Life in infinite dimensions (was: ∞)

[Apeironian]

My brain ran away from me, and I wondered.
What would life be like in infinite-dimensional (A.K.A. Hilbert) space?

[papernuke]

To the infinete-dimensional beings (i dont know if infinete dimensional spaces are real ), the dimension would be normal, much like us in our own dimension. except there would be an infinete amount of directions to move in
.... ok thats not normal at all .
If we were in the infinite-dimensional space, we would see stretched and contorted things floating around. Much like being in a black hole without the "you dying" part. and being able to see everything.

I have my own quesiton. if a being from a higher dimension spoke to us, what would we hear. assuming they could speak english.

[Apeironian]

We already have an infinite # of directions to move in, as is the case for any (n>1)-onian:

1 degree
0.01 degrees
0.001 degrees...

They would have an infinite # of perpendicular directions.

[Apeironian]

Apeironians, spheres, and apeireracts
OH MY!

[houserichichi]

Not all Hilbert spaces are infinite dimensional. If you really did mean Hilbert space though then it would depend whether the space was separable or not so as to determine whether a basis is even defined. Hilbert space is an abstract mathematical construct not a real place you can live in.

[Apeironian]

Whatever. Infinite-dimensional space then.

[TimeStopper]

Would you be able to see the insides of another inf-being?

Secondly hide-and-seek would be a hard game.

[zero]

I'm still learning what life is like is 3-dimensional space. Higher dimensions add to the potential complexity in various ways, but infinitely many dimensions is a whole new "extra" level of convolution, due to convergence and other issues related to the metric.

Still, if you're going to wonder, will you consider both denumerable and non-denumerable infinities?

[jadaco]

Would an infinite being exist everywhere at once? Can it interact with everything at the same time? If so, does it have to? Can more than one infinite being exist? That is, would two infinite beings overlap each other in some way?

[zero]

If we are thinking in geometric terms, then one can have infinitely long lines parallel to one another in 2-D or skew in a manifold of at least three dimensions. Likewise, two unbounded planes of infinite extent could be parallel in 3-D or skew in a manifold of at least four dimensions.

Thinking of infinity outside the context of geometry, one might consider infinite collections that intersect in various ways. The infinite set of positive numbers is separate from the infinite set of negative numbers. There is no intersection, either, between the sets of even versus odd integers. Infinite sets of this nature could have intersections that contain a finite number of elements or an infinite number. You could even have infinitely many non-intersecting infinite sets of positive integers.

[malkuth]

Would an infinite being exist everywhere at once? Can it interact with everything at the same time? If so, does it have to? Can more than one infinite being exist? That is, would two infinite beings overlap each other in some way?

The quantum theory states that the universe is but a large melting pot of particles. And mater is but superpositions of particles defined by an outside observer.
What this means is that everything is everything and everything overlaps with everything. So, in a way, we are all infinite beings. And actual physical mater is such because we choose to percept it as such.

http://en.wikipedia.org/wiki/Introducti ... _mechanics

[papernuke]

Would an infinite being exist everywhere at once? Can it interact with everything at the same time? If so, does it have to? Can more than one infinite being exist? That is, would two infinite beings overlap each other in some way?

What do you mean by an "infinite being"?
Do you mean infinite dimensioned being? In that case, they wouldnt exist everywhere,
but a part of themselves would exist in all the dimensions of the "multi-verse." they live in.

[kingmaz]

Secondly hide-and-seek would be a hard game.

LOL
I'm off to hide in an infinteract then.

[quickfur]

The problem with making the quantum leap from a finite number of dimensions (even if the number if really, really, large, such as Graham's Number, say) to an actual, infinite number of dimensions, is that many geometrical properties break down, and if you wish to conserve them, you would need an extended number system that can deal with infinite quantities in a geometrically-consistent way.

A very simple example is this: what is the distance from the center of an infinite-dimensional cube to one of its vertices? Assuming a unit cube, the coordinates of the vertex would be (1,1,1,1,...). The distance is thus infinite: the square root of 1+1+1+1+... . OK, this is to be expected, since we have an infinite number of dimensions. No problem, right? Now consider what happens if we rotate this cube, such that the vector from its center to its vertex is now parallel to the vertical axis. What are the new coordinates of this vertex now? According to the geometrical interpretation, it should be of the form (X,0,0,0,0, ...), assuming we assign the first coordinate to be vertical. It should be clear that X has to be infinite. Now we have a problem: since X is infinite, that means if we want our space to be geometrically consistent, we need to allow infinite-valued coordinates, since otherwise, we've somehow managed to rotate part of the cube out of the space it resides in!

So this means that simply allowing an infinite number of real-valued coordinates does not yield a space with consistent geometrical properties. The only way to have a consistent infinite-dimensional space is to also allow infinite values in its coordinates. No problem; since we're going infinite-dimensional, why not also go for infinite coordinates, right?

The only problem is, now we get ourselves into a much more messy tangle. We cannot simply use any system of infinite numbers for our coordinates; in order to get consistent geometrical properties, the number system must satisfy certain criteria. First of all, the number system must be a field, since otherwise we get a lot of strange properties that precludes a consistent geometry. Another particularly important property is that infinite sums of positive numbers must always have a limit. However, here we run into a roadblock: a theorem by G.A. Edgar shows that if commutativity of addition holds, then if a sequence N of natural numbers converges to some infinite L, the sequence (N+1) must also converge to L. So we have L = L+1 by continuity of addition, and, since we are in a field, we can subtract L from both sides and get a contradiction: 0 = 1.

In other words, there is no possible number system with infinite quantities that also has the properties we need to have a consistent geometry of space. Consequently, any space that has an infinite number of dimensions will also have some counterintuitive, bizarre property that we don't normally admit in a geometrical system. An example of such a property is that some lengths are incomparable: i.e., in finite-dimensional space, if you are given two line segments L1 and L2, either L1 is shorter, or L2 is shorter, or they are equal in length. However, once infinite coordinates come into play, it's possible that L1 is neither longer nor shorter than L2, but they aren't the same length either! This means that it's impossible to rotate certain vectors to be parallel to another vector (since otherwise we simply rotate the endpoints of L1 and L2 to be parallel to (1,0,0,0,...) and then compare the first coordinates). In other words, rotation is defective in such a space. Another, even more pathological property, is that the length of a line segment changes depending on which coordinate you sum first when computing its norm. Or, geometrically-speaking, reflecting the line across a certain set of coordinate planes causes its length to change. This is very un-geometrical, since reflections are supposed to preserve length!

The upshot of all this, is that while the idea of an infinite-dimensional space is certainly tempting, it is either impossible, or has such bizarre properties that we can hardly call it a geometry.

[zero]

You could always restrict yourself to points that are a finite distance from the origin according to some metric. This would mean that you have to be careful to consider convergence properties for allowable positions that have a non-zero value in infinitely many dimensions.

[quickfur]

You could always restrict yourself to points that are a finite distance from the origin according to some metric. This would mean that you have to be careful to consider convergence properties for allowable positions that have a non-zero value in infinitely many dimensions.

Yes, you could, for example, use what's commonly known as L^2 space.

However, the major problem in all of these finite-distance subsets is that it only includes objects of zero content (volume). For one, it excludes the unit measure polytope (unit hypercube), since its vertices are infinitely distant, and yet it only occupies a unit volume in the space! The unit hypercube has volume 1, but the half hypercube (hypercube with edge length 1/2) has zero volume (1/2 * 1/2 * 1/2 * ...), and any hypercube with larger edge length than 1 has infinite volume (e.g., 2*2*2*...). Those objects that do not have this kind of explosive/implosive behaviour are those that are effectively of a lower, finite dimension: i.e., their size along most of their dimensions are effectively zero (since otherwise the vertices will lie at an infinite distance). So one can effectively consider them as merely N-dimensional objects where N is a finite, albeit potentially very large, number. (For example, if N is, say, Graham's number, then pretty much any object you can name in L^2 space would fit into finite N-dimensional space.) They do not really give much insight into full-dimensioned objects in infinite-dimensional space.

[Halfbaker]

An infinite-dimensional creature would need infinitely many legs to support it.

I had a *much* longer reply but i was auto logged out while writing it and I'm too lazy to write it again.