That is application. Proof is something else entirely.
Mathematical proofs exist. But their subject matter are mathematical entities. Whether they correspond to anything in the real world is a human burden to deal with.
The "real" world compared to the "mental" abstraction is like "hand-in-glove". It's best evidenced in olfactory perception. The model is "key-in-lock". It used to be considered the geometric configuration of the compound which "fit" with the receptors in the nose. But now it is considered the chemical bonds themself which "fit". It's a quantum phenomena which is "congruent" with the quantum phenomena in the brain. It's "hand-in-glove" or a "cinderalla-fit". This explains the strong connection between scent and memory.
Primitve thinkers did not have the tools, knowledge, understanding, that modern people have in this regard. It was considered "wise" and "enlightened" to "realize" that the mind is abstracting sensory perception. And in a way it is. But the truth is much more complex than those primitive ideas. Some take these primitve ideas to extremes and apply them literally and make fools of themself.
Anytime a person drives a car safely in traffic they are confirming the near miraculous accuracy of sensory perception in real-time. Anytime a toddler learns to speak proves it. Teaching them to play catch. Watching them develop. If one has not had the opportunity to raise a child, then, these sorts of analogies will be lost on them. The primitive philosophers who were strongly
attached to these notions didn't seem to value procreation.
And that's why "how many fingers?" is a test for lucidity.
