Infinity and the Finite world - A Sketch

The world was observed by man as a volume filled with experience. All his experience was set in space, a 3D space. It was in fact these experience that made sure a 3D spatial theory evolved from the 2D possibility.
In this volume, things were stretched in 3 dimensions. From this, man exctracted the 2-dimensional plane. This became the flat surface. And from this came the notion of the line: the 1-dimensional edges of the plane. Of course, this was only an abstract idea, since such a line had never, and in fact could have never, been seen. The eye was incapable of seeing the line and even thinking about it. But as an abstract idea, it was there. It was the edges of the plane. And from this abstract notion came the second abstract idea: the point. The point was where two lines met, or crossed over one another. Since the line had no dimension but stretched, and the point was not stretched, thus the point had no dimension.
And from this point came the idea of a separate identity from all other points or identities upon that line. Here, mathematics became a slave of its own progress, its own abstractions, its own ideas - just as language had become a slave of its own progress, its own abstractions, its own ideas and semantics and grammar and syntax.
Whereas there is no evidence in the world of experience of a line or a point, then such abstractions were arbitrary and therefore, not necessarily applicable to the real world.
So, to say we are at a point in the real world or such-and-such is bound by a line is to make a semantical mistake.
For the world is either perceived as volume or plane. In fact which one it is can be debated and we shall debate it. But for now, let us discuss the more abstract ideas of maths.
If the world is either volume or plane, then we should use models that are described in those terms. Indeed, there is no such thing as a line. A line is only the edges of a plane. But there is no need for the abstract line. A plane is a plane. It is not bound by lines at all. All it is is a finite plane.
For example this: O.
In reality, we do not see the binding line. We just see the white bits and where the white bits end, we call a line. In fact, all that happens is that the plane ends. A plane is not defined by lines at all, but defined by volume. And volume is defined by perception and experience which we have set is the basis of our method towards comprehension and understanding.
So, we have no reason to use the idea of the line except in the same way we use "border", which is nothing but an arbitrary idea. For do borders really exist to divide nations? Or are they simply made-up to show this division?
Indeed, the different planes exist, but are not defined from lines at all. Indeed, lines are defined by them. It is from knowing the semantics of "plane" and "edge" that we reach the idea of "line".
But how can we reach the idea of "plane" from "line"? To say that a plane is an area bound by lines is to say that a plane is a plane bound by lines. So you then include the word "plane" in it which is wrong, for then you will have a regress of definitions.
But to say that a line is the "boundries of a plane" is quite alright. One may object that a line is the same as a boundry. If that is so, then we simply say that "lines are what distinguish planes from each other." By this definition, we have kept the plane as our base unit and extracted the idea of the line from it.

The next part we will try to discuss is the numbers. Numbers came about from "finiteness".
So, a plane that was distinguished from another became the basis for a different identity. Thus, each identity became a singularity and each singularity became known as 1. And therefore, all other numbers were extracted from this notion.
0 or zero was a later concept and meant "the absence of 1". But indeed, most primitive civilisations did not regard 0 and in fact it was developed later as a sign of nothingness.
And when they did develop it, they developed it for things such as commerce, astronomy, geometry and pure algebra. In the latter two disciplines, 0 can be considered since they are abstract disciplines, dealing with the world of numbers. This was a world of abstraction where 1 meant something without referring to anything. 0 too.
In the real world, especially in trade and commerce, 0 was used as the absence of value.
Yet, during this process of advance, the nature of how maths was born was forgotten and men delved deeper into the abstract realms of this mystical world - so mystical that Pythagoras created a cult from it!
Now, had they actually cared more for its origins than its arbitrary development, they would have noticed that the number was a multiple of 1 and that 1 was the singularity perceived by the identifying mind of the human awareness.
But this 1 was known via an experience, such as the vision of an animal in a herd, a woman's voice in a crowd of men, an aromatic taste amongst the many tastes of a perfect dish.
Therefore, this was not just 1 but 1x.
And it was by not understanding the notion of 'x' completely that the algebraic functions were doomed as well as was the mathematical proceedings of the later centuries after the development of algebra.
For 'x' was the quality, the essence, the experience that was identified as 1 identity. Indeed, it was when the quantifier and the quality were divorced that mathematics sank into an ocean of absurdity and self-refuting quagmire.
Such self-refutations can be seen when mathematic is used to describe the real world, yet certain mathematical notions make little sense when applied to the real world - Zeno's paradox of motion and the Infinite past paradox are all part of this mathematical fiasco that had its origins in the separation of quantifier and quality.
So, we are yet to discuss what is a quantifier, but let us first discuss the quality, for the quality came first and the quantifier was nothing but the work of the human mind that singled out that quality amongst a collection of qualities and thus made it a separate entity - and understood from the identification, this separation, the idea of "1".
So, what is the quality? The 'x'? It is of course the experience.
But, what is it that we should know of it?
We should remember that all experiences are set in a partcular background, for example a sound comes from "somewhere", the touch is felt "somewhere", that taste is "somewhere" on the tongue, and so on. Therefore, "place" is required for each experience.
Therefore, 'x' is an experience set in place. Can an experience exist without space? This question is discussed in another work of ours and thus leave much of it out of this sketch.
But it should be known that a quality cannot exist without space (at least for us) and space cannot exist without quality (again, at least for us). Indeed, one is tempted to say that an experience is a quality plus space. That there is no such thing as simply space or quality and that they are one. However, a partcular space may have the quality of, say, black at one time and white at another. Or any other colour. This means that the same space can be seen as another colour. But, again, we know it must be seen as some type of colour.
But what if it is not?
Then it would be like air or glass - we see through it, up to the next visible thing. Therefore, space must exist for visible things to live with it. This space, some have reminded us, is relative and not absolute. We can easily accept this by saying that space is the product of the mind - indeed it is one of things it is aware of - but to be aware of space is to be aware of the qualities - and thus we should be able to see "black" which is the absence of light. But then, if black is a quality, then we have indeed reached a point that we cannot separate space and quality.
But let us return to knowing what this quality is. It is of course, finite. That is, it has been identified by us and therefore separated from the background. Therefore, it is finite and bound.
Also, it is separated from a background. That is, it's non-existence would be the non-existence of a finite entity from a background that exists no matter what. This is the experience we get from the real world and therefore, we should understand that the space would exist no matter what, if that particular experience suddenly became non-existent.
This can be explained in the following way:
If there was a tree blocking the path, you would not see the path - but if you got rid of the tree, you would see the cloudy sky. If you got rid of the cloud, you would then see the blue sky. All this is because there is a space out there and the quality in which we perceive this space depends on the light and energy and so on. Thus, the space does exist, but it is inevitable for it not to be perceived as some sort of quality.
So, the fact that space does exist at least as something within the range of our awareness has been proven in our other work.
But, if we have all experiences in the background of time, even if we don't have any experiences, we will still know that the capacity is there - that it is possible - therefore, there does remain a "potential" background for the experiences to occur within. This potential background shows actuality only in pieces and parts, here and there, with finite experiences which occupy finite areas of space with a particular quality.
This awareness has no end to what it can perceive, but say sight does have a finiteness about it. You cannot see behind you, for example. But the potentiality exists, therefore the potential background does exist. This we cannot deny, since the existence of an actuality proves the existence of the potentiality.
Therefore, we can safely conclude that space is this potentiality and the experience is the actuality within this area.
Now - what is 'x'?
Of course, an experience. Therefore, it is an actuality. So, for example an apple is 1x. If the apple did not exist, it would be 0x. But it would not mean absence of all value. It just means an absence of an actuality. What remains is the potentiality.
So, when we say 0x, all we are saying is that 'x' has not become actualised in this potentiality. But it does not negate the fact that its potential does exist.
Therefore, 0 must apply to the 'x' and without the 'x', it may be mistakenly thought that 'nothing' exists, meaning nothingness, or 'Neesti' in the Persian language.
After all, the potential of 'x' existing still remains.
Additionally, all experiences are known to us in a finite space. This means that our range of sense and perception is finite. Therefore, the potential background is also finite. The only reason we guess it might be infinite is because we "assume" that space goes on forever. This is an assumption with no basis. No basis in empiricism and no basis in rationalism. The concept of an infinite space is one so foreign to the human experience that its concoction is a strange event indeed. How could we come up with such an incredible concept? What infinity had we seen at all? None! So, what did we do to reach infinity?
Simple. We just said that our range of perception was finite. Therefore, we called this finite range a singularity since we distinguished it from another perceived range. Therefore, we called the finite range 'x' and we saw that it was 1x. We then added the finite ranges on and on endlessly.
1x + 1x + 1x +...
But why?! Why 'endlessly'?!! What was the reason? Had we any intuition that space went on forever?!!!!
Yet, all we could perceive were finite entities, all in volumes of perception (perception range).
So - there was no infinity in the world of experience at all. So - how did the infinity come to be in the mathematical realm??? This we shall see.
To get any number, one must identify a singularity from the background. This is 1x.
Then, if there was another 1x which was different in some respect from the previous, we would say there are (1+1)x or 2x. This is how we count.
But say, someone said: count but never stop.
That is (1+1+1+...)x.
Of course, like this, we will never get 'infinity'. Because however high we go, we are still in the realm of the finite entities. Infinity is simply the potentiality of these finite entities in our mind. That is, in our mind, we can accept the existence of ANY number of 'x's. Therefore, we think that this means that there can be infinite 'x's. This belief is wrong.
Infinite is not a magnitude. It is endlessness. And our mind CANNOT accept an endless chain of things. That is because our mind's can accept only a finite number in a finite range.
However, we would never be able to count up to this so-called infinity. But could this infinity exist at once, regardless of whether we could count it?
Well, yes. But we have no reason to believe it does. Therefore, why should we say it does? All evidence points that it does not, so should we get bogged down into the debate of infinite or not?
But then again, let us do bogged down in this debate - for it is a debate for human understanding, which is valued above all amongst humans.
What is true is that infinity can exist in one go.
But what is infinity 'x's? Surely that means that the entire potential has been taken up - that is not true. For something may be infinite in one dimension only, and thus finite in other dimensions. Such as the following:

__________________________________
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The ___ lines show the infinite potential backgound which is infinite in all dimensions.
The ---- lines show the infinite actual 'x' which is infinite in one dimension but finite in the other. That is, it has infinite length but a finite width.

That is, when some infinities seem 'bigger' than others, it is only the fact that the 'bigger' ones are only infinite in more than just one aspect.
There are infinite aspects to be infinite in - therefore, you can have infinite types of infinite!
But, here's the point we are trying to make - how many basic infinite categories can you have?
Three.
The one with the beginning.
The one with the end.
The one with neither end nor beginning.
That is:
1,2,3,...
...,3,2,1
...-1,0,1,...
The third type needs explanation at a later time, since it also requires us to have a look at zero and the negative numbers and so now we will deal with the first two infinite categories only.
The first type is saying that the infinite line begins from 1 and continues counting forever.
Of course, we have already said that this is not infinite because we will never reach forever. Since it has no end, then we can only say that the numbers get bigger - and never reach an end. SINCE THERE ARE INFINITE 'FINITE' NUMBERS, WE WILL ALWAYS BE IN A FINITE NUMBER!

"There is only one infinite and that is the infinite number of finite numbers."

However, the second type is strange:
...3,2,1
This implies that we do not begin, but we do end.
Of course, we believe this is paradoxical. How can something end at all if it has not begun?
To say something has ended is to say that the numbers were being counted one by one towards a final number - a final point.
But to say that these numbers were being counted is to say that they had a beginning.
Because to count, you count one number FIRST and then another number SECOND. This is what gives it SUCCESSION. An example would be Time.
So, to state succession is to imply first and second and to imply this is to imply beginning and direction. That is, the numbers are counted FROM a point in a particular DIRECTION from that point.
Some may object that to imply a direction from an arbitrary point is enough to set an infinite past into motion. But this is wrong. Because in that case, motion would start from that arbitrary point and not from before it, whereas we know that points before that arbitrary point also moved in that direction.
And to simply set a direction without saying from which point this direction moves is also wrong. After all, a vector is a translation from one point to another point and in that direction. And to say that there is a direction is not enough. Very well, there is a direction, but where is the cursor that indicates our position in the timeline, so we can move in that direction?
It is not anywhere, because there is no beginning.
And here is the basic fallacy in the reasoning of those believers in an infinite past.
The idicator/cursor can be nowhere since to imply it is anywhere at any time is to say it is not someplace else. And to say it is not someplace else is because that other place comes after or before that point in which it is.
Therefore, there must be a reason that it is there and not another place.
The reason can only be that resting on that point was the only possible outcome. That is, that there was still time left for it to reach a point further down - or that some time had passed from the point further back.
If that is the only reason, then it is the number of time-points that pass that separate the points. Therefore, if there is only 3 points between them, it will take 3 units to reach the final point.
If there is 100 points, then 100 units stand in between and 100 units must pass to reach that final point.
Therefore, we can say that the differences of points is due to their PRIORITY and since PRIORITY does exist in Time, then we can safely say that the one that is further up in the timeline is reached earlier because there is LESS time to reach it then there is to reach the point that comes after.
Therefore, to say there is less of something for A then there is for B is to say that that particular something is finite but expanding. Also, to say there is less of something for A compared to B is to say that if you add all the components of A and that of B, you will find that B has extra components.
Now, for an infinity "more" and "less" have no meaning.
Some have used Cantor's proof of different infinities as proof for some infinities being larger than others.
Example would be:
1,3,5,7,9,...
1,2,3,4,5,6,7,8,9,...
Now, there is obviously more numbers in the second - but is there?
Since all numbers are multiples of one:
1+1+1+1+1+1+...
1+1+1+1+1+1+1+1+1+1+1+...
So, if all the 1s add up forever, will there ever be a difference?
Now, what if we say there are more elements in the second 'set'?
Well, how many elements are there in the first set?
1,3,5,7,9,...
1,2,3,4,5,...

And

1,2,3,4,5,6,7,8,9,...
1,2,3,4,5,6,7,8,9,...

But as you see, the elements of both sets are equal in number.

But as for Cantor's proof that between two infinities, one may include all the others, but not include at least one of the other sets.
They use this proof to show that one infinite can have an element that another infinite does not have, even though all other elements are included.
So, for example:
1,2,3,4,5,...
0,1,2,3,4,5,...
Well - our response is that the number of elements will still be infinite for both and since we can never stop this progression (for to do so is to make it finite) then the second can never be established as larger.
Now - if we were to pair each with the one that looks the same: 1,1 and 2,2 and not 1,2:
1,2,3,4,5,...
0,1,2,3,4,5,...
But this should not be accepted, since to say how it is to pair them means you can say it should pair in a way that is internally and analytically impossible. For example, the proof of Cantor's theorem about the set of non-selfish numbers.
But that is impossible. For we are telling to the set to be paired with something and saying that the pair should not be in and therefore should be in it.
This is a logical paradox - a fallacy - I don't quite understand how he got away with it.
To say that a set should include numbers and be paired off with another set and that set should not be in it and therefore be in it is a paradox of the worst kind.

So, we can conclude that no infinite is greater than the other in "number of elements" (cardinality) :
1,2,3,4,5,6,...
0,1,2,3,4,5,...
As you see, both have '5' numbers before the infinity sign.

Also, we can conclude that in an infinity which begins one unit sooner and has at least an one extra different subset, then we can believe that all other subsets match but this one. Does this mean that this is different?
Yes - but not in a profound way. Both are still infinity and not one is greater. If they are indeed infinites, then they cannot be added to.
So 1,2,3,4,5,... cannot have anything added on to them. But that means they cannot have anything added on to them horizontally. That is, you cannot have more numbers than infinity. But, one does have one more subset then the rest - but no it does not - to say that is to imply that infinity can have more - but infinity is "something which has no more nor less - for it has no magnitude - indeed it is not a number, nor is it a number of numbers. It is just a way of saying "endless" in mathematical terms."
Therefore, nothing can be added to infinity. Nothing can be taken off it. Therefore, to say that there is a subset in infinity which another infinity does not have is to say that there is another subset added to the infinity.
But to say this is to presume the infinity. But we should come to terms with this everlasting truth that an "infinity" cannot be presumed since it is not a number. It is not a magnitude. It is not a value. It cannot increase. It cannot decrease. To presume an infinity and to presume another and to compare them is NOT to compare their number because to compare their number is to presume that an Infinity is Finite. That is, that an infinity has been counted and seen as equal to another infinity except for one particular infinite. But that is false.
An infinite is not a set of numbers - it is just an endless progression - therefore however we pair the two infinities up, there is more to pair up - and to say that except for that special subset, all others are obviously paired is a good objection.
Our response is: an infinite has no magnitude - therefore more or less does not apply to it.
Our other response is: an infinite cannot have an extra subset, for it wouldn't be an infinity and it would have to begin somewhere and that is "nothingness" and it will progress to the first singularity it identifies and that is "1" and then it continues to infinity.
To say that now we begin with say "2" is just a play with words. It's to say that "one" should be read "two" and that "two" should be understood as the first in line.
This is confusing the semantics of the numbers by playing with the numbers.
The truth is that by sinking deeper and deeper into this swamp we have made for ourselves, this intellectual swamp, the more confused we get and we think it is working, when in fact it bears so many inherent contradictions - these should not exist in maths.

So - we have asserted our belief that all progressions should have a beginning and that infinity is not a number nor a magnitude by definition and that nothing can be added to it.
We have also done the following:
> Shown that all infinities are equal in cardinality
> Shown that all potential infinities cannot exist in a progression
> Shown that all beliefs that some infinities are greater than others are a result of stopping the progression towards infinity (thus making it finite) and then comparing
> Shown that Cantor's theorem is demanding the impossible
> Shown that any progression should have a beginning and a direction and the beginning should be a basic unit of that progression (which is 1)

Some may say that the following progression rejects this last understanding:

3,4,5,6,7,...

Where 3 is (1+2) and so on.
But that is acceptable, since it does begin with 1.

And what about 1,4,9,16,25,...?
Well, that also begins with 1.

So, the 'x' must begin with 1. Must!
All numbers must be known as we have always known them - multiples of 1. If we change our understanding which was given to us directly by our minds without any effort on our part or on the part of nature, then we may fall into contradictions with reality and intuition.

But, some say that the following infinity has more than the other:

x-1 where x is a natural number
x+1 where x is a natural number

0,1,2,3,4,...
2,3,4,5,6,...

But in this case, can we say that the first infinity has two more elements? We'll say:
You cannot compare the two infinities unless you compare them by their cardinalities - because 2 is not the same 2 of the other, but is equivalent to the other's 0. This is due to the rule that all infinities begin with the first, then the second and so on and these are shown by:
1,2,3,...
And this is the rule for all progressions. Therefore:

1,4,9,25,...
1,2,3,4,...

So, you can see that by comparing this to:

0,5,10,15,...
1,2, 3, 4,...

You'll get:

1,2,3,4,...
1,2,3,4,...
And that is equal.
So - you compare with cardinality-numbers. You should not judge a number by its individual value, but by its equivalent in the other infinity - in that way, you will find that all infinities are equivalent - showing the correct understanding of mathematics.