)), showing that there are two items there, yet the two items do not combine to form one, 1. The two new label 1’a share the concept of the label, 1.

The space is a space where a point can be multiple.

Since it is logical that two items of no extent can exist together and be two items, it must be that the usual idea of a point is replaced with this new idea, so as to give it existence. See the teacup shadow diagram below. Also places can be moved off in this new dimension (can be separated) Hence they are different from each other and different from the usual 1, because they can be separated.

A point is the notion of that which has no extent or location only, labelled, p.

But since the idea of no extent is to have no dimension.

Two ‘items’ of no extent could exist together and still have no extent The two will be hidden as one.

It would be two of the same location.

Since points have location only, this could not be two points, but would have to be two of something else, call them ‘parts’.

Represent them by the letter e.

e has no extent but can also be multiple or can concept share. The concept of no extent is shared by the e’s. We could notate two of them together as e(1)(1)+e(1)(2).

When there is only the idea of no extent, without this natural extension two points can only combine to form 1 point. But with this extension 2 points can exist together as 2 points

The concept of an item of no extent is basic to geometry, but what if there were more than I item of no extent? As you shall see, this maybe non-intuitive but it’s not so far-fetched.

What if two e’s(items which are like points but not exactly points) could be placed together so that they do not form 1 e. ( otherwise they would be the same as points). Additionally points, which we could call p, do not combine with e’s (otherwise e’s would be p’s).

So I say two e’s together, but these two e’s are hidden as one. There might be 3,4,5, etc. We have to say at the outset how many we would like and eliminate 1 and all the other possibilities as well.

With just the simple plane two moving points can only combine to form one point, but now that we have extended geometry, two e’s moving in the new plane can form this new non-combination (e(1)*e(2)).

This also hints at another concept sharing, that of a concept sharing of a number. The usual number of 2 does not really apply since the two e’s are together so we take out the number 2 as the only form of 2, like we did with the simple point and seek a lower level of number of original numbers. Now let it be two instead of one so that we might label the two e’s 1(1) and 1(2). The idea being two 1’s in a tie, not forming one, 1.

How are 1(1) and 1(2) different? The two labels are different from each other and from the label 1 (usual label)

Since I can move them I can place them in a new unit of one, after I replace the 1 that’s there. Then the usual number line can be thought of as the diagonal line, with the associated 1 being non-movable (the idea of a fixed unit of 1). We are able to remove the usual 1 that’s there and replace with 2, new movable 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.

Mathematical Concept removal and sharing

Introduction:

The notion of a point, that which has no parts or no extent, is basic in math. The ancient Greeks thought about points, but what if they were not entirely correct?

They asked, what if two points were placed next to each other? They thought that this would be one point and stopped there.-Aristotle : Physics- “neither can two points be contiguous with one another”

But what if this could be regarded as two points? Since something of no extent placed next to something of no extent would have no extent- but there could be two items of no extent there! The two points could be hidden as one!

Yet, as Aristotle thought the two indivisables of no parts would overlap for the two items can be considered together, side by side or to be on top of one another. This is possible with items of no extent as we can consider them either to be on top of each other or side by side. If two dots are placed side by side, they can be considered on top of one another if we look at them from the side. Also if two dots are placed on top of one another we can look at them as being side by side from the top.

Since we want to keep this concept of no extent, or no parts it must also be possible to have two together and still have two, as long as the two parts are different entities so that they do not combine into one. That is, there must be another possible entity with no extent!

One can regard the overlapping shadow diagram below:

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, we can regard these as e( e is another possible entity of no extent)  and p and not points, p and p. If I take away one light, one shadow still remains. 

This concept cannot be visualized as one would be trying to visualize two different items in the same place, but it is still logical, nonetheless. Even the concept of no extent cannot be fully visualized as well. Only usually we represent a point with a dot, but this is not really a visual of an object with no extent.

Also, consider that we have to take a point out in order to have this e*p present. That is e, the new entity sharing position with the point, p. That means we need another level to place, this must exist since we have to have e*p somehow.

Then this is how e can be made different from p, as in an extent of e and p which can come about, we can move e through this e extent, it then has a new location of original location, making it different from p. That is, e is the next level of location.

Yet we want this concept of point removal and sharing (the point is removed and the two points are sharing the notion or concept of an object of no extent) to join up with the knowledge that is already in existence.

In doing this we have to add to and change the mathematics which already exists. This turns out not to destroy but heal mathematics, as this can be seen as something which is basic, but is missing. The math from this leads to a way to solve many old problems which are easy to state but previously hard to solve, such as:

The knot problem:https://calctutor.ca/category/mathematics/knots-mathematics/the-knottedness-and-chirality-of-the-trefoil/

The twin prime conjecture:https://calctutor.ca/category/mathematics/the-twin-prime-conjecture/

The Goldbach conjecture:https://calctutor.ca/category/mathematics/goldbachs-conjecture/

Fermat’s last theorem;https://calctutor.ca/category/mathematics/a-clearer-and-simpler-demonstration-of-fermats-last-theorem-wiles-theorem/

The Poincare conjecture:https://calctutor.ca/category/mathematics/the-poincare-conjecture/

The Riemann Hypothesis:https://calctutor.ca/category/mathematics/the-riemann-hypothesis/

The Collatz conjecture:https://calctutor.ca/category/mathematics/the-collatz-conjecture/

In fact this idea can complete mathematics:

The completion of mathematics:https://calctutor.ca/category/mathematics/the-completion-of-mathematics/

Definition and description of e:

When two points come together or start together, it seems as if there could be two there as 0+0=0.(two items of no extent could still combine to no extent). Only that points have location only, and having them together means we have one location so we have to conclude that there is one point there.

What if an object of no extent could be expressed as two different items of no extent, starting together. Since an item of no extent together with another different item of no extent would still have no extent, yet there could still be two items here and not one. The other item of no extent would have to have something more than location (it would have to be different from a point).

Yet since it is logical that two items of no extent could be together and still be two items, this must be mathematically possible, somehow. It would work if the two items were not both points, in the sense that they were not the same type of entity. For points are defined to have location only and here we are at the same location.

Call this new indivisible item “e”. Then this together with a point can be called e*p. We can separate e and p (not divide as this is a separation not a division) if we move the original e along an extent made up of e’s. This extent of e’s can be together with an extent of p’s. The e extent is different from the p extent in that it is a series of positions of the original e position. That is it is the next level of position.

Then we have to have an object of no extent which is somehow different from the usual conception. I am thinking of putting two items of zero size together. One can think of putting a dot on top or beside a dot which is there already. One cannot visualize two items of no extent being together, but one can also not visualize an item of no extent as well.

Yet they are an e( new entity) and a point (p) at the same place, if we have a preexisting space made up of ordinary points (p’s). 

In terms of symbols, one may write pop=p as the idea that for an ordinary point two points starting together or being brought in coincidence form again an ordinary point. O is the idea of coincidence.

Then e would be something different. E then cannot be in coincidence with p, if we redefine coincidence as the starting together or coming together of two or more points. so we invent a way for the entities to be together and call it sharing, E*p is e sharing with p . Sharing is a way for two items to come together or be defined together and not combine. This should be possible as seen in the analogy of the overlapping shadows. This must exist somehow in mathematics, since we must have the notion of something with no parts or no extent and we can put two together and still have two. So we have p and e as different entities which are not identical. 

Additionally we must remove p for p*e( p and e together) to exist otherwise there is confusion. If we start with a space with P’s everywhere we have to remove a p to put in an e*p. So there must be another level to space. A container of points and e’s. This would then be a new place of original places, another level to place. Then we can see that e can be different from p, as in an extended space of e’s, I could move an e to another new place of the original place of e, along the e continuum. The e continuum can coexist with the p continuum.

So e is another type of point. It is able to be with another entity which has no extent and not be equal to that entity. P doesn’t share with p, as this is already seen as coincidence. Since points have location only this is the same location and so the same point.

Looking at e, e(1)oe(1)=e(1). But also possible is e(1)(e(2)), if we have an extent of e outwards from p(1). e(2) is some e on the e extent outwards from p(1). This is what makes e different from p, with p we only have pop=p. That is an e can act as a type of container for other e’s, as two different e’s do not combine similar to a p and an e. That means we can have a shifting movement of one e along another continuum of e’s. Then e’s can have a new location of the original location of an e which makes them different from p’s, which have location only.

e(1)*p(1) is e and p sharing a location. We can separate a copy of e(1) away, since 0-0=0. Think of a jigsaw puzzle of a landscape being taken apart. e(1)*p(1) =e(1)oe(1)*p(1). Let e(0) be a bridge into the other dimension. Then e(0)(e(1))oe(1)*p(1)=(e(0)*e(1))o*p(1) and p(1) falls away leaving e(0)(e(1)). Then we have e(2)(e(1)) with e(1) moving to an e(2). There might also be an extent of p, along with the e, In which case we can have e(1)*e(2)*p(2).

The e can be multiple, also e can move along another extent of e’s. This is how e is different from p.

To be complete…=p(1)op(1)op(1)op(1) =p(1)op(1)op(1)=p(1)op(1)=p(1), and also….=e(1)oe(1)oe(1)oe(1)=e(1)oe(1)oe(1)=e(1)oe(1)=e(1) and also.. e(1)*e(2)*e(3)*e(4) is not equal to e(1)*e(2)*e(3) is not equal to e(1)*e(2) is not equal to a single e .Also e(0)(e(1)*e(2)*…)). That is, e or multiple e’s can travel along an extent of e’s.

These would not be points, then, as points have location only. Two moving points could come together to form one point. In Physics or mathematics if we have a plane of points(fixed) there can be a point moving in this plane which takes up the locations of these planar points as it moves, and then we have pop=p as it moves.

But also e’s can move as well and we could have two e’s come together to form e*e when the initial location is moved to the extent of e’s.

A point is the notion of that which has no extent or location only, labelled, p.

But since the idea of no extent is to have no dimension.

Two ‘items’ of no extent could exist together and still have no extent. The sum of any number of these new items of no extent is also no extent. The two will be hidden as one. They are hidden as one, at the ordinary point in the pre-existing space. 

It would be two of these new locations at a regular location. By regular location, we mean an ordinary point of the pre-existing space. 

Since points have location only, and not this multiplicity, this could not be two points, but would have to be two of something else, call them ‘parts’.

Represent them by the letter e.

e has no extent but can also be multiple. E’s have the position to them given by the pre-existing points of the pre-existing space.

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. 

These two items could be multiple. This is how we can hide! In order to do this we would have to take out the usual concept of a point as being the only entity with no extent and replace it with these new conceptions as other entities which can have no extent. There are these little entities that don’t combine to form a point. This is because a point is already there as the only idea of no extent. We need some room!

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! The concept of no extent is an example of a concept.

To remove the concept of a point as the only item of no extent, we need to have a concept space, consisting of a concept of multiplicity. 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 as the only item of no extent to have it appear somewhere. Also we must take out any other possibility of any number of items, except 2.

Extension to other concepts:

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.

We can replace e and p, as two items sharing the concept of no extent with two items sharing the concept of a number as well. Or the concept of a set, or group, etc.

Math concepts are exact so that they don’t change from person to person or over time or space. A number, point, function or group, etc. is unchanging, eternal, unalterable.

So also they all can be multiple as the basic ideas of math are all based on geometry and numbers. Functions map inputs to outputs based on formulas which are algebraic expressions of variables(numbers) to points on a graph. Groups are collections of the rotations or flips of geometric objects. Elements of groups are exact in the end numerical or geometric. So numbers, sets, groups, functions, etc. can all have concept spaces!

The idea is to extend the concept using the idea of the concept and the extender “of”. So for example; location of location, number of numbers.

The idea is to go backwards into the idea.

Look at the example of location. For a new location of an initial location we need an extended location, that is there must be an extended location, somewhere to put one of the entities, the extended entity.

Then the extended entity has the location of an initial location, which makes it different from a point, having only location.

The concept sharing of a number:

We can start with a line of p’s with two lines of e’s sharing locations. All in a plane of e’s as shown below:

The numbers 1, 1(1) and 1(2) can be associated as shown, an e is removed and replaced with an e*e and we can associate the new numbers 1(1) and 1(2), concept sharing with the number 1. The number of numbers is 2 and not 1. We can then unfold the two lines of e’s to make an axis where we can have the numbers 1(1) and 1(2). The two numbers 1(1) and 1(2) are sharing the concept of 1, with 1 on the number line. We have the usual number line on the midline of the new plane.

This, then is how the numbers move, they are associated with moving e’s. See below:

The locations move off into the new plane. The numbers come along with the moving e’s. We can have one e at 1, two e’s at 2, three e’s 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.

Concept Sharing and a New View of Knots

Abstract: Here I introduce concept sharing. In uncovering extended space, I show a new way of understanding knots.

Mathematical Concept removal and sharing

Introduction:

The notion of a point, that which has no parts or no extent, is basic in math. The ancient Greeks thought about points, but what if they were not entirely correct?

They asked, what if two points were placed next to each other? They thought that this would be one point and stopped there.-Aristotle : Physics- “neither can two points be contiguous with one another”

But what if this could be regarded as two points? Since something of no extent placed next to something of no extent would have no extent- but there could be two items of no extent there! The two points could be hidden as one!

Yet, as Aristotle thought the two indivisables of no parts would overlap for the two items can be considered together, side by side or to be on top of one another. This is possible with items of no extent as we can consider them either to be on top of each other or side by side. If two dots are placed side by side, they can be considered on top of one another if we look at them from the side. Also if two dots are placed on top of one another we can look at them as being side by side from the top.

Since we want to keep this concept of no extent, or no parts it must also be possible to have two together and still have two, as long as the two parts are different entities so that they do not combine into one. That is, there must be another possible entity with no extent!

One can regard the overlapping shadow diagram below:

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, we can regard these as e( e is another possible entity of no extent)  and p and not points, p and p. If I take away one light, one shadow still remains. 

This concept cannot be visualized as one would be trying to visualize two different items in the same place, but it is still logical, nonetheless. Even the concept of no extent cannot be fully visualized as well. Only usually we represent a point with a dot, but this is not really a visual of an object with no extent.

Also, consider that we have to take a point out in order to have this e*p present. That is e, the new entity sharing position with the point, p. That means we need another level to place, this must exist since we have to have e*p somehow.

Then this is how e can be made different from p, as in an extent of e and p which can come about, we can move e through this e extent, it then has a new location of original location, making it different from p. That is, e is the next level of location.

Yet we want this concept of point removal and sharing (the point is removed and the two points are sharing the notion or concept of an object of no extent) to join up with the knowledge that is already in existence.

In doing this we have to add to and change the mathematics which already exists. This turns out not to destroy but heal mathematics, as this can be seen as something which is basic, but is missing. The math from this leads to a way to solve many old problems which are easy to state but previously hard to solve, such as:

The knot problem:https://calctutor.ca/category/mathematics/knots-mathematics/the-knottedness-and-chirality-of-the-trefoil/

The twin prime conjecture:https://calctutor.ca/category/mathematics/the-twin-prime-conjecture/

The Goldbach conjecture:https://calctutor.ca/category/mathematics/goldbachs-conjecture/

Fermat’s last theorem;https://calctutor.ca/category/mathematics/a-clearer-and-simpler-demonstration-of-fermats-last-theorem-wiles-theorem/

The Poincare conjecture:https://calctutor.ca/category/mathematics/the-poincare-conjecture/

The Riemann Hypothesis:https://calctutor.ca/category/mathematics/the-riemann-hypothesis/

The Collatz conjecture:https://calctutor.ca/category/mathematics/the-collatz-conjecture/

In fact this idea can complete mathematics:

The completion of mathematics:https://calctutor.ca/category/mathematics/the-completion-of-mathematics/

Definition and description of e:

When two points come together or start together, it seems as if there could be two there as 0+0=0.(two items of no extent could still combine to no extent). Only that points have location only, and having them together means we have one location so we have to conclude that there is one point there.

What if an object of no extent could be expressed as two different items of no extent, starting together. Since an item of no extent together with another different item of no extent would still have no extent, yet there could still be two items here and not one. The other item of no extent would have to have something more than location (it would have to be different from a point).

Yet since it is logical that two items of no extent could be together and still be two items, this must be mathematically possible, somehow. It would work if the two items were not both points, in the sense that they were not the same type of entity. For points are defined to have location only and here we are at the same location.

Call this new indivisible item “e”. Then this together with a point can be called e*p. We can separate e and p (not divide as this is a separation not a division) if we move the original e along an extent made up of e’s. This extent of e’s can be together with an extent of p’s. The e extent is different from the p extent in that it is a series of positions of the original e position. That is it is the next level of position.

Then we have to have an object of no extent which is somehow different from the usual conception. I am thinking of putting two items of zero size together. One can think of putting a dot on top or beside a dot which is there already. One cannot visualize two items of no extent being together, but one can also not visualize an item of no extent as well.

Yet they are an e( new entity) and a point (p) at the same place, if we have a preexisting space made up of ordinary points (p’s). 

In terms of symbols, one may write pop=p as the idea that for an ordinary point two points starting together or being brought in coincidence form again an ordinary point. O is the idea of coincidence.

Then e would be something different. E then cannot be in coincidence with p, if we redefine coincidence as the starting together or coming together of two or more points. so we invent a way for the entities to be together and call it sharing, E*p is e sharing with p . Sharing is a way for two items to come together or be defined together and not combine. This should be possible as seen in the analogy of the overlapping shadows. This must exist somehow in mathematics, since we must have the notion of something with no parts or no extent and we can put two together and still have two. So we have p and e as different entities which are not identical. 

Additionally we must remove p for p*e( p and e together) to exist otherwise there is confusion. If we start with a space with P’s everywhere we have to remove a p to put in an e*p. So there must be another level to space. A container of points and e’s. This would then be a new place of original places, another level to place. Then we can see that e can be different from p, as in an extended space of e’s, I could move an e to another new place of the original place of e, along the e continuum. The e continuum can coexist with the p continuum.

So e is another type of point. It is able to be with another entity which has no extent and not be equal to that entity. P doesn’t share with p, as this is already seen as coincidence. Since points have location only this is the same location and so the same point.

Looking at e, e(1)oe(1)=e(1). But also possible is e(1)(e(2)), if we have an extent of e outwards from p(1). e(2) is some e on the e extent outwards from p(1). This is what makes e different from p, with p we only have pop=p. That is an e can act as a type of container for other e’s, as two different e’s do not combine similar to a p and an e. That means we can have a shifting movement of one e along another continuum of e’s. Then e’s can have a new location of the original location of an e which makes them different from p’s, which have location only.

e(1)*p(1) is e and p sharing a location. We can separate a copy of e(1) away, since 0-0=0. Think of a jigsaw puzzle of a landscape being taken apart. e(1)*p(1) =e(1)oe(1)*p(1). Let e(0) be a bridge into the other dimension. Then e(0)(e(1))oe(1)*p(1)=(e(0)*e(1))o*p(1) and p(1) falls away leaving e(0)(e(1)). Then we have e(2)(e(1)) with e(1) moving to an e(2). There might also be an extent of p, along with the e, In which case we can have e(1)*e(2)*p(2).

The e can be multiple, also e can move along another extent of e’s. This is how e is different from p.

To be complete…=p(1)op(1)op(1)op(1) =p(1)op(1)op(1)=p(1)op(1)=p(1), and also….=e(1)oe(1)oe(1)oe(1)=e(1)oe(1)oe(1)=e(1)oe(1)=e(1) and also.. e(1)*e(2)*e(3)*e(4) is not equal to e(1)*e(2)*e(3) is not equal to e(1)*e(2) is not equal to a single e .Also e(0)(e(1)*e(2)*…)). That is, e or multiple e’s can travel along an extent of e’s.

These would not be points, then, as points have location only. Two moving points could come together to form one point. In Physics or mathematics if we have a plane of points(fixed) there can be a point moving in this plane which takes up the locations of these planar points as it moves, and then we have pop=p as it moves.

But also e’s can move as well and we could have two e’s come together to form e*e when the initial location is moved to the extent of e’s.

A point is the notion of that which has no extent or location only, labelled, p.

But since the idea of no extent is to have no dimension.

Two ‘items’ of no extent could exist together and still have no extent. The sum of any number of these new items of no extent is also no extent. The two will be hidden as one. They are hidden as one, at the ordinary point in the pre-existing space. 

It would be two of these new locations at a regular location. By regular location, we mean an ordinary point of the pre-existing space. 

Since points have location only, and not this multiplicity, this could not be two points, but would have to be two of something else, call them ‘parts’.

Represent them by the letter e.

e has no extent but can also be multiple. E’s have the position to them given by the pre-existing points of the pre-existing space.

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. 

These two items could be multiple. This is how we can hide! In order to do this we would have to take out the usual concept of a point as being the only entity with no extent and replace it with these new conceptions as other entities which can have no extent. There are these little entities that don’t combine to form a point. This is because a point is already there as the only idea of no extent. We need some room!

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! The concept of no extent is an example of a concept.

To remove the concept of a point as the only item of no extent, we need to have a concept space, consisting of a concept of multiplicity. 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 as the only item of no extent to have it appear somewhere. Also we must take out any other possibility of any number of items, except 2.

Extension to other concepts:

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.

We can replace e and p, as two items sharing the concept of no extent with two items sharing the concept of a number as well. Or the concept of a set, or group, etc.

Math concepts are exact so that they don’t change from person to person or over time or space. A number, point, function or group, etc. is unchanging, eternal, unalterable.

So also they all can be multiple as the basic ideas of math are all based on geometry and numbers. Functions map inputs to outputs based on formulas which are algebraic expressions of variables(numbers) to points on a graph. Groups are collections of the rotations or flips of geometric objects. Elements of groups are exact in the end numerical or geometric. So numbers, sets, groups, functions, etc. can all have concept spaces!

The idea is to extend the concept using the idea of the concept and the extender “of”. So for example; location of location, number of numbers.

The idea is to go backwards into the idea.

Look at the example of location. For a new location of an initial location we need an extended location, that is there must be an extended location, somewhere to put one of the entities, the extended entity.

Then the extended entity has the location of an initial location, which makes it different from a point, having only location.

The concept sharing of a number:

We can start with a line of p’s with two lines of e’s sharing locations. All in a plane of e’s as shown below:

The numbers 1, 1(1) and 1(2) can be associated as shown, an e is removed and replaced with an e*e and we can associate the new numbers 1(1) and 1(2), concept sharing with the number 1. The number of numbers is 2 and not 1. We can then unfold the two lines of e’s to make an axis where we can have the numbers 1(1) and 1(2). The two numbers 1(1) and 1(2) are sharing the concept of 1, with 1 on the number line. We have the usual number line on the midline of the new plane.

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. 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.

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.

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.

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”.

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.

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.

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.

Then this also means I can separate exe=exe in the new space and move in a space between two of the same e, like so, the displacements from the 2e’s are shown.

Then we can have the idea of a multiple point or two tangent points.

With the tangent point we measure the diameters from the point of tangency outward. These can be separated as usual with the usual distance appearing between them. The exe=exe points can be separated as well, with the new space appearing between them.This is the space of places of new places. This is the negative space.

So we must have a plane or a space in which the ordinary places e or p, take on other places.

A picture of this would look like below if we have only two e’s at the origin and I move one e off up and to the right.: The notation is (()) are places of new places and () are places.

This is a movement of one piece of a doubled origin, a single e.

Not only the origin but each identified exe=p of the new space can act as its own centre, The two e’s can  move away from each other.

We could have a closed loop of these points all moving together as shown in the diagram above. As well, this loop could be knotted, if instead of a plane we consider a three dimensional space..

Then this is also the entry into Concept Sharing as math concepts such as number, set, group, ect. Can all be thought of as point-like. That is to say they are all ideas which could have multiple expressions. They can all have sharings.

They are all exact and have no physical reality, they are just ideas.

Since they can all be multiple, there must exist lower concept spaces.

The Concept Sharing of a Number:

Numbers are exact concepts. In the above case, we can think of them as the number of shadows at the center. They have exact boundaries and some way of showing we have two there or three there, ect.

Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two hidden as one” as well. (concept sharing) if the mathematical objects represented by the numbers had the same boundaries, like the shadows at the center.

Other than the further darkness of the overlapping shadows, we cannot see or imagine that there are two separate shadows there. Two or more numbers can be hidden as one since natural numbers represent exact positions.  Similarly with two numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Then to this end let us create another number dimension, a dimension of number of numbers. This must already exist since it should be possible to put two or more numbers together at a beginning. Let the usual case be that the number of numbers is only 1. But now let us expand into the next dimension and allow the number of numbers to be 2.

So for example with the number 1, let us take away the original number 1 (since we have another underlying dimension, we can do this) and replace it with two new numbers 1’(1) and1’(2). These are together like the two shadows but do not form one number.

The two numbers together can be notated ((1’(1)((1’(2)). 1’(1) is “peeking out” from behind 1’(2). Shown by the use of a half parentheses. Seen more clearly here: (a(a).

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects, also concept sharing, hidden as one.

The objects are concept sharing as well and are somehow different from each other. We give the two hidden objects two new numbers 1’(1) and 1’(2). 

In the case of mathematical objects there is no external way of telling how many objects there are, previously it was assumed it was only one. We can state how many we wish at the onset thus fixing a certain mathematical system. Then we need the concept sharing of a number to indicate how many objects we wish to be there.

A New Plane:

Points are also exact concepts. In the Euclidean plane they are places, with the notion of no extent, in the plane. We should be able to place two together using two new numbers 0’(1) and 0’(2) identifying that we have two points. (0 is indicating an origin)

An object of no extent created together with another object of no extent, would still have no extent- but there could be two objects here, under another mathematical system. 

The two points 0’(1) and 0’(2) can be different by first uncovering a new place dimension, a place of places. This must already exist because there must be some way to have two points exist together and still be two points.

In a similar way as we uncovered the new number dimension (the number of numbers) we can uncover the new place dimension. 

Take the original point out (we can do this since we have a new underlying dimension of place, a place of places) and replace it with the two new points. This can be done for the whole plane of points.

That is, there is nothing special about the origin, so each point of interest of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a subset of sharing points co-existing with a plane of places of places. 

One of the new points can be fixed, while the other one is capable of “shifting” away in this new dimension of place. In this way these two can be different. Then all of the sharings in the new plane can become new origins-one point being fixed while the other point is capable of shifting away.

Mathematical Concept removal and sharing

Introduction:

The notion of a point, that which has no parts or no extent, is basic in math. The ancient Greeks thought about points, but what if they were not entirely correct?

They asked, what if two points were placed next to each other? They thought that this would be one point and stopped there.-Aristotle : Physics- “neither can two points be contiguous with one another”

But what if this could be regarded as two points? Since something of no extent placed next to something of no extent would have no extent- but there could be two items of no extent there! The two points could be hidden as one!

Yet, as Aristotle thought the two indivisables of no parts would overlap for the two items can be considered together, side by side or to be on top of one another. This is possible with items of no extent as we can consider them either to be on top of each other or side by side. If two dots are placed side by side, they can be considered on top of one another if we look at them from the side. Also if two dots are placed on top of one another we can look at them as being side by side from the top.

Since we want to keep this concept of no extent, or no parts it must also be possible to have two together and still have two, as long as the two parts are different entities so that they do not combine into one. That is, there must be another possible entity with no extent!

One can regard the overlapping shadow diagram below:

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, we can regard these as e( e is another possible entity of no extent)  and p and not points, p and p. If I take away one light, one shadow still remains. 

This concept cannot be visualized as one would be trying to visualize two different items in the same place, but it is still logical, nonetheless. Even the concept of no extent cannot be fully visualized as well. Only usually we represent a point with a dot, but this is not really a visual of an object with no extent.

Also, consider that we have to take a point out in order to have this e*p present. That is e, the new entity sharing position with the point, p. That means we need another level to place, this must exist since we have to have e*p somehow.

Then this is how e can be made different from p, as in an extent of e and p which can come about, we can move e through this e extent, it then has a new location of original location, making it different from p. That is, e is the next level of location.

Yet we want this concept of point removal and sharing (the point is removed and the two points are sharing the notion or concept of an object of no extent) to join up with the knowledge that is already in existence.

In doing this we have to add to and change the mathematics which already exists. This turns out not to destroy but heal mathematics, as this can be seen as something which is basic, but is missing. The math from this leads to a way to solve many old problems which are easy to state but previously hard to solve, such as:

The knot problem:https://calctutor.ca/category/mathematics/knots-mathematics/the-knottedness-and-chirality-of-the-trefoil/

The twin prime conjecture:https://calctutor.ca/category/mathematics/the-twin-prime-conjecture/

The Goldbach conjecture:https://calctutor.ca/category/mathematics/goldbachs-conjecture/

Fermat’s last theorem;https://calctutor.ca/category/mathematics/a-clearer-and-simpler-demonstration-of-fermats-last-theorem-wiles-theorem/

The Poincare conjecture:https://calctutor.ca/category/mathematics/the-poincare-conjecture/

The Riemann Hypothesis:https://calctutor.ca/category/mathematics/the-riemann-hypothesis/

The Collatz conjecture:https://calctutor.ca/category/mathematics/the-collatz-conjecture/

In fact this idea can complete mathematics:

The completion of mathematics:https://calctutor.ca/category/mathematics/the-completion-of-mathematics/

Definition and description of e:

When two points come together or start together, it seems as if there could be two there as 0+0=0.(two items of no extent could still combine to no extent). Only that points have location only, and having them together means we have one location so we have to conclude that there is one point there.

What if an object of no extent could be expressed as two different items of no extent, starting together. Since an item of no extent together with another different item of no extent would still have no extent, yet there could still be two items here and not one. The other item of no extent would have to have something more than location (it would have to be different from a point).

Yet since it is logical that two items of no extent could be together and still be two items, this must be mathematically possible, somehow. It would work if the two items were not both points, in the sense that they were not the same type of entity. For points are defined to have location only and here we are at the same location.

Call this new indivisible item “e”. Then this together with a point can be called e*p. We can separate e and p (not divide as this is a separation not a division) if we move the original e along an extent made up of e’s. This extent of e’s can be together with an extent of p’s. The e extent is different from the p extent in that it is a series of positions of the original e position. That is it is the next level of position.

Then we have to have an object of no extent which is somehow different from the usual conception. I am thinking of putting two items of zero size together. One can think of putting a dot on top or beside a dot which is there already. One cannot visualize two items of no extent being together, but one can also not visualize an item of no extent as well.

Yet they are an e( new entity) and a point (p) at the same place, if we have a preexisting space made up of ordinary points (p’s). 

In terms of symbols, one may write pop=p as the idea that for an ordinary point two points starting together or being brought in coincidence form again an ordinary point. O is the idea of coincidence.

Then e would be something different. E then cannot be in coincidence with p, if we redefine coincidence as the starting together or coming together of two or more points. so we invent a way for the entities to be together and call it sharing, E*p is e sharing with p . Sharing is a way for two items to come together or be defined together and not combine. This should be possible as seen in the analogy of the overlapping shadows. This must exist somehow in mathematics, since we must have the notion of something with no parts or no extent and we can put two together and still have two. So we have p and e as different entities which are not identical. 

Additionally we must remove p for p*e( p and e together) to exist otherwise there is confusion. If we start with a space with P’s everywhere we have to remove a p to put in an e*p. So there must be another level to space. A container of points and e’s. This would then be a new place of original places, another level to place. Then we can see that e can be different from p, as in an extended space of e’s, I could move an e to another new place of the original place of e, along the e continuum. The e continuum can coexist with the p continuum.

So e is another type of point. It is able to be with another entity which has no extent and not be equal to that entity. P doesn’t share with p, as this is already seen as coincidence. Since points have location only this is the same location and so the same point.

Looking at e, e(1)oe(1)=e(1). But also possible is e(1)(e(2)), if we have an extent of e outwards from p(1). e(2) is some e on the e extent outwards from p(1). This is what makes e different from p, with p we only have pop=p. That is an e can act as a type of container for other e’s, as two different e’s do not combine similar to a p and an e. That means we can have a shifting movement of one e along another continuum of e’s. Then e’s can have a new location of the original location of an e which makes them different from p’s, which have location only.

e(1)*p(1) is e and p sharing a location. We can separate a copy of e(1) away, since 0-0=0. Think of a jigsaw puzzle of a landscape being taken apart. e(1)*p(1) =e(1)oe(1)*p(1). Let e(0) be a bridge into the other dimension. Then e(0)(e(1))oe(1)*p(1)=(e(0)*e(1))o*p(1) and p(1) falls away leaving e(0)(e(1)). Then we have e(2)(e(1)) with e(1) moving to an e(2). There might also be an extent of p, along with the e, In which case we can have e(1)*e(2)*p(2).

The e can be multiple, also e can move along another extent of e’s. This is how e is different from p.

To be complete…=p(1)op(1)op(1)op(1) =p(1)op(1)op(1)=p(1)op(1)=p(1), and also….=e(1)oe(1)oe(1)oe(1)=e(1)oe(1)oe(1)=e(1)oe(1)=e(1) and also.. e(1)*e(2)*e(3)*e(4) is not equal to e(1)*e(2)*e(3) is not equal to e(1)*e(2) is not equal to a single e .Also e(0)(e(1)*e(2)*…)). That is, e or multiple e’s can travel along an extent of e’s.

These would not be points, then, as points have location only. Two moving points could come together to form one point. In Physics or mathematics if we have a plane of points(fixed) there can be a point moving in this plane which takes up the locations of these planar points as it moves, and then we have pop=p as it moves.

But also e’s can move as well and we could have two e’s come together to form e*e when the initial location is moved to the extent of e’s.

A point is the notion of that which has no extent or location only, labelled, p.

But since the idea of no extent is to have no dimension.

Two ‘items’ of no extent could exist together and still have no extent. The sum of any number of these new items of no extent is also no extent. The two will be hidden as one. They are hidden as one, at the ordinary point in the pre-existing space. 

It would be two of these new locations at a regular location. By regular location, we mean an ordinary point of the pre-existing space. 

Since points have location only, and not this multiplicity, this could not be two points, but would have to be two of something else, call them ‘parts’.

Represent them by the letter e.

e has no extent but can also be multiple. E’s have the position to them given by the pre-existing points of the pre-existing space.

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. 

These two items could be multiple. This is how we can hide! In order to do this we would have to take out the usual concept of a point as being the only entity with no extent and replace it with these new conceptions as other entities which can have no extent. There are these little entities that don’t combine to form a point. This is because a point is already there as the only idea of no extent. We need some room!

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! The concept of no extent is an example of a concept.

To remove the concept of a point as the only item of no extent, we need to have a concept space, consisting of a concept of multiplicity. 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 as the only item of no extent to have it appear somewhere. Also we must take out any other possibility of any number of items, except 2.

Extension to other concepts:

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.

We can replace e and p, as two items sharing the concept of no extent with two items sharing the concept of a number as well. Or the concept of a set, or group, etc.

Math concepts are exact so that they don’t change from person to person or over time or space. A number, point, function or group, etc. is unchanging, eternal, unalterable.

So also they all can be multiple as the basic ideas of math are all based on geometry and numbers. Functions map inputs to outputs based on formulas which are algebraic expressions of variables(numbers) to points on a graph. Groups are collections of the rotations or flips of geometric objects. Elements of groups are exact in the end numerical or geometric. So numbers, sets, groups, functions, etc. can all have concept spaces!

The idea is to extend the concept using the idea of the concept and the extender “of”. So for example; location of location, number of numbers.

The idea is to go backwards into the idea.

Look at the example of location. For a new location of an initial location we need an extended location, that is there must be an extended location, somewhere to put one of the entities, the extended entity.

Then the extended entity has the location of an initial location, which makes it different from a point, having only location.

The concept sharing of a number:

We can start with a line of p’s with two lines of e’s sharing locations. All in a plane of e’s as shown below:

The numbers 1, 1(1) and 1(2) can be associated as shown, an e is removed and replaced with an e*e and we can associate the new numbers 1(1) and 1(2), concept sharing with the number 1. The number of numbers is 2 and not 1. We can then unfold the two lines of e’s to make an axis where we can have the numbers 1(1) and 1(2). The two numbers 1(1) and 1(2) are sharing the concept of 1, with 1 on the number line. We have the usual number line on the midline of the new plane.

The concept sharing of a number:

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. 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.

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 as the only entity of no extent.

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)

Then we can separate the two e’s, but the only way this can be different from the usual idea of separation in points,

T

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.

Wait, the little entities say, how can this help anything mathematical? We can just scatter anywhere! Yes but you could also stay together as a line!

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?

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.