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It is often said humans can't imagine 4D space due to limitations of our mind, but is this really the case or is 4D (and other n-dimensions greater than 3) truly metaphysically impossible, meaning that a universe could not exist with 4D space. The same question could be asked for 0 to 2 dimensional space, but I want to focus this question on 4D space.

I have seen tesseracts and other hyper-shapes, but these are just projections to either a 3D model or a 2D picture. N-D matrix mathematics also is an abstraction that is useful, but might not be meaningful to this discussion. And space-time itself is 4D, but the spatial component is 3D.

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    Welcome to SE Philosophy. I suggest that whoever often claims that humans can’t imagine 4D space are generalising their own limitations. It is quite possible to do although less easy to describe how to do it. Several branches of mathematics and physics require us to be able to deal with n-space at various levels
    – Frog
    Commented Feb 6, 2022 at 5:05
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    Considering that our spacetime is 4D it is not just possible, it is actual. There is no "spatial component", those dimensions are not separable in relativity. But generally, questions of this sort need specifying what "metaphysically possible" means. There is no standard definition, or even specific candidates for one.
    – Conifold
    Commented Feb 6, 2022 at 6:27
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    Kaluza–Klein theory is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time ...important precursor to string theory. In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of a very small radius, so that a particle moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension... Commented Feb 6, 2022 at 21:39
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    What do you mean by "possible"? Technically anything that would be compatible with all existing observations of reality would be "possible" (even if it fundamentally redefines every law we've defined to describe reality). But that's is not a very useful measure, especially when talking about the nature of reality itself and reality outside of space as we know it. It is probably "possible" that Cthulhu exists somewhere outside of space and it's making minor tweaks to reality in order to enact a personal grudge it has against you, but that isn't saying much.
    – NotThatGuy
    Commented Feb 7, 2022 at 9:38
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    The configuration space of the bones in your hand has dimension at least 14: three dimensions for each regular finger (knuckle plus two joints) and two more dimensions for your thumb. Can't get more hands-on than that!
    – Lee Mosher
    Commented Feb 8, 2022 at 4:48

9 Answers 9

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Note: To make the exposition simpler I am going to ignore time.

The world we see around us seems to have three space dimensions. This is physics as we know it.

WE don't know how to make 4D objects, which is why the tesseracts you have seen are only 2D or 3D projections. That doesn't prove that true 4D objects don't exist, somewhere.

Let me detour and ask "What exists?" (Some) physicists have a wonderfully simple answer to that: If it can affect us in some way, it exists. If it can't, it doesn't. (It gets a bit murkier when you consider the details.)

There is a very good chance that 4D space and objects do not exist by this definition.

However, metaphysics has a wider scope. It concerns all the things that might exist somewhere beyond the places we can reach.

What is to say a 4D cube doesn't exist somewhere?

Nothing says that. We can set up a set of laws for physics describing a 4D space with 4D objects interacting. We can set up many different such sets of laws, which may all exist, somewhere over the rainbow.

If you want to claim that 4D objects cannot exist, you will have to argue hard.

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  • "The world we see around us seems to have three space dimensions." We can see space is curved in our vicinity of the universe. That suggests to me that reality has at least 4 dimensions based on empirical evidence.
    – JimmyJames
    Commented Feb 7, 2022 at 15:40
  • The word "orthogonal" implies no impact. Does that mean 3D does not exist to 2D observers? Commented Feb 7, 2022 at 20:54
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    @JimmyJames Note that the fact that our space is curved does not require some external space dimensions oru space is "curved into". That is an intuition that is reinforced by the common illustrations of a 2D sheet curvced in 3D space, but there is an entirely intrinsic notion of curvature.
    – Toffomat
    Commented Feb 8, 2022 at 8:50
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    @JimmyJames That's correct. This is now leading very far astray, but note that, e.g., the two-dimensional surface of a cylinder in 3D has extrinsic curvature, but is intrinsically flat, so the whole isue is a bit involved. But when we talk about curvbature of spacet8ime, it's always about the intrinsic curvature. The field would be Riemannian geoemtry, if you want to follow up.
    – Toffomat
    Commented Feb 10, 2022 at 8:23
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    @DifferentialPleiometry Ay, by cylinder I mean (the surface of) an infinitely long cylinder. For a finite cylinder with end faces, of course the edges are singular, and if they were "smoothed out", they would be curved.
    – Toffomat
    Commented Feb 11, 2022 at 15:10
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Your question touches a series of topics, which possibly can be handled separately:

  • Is N-space mathematically possible? Yes it is. For example, there is no problem in generalizing the usual 3-dimensional Euclidean space to Euclidean spaces with arbitrary many finite dimensions. E.g. you mention hyperspace.

  • It is difficult to visualize Euclidean N-space for N>3. I assume we humans are restricted due to our mental wiring. The latter has an evolutionary origin and developed due to our experiences within our ecological niche.

  • In some domains of science it is helpful to take higher-dimensional spaces as the basis of a scientific theory. E.g., quantum mechanics is based on Hilbert space, which is an infinite-dimensional space.

  • It is important to discriminate between the two question: Can we visualize higher-dimensional space? (Answer: No). Versus: Can we develop science on the conceptual basis of infinite dimensional spaces? (Answer: Yes).

  • I consider the topic of higher-dimensional space not a question for metaphysics. For me it is a topic for mathematics and science. If it helps to explain the phenomena, then use the concept of higher-dimensional space.

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  • There are physical problems which require math in four dimensions which are less abstract than quantum mechanics. For example, calculating the mass of a 3-dimensional object with non-uniform density. In that case density becomes a 4th dimension.
    – Philipp
    Commented Feb 7, 2022 at 12:46
  • @Philipp: Indeed, even something as mundane as calculating a rotation in 3D space is significantly easier if you use quaternions (it avoids the gimbal lock problem).
    – Kevin
    Commented Feb 7, 2022 at 15:57
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    1) As a mathematician, I can say that we can train our mind to develop a certain intuition about 4D space. 2) About 3D, when I was teaching, I have seen a lot of students who had tremendous difficulties at perceiving phenomenas in 3D 3) The etymology of Metaphysics is "beyond physics" or more exactly "beyond physical perception" : therefore I don't agree when you say "I consider the topic of higher-dimensional space not a question for metaphysics"? Commented Feb 7, 2022 at 16:45
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    @Jean Marie Becker ad1) 2): OK – ad3) I take the term „metaphysics“ as referring to Aristotle‘s lecture - to its content, not necessarily to its later title. Then metaphysics means „first science“, comprising logic and ontology. „First“ because it is considered the basis of all other specific science. Contrary to Aristotle, I am sceptical that one can base science on an armchair concept of space, or on a concept from everyday experience. Instead, the concept of space has to be developed and adapted when doing concrete physics. And mathematics provides the tools for its formalization.
    – Jo Wehler
    Commented Feb 7, 2022 at 17:54
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    +1 I'd go a step further on the last point and say metaphysics is dead. There are no more questions for it. It has been entirely supplanted by science and mathematics. There really isn't anything it can credibly contribute to in this day and age.
    – J...
    Commented Feb 7, 2022 at 21:23
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It depends upon what you mean by space. Metaphysics after all means thinking about the basic constituents of what is physical: space, time, matter etc. It is about what is necessarily the case. However, such thinking often finds a place for what is not the case, because we can ask why is this not the case.

For example, space is 3d. But we have established consistent descriptions for a geometry of any dimension. So why is it space is not 4d or 5d or higher but actually 3d?

This has turned out to be a very good question. And there may be very good reasons for it to be 3d. We just don't know yet.

Until recently, no physical theory determined the dimensionality of space. It was taken as an empirical given. It's a physical constant that is not usually taken to be one.

One clue, however, is that string theory determines the space dimension to be 25d (+1 of time). Of course it would be much nicer if it was the value we know, 3d. It may be that other ideas can bring it down. In fact, one does, supersymmetry. In that case string theory says space must be 9d (+1 of time). But of course, the jury is still out on whether supersymmetry is realised in our universe.

Now, there are many kinds of higher dimensional spaces. No mathematician actually visualises these. What they do is invent and discover tools that help them work these spaces. When they imagine spaces, it is the low-dimensional spaces that they imagine: 1, 2 & 3d.

This is one area where popular science books fall down on. They don't make this clear, instead relying on visualisations. For example, one tool we have for building spaces is by multiplying them: a line segment multiplied by another one gives a square. A line segment multiplied by a circle gives a cylinder. Whereas a circle times a circle is a torus. We can also add them, a circle plus another circle - is, drum roll, just two circles!

Consider an analogy: Since mass education became widespread, most people can add 25,667,778 to 3,445,556 but no-one actually imagines either of these two numbers. What they do is use an algorithm taught at school. However, ask them to add 2 to 3, and then they can easily imagine these two numbers and they can imagine - that is directly visualise - adding them together too. Moreover, the properties we can establish here also carry on for much bigger numbers. This shows the utility of thinking about 'small' cases.

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    There is a reason why space is 3D: the fields of empirical physics have inverse-square relationships with distance, which implies that only three dimensions manifest macroscopically.
    – Corbin
    Commented Feb 6, 2022 at 17:45
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    We can be more precise about string theory's predictions. There are two routes; either we use the Cayley-Dickson ladder, or we use the curiously unique Leech lattice. On the former, we can have 8D (2D isn't enough, QM rules out 4D, and 16D loses properties); the latter is 24D. Then string theory adds one spatial and one temporal dimension in every case. Baez 2008 is a great introduction.
    – Corbin
    Commented Feb 6, 2022 at 17:52
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    @Corbin: Its possible to have gravity and electromagnetism in higher dimensions. We don't need to check that physical forces follow inverse square laws to determine the dimension of space, we can determine this directly. Commented Feb 6, 2022 at 18:11
  • @Corbin: There's no need to be more more precise. This isn't a physics site - its a philosophy site. It's enough to explain the background without getting into the mathematical details which is only of interest to physicists amd maybe mathematicians. Commented Feb 6, 2022 at 19:11
  • @Corbin: I thought M-Theory had unified the different String Theory approaches and settled on 11D, related to the mathematics of octonions?
    – CriglCragl
    Commented Feb 7, 2022 at 12:36
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In the absence of closed timelike curves which could enable multiple times to exist at the same time (go figure), we have just the present 3D universe existing, albeit it changes shape from moment to moment. That is to say, only one of those moments actually exists.

Addendum

I understand by the OP's question he is asking if a pan-time universe can be imagined: a 4D block model, in the physical, natural world. An alternative point of view would be the existential reality of being - which of course embodies time. But I don't think that is what was asked.

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We might even consider the 3D space we live in as a subspace of a 4D space. If this space is curved negatively, it even offers a solution of dark energy and inflation.

If we consider matter to be confined to 3D space (like in brane models) and consider 4D substrate space with an appropriate topology, then two 3D universes, a matter one and an antimatter one on the other side, can emerge from a common singularity and move and expand on this structure. General relativity is considered intrinsically curved but there is nothing that prohibits such an immersion.

Three dimensions are the minimum to stay one whole and let food come in and shit go out.

I read:

"And space-time itself is 4D, but the spatial component is 3D."

The time component, sometimes given as it (imaginary i multiplied by t), is not a real existing coordinate. There simply is no dimension of time on which a particle can move. Of course, if we place clocks everywhere than these will show a value. The perfect clock (with constant period time) exists in the mind only, and it's not stuff moving in time, but time moving besides that stuff.

So a 4D space is metaphysically as well as physically possible, and the latter can be even the case.

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    You should check out "The Planiverse" by A. K. Dewdney. There are explicit descriptions of how life could exist in 2D as well as how computer circuits could work.
    – JimmyJames
    Commented Feb 7, 2022 at 15:35
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I would ask ask counter questions. What do you think a dimension is? What does imagining mean?

We discussed a very similar question, and I give my answers there: Is it possible to visualize higher dimensional space?

Physicists work with additional dimensions all the time. Even engineers do! Strain like on blocks of concrete or steel beams, is calculated using tensors, that is a 3D field of vector forces, directional forces, this is essential to understanding fracture propagation.

Spacetime, is also 4D, a 3D field of gravitational force vectors. A 2D sheet that is curved, is a 3D structure. Our 3D space curves, in time, making it 4D.

Holographic theory helps us think about dimensionality, as emerging from locality relations. The total entropy possible in a space is defined by the size of the surface of the space, a 2D shape.

Noether's theorem shows is that continuous symmetries are formally equivalent to conservation laws. We can think of dimensions as sets of symmetries. That is, conservation laws that have functionally reduced dimensionality, like entropy, ARE reductions in degrees of freedom of the involved forces (ie, uncurving the paper).

We can then apply this to higher dimensions, including partial ones like treatment of Turbulent black holes grow fractal skins as they feed

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Not a philosopher, and apologies if you've already read it --- I think it's very well known --- but if you are struggling with the concept of multiple dimensions then the book Flatland by Edwin Abbott is a great place to start. It's a deceptively simple read, but if you engage with it then I found you end up with a much better human conception of how multiple dimensions work, and how we might experience them.

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In the overall scheme of things, I think it would be very sapiens-centric to assume there is any limitation at all on the number of dimensions.

In the spirit of Abbot's Flatland, imagine a world of cartoons living on a flat paper, floating around in a 3D world, and they assume the universe extends infinitely north, south, east, and west, but have no concept of up and down.

So are we.

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Well, we can use some version of the anthropic principle to defend that our minds only could be developed in three dimensional space. We leave in the surface of a sphere. For most purposes, we think in two dimensional terms. We can easily travel North or West for thousand kilometers. Most days we only move up or down a couple of hundred meters. Most people have great difficulty in imagining three dimensional objects. You just have to go to class on Advanced Calculus to understand that quite well. Height is a bit less important than the other two dimensions, but essential anyway. If we were living in space ships and the three dimensions were equally important, it would be really hard to find someone else. Our social life would be quite different. We are social animals, hence our minds would works in quite different ways.

To leave in a 4 dimensional world would create completely different minds. There is "too much space", too much freedom to move. We can perform a sphere eversion without self intersections. We could not contain a liquid inside a bottle. In four dimensional space it would be very hard for a being to bump into another being. Just that would change too many things.

Other questions not related to the anthropic principle. There are problems of classification of mathematical objects like topological manifolds that are only really hard to solve (I mean: complex enough) in dimensions three and four (dimensions of space and space time in our universe). We would expect things to get more complex each time the dimension increases but we get more (too much) space to move things and everything can be trivialized.

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