The Collatz conjecture
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 placed 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!
It would be like being next to each other, as we have two here, but unlike being next to each other since these both are in one location. The two items would have to be different enough so that they would not combine. We would need an underlying space to be able to make these two different.
One can regard the overlapping shadow diagram below:
Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:
Now as seen in the overlap, two different points of the shadows can take up the space of one location, we can regard these as e(e is another possible entity of no extent) and p(a point) and not points, p and p combining to a single point . If I take away one light, one shadow still remains. The table is like an underlying space.
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.
The two points would have to touch in their entirety.
Yet what if we didn’t have two points, but two different geometric entities of no extent?
A point has location only but a location does not have to have a point only. Usually a point and location have the same meaning, but now I can split these two as I imagine another entity to no extent, sharing the location with a point.
Because we want to keep the notion of an item of no extent we have to accept that two different items of no extent can be together since they can take the space of one point. Then remove the point as the only item of no extent so this other structure can exist.
Start with a continuous space of points. Take out one point and place an e here. To do this I need an underlying space. Since I have this it changes all the items we had as p’s into places which are fixed in the underlying space. They are transformed to q’s. We can call them q’s.
E is different from q in that it can move, unchanging in an extent of q’s to share with different q’s.
Unlike a p moving in an extent of p’s which is constantly changing with changing position, since p has position only.
E has a new location away from its original position
Also we can have an extent of e’s to go along with the q’s so can have a sharing like e(1)*q(2)*e(2).
E has multiple locations where it could be. Q has one location only, at a time.
E can be different from q as e has the extent of q it can move in. Before we only had p alone so a moving p could only be in coincidence with a p in the plane of p’s. E can move unchanged in a plane of q’s or in a combined plane of e’s and q’s.
Now, we have in the same new places of original places two different entities. How do we number these?
It is not the number 2, as we know it, since the two entities are in the same position. Since we have concept shared a point so too can we concept share the concept of a number. So I can take out the number one and label the entities 1(1) and 1(2). The next level of numbers is the new number of different numbers. This can be 2 instead of 1. So I may have the new numbers 1(1) and 1(2) representing the new entities.
The two “points” have to be in the space of one location, so they must overlap as well as be separate. It means that the “points” don’t combine into one point. The two “points” must be different enough so that they don’t combine.
There could be at least two different geometric entities of no parts. For then these two could be together and still be two, hidden as one, since this would still have no extent. This could be possible, somehow.
If we want the notion of an object of no extent we could have another object of no extent as well, since two could be hidden as one. The two entities would have to be different enough so that they would not combine.
Because there is an entity of no extent, a point, and we base geometry on this, there might be another entity of no extent we could pair it with, still have a non-combined entity of no extent and have another geometry, which would fit and extend the geometry which is already there. This is because we are still keeping to the items to no extent as a basis. We would just need to make the other item different.
The two items could be two different items of no parts, if the two items were different entities. That way we could not conclude that the combination was not a point or the other entity but we would have to leave it as these two, together, since now we have two entities of no extent.
E’s and q’s don’t combine. E’s share the concept of no extent with q’s. E’s are also shared with different e’s. So e’s have to share with other e’s so they can all be sharing a single location. Q’s are still at the level of places and e’s can move, this is what makes e’s different from q’s. E’s can have new locations of original locations and can share with q’s and other e’s. Q keeps its original position, wherever it is. The idea of a moving point is just that this is a point in coincidence with another point.
We can keep both of these ideas if two points together are defined to be one point, yet e(the new entity) and q placed together do not combine in the new underlying space.
Since we want to keep this concept of no extent, or no parts it must also be possible to have two together and still have two, as long as the two parts are different enough so that they do not combine into one. That is, since we want to keep this notion of an entity with no extent and since two could be together and still be two if they were different entities, or different individual entities.
Yet let’s keep the math we already have just add a new item to the concept of “no extent”. That way the usual idea of two points together being one, still is valid, just create another item for the other case. In the original case the two items lead to an item of no extent which we say is the same as the original two. In the new case there are two items together which together have no extent, but they do not combine, being different entities.
Each e can be considered an origin and can move along the extent of e which is defined as being with the extent of q.
Then this is how e can be made different from q as in an extent of e and q which can come about( we can extend e*q outwards to have a whole plane of this sharing idea), we can move e through this e and q extent, it then has a new location of original location, making it different from q. That is, e is the next level of location. Each e can act as an origin and then the origins are moved. Each e is different from the other e’s it encounters. The e’s are then sharing location like e shares with q.
Yet we want this concept of point removal and sharing (the point is removed and the two new items are sharing the notion or concept of an object of no extent) to join up with the knowledge that is already in existence.
In doing this we have to add to and change the mathematics which already exists. This turns out not to destroy but heal mathematics, as this can be seen as something which is basic, but is missing. The math from this leads to a way to solve many old problems which are easy to state but previously hard to solve, such as:
The knot problem:https://calctutor.ca/category/mathematics/knots-mathematics/the-knottedness-and-chirality-of-the-trefoil/
The twin prime conjecture:https://calctutor.ca/category/mathematics/the-twin-prime-conjecture/
The Goldbach conjecture:https://calctutor.ca/category/mathematics/goldbachs-conjecture/
Fermat’s last theorem;https://calctutor.ca/category/mathematics/a-clearer-and-simpler-demonstration-of-fermats-last-theorem-wiles-theorem/
The Poincare conjecture:https://calctutor.ca/category/mathematics/the-poincare-conjecture/
The Riemann Hypothesis:https://calctutor.ca/category/mathematics/the-riemann-hypothesis/
The Collatz conjecture:https://calctutor.ca/category/mathematics/the-collatz-conjecture/
In fact this idea can complete mathematics:
The completion of mathematics:https://calctutor.ca/category/mathematics/the-completion-of-mathematics/
Definition and description of e:
When two points come together or start together, it seems as if there could be two there as 0+0=0.(two items of no extent could still combine to no extent). Only that points have location only, and having them together means we have one location so we have to conclude that there is one point there.
What if an object of no extent could be expressed as two different items of no extent, starting together. Since an item of no extent together with another different item of no extent would still have no extent, yet there could still be two items here and not one. The other item of no extent would have to have something more than location (it would have to be different from a point).
Yet since it is logical that two items of no extent could be together and still be two items, if they were two different entities, this must be mathematically possible, somehow. It would work if the two items were not both geometric points, in the sense that they were not the same type of entity. For points are defined to have location only and here we are at the same location.
Call this new indivisible item “e”. Then this together with a point can be called e*q. We can separate e and q (not divide as this is a separation not a division) if we move the original e along an extent made up of e’s. This extent of e’s can be together with an extent of q’s. The e extent is different from the q extent in that it is a series of positions of the original e position. That is it is the next level of position.
Then we have to have an object of no extent which is somehow different from the usual conception. I am thinking of putting two items of zero size together. One can think of putting a dot on top or beside a dot which is there already. One cannot visualize two items of no extent being together, but one can also not visualize an item of no extent as well.
Yet they are an e( new entity) and a q at the same place, if we have a preexisting space made up of ordinary points (p’s).
In terms of symbols, one may write pop=p as the idea that for an ordinary point two points starting together or being brought in coincidence form again an ordinary point. O is the idea of coincidence.
Then e would be something different. E then cannot be in coincidence with p, if we redefine coincidence as the starting together or coming together of two or more points. so we invent a way for the entities to be together and call it sharing, E*q is e sharing with q . Sharing is a way for two items to come together or be defined together and not combine. This should be possible as seen in the analogy of the overlapping shadows.
This must exist somehow in mathematics, since we must have the notion of something with no parts or no extent and we should be somehow able to put two objects together and still have two. So we have q and e as different entities which do not combine.
Then we can see that e can be different from q, as in an extended space of e’s, I could move an e to another new place of the original place of e, along the e continuum. The e continuum can coexist with the q continuum.
So e is another type of point. It is able to be with another entity which has no extent and not be equal to that entity. P doesn’t share with p, as this is already seen as coincidence. Since points have location only this is the same location and so the same point.
Looking at e, we can’t have e(1)oe(1)=e(1) as then e=p. But possible is e(1)(e(2)), if we have an extent of e outwards from q(1). e(2) is some e on the e extent outwards from q(1). This is what makes e different from q, with p we only have pop=p. That is an e can act as a type of container for other e’s, as two different e’s do not combine similar to a q and an e. That means we can have a shifting movement of one e along another continuum of e’s and q’s. Then e’s can have a new location of the original location of an e which makes e’s different from q’s, which have location and the underlying space only.
e(1)*q(1) is e and q sharing a location. We can separate a copy of e(1) away, since 0-0=0 and we have two extents. Think of a jigsaw puzzle of a landscape being taken apart. e(1)*q(1) =e(1)oe(1)*q(1). Then e(0)(e(1))oe(1)*q(1)=(e(0)*e(1))o*q(1) and q(1) falls away leaving e(0)(e(1)). Then we have e(2)(e(1)) with e(1) moving to an e(2). There might also be an extent of q, along with the e, In which case we can have e(1)*e(2)*q(2).
This is how e is different from q.
To be complete…=p(1)op(1)op(1)op(1) =p(1)op(1)op(1)=p(1)op(1)=p(1), and also… e(1)*e(2)*e(3)*e(4) is not equal to e(1)*e(2)*e(3) is not equal to e(1)*e(2) is not equal to a single e .Also e(0)(e(1)*e(2)*…)). That is, e or multiple e’s can travel along an extent of e’s and q’s.
These would not be points, then, as points have location only. Two moving points could come together to form one point. In Physics or mathematics if we have a plane of points(fixed) there can be a point moving in this plane which takes up the locations of these planar points as it moves, and then we have pop=p as it moves.
But also e’s can move as well and we could have two e’s come together to form e*e when the initial location is moved to the extent of e’s.
Two little nothings still add up to nothing, yet there can be two little nothings there. The two items would just be hidden as one. Let’s hide!-they say. Who can see us? The two items would not be points, for points have a singularity to them. That is, when I combine two moving points, in the usual geometry, it leads to one point, but this doesn’t have to be so with another possible entity.
These two items could be multiple. This is how we can hide! In order to do this we would have to take out the usual concept of a point as being the only entity with no extent and replace it with these new conceptions as other entities which can have no extent. There are these little entities that don’t combine to form a point. This is because a point is already there as the only idea of no extent. We need some room!
If you accept the notion of an entity with no extent you have to accept this new possibility as well. If you want a point, you have to have us too! The new entities say! The notion of no extent leads naturally to concept sharing! The concept of no extent is an example of a concept.
To remove the concept of a point as the only item of no extent, we need to have a concept space, consisting of a concept of multiplicity. This must exist because I must have this further concept of a doubled item of no extent somewhere.
Extension to other concepts:
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 backwards into the idea.
Look at the example of location. For a new location of an initial location we need an extended location, that is there must be an extended location, somewhere to put one of the entities, the extended entity.
Then the extended entity has the location of an initial location, which makes it different from a point, having only location.
The concept sharing of a number:
We can start with a line of p’s with two lines of e’s sharing locations. All in a plane of e’s as shown below:
The numbers 1(3), 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(3). 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 two numbers 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.
We can create a combined number (10(5(16(8(4(2(1) if we bring back coincidence of the ordinary numbers and then allow sharing of the one number so that all ordinary numbers appear once and are shared here.
But then we can see (10(5)=(5(16), as well as all the other equalities, as all these ordinary numbers are sharing together.
The other thing to realize is that these partially sharing numbers are rules and the rules are partially sharing numbers. So that all rules must be represented by partially sharing numbers. Which means all ordinary starting numbers with associated ordinary output numbers are represented and are all equal. Since One set leads to the loop (4(2), (2(1), (1(4), they all do.