A mathematical proof that our world is in three dimensions

Everyone knows that our world is three-dimensional. It seems indeed obvious that there are three independent spatial directions, no less, no more. But can we really be sure? After all, everything we know about our world comes from our senses and from what our brains interprete. Does the fact that we see in 3D really proves that the world is in 3D? Can we have a mathematical proof, based on experimental observations, that the world is three-dimensional? This question is totally relevant, and several scientists and philosophers, like Immanuel Kant, have been interested in it (see for example this blog post in French for a nice review).

Here, I detail Kant's idea of a rigourous proof that we live in a 3D world. There are three steps:

1. We model the world as an N-dimensional Euclidean space and we study a mathematical object called a vector field that verifies some natural properties.
3. We link these properties to the number of spatial dimensions and we conclude that $$N=3$$.

The physical notions behind this proof are about conservative forces and potential theory. The mathematical notions concern the Laplace operator and harmonic functions.

First, here is an intuitive explanation of this proof (quoting Wikipedia):

The inverse-square law generally applies when some force, energy, or other conserved quantity is radiated outward radially in three-dimensional space from a point source. Since the surface area of a sphere (which is $$4\pi r^2$$) is proportional to the square of the radius, as the emitted radiation gets farther from the source, it is spread out over an area that is increasing in proportion to the square of the distance from the source. Hence, the intensity of radiation passing through any unit area (directly facing the point source) is inversely proportional to the square of the distance from the point source.

1. Mathematical model

We model the world as an open set $$U$$ of $$\mathbb R^N$$, where $$N$$ is the number of dimensions. We consider a vector field $$F$$, that is, a function $$F: U \longrightarrow \mathbb R^N$$ that maps any point $$p$$ in $$U$$ to a $$N$$-dimensional vector attached to that point. This vector field can represent a force field, like a gravitational field generated by a massive object, or an electrostatic field generated by a charged body. It can also represent a flux, like in transport phenomena: heat transfer, mass transfer or fluid dynamics.

We will assume three natural hypotheses: that this vector field is incompressible (or solenoidal), conservative (or irrotational), and isotropic. One also uses the terminology of Laplacian vector field when the first two hypotheses are satisfied. These hypotheses correspond to a conserved quantity that is radiated outward radially and isotropically from a point source.

Incompressibility occurs when the divergence of the vector field is zero: $$\nabla \cdot F=0$$. It means that there is no local source or sink, no local creation or annihilation. If the vector field represents a mass transfer, incompressibility means that there is conservation of mass: mass cannot be created or annihilated at any point in space. For a gravitational field, the divergence is zero outside mass sources. Similarly, for an electrostatic field, the divergence is zero outside charge sources.

Irrotationality occurs when the curl of the vector field is zero: $$\nabla \times F=0$$. It means that the field does not spin locally (in fluid dynamics terms, there is no vorticity). An equivalent property is that the field is conservative, so that the line integral from one point to another is independent of the choice of path connecting the two points (in fact, one has to assume the simple connectedness of $$U$$ for the equivalence). A conservative field is the gradient of a scalar field: there exists a scalar field $$u : U \longrightarrow \mathbb R$$ such that $$F = \nabla u$$. The gravitational force and the electrostatic force are examples of conservative forces. The gravitational vector field is the gradient of the gravitational potential field, and similarly for the electrostatic field.

Finally, isotropy means uniformity in spatial directions, that is, we assume that the vector field $$F$$ is rotational invariant. For example, the gravitational or electrostatic field generated by a perfectly spherical source is rotational invariant.

Under the two last hypotheses, we can write $$F=\nabla u(r)$$ where $$r$$ is the distance to the origin: $$r^2 = \sum x_i^2$$. The vector field $$F$$ is the gradient of a scalar field $$u$$ that depends only on $$r$$ since it is isotropic. The first hypothesis, $$\nabla \cdot F=0$$ (outside field sources), implies that $$\nabla \cdot \nabla u=0$$, that is, $$\Delta u=0$$ where $$\Delta=\sum \frac{\partial^2}{\partial x_i^2}$$ is the Laplacian.

The equation $$\Delta u=0$$ is a partial differential equation (PDE) called the Laplace equation. A function with a null Laplacian is called an harmonic function, so we're looking for an harmonic function $$u(r)$$.

2. Experimental observations

It is known since Newton and Coulomb that the laws of gravitation and electrostatics follow inverse-square laws, where the strength of interaction forces is proportional to the inverse square of the distance. This observation has been validated by a number of experiments during the last centuries. The same inverse-square law is observed for light, sound and radiation phenomena.

We will now show that if a vector field satisfying the three preceding properties (like the gravitation law) verifies the inverse-square law, then it implies necessarily that $$N=3$$, i.e. the world has three dimensions.

3. Proof

The proof is straightforward once one knows the following expression for the Laplacian in N dimensions and in spherical coordinates:

$$\Delta u(r) = \frac{1}{r^{N-1}} \times \frac{\partial}{\partial r} \left( r^{n-1} \frac{\partial u}{\partial r} \right) + \frac{1}{r^2} \times \Delta_{S^{N-1}} u$$

where $$\Delta_{S^{N-1}}$$ is the Laplace-Beltrami operator on the (N-1)-hypersphere (an (N-1)-dimensional Riemannian manifold), also called spherical Laplacian.

Since $$u$$ is isotropic and depends only on $$r$$, it is constant on hyperspheres (defined by $$r=\textrm{constant}$$), so that the restriction of $$u$$ on $$S^{N-1}$$ is constant, its gradient is null, and $$\Delta_{S^{N-1}} u=0$$. Therefore:

$$\frac{1}{r^{N-1}} \times \frac{\partial}{\partial r} \left( r^{n-1} \frac{\partial u}{\partial r} \right) = 0$$

hence:

$$\left( r^{n-1} \frac{\partial u}{\partial r} \right) = \textrm{constant}$$

and finally we find that:

$$u(r) = \frac{C}{r^{N-2}}.$$

We omit the integration constant since a potential field is defined up to an additive constant. Hence, the intensity of the vector field $$\nabla u$$ is proportional to $$1/r^{N-1}$$. For an inverse-square law, it means that $$N-1=2$$ so that $$N=3$$.

Finally, we should add that some modern theories, like string theory, assume a higher number of spatial dimensions, where the extra dimensions are of a much smaller scale, such that, at common scales, the three dimensions of space are still a reasonable approximation.