… do we also rely on proof to believe ‘proof’ exists?
I suppose the “easy” answer is within the “correspondence theory” of truth – I.e., that the truth of any claim lies in its correspondence to the way things actually are.
Aristotle’s definition of the truth from his Metaphysics, (1011b25) is as good a starting point as any: “To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true”.
The implication is that providing a proof is only necessary for things which are not obviously or self-evidently true or false. We would not require a proof, for example, for snow being white or humans being mortal.
Some things have a permanence about them and are unchanging by nature so those would not be up for debate, once properly understood – well except in post-modern times such as ours. However, some things are changing and may or may not be in the same state at all times. Proofs for such things wouldn’t work because there is not an internal consistency or essential nature about them.
Uncovering the underlying or eternal (timeless) principles of being is the domain of metaphysics. As Aristotle pointed out in Metaphysics (Book II 993b30-31) “The principles of eternal things must be always most true (for they are not merely sometimes true, nor is there any cause of their being, but they themselves are the cause of the being of other things), so that as each thing is in respect of being, so is it in respect of truth.”
The problem isn’t so much the changing nature of being or truth, but the relative capacity or incapacity, disposition or lack of any proper disposition, on the part of those who are self-ascribed “seekers” of truth to apprehend the truth. Motives, preconceptions, vested-interests, fears, pride, etc., all prevent a proper disposition towards the truth. Some may ostensibly require proofs for things which may be self-evident when really they have no inclination to know the actual truth and are merely deflecting or worse. Others may take as obvious to them that which is just very appealing to their self-interest or preconceived notions.
The best you can do is cultivate a proper disposition in yourself and seek to know the truth, the whole truth and nothing but the truth with all sincerity and use others around you as sounding boards to test your tentative ideas, not assuming necessarily that they have all the answers nor intimacy with the truth that you might lack.
Developing a commitment to the truth above all else is not easy but it beats the alternatives by an infinite degree.
To answer your question…
A proof is, in simple terms, a clarifying statement of the self-evidence of the ontological permanence (truth) of a thing. By stating clearly the “whatness” of a thing and its necessary relationship to being, it’s “thatness” is demonstrated in a way that cannot be denied by any reasonable being but, of course, will always be challenged by those who have no inclination to be rational or truthful.
