The Universe’s Ultimate Complexity Revealed by Simple Quantum Games
quantamagazine.org
quantamagazine.org
Hmm, yeah, I thought there will be, like with all of those quantum tests trying to say if space is continuous.
> “In the real world you’re limited by your experimental setup,” Yuen said. “How do you distinguish between 100 percent and 99.9999 percent?”
And that is exactly the problem with idea of discrete space. Once you try to pin-down those pixels, it appears universe is blurry before you can go down to that resolution. "In the real world, you are limited by real world".
If you can transmit 1 real number you can transmit any number of real numbers.
For any finite n, the cardinality of R^n and R are the same.
I'm guessing that those bijections break down once you step into infinite dimensional spaces, but I haven't found a simple reference for this. Can someone verify this?
From the article:
> Researchers talk about a state space as having a certain number of dimensions, reflecting the number of independent characteristics you can adjust in the underlying system.
> For example, even a sock drawer has a state space. Any sock might be described by its color, its length, its material, and how raggedy and worn it is. In this case, the dimension of the sock drawer’s state space is four.
That's meaningless IMHO, 4 is the same as 1.
Your mistake is to think that 4 is the same as 1. It is not. You are putting a hand up to one eye and complaining that stereoscopy is meaningless. You are pretending that only set theory exists, and claiming that therefore linear algebra is meaningless. It is rhetorical nonsense!
Why? If I can describe a system with 1 number why invent 4 parameters to describe it? And why not 100? The choice seems arbitrary.
You are like a blind man feeling the tail of an elephant and claiming the elephant's nature is the same as a rope!
I suppose you could encode it all to one number so that structure is preserved, but dimensions themselves are not going anywhere, they are in your choice of encoding that must reside outside of your one coordinate.
Sets with the same cardinality can have wildly different topologies, which describe the possible paths through the space, which in turn describes the nature of functions into and out of that space, as well as transformations and relationships within that space.
A line through the center of a sphere doesn't necessarily touch the sphere in a "4 dimensional" connectivity structure, must touch the sphere in a "3 dimensional" connectivity structure, while a sphere doesn't even exist in a "1 dimensional" connectivity structure.
The main reason that we use "4 dimensional" spacetimes for particle physics is that there are three independent components of motion with respect to time, and so it makes the math simpler to allow for independent variation of those parameters to functions, instead of compressing them into a single encoded number, eg, derivatives wouldn't function correctly (or likely even be defined) on a single parameter model. An additional reason that we don't use a "1 dimensional" spacetime is because the product of a "1 dimensional" time and a "1 dimensional" space is a "2 dimensional" spacetime -- and we don't have any models of reality that lack either space or time. (Which is good: any model needs to explain both, if only as an emergent feature.)
Topology is about connectivity, not scale; dimension is a topological fact.
Just wanted to add that if you'd like to take a more set-theoretic approach to the topic of dimensionality, you might be interested in Matroid theory, which is a generalization of the notion of linear independence in linear algebra.
I found this video to be helpful in getting a basic sense for what Matroids are: https://www.youtube.com/watch?v=FKTPFy-MGp0
Start with a simulation using the best known but complex formulas. Then use the genetic algorithm (mutation and cross breeding) to find the most compact formulas/algorithms that can mirror/predict the complex one. Who knows, the whole shebang may be reducible to something as simple as E = mc^2 or Newton's F=ma.
Many consider him to be wrong.