10

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.

user289980
  • 315
  • 2
  • 10
  • 9
    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 Feb 06 '22 at 05:05
  • 12
    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 Feb 06 '22 at 06:27
  • We do not experience time as a spatial dimension but that does not imply that it isn't a spatial dimension or cannot be thought of as one, that would be a physics question that can be empirically and mathematically tested, and one general relativity has implications for. Perhaps https://physics.stackexchange.com/ may have useful questions related to this – Tom J Nowell Feb 06 '22 at 15:32
  • 2
    [Kaluza–Klein theory](https://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_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...* – Double Knot Feb 06 '22 at 21:39
  • *...This extra dimension is a compact set, and construction of this compact dimension is referred to as compactification. In modern geometry, the extra fifth dimension can be understood to be the circle group U(1), as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the circle bundle, with gauge group U(1). In Kaluza–Klein theory this group suggests that gauge symmetry is the symmetry of circular compact dimensions...* – Double Knot Feb 06 '22 at 21:39
  • Do you use “metaphysicallly possible” in the strict Aristotelian sense? – Dave Feb 07 '22 at 02:52
  • 1
    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 Feb 07 '22 at 09:38
  • Given that scientific consensus is that we can observe that (3D) space is curved, how would that be possible, if there are no other dimensions for it to curve in? – JimmyJames Feb 07 '22 at 15:28
  • @JimmyJames There is such thing as intrinsic curvature. Surfaces (including spacetime) can be curved without being embedded in a higher-dimensional space. – gardenhead Feb 08 '22 at 01:23
  • 1
    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 Feb 08 '22 at 04:48
  • 1
    @JimmyJames [Does space curvature automatically imply extra dimensions? \[No.\]](https://physics.stackexchange.com/questions/99511/does-space-curvature-automatically-imply-extra-dimensions) – NotThatGuy Feb 08 '22 at 09:22
  • @NotThatGuy I'm not an expert on this but the idea that the ant can't figure out if his world is curved is not true in general. Simple example, start at the north pole, walk south to the equator, walk sideways (without turning) for a distance then walk backwards to the north pole. You will be pointed in a different direction without turning. In that way, you can determine that the surface is curved without 'leaving' the 2D space. – JimmyJames Feb 08 '22 at 15:05
  • @NotThatGuy After reading more on the difference between intrinsic and extrinsic curvature, I think you are reading into it. See [this definition](https://mathworld.wolfram.com/IntrinsicCurvature.html): "A curvature such as Gaussian curvature which is detectable to the "inhabitants" of a surface and not just outside observers." It's not saying that the space is not curved in another dimension at all. – JimmyJames Feb 08 '22 at 15:13
  • @NotThatGuy Now that I think about, that answer you link to is pretty obviously wrong, at least about the ant. If you wrap the stick into a circle and fact that the ant can return to the same point by going 'straight' proves the space it lives in is curved. – JimmyJames Feb 08 '22 at 18:17
  • @gardenhead That's not what intrinsic curvature means. In fact extrinsic curvature is completely irrelevant here because we are talking about something that is observable within 3-dimensions. – JimmyJames Feb 08 '22 at 18:19
  • There is no reason to believe that our universe is limited to 3 dimensions + time. This order is not accounted for in physics, and several things suggest possibilities which we just don't know how to visualize or imagine (cf. Ed Abbott). So there is no such thing as "metaphysically impossible". – Marxos Feb 11 '22 at 17:30

9 Answers9

10

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.

Jo Wehler
  • 20,817
  • 2
  • 26
  • 77
  • 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 Feb 07 '22 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 Feb 07 '22 at 15:57
  • 3
    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"? – Jean Marie Becker Feb 07 '22 at 16:45
  • 3
    @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 Feb 07 '22 at 17:54
  • 1
    +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... Feb 07 '22 at 21:23
3

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.

Mozibur Ullah
  • 1
  • 14
  • 88
  • 234
  • 3
    There is a reason why space is 3D: the fields of empirical physics have [inverse-square](https://en.wikipedia.org/wiki/Inverse-square_law) relationships with distance, which implies that only three dimensions manifest macroscopically. – Corbin Feb 06 '22 at 17:45
  • 1
    We can be more precise about string theory's predictions. There are two routes; either we use the [Cayley-Dickson ladder](https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction), or we use the curiously unique [Leech lattice](https://en.wikipedia.org/wiki/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](https://math.ucr.edu/home/baez/numbers/) is a great introduction. – Corbin Feb 06 '22 at 17:52
  • 1
    @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. – Mozibur Ullah Feb 06 '22 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. – Mozibur Ullah Feb 06 '22 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 Feb 07 '22 at 12:36
  • @CriglCragl: Yes, [bosonic string theory](https://en.wikipedia.org/wiki/Bosonic_string_theory) is probably not physical, and that's the only 24D approach I know about. Your 11D is my 8D; we're just counting worldsheets differently. The linked talks by Baez should cover all of this. However, until we actually observe or falsify the "curled-up" dimensions, whether they exist is a metaphysical question. – Corbin Feb 08 '22 at 16:06
  • @Corbin: The mathematical details of string theory are not relevant to my post. This is a philosophy site and not a physics site. Simply because I mentioned string theory in my post, does not mean that this is an invitation to throw in this detail. – Mozibur Ullah Feb 08 '22 at 16:09
  • @criglcragl: See above. – Mozibur Ullah Feb 08 '22 at 16:10
  • @MoziburUllah: Well, I'm glad to have this information. – CriglCragl Feb 08 '22 at 21:58
3

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.

user985366
  • 105
  • 3
Stig Hemmer
  • 392
  • 1
  • 5
  • "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 Feb 07 '22 at 15:40
  • The word "orthogonal" implies no impact. Does that mean 3D does not exist to 2D observers? – Victor Grazi Feb 07 '22 at 20:54
  • 5
    @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 Feb 08 '22 at 08:50
  • @Toffomat So what does it mean to be curved, then? – JimmyJames Feb 08 '22 at 15:14
  • @Toffomat if you mean ["A curvature such as Gaussian curvature which is detectable to the "inhabitants" of a surface and not just outside observers."](https://mathworld.wolfram.com/IntrinsicCurvature.html) that by no means contradicts my point. In fact, it is exactly my point. – JimmyJames Feb 08 '22 at 18:27
  • @JimmyJames OK, I may have misunderstood. My point is that "reality is 4D" in the sense that ist is a curved 4D spectime with (non-unique) curved 3D spatial slices, and the curvature is intrinsic, i.e. does not require an embedding into some higher-dimensional space(time). – Toffomat Feb 09 '22 at 09:46
  • @Toffomat As an analogy, suppose we were 2D beings 'trapped' in a surface which has been assumed to be 'flat'. Through various observations, we can see that our plane has an intrinsic shape. Eventually, we determine that this intrinsic shape of our plane is exactly like it would be if it were the surface of a sphere embedded in 3 dimensions. You could argue that this is just a coincidence and that it's a very complex flat space or we could assume that it's a simple shape in a higher dimension. Perhaps there's no way to prove which is true but I would tend to favor the 'simpler' answer. – JimmyJames Feb 09 '22 at 14:29
  • @Toffomat And with regard to the original question, I would think it's a reasonable enough interpretation to consider it "metaphysically possible" unless I misunderstand the phrase as it is meant here. – JimmyJames Feb 09 '22 at 14:33
  • @JimmyJames I'd say the notion that an embedded submanifold is 'simpler' than a curved space is wrong. Under some assumptions, any curved $d$-dimensional space can be embedded into a $2d$-dimensional flat space (one extra dimension is not enough in general; this is Whitney's embedding theorem), but you don't gain anything from the embedding. Note that in general relativity, you usually only consider a 4D spacetime with metric (and curvature etc), not an embedding into some auxiliary 8D flat space. E.g., in the usual cosmological metric, spatial slices are just constant-curvature spaces,.... – Toffomat Feb 09 '22 at 16:37
  • ... but are not embedded into higher-dimensional space. Also, you would have to embed the 4D spacetime, not the 3D spatial slices. In that sense, curved space does not in any obvious sense imply a a four-dimensional reality. (Regarding the original question, I don't see any reason why 4D space would not be "metaphysically possble", I don't even see what that's supposed to mean.) – Toffomat Feb 09 '22 at 16:39
  • @Toffomat OK, I'll admit that's mostly over my head. I'm seeing assertions on this thread and others that 'extrinsic' means 'embedded in a higher dimension' and 'intrinsic' means that it is not. But that's not what I understand from what I have read. For example, the surface of a sphere is one of the most common intrinsic 2D examples and obviously the surface of a sphere can be embedded in 3D. Do I at least have that right? – JimmyJames Feb 09 '22 at 17:04
  • 1
    @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 Feb 10 '22 at 08:23
  • @toffomat A cylinder has sharp edges between the circular and rectangular faces. Would the tangent space exist at those points? If not, then in what sense do you mean "intrinsically flat"? – Galen Feb 10 '22 at 16:34
  • 1
    @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 Feb 11 '22 at 15:10
2

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.

Chris Degnen
  • 4,780
  • 1
  • 14
  • 21
  • Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/133942/discussion-on-answer-by-chris-degnen-is-4d-space-metaphysically-possible). – Philip Klöcking Feb 07 '22 at 18:56
1

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.

Pathfinder
  • 1
  • 8
  • 1
    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 Feb 07 '22 at 15:35
0

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

CriglCragl
  • 19,444
  • 4
  • 23
  • 65
0

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.

Kieran
  • 1
0

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.

Victor Grazi
  • 101
  • 1
0

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.

Just me
  • 129
  • 8