This question – how do you “prove” something – has to go to the department of “epistemology.” That word means “how you know what you know.” Vedic epistemology is very thorough: There are 10 primary “proofs” (pramaana), but of those 10 there are really 3 that are crucial. One is by direct empirical observation (pratyaksha), the second is by mental deduction (anumaan), and the third is by declaration of authority (shabda).
Different subjects require different proofs. Something that fits within the range of the senses can be proven by direct observation. Something that is beyond the immediate perception but is still within the grasp of ordinary logic can be proven by mental deduction. Something which is beyond all of these cannot be proven except by the statement of an authority on the topic.
That’s a nutshell synopsis of Vedic epistemology. Here are some examples to clarify:
I can prove that a glass is on the table by looking at it. the glass, the table and their relationship are all within the range of my senses – therefore I can validate their relationship with an observation (pratyaksha).
I cannot prove that there is fire on the hill by direct observation, because the hill is too far away and I cannot see the fire directly. But the smoke billows up and I can deduce that every time I see smoke it is associated with fire. Therefore by mental deduction (anumaan) I can prove that there is fire on the mountain. Fire, smoke and their relationship are all within the purview of our experience and therefore we can make rational deductions and validate fire via the observation of smoke.
That God is dark blue and has a peacock feather in his hair, for example, has absolutely no reference point within our experience. What to speak of these fine details of how God looks in his most intimate and unguarded moments, everything pertaining to God is by definition beyond the limitations of our ignorant materially confined capacity of experience and understanding. Therefore I can conclusively neither prove nor disprove anything about God via direct perception or mental deduction alone. Only if I have the blessing from the infinite to be able to hear directly from the divine an authoritative statement, only then can any proof be offered and considered regarding such subjects.
The three proofs are not mutually exclusive. In fact they all work together. It is just that in divine, transcendental matters the statement of authority is the primary proof – and logical deduction and practical observation play a role of elaborating and substantiating and clarifying the statement of authority. In all three pramaana, all three work together. We simply name the pramaana for the more prominent of the three. So the important point to take away here is that it is not that observational validation and logical deduction play no role at all in congizing the reality of God. They do play a role, but the primary role – for the soul in ignorance, without access to a transcendental mind and senses – is the statement of authority.
The Vedic body is broadly accepted to represent the transmission of Godhead Vishnu to the world for the primary sake of giving authoritative statements on supra-ordinary phenomena. Those of sufficient rational mettle find that the authoritative statements in these scriptures are thoroughly logical, rigorously sensible, and bear out against empirical observation whenever possible.
There is at least one school of Vedic thought which attempts to prove and explain the transcendental divine reality without reference to any authoritative statements – only via empiric observation and logical deduction. This school is called nyaaya. Its main target is to counteract the atheistic influence of Buddhism without being too unlike the basic tenor of Buddhism (which rejects the Vedas and thus rejects the sources of authoritative statements). It was quite successful, and thus you find very, very little Buddhism surviving in its homeland, where it had to content with Nyaya. If a person is very against the concept of “authoritative statements” as a method of proof, you could make some headway with them by presenting the systematic arguments developed by the nyaya school.
The stripped down fundamentals of their school is that every object is created by something else – therefore there must be a creative intelligence of some supra-mundane type behind the creation that we directly see all around us. It is essentially the Shelter lyric, “A creation without a creator, can it be? It’s like a painting without a painter – an absurdity.” Although this is a strong logic it does have several holes, which were thoroughly attacked by other Vedic schools of thought, and thoroughly counter-argued by the proponents of Nyaya.
It is a fascinating subject, and useful for clearly demonstrating that Vedic based philosophy is not philosophically shallow at all or from any angle – emperic, logical, or mystical. However in the end, I agree with the Vedantic and other schools that say when all is said and done that which is beyond the mind and senses cannot be validated thoroughly without reference to authoritative statements.