“Proof,” in the full sense, is the process of deriving a conclusion step by step from the evidence of the senses, each step being taken in accordance with the laws of logic.
“You cannot prove that you exist or that you’re conscious,” they chatter, blanking out the fact that proof presupposes existence, consciousness and a complex chain of knowledge: the existence of something to know, of a consciousness able to know it, and of a knowledge that has learned to distinguish between such concepts as the proved and the unproved.
When a savage who has not learned to speak declares that existence must be proved, he is asking you to prove it by means of non-existence — when he declares that your consciousness must be proved, he is asking you to prove it by means of unconsciousness — he is asking you to step into a void outside of existence and consciousness to give him proof of both — he is asking you to become a zero gaining knowledge about a zero.
An axiomatic concept is the identification of a primary fact of reality, which cannot be analyzed, i.e., reduced to other facts or broken into component parts. It is implicit in all facts and in all knowledge. It is the fundamentally given and directly perceived or experienced, which requires no proof or explanation, but on which all proofs and explanations rest.
The first and primary axiomatic concepts are “existence,” “identity” (which is a corollary of “existence”) and “consciousness.” One can study what exists and how consciousness functions; but one cannot analyze (or “prove”) existence as such, or consciousness as such. These are irreducible primaries. (An attempt to “prove” them is self-contradictory: it is an attempt to “prove” existence by means of non-existence, and consciousness by means of unconsciousness.)