Proof that Vectors Describe Reality Correctly

Physics is an empirical science. The only things that have physical meaning are experimental measurements. One of our goals as physicists is to write down mathematical models, the outputs of which correspond to these measurements. The only way to check the validity of a model is to compare the outputs of the model to the measurements. There is no way to ever guarantee or prove that a given model is correct, in the sense of a mathematical proof. The best we can ever do is acquire evidence that our model agrees with experiments.

Since the only things with physical meaning are measurements, the mathematical objects and techniques we use within our models have no meaning in the physical world. They are only tools for calculating outputs that we can compare with experiments.

If we use a model like newtonian mechanics that uses vectors to produce predictions that agree with experiments, we accept the validity of that model, including the mathematical tools it applies. But this is absolutely not a proof that vectors provide a description of reality, nor can we ever hope to have such a proof — because experiments cannot prove anything, and because experimental measurements are the only description of reality.

Having said all that, vectors are very natural and generic mathematical objects that show up any time we use a model that is linear. Linear models are typically the simplest models to understand and compute with. (Perhaps this is because of the way our brains evolved.) Much of what we do in physics is to reduce complicated problems to something linear, in which case vectors will always show up.

For example, in general relativity, spacetime is not a linear space. But if we zoom in to an infinitesimally small region of spacetime, we can model this small region by a linear space, which we can describe using vectors. Then we can describe the complicated spacetime essentially by gluing together these linear spaces.

Another example is that in physics we are interested in the action of symmetries on our model. To study how these symmetries act, we can look in particular at how they act on linear spaces — that is, on vectors. This is called representation theory. Representation theory is essential to our current understanding of quantum mechanics.

In spite of the fact that vectors do not and cannot have any physical meaning, it is likely that vectors will always show up in our physical models, because we will always want to be able to reduce complicated problems to more easily understood linear problems.