How to prove that we are living in a 3+1D world?

Question

Is there any scientific experiment that can lead us to conclude we live in 3 spatial dimensions without the premise of the conception of limited dimensions?

Thank you all who helped in the improvement of this question (which was not clear at first).

EDIT:

I know that this can be a little philosophical, but it is also a scientific question.

Let's consider the scenario where the mankind was not ever able to see.

Let's also consider that this limitation could be surpassed thus not limiting us to reach a scientific and technological knowledge "similar" to what we have today.

Would this civilization of blind people reach the conclusion that they are living in a 3D spatial world?

Is the sense of touch enough to reach that conclusion? Is there any scientific experiment that can lead us to that conclusion without the premise of the conception of limited dimensions?

Would it be easier, harder, or just different to reach a conclusion predicted by the M-Theory? (please do not focus only on this last question)

Answer 1

This question has changed in such a way that my answer (previously here) didn't seem even related anymore. I therefore came up with something new, gladly inheriting 4 upvotes, but much less confident. In fact, I can plainly state that I'm fully incompetent in these matters.

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With that out of the way, another introductory remark. Science doesn't prove things. Dimensions (in this case) are part of mathematical models that aim to describe nature. Whether dimensions "exist" or not is a philosophical question.

Now, let's see...

Anthropic arguments are frowned upon by many, but that is when they are used to explain something. Here one can use anthropic arguments just to reason our way to, not why, but rather, that something is the case. (I guess these arguments aren't really anthropic, but rather anthropic-like.)

My main source, as usual, is Wikipedia. The thing in which we are interested is spacetime (note: a mathematical model). From there we pick up this:

How many dimensions are needed to describe the universe is still an open question. Speculative theories such as string theory predict $10$ or $26$ dimensions (with M-theory predicting $11$ dimensions: $10$ spatial and $1$ temporal), but the existence of more than four dimensions would only appear to make a difference at the subatomic level.

Then the section "Privileged character of 3+1 spacetime" gives us a whole bunch of reasons of the following form:

"if $N\neq3$, then the world (or stuff in it) as we know it wouldn't exist",

where $N$ does not include compactified dimensions invoked by string theory and undetectable to date. (See, e.g., Calabi-Yau manifolds.)

Planets wouldn't have stable orbits. Stars wouldn't have stable orbits. Electromagnetism wouldn't work (at all). Electrons would either fall into the nucleus or disperse. Nerves cannot cross without intersecting. (Some of these arguments apply to $N<3$, $N>3$, or both.)

Hence anthropic and other arguments rule out all cases except $N = 3$ and $T = 1$ [...]—which happens to describe the world about us.

And then there is @Dilaton's comment to the question: "[M]easure how the gravitaional force depends on the distance". That may be expanded upon a little:

"[I]t is the three-dimensionality of space that explains why we see inverse-square force laws in Nature [...]" (Barrow 2002: 204). This is because the law of gravitation (or any other inverse-square law) follows from the concept of flux and the proportional relationship of flux density and the strength of field. If $N = 3$, then $3$-dimensional solid objects have surface areas proportional to the square of their size in any selected spatial dimension. In particular, a sphere of radius $r$ has area of $4\pi r ^2$. More generally, in a space of $N$ dimensions, the strength of the gravitational attraction between two bodies separated by a distance of $r$ would be inversely proportional to $r^{N-1}$.

For completeness' sake, I mention also i) the model with large extra dimensions and ii) causal dynamical triangulation, both of which I know even less of. The latter does apparently do something that might appeal to you:

It does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves. It shows spacetime to be two-dimensional near the Planck scale, and reveals a fractal structure on slices of constant time, but spacetime becomes $3+1$-d in scales significantly larger than Planck. So, CDT may become the first theory which does not postulate but really explains observed number of spacetime dimensions.

I would bet that the status of this approach (CDT) is controversial.

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Now, you stipulated: "without the premise [or] the conception of limited dimensions". I don't know what that really means. I hope that it doesn't mean "without the idea that mathematics can describe nature".

I also hope that I got at least something right. :)

Answer 2

Verify any inverse-square law process, like gas diffusion or classical forces.

Answer 3

First , human brain would not be able to benefit from the great amount of information that electromagnetic force provide about the natural world . Human beings will then resort to mechanical means to interact with the environments .They would develop more sophisticated mechanisms that can sense chemical and mechanical stimuli more effectively . I think that they will be required to be more smart too, otherwise , they wouldn't be able survive

Humans would be able to understand that they live in a 3D world because they will understand that there are three perpendicular direction that things can move in . They will differentiate simply between two universes (or places ) that are different in spatial dimensions even if they could't benefit from light .

Anyway , what does string theory and quantum mechanics have to do with this ?

Answer 4

According to the holographic principle, the information in a bounded three dimensional space can be stored in the two dimensional boundary. This is in accordance with black hole entropy, the information that can be stored in a black hole. This is a quantum gravitational effect, which states that the entropy of a black hole is proportional to the area of its horizon.

Thus, it may be misleading to think that we are living in a three spatial dimensional universe.

Answer 5

The realization that we live in a world with three spatial dimensions is deeply rooted in human perception. We can walk in the North-South direction, the East-West direction, an also move in the Up-Down direction by jumping. No one has experienced the additional freedom of a fourth spatial dimension independent to the other three.

The key word here is the word 'independent'. You could say that it is Einstein who has made the meaning of 'independent' precise in an observable way. Bottom line is that we can monitor and keep track of physical motion by a three dimensional grid of synchronized and mutually fixed clocks. No single experiment has ever been performed that tells us the we need more spatial dimensions than three.

The most precise confirmation of the number of spatial dimensions being three is the large number of experiments on aging. We all perceive aging, albeit in a very imprecise way. Yet, aging is the single observable in physics we can measure amazingly precise, with a relative accuracy well below the $10^{-15}$ level. All this requires is to attach a highly accurate clock to the object the aging of which we want to measure.

Einstein's relativity theory identifies aging to a distance in space-time. For an object moving through above mentioned 3D grid of clocks, undergoes an aging $\Delta \tau$ given by

$$(\Delta \tau)^2 = (\Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2$$

Here, we work in units in which $c=1$. The quantity $\Delta t$ denotes the aging of the grid of clocks, and $\Delta x$, $\Delta y$ and $\Delta z$ the movement vector of the object as seen from the grid of clocks.

Using highly accurate clocks as objects moving through a 3D grid of synchronized clocks at fixed distances, the above 'semi-Pythagorean' (Lorentzian) concept of distance has been confirmed time and again. There is 3+1 terms on the right hand side of the above equation. nothing more, nothing less. I am not aware of a single experiment that casts doubt on the local 3+1 Lorentzian (3 spatial and 1 time dimension interlinked by above concept of distance) nature of spacetime.

Of course, this need not be the end of it. Quantum gravity research might direct us to a deeper reality that can be described by fewer (holographic) spatial dimensions. Such a holographic reality, however, can only manifest itself at very small length scales and correspondingly high energies. If such were to be the case, at length and time scales of human perception, our universe would still have 3 spatial dimensions, not one more, and not one less. And the deeper (holographic) theory would tell us exactly why such is the case and how 3D reality is emergent from a holographic description.