A model of Smolin's postulate

[Mighara Sovmadhi]

I'm referring to Lee Smolin and his idea of changing laws of physics.

So there is an article in the Stanford Encyclopedia of Philosophy on something called infinitary logic. The idea is to have a language allowing conjunctions, variables, and quantifiers with ranges indexed by transfinite numbers. The format reads (roughly) as L(k,l), where k,l are infinite ordinals like omega or perhaps very tall ordinals associated with very large cardinals.

Let us suppose that the universe has an L assigned to it. At t = 0, k,l are 0. But due to the passage of time, these values increase. The initial expansion of space then reflects this shift, in fact space is correlated with l while time itself corresponds to k. I'm writing this on a phone so unfortunately I can't justify this particular correspondence right now as it requires an equation I can't write via phone, but anyway the idea is that by now, time's ordinal is the initial ordinal for aleph-2. The rules of dimensional perception say that we can see complete enclosures in the next dimension down from the one we're "on," like we can see all sides of polygons at once and if we were four-dimensional we could observe polyhedra likewise. So if the Continuum is aleph-1, then we can "see" continuity due to us being in time that is higher-dimensional.

Now, this model is supposed to explain the accelerated expansion of space as a result of a shift in L for our universe. Modulo Smolin's postulate, the model predicts, therefore, an infinite sequence of shifts, each of which "sort of" alters the "laws of physics."

At worst, this could make for a good hard scifi novel called THE SHIFT

EDIT: according to the equation, btw, the next switch would be to aleph-4, not aleph-3, and the one after would be to aleph-(omega-4). So perhaps events much more extreme than accelerated expansion. Does this make the model into a testable hypothesis? I hope so!

EDIT 2: Here's the equation, partly:

{{{{0 ↑⁰ ℵ₀} + ℵ₁} × ℵ₂}^ℵ₃} ↑↑ ℵ₄}

[Cord Hurn]

The rules of dimensional perception say that we can see complete enclosures in the next dimension down from the one we're "on," like we can see all sides of polygons at once and if we were four-dimensional we could observe polyhedra likewise. So if the Continuum is aleph-1, then we can "see" continuity due to us being in time that is higher-dimensional.

The only way we could see all sides of polygons at once is if all sides were transparent, and that goes for polyhedra in general.

EDIT: according to the equation, btw, the next switch would be to aleph-4, not aleph-3, and the one after would be to aleph-(omega-4). So perhaps events much more extreme than accelerated expansion. Does this make the model into a testable hypothesis? I hope so!

I don't believe it's testable, because the whole idea is fantastical nonsense.

[Mighara Sovmadhi]

The only way we could see all sides of polygons at once is if all sides were transparent, and that goes for polyhedra in general.

You've never read Flatland? They explain this principle very well there. It's an established fact...

[Fist and Faith]

I'm not sure the story actually establishes that observers from any dimension can observe the totality of things in the dimension immediately below it as fact.

[Mighara Sovmadhi]

Not the totality of all things, but a given thing. Like, we can see all the sides of a square at the same time, because we're three-dimensional; if we were four-dimensional, we could see all the sides of a cube at the same time; etc. I don't actually know for sure that we can transplant this into the aleph-realm; that idea is part of my attempt at a hypothesis.

[Fist and Faith]

No, I didn't mean all things, either. I just don't know that we know enough about extra spacial dimensions to know if the principal holds above the 3rd.

[Mighara Sovmadhi]

We do know this.

A 4-D being would be a god to us. It would see everything in our world. It could even look inside your stomach and remove your breakfast without cutting through your skin, just like you could remove a dot inside a circle by moving it up into the third dimension, perpendicular to the circle, without breaking the circle.

A hyperbeing can effortlessly remove things before your very eyes, giving you the impression that the objects simply disappeared. The hyperbeing can also see inside any 3-D object or life form, and if necessary remove anything from inside. The being can look inside our intestines, or remove a tumor from our brain without ever cutting through the skin. A pair of gloves can be easily transformed into two left or two right gloves. And 3-D knots fall apart in the hands of a hyperbeing, much as a 2-D knot (a loop of string lying on a plane) can easily be undone by a 3-D being simply by lifting the end of the loop up into the third dimension.

[Fist and Faith]

We don't know that. It might be that way, assuming the 4th dimension is to the 3rd as the 3rd is to the 2nd. But that might not be the case. What if the theories Brian Greene discusses here more accurately describe things:
https://www.ted.com/talks/brian_greene_ ... anguage=en
The whole thing is great, but the particularly relevant part begins at 5:53.

[Mighara Sovmadhi]

Yeah, he's talking about dimensions in a different sense. The stuff I was referencing is provable in pure geometry, has nothing direct to do with dimensions being curled up or varying in size or what. See here for an example.

[Fist and Faith]

What you're talking about is only an internally consistent theory. The math works. It works for String Theory, also. Neither has any actual empirical evidence to support it. There might not be a 4th dimension in the sense you are talking about. Mathematical theory, sound and consistent, is proof only of its own soundness and consistency. Math can describe reality. But it cannot prove the existence of something for which there is no proof other than the math. We can calculate the shape of a hypercube, but that doesn't mean there is any such thing.

Now, I would sure love to see one of those four-dimensional objects moving nearby! But I haven't. And nobody else has. And in all our examinations of everything we can examine, we have not seen one. Nor have we seen the results of one. If I moved through Flatland, I'd destroy it. The 2-D beings wouldn't see a cross section of me, because I'd tear their 2-D world apart. It happens all the time. If anything 2-Dimensional exists in the air I'm walking through, it's not going to survive my passing. The air is in this spot one moment, and my body occupies that same spot the next. Is there reason to believe a 4-D object passing through my house and my body would not destroy it and kill me? But we don't see the wreckage of such things. A supernova has a known cause. As does a tornado. Where are the unexplainable paths of destruction in the wake of 4-D motion?

[Mighara Sovmadhi]

What you're talking about is only an internally consistent theory. The math works. It works for String Theory, also. Neither has any actual empirical evidence to support it.

The equations of string theory aren't "alternatives" to the equations I started out with. There's no conflict. So it's not like, "Well, either could be true, so..." That's not the issue.

I'm not trying to get into a debate about the merits of empiricism or rationalism or skepticism, at any rate. It's common knowledge among mathematicians and physicists that if we were four-dimensional, we would perceive all the sides of polyhedra at once. As for non-mathematicians and non-physicists, they're free to doubt whatever they want, I suppose.

[Fist and Faith]

Although, if I was four-dimensional, I would not have to see all sides of a polyhedra at once. After all, I am three-dimensional. But if I look at a 2-D square on-edge, I see nothing. Not if it's truly 2-D. It's only when I look at it from an angle that I see something in 2-D, eh? The shape would change as my angle of view changed. I would only see the actual square from a certain angle.

I assume it's possible that, if I was four-dimensional, I could look at a cube "on edge", and see only a 2-D shape. The shape depending on what angle I was viewing it from. As I change my viewing angle, I might see more and more of all six sides come into view, until I saw the whole cube, all six sides, at once.

[Mighara Sovmadhi]

That sounds about right. And to reiterate, I don't know that these principles carry over into the transfinite realm. We can show that any n-dimensional space, for n greater than 0, has Continuum-many points. So my working assumption is that ℵ0-dimensional space has Continuum-many points (I'm not quite sure this is so but we know, for example, that n to the power of ℵ0 = the cardinality of the Continuum, and so does ℵ0 to the power of ℵ0, so n^ℵ0 = ℵ0^ℵ0). I favor the Continuum Hypothesis (℘(ℵ0) = ℵ1) and the Generalized CH (℘(ℵa) = ℵ(a+1)), so then I assume that ℵa-dimensional space always has ℵ(a+1)-many points. But here I run the risk of pseudomathematics, and I don't want to commit to these statements too strongly (for now).

[Fist and Faith]

As I change my viewing angle, I might see more and more of all six sides come into view, until I saw the whole cube, all six sides, at once.

I'm probably wrong there. If 4>3 works the same as 3>2, I would see all six sides of the cube as soon as I changed my viewing angle from "on edge". But I wouldn't see it as a cube right away, just as I don't currently see the square as a square the moment I move away from "on edge".