# If all the information about the universe is stored on some 2-dimensional boundary, why is a 3-dimensional “hologram” of the information needed?

If all the information about the universe is stored on some 2-dimensional boundary, why is a 3-dimensional "hologram" of the information needed? Certainly as the good Mr. William Ocam of Orange tells us, quite sensibly I think, "entities ought not be multiplied beyond necessity," so I doubt he would consider 3-dimensional re-representation of the data to be needed or warranted.

We know that one can represent, at least in principle, even an infinite amount of 2-dimensionally arrayed information in a 3-dimensional space. And it certainly seems just as likely that an infinite 3-dimensional data bank can also be stored on a 2-dimensional surface; it's the peculiarities of "infinite sets" that makes it possible. And of course, the information from the 3-dimensional virtual world we are supposedly living in could also be stored in even just a (very long) 1-dimensional world, à la Cantor's astounding "I see it but I can't believe it revelation."

So then, if one, as demiurge, wants to have a maximally nice and tidy universe, why not just store all the data, for however many dimensions you might want to have in your world, in one dimension, and then have a very fast Turing Machine, -- why maybe even a couple, three of them, -- to calculate everything that "happens" from the big bang to the great rip?

And, by the way, does the 2-dimensional, all-information-holding, boundary, if it exists, exist inside or outside of the 3-dimensional "virtual" universe itself? If it is inside, then surely it too has a "correlate" in some other 2-space, and one which is not just a subset of itself, and thus then also another subset of itself, ad infinitum? And if, on the contrary, the boundary is actually outside the universe, well then, where is it?

Hmm, didn't those smart old Greeks worry about that kind of thing about 2,500 years ago? What has it got to do with physics in our 21st century world other than to provide a truly ad hoc "solution" to the old quantum information paradox? I think the whole holographic universe theory, however well accepted, and even useful as it may be, really needs to confront some very basic metaphysical problems.

Perhaps it has. If so, I've never seen the discussions. Any pointer?

• Have a look at en.wikipedia.org/wiki/Holographic_principle – Bruce Lee May 10 '18 at 18:28
• Yes I have read that entry and several similar to it. But the still formidable metaphysical problems aboud. even in the case of just a blackhole event horizon, which at least is conceivably observable, There is nothing special about the event horizon -- leaving aside the possible heat. You would fall right through it without even noticing. Caettainly you wouldn't see a sudden bright flash of neatly organized data!! So if there is actuallyan huge mass of data there, HOW is it stored and ever accessible again, both of which things would also require energy and creat entropy in the process. – Wd Fusroy May 10 '18 at 18:50

## 1 Answer

I presume that you are referring to $\mathbb{R}$, $\mathbb{R^2}$, $\mathbb{R^3}$, and even $\mathbb{R^4}$ having the same cardinality. In other words, there are bijections (one to one and onto maps) between them. So, in a sense, they are the same size. However, these maps are not continuous, these spaces (with the usual topologies) are not homeomorphic Wikipedia. So, if you map a nice object (e.g. a sphere) in $\mathbb{R^3}$ to $\mathbb{R^2}$ with one of these maps, you will not get a nice result.

The real numbers appear to be a very good model of reality but there is no reason to suppose that they are perfect. In particular, it is far from clear whether the large infinities studied in mathematics have any relevance to reality. I guess that you know that the real numbers have a higher cardinality than the natural numbers. Do you know the Continuum Hypothesis Wikipedia? It states that there is no cardinality between that of the natural numbers and the real numbers. This remained neither proved nor disproved for a long while but was finally proved to be unprovable. This is surprising but it would be even more surprising if these large cardinals had any relevance to the real word.

I would recommend The Road To Reality by Roger Penrose. It is a great for a fairly serious amateur. It has a chapter on infinity. Penrose seems to find it surprising that uncountable infinities are not useful as so much of maths is useful.

I guess that you like both maths and physics (I do) but I resist the temptation to try to force fit the weirder bits of maths onto physics.

• Sorry to respond so late, but I didn't see your answer until now. Yes, I certainly know about the continuum hypothesis and the other basic axioms of elementary set theory. I had not heard before that there are no continuous homeomorphisms from R3 to R2, although I was aware that there are none between R1 and R2 even though it is quite easy to go from one space to the other by simply interweaving the digits of R2, a discovery which Cantor is famous for tellng a friend, "I see it but I don't believe it!" I will definitely check out the Penrose book! Thanks for your answer! – Wd Fusroy Nov 16 '19 at 23:48