Foundational Preface: Universal Concept Theory (UCT) framework:
Abstract:
A framework for the structural completion of mathematics
Objective: to propose a unified foundation framework-Universal Concept Theory-(UCT)-that resolves long-standing mathematical conjectures (e.g.., the Collatz Conjecture and Fermat’s Last Theorem) by redefining the nature of mathematical identity and coincidence.
Methodology: UCT departs from standard axiomatic set theory by introducing “Conceptual Engineering”. This process involves three primary stages.
Scaffolding: The construction of higher level “Places of places” and “Number of numbers” that exist as containers for lower level concepts.
Concept Removal: The systematic removal of the single occupant rule, allowing a single placement to support multiple entities.
Concept Sharing and Separation: The introduction of a variable “Coincidence Switch”. In the 1-sharing state, the distance between distinct concepts( such as the steps in the Collatz sequence) is reduced to zero, creating a unified identity. In the 0-sharing state, concepts are “separated” into the discrete non-overlapping values found in standard arithmetic.
Structural Capacity: UCT demonstrates that the transition from sharing to separation is governed by the “structured capacity” of the engineered space.
The Fermat Limit: The theory explains Fermat’s Last Theorem as a geometric mismatch: while 2D squares possess the directional capacity to support 1-sharing, higher dimensional cubes (n>2) do not, forcing the coincidence switch to 0 and precluding integer solutions.
Collatz Conjecture: By applying 1-sharing, the entire Collatz tree is revealed as a single, folded singularity where all integers are conceptually equal to 1.
Conclusion:
Universal Concept Theory provides the “missing layer” of mathematics, transitioning the field from a collection of isolated rules to a complete, structural hierarchy. By understanding the “backstage” of concept sharing, the paradoxes of standard math are revealed as simple logical certainties.
The Foundations of Universal Concept Theory: The Host and the Guest
In standard mathematics, a “point” or a “number” is an isolated entity. It is a lonely occupant of a single location, and standard rules dictate that no two distinct entities can occupy the same spot simultaneously. Universal Concept Theory (UCT) engineered a more sophisticated foundation by introducing the Host.
1. The Host (The Higher-Level Scaffolding)
Before we can understand how concepts interact, we must first build the environment. We define a Host (represented as; r in geometry or A’ in arithmetic).
The Host is not a “container” that is larger than its contents. Instead, the Host is the fundamental environment that shares the exact same space as the concepts themselves. It is the “scaffolding” that grants permission for multiple concepts to coexist. Without a Host, there is no room for sharing; with a Host, the capacity of a single location can expand.
2. The Guests (Fixed and Mobile Entities)
Once the Host environment is established, we perform Concept Removal—removing the old rule that a location must have only one occupant. This allows us to introduce our “Guests”:
The Fixed Guest ( p or A): This is the original concept. It remains anchored to its identity, providing the base reference for the location.
The Mobile Guest (e or B) This is the new entity (like the e iin our geometric work). Because the Host provides the room, the Mobile Guest can move or shift within the extended space while still “sharing” the same fundamental location as the Fixed Guest.
3. The 1-Sharing State (The Social Connection)
When the Host is active, we enter the 1-Sharing state. In this state, the distance between the Fixed Guest and the Mobile Guest is defined as zero. They are distinct characters, but they “coincide” perfectly.
This is the “Natural State” of mathematics. It explains why a Collatz sequence is actually a single, unified chain: every step is a different Guest sharing a seat at the same Host’s table. The sequence only looks like 111 steps long because we have “separated” the Guests.
4. The 0-Sharing State (The Standard Restriction)
What we call “Standard Math” is simply the state where the Host has restricted access. When we set the coincidence switch to 0, the Guests are no longer allowed to share the same seat. They are forced to separate into the discrete, isolated points and numbers we use for everyday arithmetic.
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 shadows combine to form a darker shadow.
Now think about points, mathematical objects which have no extent.
If we had two objects, one in front of the other illuminated by a single light, the object closer to the light would cast part of it’s shadow on the object further away, but it could extend past the closer object in both directions. The resulting shadow cast on a table below would have one “point” everywhere, it would be similar to the situation of coincidence in mathematics.
The two overlapping shadows show a new situation of two “points” being placed together as the shadows have no height so can be thought of as points which have no extent.
This can be thought of as ‘sharing’. The two “points” are sharing one location.
There can be hidden items of no extent due to the nature of the notion of no extent.
Something of no extent can be multiple, for example doubled, there could be two items of no extent there. They would just appear to be one item there as both items have no extent. This is certain. So it is possible there could be more structure.
So we need another item to no extent. But the only items we know about are points. We know that they have no extent and also that if I place two together the result is a single point.
So what if there is another entity of no extent but if we place two of these types of items together they do not merge into a single point but share position as in the overlapping shadows.
We can call points p’s and the other new entities e’s. If I place two points together in overlap they coincide and we say we have one point. Also if two points are placed next to each other they have to touch in their entirety so we also have one point.
But now we have e’s as well which are zero-dimensional but not points. So I can place a point together with an e as poe and this is a point and an e overlapping.
This has a form of zero distance like pop, but e and p do not merge, being different entities. Yet there is another form of zero distance as we have an e and a p and not two p’s. The only other way two entities of no extent can come together is contiguously, so we write e*p as e and p together with the new zero distance in mind. E is contiguous with p and also overlapping with p. This is how e is different from p.
We can think of the analogy given near the top of one object in front of and extending past another object, both objects being lit by a single light. Then there is that shadow cast on a table. But there is some of the shadow cast on the further object from the light from the nearer object to the light. We can think of the shadow being lifted off of the combined shadow onto the further object. This is an analogy of e being both overlapping and contiguous with p. As it shows e lifted off of p.
Two items of no extent placed together can be thought of as a single position as in points, p or two e’s next to each other or overlapping like in e’s. This is how p’s and e’s are different.
Two e’s cannot be in coincidence. If we imagine a space of only e’s, eoe does not equal e otherwise e=p. Also e*e does not equal e otherwise *=o and e=p. So e*e=e*e, that is to say e*e does not resolve to a single e.
The two e’s “touch in their entirety” (since they are both at p but they are together in a serial sense or an overlapping sense).
When we tried to put two points together there was no choice but to resolve it to a single p, since p’s were all we thought of that had no extent. Since we open up the door for another possibility, already having p’s, we can have e’s here now.
At p we can have two of the same e. Or just one e, to begin with. When one e combines with a p it does so either serially or as in coincidence (overlapping) as e and p are different entities.
Two items of no extent could be coincident, as in points, or also share as in e’s. The e’s are in the same position so that they are both there. Yet they share as in the overlapping shadows, since something of no size added serially or in overlapping to something else of no size still has no size. Yet there are two types of zero size so these can all fit together.
So the idea is that points, p and items e are sharing the concept to no extent but are different in another way. The other way is that e’s share while p’s coincide.
To sum up, two p’s can be in coincidence forming a single p. E can share with p either serially or in the sense of coincidence, this can work since e’s and p’s are different entities. E’s can exist sharing with each other in a serial way (contiguous) or overlapping. E’s can also share with other e’s, serially or overlapping.
At the beginning we can have a plane of these new entities “e”, coexisting with points. Identify which e’s are in the set as is done with points. Let us start with the whole plane then if I move an e I can put it in sharing with another e, leaving a hole.
But let’s examine e, further. Suppose I think of a single e. It can’t exist singularly like p exists singularly. We can think of p being in coincidence with any number of copies of itself, leading to just a singular p. But e doesn’t coincide with itself. So it can share with itself any number of times, sharing being associated with e’s. This is not the same as coincidence as I can separate e’s away from copies of themselves but I cannot do this with p’s.
Then instead of leaving a hole, let e share with itself once and then have a copy of e, leave to share with other e’s. We can have a subset of another coexisting plane of e’s. This way we can have a fixed plane of e’s and a moving set of e’s.
So how does e move away from p? First put in a plane of e’s then add to it a plane of p’s. Then we can have a subset of another sharing plane of e’s. Then e(1) at p(1) can move off into the fixed plane so the fixed plane becomes a place of places, a next dimension of place. So we have a dimension of place co-existing with a dimension of places of places. e(1) is sharing with itself so it both moves in the e plane and stays still in the p plane. Its place stays the same but the place’s place of places changes.
So this fixed plane may be regarded as a place of places and the set of moving e’s as moving places. The fixed plane is a new dimension or level of places, places of places.
We only need the space of a point to build this and we have to have a point to start with. Then also the still e and the moving e can share position. So if I have the concept of a point, I can further build this and I have my new item to no extent. I am able to remove the moving e from the combination.
Then e and e’ are different. E’ can be fixed e’s. (e’[(e)]]. E is contained in e’.
We can then form new structures with e, which have new properties. The space is re-engineered.
Foundational Preface: Universal Concept Theory (UCT) framework:
Abstract:
A framework for the structural completion of mathematics
Objective: to propose a unified foundation framework-Universal Concept Theory-(UCT)-that resolves long-standing mathematical conjectures (e.g.., the Collatz Conjecture and Fermat’s Last Theorem) by redefining the nature of mathematical identity and coincidence.
Methodology: UCT departs from standard axiomatic set theory by introducing “Conceptual Engineering”. This process involves three primary stages.
Scaffolding: The construction of higher level “Places of places” and “Number of numbers” that exist as containers for lower level concepts.
Concept Removal: The systematic removal of the single occupant rule, allowing a single placement to support multiple entities.
Concept Sharing and Separation: The introduction of a variable “Coincidence Switch”. In the 1-sharing state, the distance between distinct concepts( such as the steps in the Collatz sequence) is reduced to zero, creating a unified identity. In the 0-sharing state, concepts are “separated” into the discrete non-overlapping values found in standard arithmetic.
Structural Capacity: UCT demonstrates that the transition from sharing to separation is governed by the “structured capacity” of the engineered space.
The Fermat Limit: The theory explains Fermat’s Last Theorem as a geometric mismatch: while 2D squares possess the directional capacity to support 1-sharing, higher dimensional cubes (n>2) do not, forcing the coincidence switch to 0 and precluding integer solutions.
Collatz Conjecture: By applying 1-sharing, the entire Collatz tree is revealed as a single, folded singularity where all integers are conceptually equal to 1.
Conclusion:
Universal Concept Theory provides the “missing layer” of mathematics, transitioning the field from a collection of isolated rules to a complete, structural hierarchy. By understanding the “backstage” of concept sharing, the paradoxes of standard math are revealed as simple logical certainties.
The Foundations of Universal Concept Theory: The Host and the Guest
In standard mathematics, a “point” or a “number” is an isolated entity. It is a lonely occupant of a single location, and standard rules dictate that no two distinct entities can occupy the same spot simultaneously. Universal Concept Theory (UCT) engineered a more sophisticated foundation by introducing the Host.
1. The Host (The Higher-Level Scaffolding)
Before we can understand how concepts interact, we must first build the environment. We define a Host (represented as; r in geometry or A’ in arithmetic).
The Host is not a “container” that is larger than its contents. Instead, the Host is the fundamental environment that shares the exact same space as the concepts themselves. It is the “scaffolding” that grants permission for multiple concepts to coexist. Without a Host, there is no room for sharing; with a Host, the capacity of a single location can expand.
2. The Guests (Fixed and Mobile Entities)
Once the Host environment is established, we perform Concept Removal—removing the old rule that a location must have only one occupant. This allows us to introduce our “Guests”:
The Fixed Guest ( p or A): This is the original concept. It remains anchored to its identity, providing the base reference for the location.
The Mobile Guest (e or B) This is the new entity (like the e iin our geometric work). Because the Host provides the room, the Mobile Guest can move or shift within the extended space while still “sharing” the same fundamental location as the Fixed Guest.
3. The 1-Sharing State (The Social Connection)
When the Host is active, we enter the 1-Sharing state. In this state, the distance between the Fixed Guest and the Mobile Guest is defined as zero. They are distinct characters, but they “coincide” perfectly.
This is the “Natural State” of mathematics. It explains why a Collatz sequence is actually a single, unified chain: every step is a different Guest sharing a seat at the same Host’s table. The sequence only looks like 111 steps long because we have “separated” the Guests.
4. The 0-Sharing State (The Standard Restriction)
What we call “Standard Math” is simply the state where the Host has restricted access. When we set the coincidence switch to 0, the Guests are no longer allowed to share the same seat. They are forced to separate into the discrete, isolated points and numbers we use for everyday arithmetic.
Mathematical concept removal, subsequent sharing and separation
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 something like the contiguousness of two points could be possible. Since something of no extent combined with something of no extent would still have no extent- but there could be two items of no extent there! The two objects could be hidden as one! The two items of no extent would have to be different from points in some way so that they would not combine to be one singular item.
One can regard the overlapping shadow diagrams below:
Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:
The shadows can be regarded as two points overlapping which could be thought of as at a common point, but also as two items which are both not points being in a contiguous state. Now as seen in the overlap, two different shadows can take up the space of one location, we can regard these as e and p. e is another possible entity of no extent, p is a point. If I take away one light, one shadow still remains. The table can be like an underlying, hosting space.
If e and p are sharing a position, they do not combine because they are different. It must be that they are either next to each other, or that they overlap and do not merge because they are different entities, as they are both not p’s. Or overlap, creating one point if they were both points or creating one point if they were side by side as was Aristotle’s way of thinking of things. An item of no extent put with a different item of no extent would still have no extent but there would be two items there.
We can consider 2 e’s at a point. The two e’s would be in a sense the same, since they are at the same point, yet they are different from points, in that they can appear multiply in a contiguous state..
Think of a mathematical point not as a single solid object but as a Russian Nesting Doll. Even when the dolls are tucked inside one another and appear to occupy a single spot, their individual identities remain perfectly intact and shared. My new language allows us to unfold these nested layers, revealing the hidden structures and connections that standard math accidentally flattens out. This analogy was developed in collaboration with an AI assistant.
Start with a plane of places and a horizontal line of points labelled 1,2,3.. So that we have a unit length.
We wish to show that there is a next level to this plane and set of numbers on a line.
Let’s build the next level.
What is scaffolding?
It is the creation of a new concept based on a pre-existing concept. The new concept is said to host the pre-existing concept. It has the same basic notion but exists at the next level of conceptual space.
Since the concepts have the same basic notion there is an exact fit of the next level concept and the lower level concept sharing space together. So there is “room” for this. But then also since the two concepts share the same basic notion there is a ‘building” possible. One can build upwards.(Since the host is the next level of the original concept in the concept space as I can remove the guest. Also the host itself might also have a host). Since we only have e’s and p’s. The host level is made of e’s and can host p’s or other e’s.
Also,since there is more than one concept, there is more than one next level of concepts, so we can build outwards as well.
In points, call this new item “r” This is a set of e’s. Then this together with a point can be written r(p). We can separate r and p (not divide as this is a separation not a division) if we take p out, the lower concept can be removed. We can place a different kind of lower concept which can have freedom of motion in the higher concept, while the original lower concept remains fixed. Then the two lower concepts are also sharing space and don’t coincide.
One entity of no extent is able to host the guest entity of no extent. So that the two exist together without combining. The guest entity doesn’t host so it is different from the host entity.
We can remove the guest entity of no extent ‘p’ and replace with ‘e’ a new entity of no extent which is capable of moving through an extent of hosts. Also we could put p back in and e would exist together with p and r. All these would be different. P is fixed while e is mobile.
So the host is a next level place of e’s, hosting guest places p or e. Where hosting is defined as r and p or e sharing ( together but not combining) position, with p or e capable of being removed from r and e moving in r. R is the hosting space coexisting with p and e. Briefly (r[(p(e)].
I can place a point on another point, but let’s not have them coincide but let the new point be a point where points could be. So I am thinking of an e placed on a different concept of e. Rather than just a plain point (1st level). This would be a second level, possible since the new point and the original point carry the same basic notion, that of a place of no extent, so that the upper concept becomes a place of places of no extent-containing the lower places of no extent. It fits exactly with the lower concept. These concepts are co-existing. This is concept building.
Since e is not p, e can have parts. But since it is of zero-dimension it has to have zero-dimensional parts which add to itself. These can’t be p’s since p’s add to a single p.
But must be on the same level as e’s and p’s, so must be e’s. Therefore e can have parts of itself so that e(1)=(e(1)(e(1)).
So e is able to self-replicate and split up or recombine in a space of r.
In concept sharing we have the same basic notion but two different ideas. So two ideas are sharing the same space, as they are only concepts, this is possible. It must be that I can take one away from the other. So one exists at the next level from the other, as there is no other option at the beginning.
That there is a next level is granted because any concept can be continued. Hosting can be considered as exact containment. The concept can fit exactly into the next level of itself. Since this is an exact fit we can move into the next level.
In geometry the host becomes the place of a place at the position of the place. The shared concept is “no extent”
In numbers the host becomes the number of a number at the position of the number. The shared concept is a natural number.
Since points and numbers both have positions these ideas can be combined into a new plane.
We must have more than one place at this place of lower places so that one lower place may be fixed to keep contact with the first level and the other places would be mobile in the new dimension as I can have multiple places of lower places of no extent.
Then a new number dimension is co-created. Number the lower places (1(1) and (1(2) or (2(1)-for two lower places. These are two new higher numbers where the number of lower numbers is 2 and not 1.
We can think of a jigsaw puzzle of a landscape being taken apart.
This is how it would be for the concept of a point and a number. In general then we can regard any mathematical concept as capable of this sharing. Coincider a concept A and related concept B.(Shown as circles in the diagrams below).
We can have an axiom of concept sharing if I introduce the notation ((). This is showing that the combined concepts are not in coincidence. Like (<—() The left bracket is moved away showing the revealed concepts. The extra structure is shown as a revelation of one labeled concept away from the combination which does not coincide as the concepts are different in some way to be revealed, it is an expanded notation and makes a sentence. So for example (A(B) means A is not in coincidence with B, but sharing the same space.
So the axiom is expressed as (A’[(A(B)] or for a specific concept (A’(1)[(A(1)(B(1)}. A(1) is a specific, fixed instance of a concept. B(1) is another instance of the concept yet different in some way. In order for the two not to be in coincidence another level of the concept is opened up, overlying but co-existing, so that A(1) and B(1) can be in the same lower space, able to separate.[,] is showing containment. The spaces are co-existing so that they can be in the new space to begin with. This creates a concept space with the next level concept denoted A’(1). Otherwise there is no “room” and A(1) and B(1) just coincide.
To have (A(B) I have to take out A. A(1) can only be taken out if there is an overlying concept A’ which copies the notion of the original concept A but exists at a higher level such that A’ is the concept of the concept A taken to a new expanded level. Such as for points a “ new place of lower places B”. This creates a leveled concept space …(A’’[(A[’(A(B)]] with A’’ even higher than A’. All these A’, A’’, … coexist with A as the (() sentence shows.
The concept B can move in the expanded concept level as B can overlap with A’ since A’ is the expanded notion of A and B is shared with A. Making the logic of A’ and B consistent as B must be different from A, being able to move in A’ while A is fixed.
The next level concept carries the same basic notion of the original concept, so that it can overlap and connect with the original concept; this is how a concept can build upon itself. Yet it is overlying and co-exists with the original concept.
Then if I have another instance of this concept say A(2) I can have (A’(2)[(A(2)(B(2))]
Then (A’(1)[(A(1)(B(1))] and (A’(2)[(A(2)B(2))] allows (A’(1)[(A(1))] and (A’(2)[(A(2)B(1)(B(2))].
This is the movement of B(1) making B(1) different from A(1) as A(1) is fixed. Then the axiom is self-consistent as A(1) is different from B(1) as required.
I have to take out A to have (A(B) so this necessitates A’ Which then A and B are defined to exist in. A’ has to have the same notion as A and B, but extended.
B has to travel in A’ so that A’ has to be a new dimension of concept A, an extension of concept A. The only way it can be extended is to have another level of concept A a concept of concept. So for example in points a place of places or in numbers a number of numbers.
If we take out a place, we have a place of lower places, where more than one place can be. This has to be built first. If we take out a number we have a number of lower numbers, a number which indicates how many numbers there are present.
These can number the places in the place of lower places.
We can have more than one number or place because we are opening up another dimension of the concept. The concept can now move away from its singularity.
There has to be more than one in order to move the other type of concept along the new dimension. Then the new concept dimension has to copy the notion of the original concept but be at its next level higher, so that the original concept doesn’t change when moved along the new concept dimension.
So A’ is an overlying , coexisting space. A new level of concept A.
In standard math, we are taught that a “point” is the only thing that can exist at a specific co-ordinate. But think about the pixels on your screen. A single pixel isn’t just one “thing”-it is actually a shared space where Red, Green and Blue exist together.
When these colors share the pixel, they create a new result (like white light), but they don’t lose their individual identities. They are nested within that single coordinate. My notation (A’[(A(B)] works the same way:
The pixel is the higher concept level (A’). The Red is the fixed anchor(A) and the Blue is the second entity(B) that shares that space
The magic hapens in the separation. In a standard co-ordinate system, if you move the “point” the whole pixel moves. But in this new language we can “unfold” the pixel We can keep the Red fixed while moving the Blue to a new placement. This allows us to see how complex structures-like prime numbers or intricate knots-are actually built from multiple “colors” sharing a single origin.-This analogy was created in collaboration with an AI assistant.
In the case of two ordinary points we would have two of the same item, leading to the same item.
In order to have the two different items together we have to be able to remove the concept of a point as the only item of no extent and the removal of the point itself. It requires the construction of an overlying space of higher places of lower places where points can be, coexisting with the space of points.
I’m deleting the concept of a point as the only item of no extent in geometry. As well, I can delete it physically as I now have an overlying level.
This then allows the creation of other new entities of no extent.
In particular, we can think of removing a point from space, necessitating an overlying, co-existing plane of new places of original places, by extending the concept of a place or a point, into a plane. This also gives room for a new entity ‘e’.
We can create another new entity called e, which can move in this new dimension making it different from p which can still be fixed in this new overlying, co-existing dimension. This keeps the connection with the geometry which already exists.
This creates a mixed space of e and p in an overlying co-existing space of r.
We can think of e’s as having the ability to move, unchanging. An analogy is that of a jigsaw puzzle of a landscape being taken apart. The e’s can be thought of as pieces of the puzzle, moving but unchanging.
Here is then another dimension where e can move off into making it different from p as required. See the diagram below:
Since in mathematics we wish to have this concept of no parts or no extent, so too can we have this further notion. The further notion is that two items could be together as one, if they both had no extent. They would have to be two different items to no extent. But how do we number these? We can think of removing the number one and replacing it with 1(1) and 1(2). Similar to removing a point to make room for e.
Then the two items touch in their entirety both being of no extent. But they do not combine being different entities.
They could not both be points as we know them now, as they would just become one point as Aristotle thought. Aristotle thought that two points could not be contiguous since there would have to be no line in between them and we would have no extent.-Aristotle : Physics- “as there are an infinite number of points between two selected points”. The idea is that the two points would combine into one. They would touch in their entirety. But he only had the idea that there could only be one item of no extent.
The new entity e(1) has a location but e(1)1 and e(1)2 if we start off with two e’s together also sharing with p) can also have new locations of their initial locations, if they move in the new dimension of new locations of original locations opened up by the creation of the new level.
E(1) can be fixed for now, but could move, so its new location of its original location is the same as its initial location. This is the difference between e(1)1 and e(1)2. This also makes it different from a point which we can say still has its location fixed in r.
Furthermore, all math concepts are point-like in that they are exact, universal 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 universal 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 forwards into the idea. To create higher spaces.
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 3 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 three numbers 1, 1(1) and 1(2) are sharing the concept of 1. 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.
If we look at a set of numbers taken from two neighbors such as { (5(16), (16(8) } we can extend this set to a set of four numbers like so { (5(16), (16(8) (8(16) (16(5) } if we use the property of partially shared numbers for example that (16(8)=(8(16) and (5(16)=(16(5) I can fold over the last two numbers into the first two numbers. Then we can also fold over 5(16) into (16(8) as well as we have the concept of sharing numbers.
But then we can see (5(16)=(16(8), 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.
In the framework, the reason every number eventually leads back to the main trunk is a combination of concept sharing and the structural nature of the equations engineered.
Here’s how these work together to ensure no number is left floating alone.
The Collatz tree is defined by two primary equations that represent the “mapping” and “growth” of every number in existence.
My framework defines the entire set of positive integers through the union of these two functions:
Mapping equation: C(x,0) =3x+1
The equation describes how a new child branch attaches to an existing parent branch. It maps odd numbers(x) into their position on the tree.
Branch growth equation: C(x,n)=x*2^n
This describes the infinite extent of a branch. Every branch starts with an odd number(x) and doubles indefinitely as n increases from o to infinity.
The sharing process acts as a logical fold that forces connection. Because I define the distance between shared pairs like (x(3x+1) or (x(x/2) as zero I am saying that every number is “welded” to it’s successor in the higher dimension place of places.
If every step has zero distance then the entire chain-no matter how long-is effectively a single point. Since the 1-2-4 loop is the only stable “place” where these shared concepts can rest, all other shared chains are pulled into it like a magnet.
While sharing provides the connection, the equations provide the direction.
C(x,n)=x*2^n (the branch) this ensures every number belongs to a specific “growth line”.
C(x,0)=3x+1 (the bridge). This is the mechanism that moves a number from it’s current branch to a different branch.
Standard math worries about a sequence that might go to infinity or get stuck in a separate loop. The framework addresses this through concept removal.
By removing the concept of a point as a fixed, isolated island, I make it impossible for a number to exist outside the engineered scaffolding.
Since the place of places is built first as a unified container, there is no “room” for a second , disconnected tree to exist, it must be a part of the single shared structure I’ve constructed.
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.
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 shadows combine to form a darker shadow.
Now think about points, mathematical objects which have no extent.
If we had two objects, one in front of the other illuminated by a single light, the object closer to the light would cast part of it’s shadow on the object further away, but it could extend past the closer object in both directions. The resulting shadow cast on a table below would have one “point” everywhere, it would be similar to the situation of coincidence in mathematics.
The two overlapping shadows show a new situation of two “points” being placed together as the shadows have no height so can be thought of as points which have no extent.
This can be thought of as ‘sharing’. The two “points” are sharing one location.
There can be hidden items of no extent due to the nature of the notion of no extent.
Something of no extent can be multiple, for example doubled, there could be two items of no extent there. They would just appear to be one item there as both items have no extent. This is certain. So it is possible there could be more structure.
So we need another item to no extent. But the only items we know about are points. We know that they have no extent and also that if I place two together the result is a single point.
So what if there is another entity of no extent but if we place two of these types of items together they do not merge into a single point but share position as in the overlapping shadows.
We can call points p’s and the other new entities e’s. If I place two points together in overlap they coincide and we say we have one point. Also if two points are placed next to each other they have to touch in their entirety so we also have one point.
But now we have e’s as well which are zero-dimensional but not points. So I can place a point together with an e as poe and this is a point and an e overlapping.
This has a form of zero distance like pop, but e and p do not merge, being different entities. Yet there is another form of zero distance as we have an e and a p and not two p’s. The only other way two entities of no extent can come together is contiguously, so we write e*p as e and p together with the new zero distance in mind. E is contiguous with p and also overlapping with p. This is how e is different from p.
We can think of the analogy given near the top of one object in front of and extending past another object, both objects being lit by a single light. Then there is that shadow cast on a table. But there is some of the shadow cast on the further object from the light from the nearer object to the light. We can think of the shadow being lifted off of the combined shadow onto the further object. This is an analogy of e being both overlapping and contiguous with p. As it shows e lifted off of p.
Two items of no extent placed together can be thought of as a single position as in points, p or two e’s next to each other or overlapping like in e’s. This is how p’s and e’s are different.
Two e’s cannot be in coincidence. If we imagine a space of only e’s, eoe does not equal e otherwise e=p. Also e*e does not equal e otherwise *=o and e=p. So e*e=e*e, that is to say e*e does not resolve to a single e.
The two e’s “touch in their entirety” (since they are both at p but they are together in a serial sense or an overlapping sense).
When we tried to put two points together there was no choice but to resolve it to a single p, since p’s were all we thought of that had no extent. Since we open up the door for another possibility, already having p’s, we can have e’s here now.
At p we can have two of the same e. Or just one e, to begin with. When one e combines with a p it does so either serially or as in coincidence (overlapping) as e and p are different entities.
Two items of no extent could be coincident, as in points, or also share as in e’s. The e’s are in the same position so that they are both there. Yet they share as in the overlapping shadows, since something of no size added serially or in overlapping to something else of no size still has no size. Yet there are two types of zero size so these can all fit together.
So the idea is that points, p and items e are sharing the concept to no extent but are different in another way. The other way is that e’s share while p’s coincide.
To sum up, two p’s can be in coincidence forming a single p. E can share with p either serially or in the sense of coincidence, this can work since e’s and p’s are different entities. E’s can exist sharing with each other in a serial way (contiguous) or overlapping. E’s can also share with other e’s, serially or overlapping.
At the beginning we can have a plane of these new entities “e”, coexisting with points. Identify which e’s are in the set as is done with points. Let us start with the whole plane then if I move an e I can put it in sharing with another e, leaving a hole.
But let’s examine e, further. Suppose I think of a single e. It can’t exist singularly like p exists singularly. We can think of p being in coincidence with any number of copies of itself, leading to just a singular p. But e doesn’t coincide with itself. So it can share with itself any number of times, sharing being associated with e’s. This is not the same as coincidence as I can separate e’s away from copies of themselves but I cannot do this with p’s.
Then instead of leaving a hole, let e share with itself once and then have a copy of e, leave to share with other e’s. We can have a subset of another coexisting plane of e’s. This way we can have a fixed plane of e’s and a moving set of e’s.
So how does e move away from p? First put in a plane of e’s then add to it a plane of p’s. Then we can have a subset of another sharing plane of e’s. Then e(1) at p(1) can move off into the fixed plane so the fixed plane becomes a place of places, a next dimension of place. So we have a dimension of place co-existing with a dimension of places of places. e(1) is sharing with itself so it both moves in the e plane and stays still in the p plane. Its place stays the same but the place’s place of places changes.
So this fixed plane may be regarded as a place of places and the set of moving e’s as moving places. The fixed plane is a new dimension or level of places, places of places.
We only need the space of a point to build this and we have to have a point to start with. Then also the still e and the moving e can share position. So if I have the concept of a point, I can further build this and I have my new item to no extent. I am able to remove the moving e from the combination.
Then e and e’ are different. E’ can be fixed e’s. (e’[(e)]]. E is contained in e’.
We can then form new structures with e, which have new properties. The space is re-engineered.
Foundational Preface: Universal Concept Theory (UCT) framework:
Abstract:
A framework for the structural completion of mathematics
Objective: to propose a unified foundation framework-Universal Concept Theory-(UCT)-that resolves long-standing mathematical conjectures (e.g.., the Collatz Conjecture and Fermat’s Last Theorem) by redefining the nature of mathematical identity and coincidence.
Methodology: UCT departs from standard axiomatic set theory by introducing “Conceptual Engineering”. This process involves three primary stages.
Scaffolding: The construction of higher level “Places of places” and “Number of numbers” that exist as containers for lower level concepts.
Concept Removal: The systematic removal of the single occupant rule, allowing a single placement to support multiple entities.
Concept Sharing and Separation: The introduction of a variable “Coincidence Switch”. In the 1-sharing state, the distance between distinct concepts( such as the steps in the Collatz sequence) is reduced to zero, creating a unified identity. In the 0-sharing state, concepts are “separated” into the discrete non-overlapping values found in standard arithmetic.
Structural Capacity: UCT demonstrates that the transition from sharing to separation is governed by the “structured capacity” of the engineered space.
The Fermat Limit: The theory explains Fermat’s Last Theorem as a geometric mismatch: while 2D squares possess the directional capacity to support 1-sharing, higher dimensional cubes (n>2) do not, forcing the coincidence switch to 0 and precluding integer solutions.
Collatz Conjecture: By applying 1-sharing, the entire Collatz tree is revealed as a single, folded singularity where all integers are conceptually equal to 1.
Conclusion:
Universal Concept Theory provides the “missing layer” of mathematics, transitioning the field from a collection of isolated rules to a complete, structural hierarchy. By understanding the “backstage” of concept sharing, the paradoxes of standard math are revealed as simple logical certainties.
The Foundations of Universal Concept Theory: The Host and the Guest
In standard mathematics, a “point” or a “number” is an isolated entity. It is a lonely occupant of a single location, and standard rules dictate that no two distinct entities can occupy the same spot simultaneously. Universal Concept Theory (UCT) engineered a more sophisticated foundation by introducing the Host.
1. The Host (The Higher-Level Scaffolding)
Before we can understand how concepts interact, we must first build the environment. We define a Host (represented as; r in geometry or A’ in arithmetic).
The Host is not a “container” that is larger than its contents. Instead, the Host is the fundamental environment that shares the exact same space as the concepts themselves. It is the “scaffolding” that grants permission for multiple concepts to coexist. Without a Host, there is no room for sharing; with a Host, the capacity of a single location can expand.
2. The Guests (Fixed and Mobile Entities)
Once the Host environment is established, we perform Concept Removal—removing the old rule that a location must have only one occupant. This allows us to introduce our “Guests”:
The Fixed Guest ( p or A): This is the original concept. It remains anchored to its identity, providing the base reference for the location.
The Mobile Guest (e or B) This is the new entity (like the e iin our geometric work). Because the Host provides the room, the Mobile Guest can move or shift within the extended space while still “sharing” the same fundamental location as the Fixed Guest.
3. The 1-Sharing State (The Social Connection)
When the Host is active, we enter the 1-Sharing state. In this state, the distance between the Fixed Guest and the Mobile Guest is defined as zero. They are distinct characters, but they “coincide” perfectly.
This is the “Natural State” of mathematics. It explains why a Collatz sequence is actually a single, unified chain: every step is a different Guest sharing a seat at the same Host’s table. The sequence only looks like 111 steps long because we have “separated” the Guests.
4. The 0-Sharing State (The Standard Restriction)
What we call “Standard Math” is simply the state where the Host has restricted access. When we set the coincidence switch to 0, the Guests are no longer allowed to share the same seat. They are forced to separate into the discrete, isolated points and numbers we use for everyday arithmetic.
Mathematical concept removal, subsequent sharing and separation
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 something like the contiguousness of two points could be possible. Since something of no extent combined with something of no extent would still have no extent- but there could be two items of no extent there! The two objects could be hidden as one! The two items of no extent would have to be different from points in some way so that they would not combine to be one singular item.
One can regard the overlapping shadow diagrams below:
Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:
The shadows can be regarded as two points overlapping which could be thought of as at a common point, but also as two items which are both not points being in a contiguous state. Now as seen in the overlap, two different shadows can take up the space of one location, we can regard these as e and p. e is another possible entity of no extent, p is a point. If I take away one light, one shadow still remains. The table can be like an underlying, hosting space.
If e and p are sharing a position, they do not combine because they are different. It must be that they are either next to each other, or that they overlap and do not merge because they are different entities, as they are both not p’s. Or overlap, creating one point if they were both points or creating one point if they were side by side as was Aristotle’s way of thinking of things. An item of no extent put with a different item of no extent would still have no extent but there would be two items there.
We can consider 2 e’s at a point. The two e’s would be in a sense the same, since they are at the same point, yet they are different from points, in that they can appear multiply in a contiguous state..
Think of a mathematical point not as a single solid object but as a Russian Nesting Doll. Even when the dolls are tucked inside one another and appear to occupy a single spot, their individual identities remain perfectly intact and shared. My new language allows us to unfold these nested layers, revealing the hidden structures and connections that standard math accidentally flattens out. This analogy was developed in collaboration with an AI assistant.
Start with a plane of places and a horizontal line of points labelled 1,2,3.. So that we have a unit length.
We wish to show that there is a next level to this plane and set of numbers on a line.
Let’s build the next level.
What is scaffolding?
It is the creation of a new concept based on a pre-existing concept. The new concept is said to host the pre-existing concept. It has the same basic notion but exists at the next level of conceptual space.
Since the concepts have the same basic notion there is an exact fit of the next level concept and the lower level concept sharing space together. So there is “room” for this. But then also since the two concepts share the same basic notion there is a ‘building” possible. One can build upwards.(Since the host is the next level of the original concept in the concept space as I can remove the guest. Also the host itself might also have a host). Since we only have e’s and p’s. The host level is made of e’s and can host p’s or other e’s.
Also,since there is more than one concept, there is more than one next level of concepts, so we can build outwards as well.
In points, call this new item “r” This is a set of e’s. Then this together with a point can be written r(p). We can separate r and p (not divide as this is a separation not a division) if we take p out, the lower concept can be removed. We can place a different kind of lower concept which can have freedom of motion in the higher concept, while the original lower concept remains fixed. Then the two lower concepts are also sharing space and don’t coincide.
One entity of no extent is able to host the guest entity of no extent. So that the two exist together without combining. The guest entity doesn’t host so it is different from the host entity.
We can remove the guest entity of no extent ‘p’ and replace with ‘e’ a new entity of no extent which is capable of moving through an extent of hosts. Also we could put p back in and e would exist together with p and r. All these would be different. P is fixed while e is mobile.
So the host is a next level place of e’s, hosting guest places p or e. Where hosting is defined as r and p or e sharing ( together but not combining) position, with p or e capable of being removed from r and e moving in r. R is the hosting space coexisting with p and e. Briefly (r[(p(e)].
I can place a point on another point, but let’s not have them coincide but let the new point be a point where points could be. So I am thinking of an e placed on a different concept of e. Rather than just a plain point (1st level). This would be a second level, possible since the new point and the original point carry the same basic notion, that of a place of no extent, so that the upper concept becomes a place of places of no extent-containing the lower places of no extent. It fits exactly with the lower concept. These concepts are co-existing. This is concept building.
Since e is not p, e can have parts. But since it is of zero-dimension it has to have zero-dimensional parts which add to itself. These can’t be p’s since p’s add to a single p.
But must be on the same level as e’s and p’s, so must be e’s. Therefore e can have parts of itself so that e(1)=(e(1)(e(1)).
So e is able to self-replicate and split up or recombine in a space of r.
In concept sharing we have the same basic notion but two different ideas. So two ideas are sharing the same space, as they are only concepts, this is possible. It must be that I can take one away from the other. So one exists at the next level from the other, as there is no other option at the beginning.
That there is a next level is granted because any concept can be continued. Hosting can be considered as exact containment. The concept can fit exactly into the next level of itself. Since this is an exact fit we can move into the next level.
In geometry the host becomes the place of a place at the position of the place. The shared concept is “no extent”
In numbers the host becomes the number of a number at the position of the number. The shared concept is a natural number.
Since points and numbers both have positions these ideas can be combined into a new plane.
We must have more than one place at this place of lower places so that one lower place may be fixed to keep contact with the first level and the other places would be mobile in the new dimension as I can have multiple places of lower places of no extent.
Then a new number dimension is co-created. Number the lower places (1(1) and (1(2) or (2(1)-for two lower places. These are two new higher numbers where the number of lower numbers is 2 and not 1.
We can think of a jigsaw puzzle of a landscape being taken apart.
This is how it would be for the concept of a point and a number. In general then we can regard any mathematical concept as capable of this sharing. Coincider a concept A and related concept B.(Shown as circles in the diagrams below).
We can have an axiom of concept sharing if I introduce the notation ((). This is showing that the combined concepts are not in coincidence. Like (<—() The left bracket is moved away showing the revealed concepts. The extra structure is shown as a revelation of one labeled concept away from the combination which does not coincide as the concepts are different in some way to be revealed, it is an expanded notation and makes a sentence. So for example (A(B) means A is not in coincidence with B, but sharing the same space.
So the axiom is expressed as (A’[(A(B)] or for a specific concept (A’(1)[(A(1)(B(1)}. A(1) is a specific, fixed instance of a concept. B(1) is another instance of the concept yet different in some way. In order for the two not to be in coincidence another level of the concept is opened up, overlying but co-existing, so that A(1) and B(1) can be in the same lower space, able to separate.[,] is showing containment. The spaces are co-existing so that they can be in the new space to begin with. This creates a concept space with the next level concept denoted A’(1). Otherwise there is no “room” and A(1) and B(1) just coincide.
To have (A(B) I have to take out A. A(1) can only be taken out if there is an overlying concept A’ which copies the notion of the original concept A but exists at a higher level such that A’ is the concept of the concept A taken to a new expanded level. Such as for points a “ new place of lower places B”. This creates a leveled concept space …(A’’[(A[’(A(B)]] with A’’ even higher than A’. All these A’, A’’, … coexist with A as the (() sentence shows.
The concept B can move in the expanded concept level as B can overlap with A’ since A’ is the expanded notion of A and B is shared with A. Making the logic of A’ and B consistent as B must be different from A, being able to move in A’ while A is fixed.
The next level concept carries the same basic notion of the original concept, so that it can overlap and connect with the original concept; this is how a concept can build upon itself. Yet it is overlying and co-exists with the original concept.
Then if I have another instance of this concept say A(2) I can have (A’(2)[(A(2)(B(2))]
Then (A’(1)[(A(1)(B(1))] and (A’(2)[(A(2)B(2))] allows (A’(1)[(A(1))] and (A’(2)[(A(2)B(1)(B(2))].
This is the movement of B(1) making B(1) different from A(1) as A(1) is fixed. Then the axiom is self-consistent as A(1) is different from B(1) as required.
I have to take out A to have (A(B) so this necessitates A’ Which then A and B are defined to exist in. A’ has to have the same notion as A and B, but extended.
B has to travel in A’ so that A’ has to be a new dimension of concept A, an extension of concept A. The only way it can be extended is to have another level of concept A a concept of concept. So for example in points a place of places or in numbers a number of numbers.
If we take out a place, we have a place of lower places, where more than one place can be. This has to be built first. If we take out a number we have a number of lower numbers, a number which indicates how many numbers there are present.
These can number the places in the place of lower places.
We can have more than one number or place because we are opening up another dimension of the concept. The concept can now move away from its singularity.
There has to be more than one in order to move the other type of concept along the new dimension. Then the new concept dimension has to copy the notion of the original concept but be at its next level higher, so that the original concept doesn’t change when moved along the new concept dimension.
So A’ is an overlying , coexisting space. A new level of concept A.
In standard math, we are taught that a “point” is the only thing that can exist at a specific co-ordinate. But think about the pixels on your screen. A single pixel isn’t just one “thing”-it is actually a shared space where Red, Green and Blue exist together.
When these colors share the pixel, they create a new result (like white light), but they don’t lose their individual identities. They are nested within that single coordinate. My notation (A’[(A(B)] works the same way:
The pixel is the higher concept level (A’). The Red is the fixed anchor(A) and the Blue is the second entity(B) that shares that space
The magic hapens in the separation. In a standard co-ordinate system, if you move the “point” the whole pixel moves. But in this new language we can “unfold” the pixel We can keep the Red fixed while moving the Blue to a new placement. This allows us to see how complex structures-like prime numbers or intricate knots-are actually built from multiple “colors” sharing a single origin.-This analogy was created in collaboration with an AI assistant.
In the case of two ordinary points we would have two of the same item, leading to the same item.
In order to have the two different items together we have to be able to remove the concept of a point as the only item of no extent and the removal of the point itself. It requires the construction of an overlying space of higher places of lower places where points can be, coexisting with the space of points.
I’m deleting the concept of a point as the only item of no extent in geometry. As well, I can delete it physically as I now have an overlying level.
This then allows the creation of other new entities of no extent.
In particular, we can think of removing a point from space, necessitating an overlying, co-existing plane of new places of original places, by extending the concept of a place or a point, into a plane. This also gives room for a new entity ‘e’.
We can create another new entity called e, which can move in this new dimension making it different from p which can still be fixed in this new overlying, co-existing dimension. This keeps the connection with the geometry which already exists.
This creates a mixed space of e and p in an overlying co-existing space of r.
We can think of e’s as having the ability to move, unchanging. An analogy is that of a jigsaw puzzle of a landscape being taken apart. The e’s can be thought of as pieces of the puzzle, moving but unchanging.
Here is then another dimension where e can move off into making it different from p as required. See the diagram below:
Since in mathematics we wish to have this concept of no parts or no extent, so too can we have this further notion. The further notion is that two items could be together as one, if they both had no extent. They would have to be two different items to no extent. But how do we number these? We can think of removing the number one and replacing it with 1(1) and 1(2). Similar to removing a point to make room for e.
Then the two items touch in their entirety both being of no extent. But they do not combine being different entities.
They could not both be points as we know them now, as they would just become one point as Aristotle thought. Aristotle thought that two points could not be contiguous since there would have to be no line in between them and we would have no extent.-Aristotle : Physics- “as there are an infinite number of points between two selected points”. The idea is that the two points would combine into one. They would touch in their entirety. But he only had the idea that there could only be one item of no extent.
The new entity e(1) has a location but e(1)1 and e(1)2 if we start off with two e’s together also sharing with p) can also have new locations of their initial locations, if they move in the new dimension of new locations of original locations opened up by the creation of the new level.
E(1) can be fixed for now, but could move, so its new location of its original location is the same as its initial location. This is the difference between e(1)1 and e(1)2. This also makes it different from a point which we can say still has its location fixed in r.
Furthermore, all math concepts are point-like in that they are exact, universal 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 universal 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 forwards into the idea. To create higher spaces.
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 3 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 three numbers 1, 1(1) and 1(2) are sharing the concept of 1. We have the usual number line on the midline of the new plane.
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.
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 shadows combine to form a darker shadow.
Now think about points, mathematical objects which have no extent.
If we had two objects, one in front of the other illuminated by a single light, the object closer to the light would cast part of it’s shadow on the object further away, but it could extend past the closer object in both directions. The resulting shadow cast on a table below would have one “point” everywhere, it would be similar to the situation of coincidence in mathematics.
The two overlapping shadows show a new situation of two “points” being placed together as the shadows have no height so can be thought of as points which have no extent.
This can be thought of as ‘sharing’. The two “points” are sharing one location.
There can be hidden items of no extent due to the nature of the notion of no extent.
Something of no extent can be multiple, for example doubled, there could be two items of no extent there. They would just appear to be one item there as both items have no extent. This is certain. So it is possible there could be more structure.
So we need another item to no extent. But the only items we know about are points. We know that they have no extent and also that if I place two together the result is a single point.
So what if there is another entity of no extent but if we place two of these types of items together they do not merge into a single point but share position as in the overlapping shadows.
We can call points p’s and the other new entities e’s. If I place two points together in overlap they coincide and we say we have one point. Also if two points are placed next to each other they have to touch in their entirety so we also have one point.
But now we have e’s as well which are zero-dimensional but not points. So I can place a point together with an e as poe and this is a point and an e overlapping.
This has a form of zero distance like pop, but e and p do not merge, being different entities. Yet there is another form of zero distance as we have an e and a p and not two p’s. The only other way two entities of no extent can come together is contiguously, so we write e*p as e and p together with the new zero distance in mind. E is contiguous with p and also overlapping with p. This is how e is different from p.
We can think of the analogy given near the top of one object in front of and extending past another object, both objects being lit by a single light. Then there is that shadow cast on a table. But there is some of the shadow cast on the further object from the light from the nearer object to the light. We can think of the shadow being lifted off of the combined shadow onto the further object. This is an analogy of e being both overlapping and contiguous with p. As it shows e lifted off of p.
Two items of no extent placed together can be thought of as a single position as in points, p or two e’s next to each other or overlapping like in e’s. This is how p’s and e’s are different.
Two e’s cannot be in coincidence. If we imagine a space of only e’s, eoe does not equal e otherwise e=p. Also e*e does not equal e otherwise *=o and e=p. So e*e=e*e, that is to say e*e does not resolve to a single e.
The two e’s “touch in their entirety” (since they are both at p but they are together in a serial sense or an overlapping sense).
When we tried to put two points together there was no choice but to resolve it to a single p, since p’s were all we thought of that had no extent. Since we open up the door for another possibility, already having p’s, we can have e’s here now.
At p we can have two of the same e. Or just one e, to begin with. When one e combines with a p it does so either serially or as in coincidence (overlapping) as e and p are different entities.
Two items of no extent could be coincident, as in points, or also share as in e’s. The e’s are in the same position so that they are both there. Yet they share as in the overlapping shadows, since something of no size added serially or in overlapping to something else of no size still has no size. Yet there are two types of zero size so these can all fit together.
So the idea is that points, p and items e are sharing the concept to no extent but are different in another way. The other way is that e’s share while p’s coincide.
To sum up, two p’s can be in coincidence forming a single p. E can share with p either serially or in the sense of coincidence, this can work since e’s and p’s are different entities. E’s can exist sharing with each other in a serial way (contiguous) or overlapping. E’s can also share with other e’s, serially or overlapping.
At the beginning we can have a plane of these new entities “e”, coexisting with points. Identify which e’s are in the set as is done with points. Let us start with the whole plane then if I move an e I can put it in sharing with another e, leaving a hole.
But let’s examine e, further. Suppose I think of a single e. It can’t exist singularly like p exists singularly. We can think of p being in coincidence with any number of copies of itself, leading to just a singular p. But e doesn’t coincide with itself. So it can share with itself any number of times, sharing being associated with e’s. This is not the same as coincidence as I can separate e’s away from copies of themselves but I cannot do this with p’s.
Then instead of leaving a hole, let e share with itself once and then have a copy of e, leave to share with other e’s. We can have a subset of another coexisting plane of e’s. This way we can have a fixed plane of e’s and a moving set of e’s.
So how does e move away from p? First put in a plane of e’s then add to it a plane of p’s. Then we can have a subset of another sharing plane of e’s. Then e(1) at p(1) can move off into the fixed plane so the fixed plane becomes a place of places, a next dimension of place. So we have a dimension of place co-existing with a dimension of places of places. e(1) is sharing with itself so it both moves in the e plane and stays still in the p plane. Its place stays the same but the place’s place of places changes.
So this fixed plane may be regarded as a place of places and the set of moving e’s as moving places. The fixed plane is a new dimension or level of places, places of places.
We only need the space of a point to build this and we have to have a point to start with. Then also the still e and the moving e can share position. So if I have the concept of a point, I can further build this and I have my new item to no extent. I am able to remove the moving e from the combination.
Then e and e’ are different. E’ can be fixed e’s. (e’[(e)]]. E is contained in e’.
We can then form new structures with e, which have new properties. The space is re-engineered.
Abstract:
A framework for the structural completion of mathematics
Objective: to propose a unified foundation framework-Universal Concept Theory-(UCT)-that resolves long-standing mathematical conjectures (e.g.., the Collatz Conjecture and Fermat’s Last Theorem) by redefining the nature of mathematical identity and coincidence.
Methodology: UCT departs from standard axiomatic set theory by introducing “Conceptual Engineering”. This process involves three primary stages.
Scaffolding: The construction of higher level “Places of places” and “Number of numbers” that exist as containers for lower level concepts.
Concept Removal: The systematic removal of the single occupant rule, allowing a single placement to support multiple entities.
Concept Sharing and Separation: The introduction of a variable “Coincidence Switch”. In the 1-sharing state, the distance between distinct concepts( such as the steps in the Collatz sequence) is reduced to zero, creating a unified identity. In the 0-sharing state, concepts are “separated” into the discrete non-overlapping values found in standard arithmetic.
Structural Capacity: UCT demonstrates that the transition from sharing to separation is governed by the “structured capacity” of the engineered space.
The Fermat Limit: The theory explains Fermat’s Last Theorem as a geometric mismatch: while 2D squares possess the directional capacity to support 1-sharing, higher dimensional cubes (n>2) do not, forcing the coincidence switch to 0 and precluding integer solutions.
Collatz Conjecture: By applying 1-sharing, the entire Collatz tree is revealed as a single, folded singularity where all integers are conceptually equal to 1.
Conclusion:
Universal Concept Theory provides the “missing layer” of mathematics, transitioning the field from a collection of isolated rules to a complete, structural hierarchy. By understanding the “backstage” of concept sharing, the paradoxes of standard math are revealed as simple logical certainties.
The Foundations of Universal Concept Theory: The Host and the Guest
In standard mathematics, a “point” or a “number” is an isolated entity. It is a lonely occupant of a single location, and standard rules dictate that no two distinct entities can occupy the same spot simultaneously. Universal Concept Theory (UCT) engineered a more sophisticated foundation by introducing the Host.
1. The Host (The Higher-Level Scaffolding)
Before we can understand how concepts interact, we must first build the environment. We define a Host (represented as; r in geometry or A’ in arithmetic).
The Host is not a “container” that is larger than its contents. Instead, the Host is the fundamental environment that shares the exact same space as the concepts themselves. It is the “scaffolding” that grants permission for multiple concepts to coexist. Without a Host, there is no room for sharing; with a Host, the capacity of a single location can expand.
2. The Guests (Fixed and Mobile Entities)
Once the Host environment is established, we perform Concept Removal—removing the old rule that a location must have only one occupant. This allows us to introduce our “Guests”:
The Fixed Guest ( p or A): This is the original concept. It remains anchored to its identity, providing the base reference for the location.
The Mobile Guest (e or B) This is the new entity (like the e iin our geometric work). Because the Host provides the room, the Mobile Guest can move or shift within the extended space while still “sharing” the same fundamental location as the Fixed Guest.
3. The 1-Sharing State (The Social Connection)
When the Host is active, we enter the 1-Sharing state. In this state, the distance between the Fixed Guest and the Mobile Guest is defined as zero. They are distinct characters, but they “coincide” perfectly.
This is the “Natural State” of mathematics. It explains why a Collatz sequence is actually a single, unified chain: every step is a different Guest sharing a seat at the same Host’s table. The sequence only looks like 111 steps long because we have “separated” the Guests.
4. The 0-Sharing State (The Standard Restriction)
What we call “Standard Math” is simply the state where the Host has restricted access. When we set the coincidence switch to 0, the Guests are no longer allowed to share the same seat. They are forced to separate into the discrete, isolated points and numbers we use for everyday arithmetic.
Mathematical concept removal, subsequent sharing and separation
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 something like the contiguousness of two points could be possible. Since something of no extent combined with something of no extent would still have no extent- but there could be two items of no extent there! The two objects could be hidden as one! The two items of no extent would have to be different from points in some way so that they would not combine to be one singular item.
One can regard the overlapping shadow diagrams below:
Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:
The shadows can be regarded as two points overlapping which could be thought of as at a common point, but also as two items which are both not points being in a contiguous state. Now as seen in the overlap, two different shadows can take up the space of one location, we can regard these as e and p. e is another possible entity of no extent, p is a point. If I take away one light, one shadow still remains. The table can be like an underlying, hosting space.
If e and p are sharing a position, they do not combine because they are different. It must be that they are either next to each other, or that they overlap and do not merge because they are different entities, as they are both not p’s. Or overlap, creating one point if they were both points or creating one point if they were side by side as was Aristotle’s way of thinking of things. An item of no extent put with a different item of no extent would still have no extent but there would be two items there.
We can consider 2 e’s at a point. The two e’s would be in a sense the same, since they are at the same point, yet they are different from points, in that they can appear multiply in a contiguous state..
Think of a mathematical point not as a single solid object but as a Russian Nesting Doll. Even when the dolls are tucked inside one another and appear to occupy a single spot, their individual identities remain perfectly intact and shared. My new language allows us to unfold these nested layers, revealing the hidden structures and connections that standard math accidentally flattens out. This analogy was developed in collaboration with an AI assistant.
Start with a plane of places and a horizontal line of points labelled 1,2,3.. So that we have a unit length.
We wish to show that there is a next level to this plane and set of numbers on a line.
Let’s build the next level.
What is scaffolding?
It is the creation of a new concept based on a pre-existing concept. The new concept is said to host the pre-existing concept. It has the same basic notion but exists at the next level of conceptual space.
Since the concepts have the same basic notion there is an exact fit of the next level concept and the lower level concept sharing space together. So there is “room” for this. But then also since the two concepts share the same basic notion there is a ‘building” possible. One can build upwards.(Since the host is the next level of the original concept in the concept space as I can remove the guest. Also the host itself might also have a host). Since we only have e’s and p’s. The host level is made of e’s and can host p’s or other e’s.
Also,since there is more than one concept, there is more than one next level of concepts, so we can build outwards as well.
In points, call this new item “r” This is a set of e’s. Then this together with a point can be written r(p). We can separate r and p (not divide as this is a separation not a division) if we take p out, the lower concept can be removed. We can place a different kind of lower concept which can have freedom of motion in the higher concept, while the original lower concept remains fixed. Then the two lower concepts are also sharing space and don’t coincide.
One entity of no extent is able to host the guest entity of no extent. So that the two exist together without combining. The guest entity doesn’t host so it is different from the host entity.
We can remove the guest entity of no extent ‘p’ and replace with ‘e’ a new entity of no extent which is capable of moving through an extent of hosts. Also we could put p back in and e would exist together with p and r. All these would be different. P is fixed while e is mobile.
So the host is a next level place of e’s, hosting guest places p or e. Where hosting is defined as r and p or e sharing ( together but not combining) position, with p or e capable of being removed from r and e moving in r. R is the hosting space coexisting with p and e. Briefly (r[(p(e)].
I can place a point on another point, but let’s not have them coincide but let the new point be a point where points could be. So I am thinking of an e placed on a different concept of e. Rather than just a plain point (1st level). This would be a second level, possible since the new point and the original point carry the same basic notion, that of a place of no extent, so that the upper concept becomes a place of places of no extent-containing the lower places of no extent. It fits exactly with the lower concept. These concepts are co-existing. This is concept building.
Since e is not p, e can have parts. But since it is of zero-dimension it has to have zero-dimensional parts which add to itself. These can’t be p’s since p’s add to a single p.
But must be on the same level as e’s and p’s, so must be e’s. Therefore e can have parts of itself so that e(1)=(e(1)(e(1)).
So e is able to self-replicate and split up or recombine in a space of r.
In concept sharing we have the same basic notion but two different ideas. So two ideas are sharing the same space, as they are only concepts, this is possible. It must be that I can take one away from the other. So one exists at the next level from the other, as there is no other option at the beginning.
That there is a next level is granted because any concept can be continued. Hosting can be considered as exact containment. The concept can fit exactly into the next level of itself. Since this is an exact fit we can move into the next level.
In geometry the host becomes the place of a place at the position of the place. The shared concept is “no extent”
In numbers the host becomes the number of a number at the position of the number. The shared concept is a natural number.
Since points and numbers both have positions these ideas can be combined into a new plane.
We must have more than one place at this place of lower places so that one lower place may be fixed to keep contact with the first level and the other places would be mobile in the new dimension as I can have multiple places of lower places of no extent.
Then a new number dimension is co-created. Number the lower places (1(1) and (1(2) or (2(1)-for two lower places. These are two new higher numbers where the number of lower numbers is 2 and not 1.
We can think of a jigsaw puzzle of a landscape being taken apart.
This is how it would be for the concept of a point and a number. In general then we can regard any mathematical concept as capable of this sharing. Coincider a concept A and related concept B.(Shown as circles in the diagrams below).
We can have an axiom of concept sharing if I introduce the notation ((). This is showing that the combined concepts are not in coincidence. Like (<—() The left bracket is moved away showing the revealed concepts. The extra structure is shown as a revelation of one labeled concept away from the combination which does not coincide as the concepts are different in some way to be revealed, it is an expanded notation and makes a sentence. So for example (A(B) means A is not in coincidence with B, but sharing the same space.
So the axiom is expressed as (A’[(A(B)] or for a specific concept (A’(1)[(A(1)(B(1)}. A(1) is a specific, fixed instance of a concept. B(1) is another instance of the concept yet different in some way. In order for the two not to be in coincidence another level of the concept is opened up, overlying but co-existing, so that A(1) and B(1) can be in the same lower space, able to separate.[,] is showing containment. The spaces are co-existing so that they can be in the new space to begin with. This creates a concept space with the next level concept denoted A’(1). Otherwise there is no “room” and A(1) and B(1) just coincide.
To have (A(B) I have to take out A. A(1) can only be taken out if there is an overlying concept A’ which copies the notion of the original concept A but exists at a higher level such that A’ is the concept of the concept A taken to a new expanded level. Such as for points a “ new place of lower places B”. This creates a leveled concept space …(A’’[(A[’(A(B)]] with A’’ even higher than A’. All these A’, A’’, … coexist with A as the (() sentence shows.
The concept B can move in the expanded concept level as B can overlap with A’ since A’ is the expanded notion of A and B is shared with A. Making the logic of A’ and B consistent as B must be different from A, being able to move in A’ while A is fixed.
The next level concept carries the same basic notion of the original concept, so that it can overlap and connect with the original concept; this is how a concept can build upon itself. Yet it is overlying and co-exists with the original concept.
Then if I have another instance of this concept say A(2) I can have (A’(2)[(A(2)(B(2))]
Then (A’(1)[(A(1)(B(1))] and (A’(2)[(A(2)B(2))] allows (A’(1)[(A(1))] and (A’(2)[(A(2)B(1)(B(2))].
This is the movement of B(1) making B(1) different from A(1) as A(1) is fixed. Then the axiom is self-consistent as A(1) is different from B(1) as required.
I have to take out A to have (A(B) so this necessitates A’ Which then A and B are defined to exist in. A’ has to have the same notion as A and B, but extended.
B has to travel in A’ so that A’ has to be a new dimension of concept A, an extension of concept A. The only way it can be extended is to have another level of concept A a concept of concept. So for example in points a place of places or in numbers a number of numbers.
If we take out a place, we have a place of lower places, where more than one place can be. This has to be built first. If we take out a number we have a number of lower numbers, a number which indicates how many numbers there are present.
These can number the places in the place of lower places.
We can have more than one number or place because we are opening up another dimension of the concept. The concept can now move away from its singularity.
There has to be more than one in order to move the other type of concept along the new dimension. Then the new concept dimension has to copy the notion of the original concept but be at its next level higher, so that the original concept doesn’t change when moved along the new concept dimension.
So A’ is an overlying , coexisting space. A new level of concept A.
In standard math, we are taught that a “point” is the only thing that can exist at a specific co-ordinate. But think about the pixels on your screen. A single pixel isn’t just one “thing”-it is actually a shared space where Red, Green and Blue exist together.
When these colors share the pixel, they create a new result (like white light), but they don’t lose their individual identities. They are nested within that single coordinate. My notation (A’[(A(B)] works the same way:
The pixel is the higher concept level (A’). The Red is the fixed anchor(A) and the Blue is the second entity(B) that shares that space
The magic hapens in the separation. In a standard co-ordinate system, if you move the “point” the whole pixel moves. But in this new language we can “unfold” the pixel We can keep the Red fixed while moving the Blue to a new placement. This allows us to see how complex structures-like prime numbers or intricate knots-are actually built from multiple “colors” sharing a single origin.-This analogy was created in collaboration with an AI assistant.
In the case of two ordinary points we would have two of the same item, leading to the same item.
In order to have the two different items together we have to be able to remove the concept of a point as the only item of no extent and the removal of the point itself. It requires the construction of an overlying space of higher places of lower places where points can be, coexisting with the space of points.
I’m deleting the concept of a point as the only item of no extent in geometry. As well, I can delete it physically as I now have an overlying level.
This then allows the creation of other new entities of no extent.
In particular, we can think of removing a point from space, necessitating an overlying, co-existing plane of new places of original places, by extending the concept of a place or a point, into a plane. This also gives room for a new entity ‘e’.
We can create another new entity called e, which can move in this new dimension making it different from p which can still be fixed in this new overlying, co-existing dimension. This keeps the connection with the geometry which already exists.
This creates a mixed space of e and p in an overlying co-existing space of r.
We can think of e’s as having the ability to move, unchanging. An analogy is that of a jigsaw puzzle of a landscape being taken apart. The e’s can be thought of as pieces of the puzzle, moving but unchanging.
Here is then another dimension where e can move off into making it different from p as required. See the diagram below:
Since in mathematics we wish to have this concept of no parts or no extent, so too can we have this further notion. The further notion is that two items could be together as one, if they both had no extent. They would have to be two different items to no extent. But how do we number these? We can think of removing the number one and replacing it with 1(1) and 1(2). Similar to removing a point to make room for e.
Then the two items touch in their entirety both being of no extent. But they do not combine being different entities.
They could not both be points as we know them now, as they would just become one point as Aristotle thought. Aristotle thought that two points could not be contiguous since there would have to be no line in between them and we would have no extent.-Aristotle : Physics- “as there are an infinite number of points between two selected points”. The idea is that the two points would combine into one. They would touch in their entirety. But he only had the idea that there could only be one item of no extent.
The new entity e(1) has a location but e(1)1 and e(1)2 if we start off with two e’s together also sharing with p) can also have new locations of their initial locations, if they move in the new dimension of new locations of original locations opened up by the creation of the new level.
E(1) can be fixed for now, but could move, so its new location of its original location is the same as its initial location. This is the difference between e(1)1 and e(1)2. This also makes it different from a point which we can say still has its location fixed in r.
Furthermore, all math concepts are point-like in that they are exact, universal 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 universal 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 forwards into the idea. To create higher spaces.
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 3 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 three numbers 1, 1(1) and 1(2) are sharing the concept of 1. 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.
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.
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 shadows combine to form a darker shadow.
Now think about points, mathematical objects which have no extent.
If we had two objects, one in front of the other illuminated by a single light, the object closer to the light would cast part of it’s shadow on the object further away, but it could extend past the closer object in both directions. The resulting shadow cast on a table below would have one “point” everywhere, it would be similar to the situation of coincidence in mathematics.
The two overlapping shadows show a new situation of two “points” being placed together as the shadows have no height so can be thought of as points which have no extent.
This can be thought of as ‘sharing’. The two “points” are sharing one location.
There can be hidden items of no extent due to the nature of the notion of no extent.
Something of no extent can be multiple, for example doubled, there could be two items of no extent there. They would just appear to be one item there as both items have no extent. This is certain. So it is possible there could be more structure.
So we need another item to no extent. But the only items we know about are points. We know that they have no extent and also that if I place two together the result is a single point.
So what if there is another entity of no extent but if we place two of these types of items together they do not merge into a single point but share position as in the overlapping shadows.
We can call points p’s and the other new entities e’s. If I place two points together in overlap they coincide and we say we have one point. Also if two points are placed next to each other they have to touch in their entirety so we also have one point.
But now we have e’s as well which are zero-dimensional but not points. So I can place a point together with an e as poe and this is a point and an e overlapping.
This has a form of zero distance like pop, but e and p do not merge, being different entities. Yet there is another form of zero distance as we have an e and a p and not two p’s. The only other way two entities of no extent can come together is contiguously, so we write e*p as e and p together with the new zero distance in mind. E is contiguous with p and also overlapping with p. This is how e is different from p.
We can think of the analogy given near the top of one object in front of and extending past another object, both objects being lit by a single light. Then there is that shadow cast on a table. But there is some of the shadow cast on the further object from the light from the nearer object to the light. We can think of the shadow being lifted off of the combined shadow onto the further object. This is an analogy of e being both overlapping and contiguous with p. As it shows e lifted off of p.
Two items of no extent placed together can be thought of as a single position as in points, p or two e’s next to each other or overlapping like in e’s. This is how p’s and e’s are different.
Two e’s cannot be in coincidence. If we imagine a space of only e’s, eoe does not equal e otherwise e=p. Also e*e does not equal e otherwise *=o and e=p. So e*e=e*e, that is to say e*e does not resolve to a single e.
The two e’s “touch in their entirety” (since they are both at p but they are together in a serial sense or an overlapping sense).
When we tried to put two points together there was no choice but to resolve it to a single p, since p’s were all we thought of that had no extent. Since we open up the door for another possibility, already having p’s, we can have e’s here now.
At p we can have two of the same e. Or just one e, to begin with. When one e combines with a p it does so either serially or as in coincidence (overlapping) as e and p are different entities.
Two items of no extent could be coincident, as in points, or also share as in e’s. The e’s are in the same position so that they are both there. Yet they share as in the overlapping shadows, since something of no size added serially or in overlapping to something else of no size still has no size. Yet there are two types of zero size so these can all fit together.
So the idea is that points, p and items e are sharing the concept to no extent but are different in another way. The other way is that e’s share while p’s coincide.
To sum up, two p’s can be in coincidence forming a single p. E can share with p either serially or in the sense of coincidence, this can work since e’s and p’s are different entities. E’s can exist sharing with each other in a serial way (contiguous) or overlapping. E’s can also share with other e’s, serially or overlapping.
At the beginning we can have a plane of these new entities “e”, coexisting with points. Identify which e’s are in the set as is done with points. Let us start with the whole plane then if I move an e I can put it in sharing with another e, leaving a hole.
But let’s examine e, further. Suppose I think of a single e. It can’t exist singularly like p exists singularly. We can think of p being in coincidence with any number of copies of itself, leading to just a singular p. But e doesn’t coincide with itself. So it can share with itself any number of times, sharing being associated with e’s. This is not the same as coincidence as I can separate e’s away from copies of themselves but I cannot do this with p’s.
Then instead of leaving a hole, let e share with itself once and then have a copy of e, leave to share with other e’s. We can have a subset of another coexisting plane of e’s. This way we can have a fixed plane of e’s and a moving set of e’s.
So how does e move away from p? First put in a plane of e’s then add to it a plane of p’s. Then we can have a subset of another sharing plane of e’s. Then e(1) at p(1) can move off into the fixed plane so the fixed plane becomes a place of places, a next dimension of place. So we have a dimension of place co-existing with a dimension of places of places. e(1) is sharing with itself so it both moves in the e plane and stays still in the p plane. Its place stays the same but the place’s place of places changes.
So this fixed plane may be regarded as a place of places and the set of moving e’s as moving places. The fixed plane is a new dimension or level of places, places of places.
We only need the space of a point to build this and we have to have a point to start with. Then also the still e and the moving e can share position. So if I have the concept of a point, I can further build this and I have my new item to no extent. I am able to remove the moving e from the combination.
Then e and e’ are different. E’ can be fixed e’s. (e’[(e)]]. E is contained in e’.
We can then form new structures with e, which have new properties. The space is re-engineered.
Foundational Preface: Universal Concept Theory (UCT) framework:
Abstract:
A framework for the structural completion of mathematics
Objective: to propose a unified foundation framework-Universal Concept Theory-(UCT)-that resolves long-standing mathematical conjectures (e.g.., the Collatz Conjecture and Fermat’s Last Theorem) by redefining the nature of mathematical identity and coincidence.
Methodology: UCT departs from standard axiomatic set theory by introducing “Conceptual Engineering”. This process involves three primary stages.
Scaffolding: The construction of higher level “Places of places” and “Number of numbers” that exist as containers for lower level concepts.
Concept Removal: The systematic removal of the single occupant rule, allowing a single placement to support multiple entities.
Concept Sharing and Separation: The introduction of a variable “Coincidence Switch”. In the 1-sharing state, the distance between distinct concepts( such as the steps in the Collatz sequence) is reduced to zero, creating a unified identity. In the 0-sharing state, concepts are “separated” into the discrete non-overlapping values found in standard arithmetic.
Structural Capacity: UCT demonstrates that the transition from sharing to separation is governed by the “structured capacity” of the engineered space.
The Fermat Limit: The theory explains Fermat’s Last Theorem as a geometric mismatch: while 2D squares possess the directional capacity to support 1-sharing, higher dimensional cubes (n>2) do not, forcing the coincidence switch to 0 and precluding integer solutions.
Collatz Conjecture: By applying 1-sharing, the entire Collatz tree is revealed as a single, folded singularity where all integers are conceptually equal to 1.
Conclusion:
Universal Concept Theory provides the “missing layer” of mathematics, transitioning the field from a collection of isolated rules to a complete, structural hierarchy. By understanding the “backstage” of concept sharing, the paradoxes of standard math are revealed as simple logical certainties.
The Foundations of Universal Concept Theory: The Host and the Guest
In standard mathematics, a “point” or a “number” is an isolated entity. It is a lonely occupant of a single location, and standard rules dictate that no two distinct entities can occupy the same spot simultaneously. Universal Concept Theory (UCT) engineered a more sophisticated foundation by introducing the Host.
1. The Host (The Higher-Level Scaffolding)
Before we can understand how concepts interact, we must first build the environment. We define a Host (represented as; r in geometry or A’ in arithmetic).
The Host is not a “container” that is larger than its contents. Instead, the Host is the fundamental environment that shares the exact same space as the concepts themselves. It is the “scaffolding” that grants permission for multiple concepts to coexist. Without a Host, there is no room for sharing; with a Host, the capacity of a single location can expand.
2. The Guests (Fixed and Mobile Entities)
Once the Host environment is established, we perform Concept Removal—removing the old rule that a location must have only one occupant. This allows us to introduce our “Guests”:
The Fixed Guest ( p or A): This is the original concept. It remains anchored to its identity, providing the base reference for the location.
The Mobile Guest (e or B) This is the new entity (like the e iin our geometric work). Because the Host provides the room, the Mobile Guest can move or shift within the extended space while still “sharing” the same fundamental location as the Fixed Guest.
3. The 1-Sharing State (The Social Connection)
When the Host is active, we enter the 1-Sharing state. In this state, the distance between the Fixed Guest and the Mobile Guest is defined as zero. They are distinct characters, but they “coincide” perfectly.
This is the “Natural State” of mathematics. It explains why a Collatz sequence is actually a single, unified chain: every step is a different Guest sharing a seat at the same Host’s table. The sequence only looks like 111 steps long because we have “separated” the Guests.
4. The 0-Sharing State (The Standard Restriction)
What we call “Standard Math” is simply the state where the Host has restricted access. When we set the coincidence switch to 0, the Guests are no longer allowed to share the same seat. They are forced to separate into the discrete, isolated points and numbers we use for everyday arithmetic.
Mathematical concept removal, subsequent sharing and separation
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 something like the contiguousness of two points could be possible. Since something of no extent combined with something of no extent would still have no extent- but there could be two items of no extent there! The two objects could be hidden as one! The two items of no extent would have to be different from points in some way so that they would not combine to be one singular item.
One can regard the overlapping shadow diagrams below:
Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:
The shadows can be regarded as two points overlapping which could be thought of as at a common point, but also as two items which are both not points being in a contiguous state. Now as seen in the overlap, two different shadows can take up the space of one location, we can regard these as e and p. e is another possible entity of no extent, p is a point. If I take away one light, one shadow still remains. The table can be like an underlying, hosting space.
If e and p are sharing a position, they do not combine because they are different. It must be that they are either next to each other, or that they overlap and do not merge because they are different entities, as they are both not p’s. Or overlap, creating one point if they were both points or creating one point if they were side by side as was Aristotle’s way of thinking of things. An item of no extent put with a different item of no extent would still have no extent but there would be two items there.
We can consider 2 e’s at a point. The two e’s would be in a sense the same, since they are at the same point, yet they are different from points, in that they can appear multiply in a contiguous state..
Think of a mathematical point not as a single solid object but as a Russian Nesting Doll. Even when the dolls are tucked inside one another and appear to occupy a single spot, their individual identities remain perfectly intact and shared. My new language allows us to unfold these nested layers, revealing the hidden structures and connections that standard math accidentally flattens out. This analogy was developed in collaboration with an AI assistant.
Start with a plane of places and a horizontal line of points labelled 1,2,3.. So that we have a unit length.
We wish to show that there is a next level to this plane and set of numbers on a line.
Let’s build the next level.
What is scaffolding?
It is the creation of a new concept based on a pre-existing concept. The new concept is said to host the pre-existing concept. It has the same basic notion but exists at the next level of conceptual space.
Since the concepts have the same basic notion there is an exact fit of the next level concept and the lower level concept sharing space together. So there is “room” for this. But then also since the two concepts share the same basic notion there is a ‘building” possible. One can build upwards.(Since the host is the next level of the original concept in the concept space as I can remove the guest. Also the host itself might also have a host). Since we only have e’s and p’s. The host level is made of e’s and can host p’s or other e’s.
Also,since there is more than one concept, there is more than one next level of concepts, so we can build outwards as well.
In points, call this new item “r” This is a set of e’s. Then this together with a point can be written r(p). We can separate r and p (not divide as this is a separation not a division) if we take p out, the lower concept can be removed. We can place a different kind of lower concept which can have freedom of motion in the higher concept, while the original lower concept remains fixed. Then the two lower concepts are also sharing space and don’t coincide.
One entity of no extent is able to host the guest entity of no extent. So that the two exist together without combining. The guest entity doesn’t host so it is different from the host entity.
We can remove the guest entity of no extent ‘p’ and replace with ‘e’ a new entity of no extent which is capable of moving through an extent of hosts. Also we could put p back in and e would exist together with p and r. All these would be different. P is fixed while e is mobile.
So the host is a next level place of e’s, hosting guest places p or e. Where hosting is defined as r and p or e sharing ( together but not combining) position, with p or e capable of being removed from r and e moving in r. R is the hosting space coexisting with p and e. Briefly (r[(p(e)].
I can place a point on another point, but let’s not have them coincide but let the new point be a point where points could be. So I am thinking of an e placed on a different concept of e. Rather than just a plain point (1st level). This would be a second level, possible since the new point and the original point carry the same basic notion, that of a place of no extent, so that the upper concept becomes a place of places of no extent-containing the lower places of no extent. It fits exactly with the lower concept. These concepts are co-existing. This is concept building.
Since e is not p, e can have parts. But since it is of zero-dimension it has to have zero-dimensional parts which add to itself. These can’t be p’s since p’s add to a single p.
But must be on the same level as e’s and p’s, so must be e’s. Therefore e can have parts of itself so that e(1)=(e(1)(e(1)).
So e is able to self-replicate and split up or recombine in a space of r.
In concept sharing we have the same basic notion but two different ideas. So two ideas are sharing the same space, as they are only concepts, this is possible. It must be that I can take one away from the other. So one exists at the next level from the other, as there is no other option at the beginning.
That there is a next level is granted because any concept can be continued. Hosting can be considered as exact containment. The concept can fit exactly into the next level of itself. Since this is an exact fit we can move into the next level.
In geometry the host becomes the place of a place at the position of the place. The shared concept is “no extent”
In numbers the host becomes the number of a number at the position of the number. The shared concept is a natural number.
Since points and numbers both have positions these ideas can be combined into a new plane.
We must have more than one place at this place of lower places so that one lower place may be fixed to keep contact with the first level and the other places would be mobile in the new dimension as I can have multiple places of lower places of no extent.
Then a new number dimension is co-created. Number the lower places (1(1) and (1(2) or (2(1)-for two lower places. These are two new higher numbers where the number of lower numbers is 2 and not 1.
We can think of a jigsaw puzzle of a landscape being taken apart.
This is how it would be for the concept of a point and a number. In general then we can regard any mathematical concept as capable of this sharing. Coincider a concept A and related concept B.(Shown as circles in the diagrams below).
We can have an axiom of concept sharing if I introduce the notation ((). This is showing that the combined concepts are not in coincidence. Like (<—() The left bracket is moved away showing the revealed concepts. The extra structure is shown as a revelation of one labeled concept away from the combination which does not coincide as the concepts are different in some way to be revealed, it is an expanded notation and makes a sentence. So for example (A(B) means A is not in coincidence with B, but sharing the same space.
So the axiom is expressed as (A’[(A(B)] or for a specific concept (A’(1)[(A(1)(B(1)}. A(1) is a specific, fixed instance of a concept. B(1) is another instance of the concept yet different in some way. In order for the two not to be in coincidence another level of the concept is opened up, overlying but co-existing, so that A(1) and B(1) can be in the same lower space, able to separate.[,] is showing containment. The spaces are co-existing so that they can be in the new space to begin with. This creates a concept space with the next level concept denoted A’(1). Otherwise there is no “room” and A(1) and B(1) just coincide.
To have (A(B) I have to take out A. A(1) can only be taken out if there is an overlying concept A’ which copies the notion of the original concept A but exists at a higher level such that A’ is the concept of the concept A taken to a new expanded level. Such as for points a “ new place of lower places B”. This creates a leveled concept space …(A’’[(A[’(A(B)]] with A’’ even higher than A’. All these A’, A’’, … coexist with A as the (() sentence shows.
The concept B can move in the expanded concept level as B can overlap with A’ since A’ is the expanded notion of A and B is shared with A. Making the logic of A’ and B consistent as B must be different from A, being able to move in A’ while A is fixed.
The next level concept carries the same basic notion of the original concept, so that it can overlap and connect with the original concept; this is how a concept can build upon itself. Yet it is overlying and co-exists with the original concept.
Then if I have another instance of this concept say A(2) I can have (A’(2)[(A(2)(B(2))]
Then (A’(1)[(A(1)(B(1))] and (A’(2)[(A(2)B(2))] allows (A’(1)[(A(1))] and (A’(2)[(A(2)B(1)(B(2))].
This is the movement of B(1) making B(1) different from A(1) as A(1) is fixed. Then the axiom is self-consistent as A(1) is different from B(1) as required.
I have to take out A to have (A(B) so this necessitates A’ Which then A and B are defined to exist in. A’ has to have the same notion as A and B, but extended.
B has to travel in A’ so that A’ has to be a new dimension of concept A, an extension of concept A. The only way it can be extended is to have another level of concept A a concept of concept. So for example in points a place of places or in numbers a number of numbers.
If we take out a place, we have a place of lower places, where more than one place can be. This has to be built first. If we take out a number we have a number of lower numbers, a number which indicates how many numbers there are present.
These can number the places in the place of lower places.
We can have more than one number or place because we are opening up another dimension of the concept. The concept can now move away from its singularity.
There has to be more than one in order to move the other type of concept along the new dimension. Then the new concept dimension has to copy the notion of the original concept but be at its next level higher, so that the original concept doesn’t change when moved along the new concept dimension.
So A’ is an overlying , coexisting space. A new level of concept A.
In standard math, we are taught that a “point” is the only thing that can exist at a specific co-ordinate. But think about the pixels on your screen. A single pixel isn’t just one “thing”-it is actually a shared space where Red, Green and Blue exist together.
When these colors share the pixel, they create a new result (like white light), but they don’t lose their individual identities. They are nested within that single coordinate. My notation (A’[(A(B)] works the same way:
The pixel is the higher concept level (A’). The Red is the fixed anchor(A) and the Blue is the second entity(B) that shares that space
The magic hapens in the separation. In a standard co-ordinate system, if you move the “point” the whole pixel moves. But in this new language we can “unfold” the pixel We can keep the Red fixed while moving the Blue to a new placement. This allows us to see how complex structures-like prime numbers or intricate knots-are actually built from multiple “colors” sharing a single origin.-This analogy was created in collaboration with an AI assistant.
In the case of two ordinary points we would have two of the same item, leading to the same item.
In order to have the two different items together we have to be able to remove the concept of a point as the only item of no extent and the removal of the point itself. It requires the construction of an overlying space of higher places of lower places where points can be, coexisting with the space of points.
I’m deleting the concept of a point as the only item of no extent in geometry. As well, I can delete it physically as I now have an overlying level.
This then allows the creation of other new entities of no extent.
In particular, we can think of removing a point from space, necessitating an overlying, co-existing plane of new places of original places, by extending the concept of a place or a point, into a plane. This also gives room for a new entity ‘e’.
We can create another new entity called e, which can move in this new dimension making it different from p which can still be fixed in this new overlying, co-existing dimension. This keeps the connection with the geometry which already exists.
This creates a mixed space of e and p in an overlying co-existing space of r.
We can think of e’s as having the ability to move, unchanging. An analogy is that of a jigsaw puzzle of a landscape being taken apart. The e’s can be thought of as pieces of the puzzle, moving but unchanging.
Here is then another dimension where e can move off into making it different from p as required. See the diagram below:
Since in mathematics we wish to have this concept of no parts or no extent, so too can we have this further notion. The further notion is that two items could be together as one, if they both had no extent. They would have to be two different items to no extent. But how do we number these? We can think of removing the number one and replacing it with 1(1) and 1(2). Similar to removing a point to make room for e.
Then the two items touch in their entirety both being of no extent. But they do not combine being different entities.
They could not both be points as we know them now, as they would just become one point as Aristotle thought. Aristotle thought that two points could not be contiguous since there would have to be no line in between them and we would have no extent.-Aristotle : Physics- “as there are an infinite number of points between two selected points”. The idea is that the two points would combine into one. They would touch in their entirety. But he only had the idea that there could only be one item of no extent.
The new entity e(1) has a location but e(1)1 and e(1)2 if we start off with two e’s together also sharing with p) can also have new locations of their initial locations, if they move in the new dimension of new locations of original locations opened up by the creation of the new level.
E(1) can be fixed for now, but could move, so its new location of its original location is the same as its initial location. This is the difference between e(1)1 and e(1)2. This also makes it different from a point which we can say still has its location fixed in r.
Furthermore, all math concepts are point-like in that they are exact, universal 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 universal 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 forwards into the idea. To create higher spaces.
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 3 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 three numbers 1, 1(1) and 1(2) are sharing the concept of 1. We have the usual number line on the midline of the new plane.
