On Paradoxes

Day 19

Paradoxes are complicated, and it is rarely clear which way is the correct way to deal with a specific paradox. I have tried here only to have a treatment of the idea of paradox within my schema which makes the ideas and issues of paradoxes clearer to the reader. I have found that the use of the Limit schema, as another way of talking about worlds, makes this discussion easier. What a Limit really denotes is the area under the lim in the area of total possibility space. Thus, if I say that Lim x < Lim y, then I am saying that the area under lim x is less than the area under lim y. Because limits of possibilities are what are denoted by limits, this means that if lim x < lim y, then lim y is stronger than lim x, and that things are possible a world where lim y holds which are impossible in a world where lim x holds.

The idea that Lim World < Lim Reality is the idea that everything that is possible in the world is possible in reality. This just says that what is, is. Thus, if we observe the world in a linear fasion, then we are constantly updating it, and so the world gets weaker and weaker over time as we 'know' more and more. A collapsed world is stronger than a stable world in this vernacular. However, this appears to present a problem. The issue is that it often seems to be the opposite, that as I grow older and understand more of the world, I see things that were once thought to be impossible become possible. Thus, men on the moon, electricity, cell phones, and AI. That what I would have once rejected from my world and perhaps thought that I was hallucinating when I saw it, is now something that I accept. Or perhaps that after learning a new language I am able to comprehend what I did not comprehend before. I heard the sounds of the language, but I did not know how to interpret them as being about the world, or I am able to distinguish tones in the language sounds that I was not able to before. Sure, sometimes a world does appear to be weaker, as when we become closed-minded, but it sometimes appears to do the opposite. How does this vibe with the idea of limits containing each other, of limits limiting possibilities?

There are several ways to possible interpret this. First of all, we could of course think of collapsing worlds. If something was 'known to be impossible' that now is known to be possible, then perhaps the world has collapsed and we just didn't know it. Or perhaps we have tossed away some of our limits and created new ones. Thus, it might be that was impossible in the new world is now possible in the new one. This suggests that the stronger limit has dissolved. Secondly, we could think that what we thought was the limit was not really the limit. After all, we have stated that there is always space within a limit, and that that space is open. That is, the World under the limits is not closed, and anything is possible in that world. It is simply that we do not believe that anything outside the limit of the world is knowable. However, when something enters that world which we thought was not possible, then our previous limit of the possible world is shown to not actually be the limit of the possible world. Something snuck in, and perhaps we did not understand what we were thinking before when we imagined our limits. We can have said that it was impossible to land a man on the moon – and yet we believe in unicorns. So, we believe that anything is possible. Thus, the limits of the world are more like guidelines than laws, and can appear to be 'bent' from time to time, though they are not actually bent. Rather, we just weren't looking closely enough at them. Thirdly, there are consequences of talking about limits which reveal some interesting things about the limits of the world.

The area under a lim is a possible world. The lim represents the limits of the world. A world is made up of gems, of things. We can try to represent a thing in a world as a point in the area under a limit. Call this point Px. We can call the thing x. we know that the thing x is a gem in W, and so we also have the gem x. Thus, under different schemas, Px = thing x = gem x. Under all these schemas we are still talking about the same x, just in different ways. Because we can loosely translate between these three ways of talking about x, we should be able to take ideas about x from one schema to another in order see what is revealed, in the same way that a physical car engine and a computer simulation of a car engine are functionally similar. That is, if knowledge of them produces the same practical knowledge about how fast the car will go.

We know that is is and all things are connected, and we assume the world to be true, or practically true. We functionally know that each gem is accessible from every other gem. So, if gem x is in W, then gem x is also in every gem y in W. We also know that all gem y is also in every gem x. Yet, gem x =/= gem y. (We aren't actually sure of this, we just assume that it is so. Because what makes gems unique is the unknown that lies within them(Ux), the idea that gem x = gem y is the idea that the unknown cores Ux and Uy → U(x/y), where → represents not a traditional logical if statement, but rather the result of an action - or that multiple unknowns we deal with might be the same unknown and we just don't realize it. The idea that they are different cores is the idea that Ux and Uy → Uy and Ux.) So, gem y is some sort of gem ~x. Yet we thought that gem x is in gem y. So, gem ix is in some gem ~x, and vise-versa. This says that if we know some point x under a lim, then we can reach some point ~x which is also under that lim. Also, that if we know some point ~x under the lim, we can reach the point x under the lim. Thus, knowledge about x and~x is functionally interchangeable. This is just what we have said, that it is the limits and not the area that is important in understanding the world. Thus, Px and P~x and Py are all under the lim, are all things in the world, if any of them are. Since we can imagine them and talk about them, that means that they exist. If they exist, then they are in the world. If they are in the world, then they are within the limits of the world, or under the lim.

This implies that paradoxes to not exist within W. That is, if x and ~x are interchangeable, then we can collapse knowledge of them, of x, ~x, and y, down to one thing – we can just talk about x. Because we are talking about x as x, as a particular point under the lim and within the world but we are talking about it as any particular point but not a certain particular point, talking about x also tells us about every point under the lim. If we know every point under the lim then we know the area under the lim, and if we know the area under the lim, then we know – for functionally yet vague powers of 'know' the lim. Thus, if we know anything, then we know the limits of the world. This is just what I have said before. How are we then to understand what a paradox is?

A paradox is supposedly a thing and its opposite existing in the same world. That is, a point that is x and a point that is the negation of that x. call this point Dx. Dx is not ~x in W, because we said that x = ~x, if x and ~x are in w. Thus, Dx cannot be in w. This is just what we have said, that a paradox creates an instability, or a hole in the field, of a world. Dx says that x does not exist. Since we belive everything, we believe that x exists, since we imagine and talk about x, then we belive that x exist, and since is is, and we consider x, then x exists. Yet we can say the same of Dx, that Dx exists. Since x and Dx cannot both exist in w, this means that if x exists and Dx exists, then the limits of w are not the limits of w. The word collapses, and we pass through the paradox to observe a new world with new limits. This world either resolves the paradox by understanding x and Dx in a different way than before, or it runs away from the paradox and ignores it until such time at it becomes important again. This means that there is a hole in the logic field of the world, and a void in the area of the world. We can understand this also as an area under lim w which is not in the area under lim w, or as an interior limit in the area of w, an area in which we cannot go, for if we go we fall through and the paradox is revealed starkly. Because this area is encircled, we can say that there is some lim paradox which is, like a bubble, stuck in the area under lim w. There are two ways to get rid of a bubble. The first is to squeeze it until it dissapears, by which we mean to reduce the limit until the limit become nothing. The second is to move the bubble to the surface of the water and let it open up. This means to move the limit paradox to the edge of lim w, and incorporate the lim paradox into the lim w. The paradox becomes impossible, because it is solved.

It is still possible to think about the paradox, but the area of the paradox is not within the area of the world. How is this? It is because the area of the paradox was never within the area of the world. Remember we said that it was a bubble, a kind of cut-off area that we couldn't think about in w without going beyond the bounds of w. Thus, once the paradox is understood to be solved, the area of the paradox is accessible to us again. Dx could not be, and so Dx was never a point but a lim Dx. Because the lim Dx has been collapsed or integrated, we can say that we have a new world, a lim w'. When we collapse or integrate lim Dx, then Dx becomes a point ~x. a new point ~x that can exist in w', but not in w.. We can think about the paradox as a thing, as a type of gem or way of looking at the world, but we do not now understand it as an actual paradox, but rather as a possible paradox, a paradox that becomes apparent when we look at the world in certain ways or when we have certain limits of the world. We can say that we understand the paradox now, when before we did not. When we talk about the paradox, the paradox seems to make sense to us now.

Thus, we show how to deal with the issue of stronger and weaker worlds when we see something that was once impossible. When we see something that we assume is impossible in w, then we generate a paradox. It is by resolving these paradoxes that we expand he world, and so the new world w' is indeed a stronger world than w, but this happens because we broke the original w. This just says that worlds are fluid. When we resolve the paradox by incorporating it into lim w, then we expand the world. W' becomes stronger than w. When we resolve the paradox by squeezing it until it becomes nothing, then we are reducing the world, and w' is weaker than w. I hope that the first is understood well enough by now, but let me talk once more about the second case.

When we squeeze a paradox, what we are doing is looking at the paradox in a new way. We are resolving the paradox not by incorporating its powers and truths into our world, but by saying that the paradox was never a paradox in the first place. We are saying that when we thought that there was a paradox, we thought that the limits of the world were such-and-such, and that there was some area bound by lim Dx in the area under lim w. When we squeeze the paradox down to nothing, what we are saying is that we misinterpreted what the actual limits of the world were. Now, after we squeeze the paradox, we resolve that there was not a lim Dx within the area of lim w. Rather, what we thought was a lim Dx was really just a point x in lim w. The world becomes weaker because what was once thought to be possible, a lim Dx is not thought to be impossible. The possibilities of the world have been reduced, for all that the possibilities are still infinite. We don't incorporate the lim Dx into the new lim w', because the lim Dx is now thought to be a mistake. When we talk about the paradox, the paradox appears to be nonsense to us now, because when we re-bubble lim Dx, we introduce an instability into our world, and so anything that appears to lead to Dx makes the world more unstable, and so appears to be a lie.