Is information lost when going from the fourth dimension to the third dimension? — Wallows

Not necessarily. Of course it depends on how you metaphysically interpret Cantor.

In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence and introduced the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined countable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space Rn has the same power as the real numbers R, as does a countably infinite product of copies of R. — Wiki

Yes, the encoding of coordinates in an n-dimensional space into a single real number X is actually easy to describe:

The first n digits of X are the first digits after the decimal place of the n coordinates.
The second n digits of X are the first digits before the decimal place of the n coordinates.
The third n digits of X are the second digits after the decimal place of the n coordinates.
The fourth n digits of X are the second digits before the decimal place of the n coordinates.
and so on...

The map is discontinuous and geometrically unintuitive. But its algebraic structure is very simple.

Indeed. And I believe there are other approaches. I vaguely remember interleaving the bits in another way. The black and seamless sea of the unit interval expressed with ones and zeroes... Very beautiful math. Cantor was a poet.

All the applications are nice, etc., but there is just a real joy in trying to capture the infinite with an exact thinking (in terms of rules that make the results not-just-subjective.)

Since this would only be a mathematical game, the only answer that would make sense would be based on how we're setting up the rules of the mathematical game we're playing.

Since this would only be a mathematical game, the only answer that would make sense would be based on how we're setting up the rules of the mathematical game we're playing. — Terrapin Station

What do you mean? Trying to figure out what...

So, information is not lost when going to a lower dimension? But, then how can "time" exist in 3-dimensions?

I think your question needs clarifying. What are you trying to ask? I expect you are aware that when we want to include time as a dimension of our physical analysis we use four dimensions and call it spacetime.

What are you trying to ask? — andrewk

I'm basically asking if you can describe n dimensions in n-1 dimensions. Does this apply from going from n to n-k dimensions also or is 1 to 1 correspondence only applicable/maintained for/to a single lower dimension?

So, information is not lost when going to a lower dimension? But, then how can "time" exist in 3-dimensions? — Wallows

That depends on the metaphysical interpretation that one gives math. The real numbers are very strange if one looks into them. 'Most' real numbers contain an 'infinite' amount of information. They can't be compressed into a program that generates arbitrarily accurate rational approximations of them. But this means most real numbers exist only as a background that can never be foregrounded. It's pretty psychedelic and yet it's mainstream math.

As far as time goes, I'm personality inclined to separate the mathematical representation of time from time itself. We can usefully spatialize time in our theories, but who is to say that this is time itself and not some handy image of time? For some philosophers time is the name of the existence. To be alive is to be time or live time, drag a memory into a desired future.

It depends on what you mean by describe. We've already established that one can construct a one-to-one correspondence ('bijection') between points in n-dimensional space and points in 1-dimensional space. That means the two sets of points have the same 'cardinality'. It follows that n-dimensional and m-dimensional Euclidean space both have the same cardinality (the same 'order of infinity', one might say), for any two positive integers n and m.

But those one-to-one mappings are not useful for most purposes. A more meaningful question is 'can we embed an n-dimensional space in a m-dimensional space, where n>m, without the former losing some of its structure?' (eg embed a solid sphere in a plane, or embed a plane in a line). It is a proven theorem of topology that the answer to that is NO. This accords with our intuition that the larger-dimensional space 'would not fit' inside the smaller one. It would have to be 'flattened' in order to put it there.

It depends on what you mean by describe. — andrewk

You mean to assert that there are no conditions that would render the truth value of embedding a larger countably infinite dimension into a smaller one as true?

It is a proven theorem of topology that the answer to that is NO. — andrewk

Could you point me this theorem. I wish to read about it.

Thanks so much.

It would have to be 'flattened' in order to put it there. — andrewk

Hmm, so is this just another way of saying that you would have to apply some compression theorem to achieve that? Or truncate it? Rounding off would be cheating.

The theorem is set out here. It is that homeomorphic manifolds have the same dimension.

I'm afraid your other two questions are not meaningful and have no answer. You are attempting to paraphrase something and losing its key features in the process. If you really want to grasp the meaning of all this, I suggest you study first topology, and then differential geometry. The study is well worth the effort. They are truly beautiful subjects - the Music of the Spheres.

I don't have the willpower to study topology and differential geometry. I failed vector calculus twice! Can't really grasp the subject. I like algebra though.

Thanks for the paper.

What do you mean? Trying to figure out what... — Wallows

Dimensionality other than three dimensions (plus time if you want to consider that a dimension) isn't real. It's just a mathematical game that we can play.

Dimensionality other than three dimensions (plus time if you want to consider that a dimension) isn't real. It's just a mathematical game that we can play. — Terrapin Station

But, the instrumentality argument of mathematics would be a backbone in asserting truth or "reality" of these "mathematical games".

Huh?

Huh? — Terrapin Station

So, I take it you don't believe that mathematics is a form of reality? Ex. Platonism?

No, I'm not a platonist. I'm somewhere between a subjectivist and social constructivist on ontology of mathematics.

And more generally I'm a nominalist in the sense where I deny that there are any real (that is, extramental) abstracts period.

But, the computer you are using and the room in which you don't expect the roof to collapse are all the results of applied maths.

If we were to try and communicate with aliens, perhaps one day, it would be through the language of numbers, no?

Where does the number two exist in? Our heads only?

Let' say for the sake of argument that we live in 4-dimensions.

Now, we want to go to 3-dimensions to describe something.

Is information lost when going from the fourth dimension to the third dimension? — Wallows

Holograms are 3D images that are encoded on a 2D surface. The holographic principle in cosmology uses this idea.

Mathematics is an invented language, initially based on how we think about relations, and then the bulk of it is akin to extrapolating how we think about relations into abstract "game" of sorts.

That's not to say that it's not useful, but so is natural language. It's just with natural language, not many people are under the illusion that it exists independent of us.

So, information is not lost or is lost? How they are represented can differ and is a side issue I suppose.

As far as I know, no information need be lost in principle. In practice, holograms created from real subjects will lose some information in terms of resolution.

In a side-by-side comparison under optimal conditions, a holographic image is visually indistinguishable from the actual subject. A microscopic level of detail throughout the recorded volume of space can be reproduced. — Holography - Wikipedia

In terms of the holographic principle, no information is lost.

In the case of a black hole, the insight was that the informational content of all the objects that have fallen into the hole might be entirely contained in surface fluctuations of the event horizon. — Holographic principle - Wikipedia

Mathematics is an invented language, initially based on how we think about relations, and then the bulk of it is akin to extrapolating how we think about relations into abstract "game" of sorts. — Terrapin Station

Aren't natural languages invented too?

I bet that English and Urdu didn't just surface from genes or something.

Aren't natural languages invented too? — ssu

Yes, of course. I'm just stressing the fact that it's invented, partially because you never know what someone is going to assume if you just say that it's a language.

Is information lost when going from the fourth dimension to the third dimension? — Wallows

I'm basically asking if you can describe n dimensions in n-1 dimensions. Does this apply from going from n to n-k dimensions also or is 1 to 1 correspondence only applicable/maintained for/to a single lower dimension? — Wallows

Lets consider if information is lost when going from 3 dimensions to 2. Can a printed map accurately represent the earth's surface? Surely the answer is no. I don't think it's valid to stipulate using a hollogram to re-display the map information in 3d. I'm assuming the question is whether we can learn as much from the 2d map as we could from a model globe which would be the 3d equivalent of the map.

Is information lost when going from the fourth dimension to the third dimension? — Wallows

Yes. How could it not be?

Where does the number two exist in? Our heads only? — Wallows

Yes.