Objective reasoning can never "prove" the subjective state of "being".
Obviously if you try to prove to yourself that you exist, you say "That's simple, I already know I exist" -- without even needing any objective proof at all. You *ARE* the proof, as far as you're concerned. And that's really the only proof necessary.
But as far as objective
Science is concerned (I know there may be sects of "subjective science", but like I said that starts to get pretty damn close to religion), there is no proof that can be seen. There's no way for you to tell whether all those people are just dream-creations, atomic holograms, complex biological "robots", or whatever else. And there is nothing they can do to "prove" it. You can use instruments on them and monitor electrical brain activity and all that, but there's no way to prove that there's a conscious "someone" behind that activity, guiding that activity.
At best, it can only be inferred. You compare yourself to them, and see similarities in form and activity, and assume that there is probably a sentient consciousness at the wheel. But "probably" is not proof, and that "probably" is only an assumption. That is not science, but guesswork.
You can say "I exist, and so must you", but where is the proof? Subjective consciousness can *NEVER* be proven in a scientific, objective manner. If I ask you to prove that you exist, and you punch me in the face, that still isn't proof of your subjective state of awareness and being -- at least as far as objective
Science is concerned.
There *IS* proof, on a very deep level, but it's not something that can be scientifically proven, because the "proof" first depends on your subjective state.
The consciousness behind your ego-mind (individual identity) and body-form is only a "concentrated point" -- a point where awareness has focused -- in a large "sea of consciousness". That "sea" flows through all life, and indeed all creation. It flows through even "mundane", seemingly-lifeless things. Rocks, bricks, metal, earth, plastic, EVERYTHING. The only difference, is that the "point of consciousness" behind those things, is lacking awareness. As consciousness develops greater awareness (which takes an unfathomable amount of time... slowly, slowly...), it takes on identification with external form that is a reflection of that awareness. This entire manifested world is only a reflection of consciousness, on all levels of awareness. This "evolution of awareness" that consciousness goes through, is the "hidden" force that drives PHYSICAL evolution. Evolution is a real process, but it doesn't happen randomly -- it's "guided" by the underlying consciousness... as that consciousness moves through progressively greater states of awareness and manifests into form that expresses that awareness. Sometimes "jumps" in awareness occur, and as a result, "jumps" in physical evolution become evident, to match that sudden leap in awareness. This is why there are "missing links" in certain evolutionary trees -- those links never existed in the first place. Awareness simply "leaped ahead", and the forms it manifested followed suit, as a reflection of that. The whole creation submits to the will of consciousness. Atoms and energy obey, and "build" whatever form consciousness desires.
Now, that underlying pair of "consciousness and awareness" is the subjective state that you can't prove to anyone. BUT, you can *RECOGNIZE* it in others, because the same basic consciousness flows through them as well. Awareness is what enables you to recognize that consciousness. Obviously a brick isn't capable of recognizing consciousness in a bowl. The brick doesn't even recognize it in itself. It has no self-awareness. It has *BEING*, but no awareness of that "being". But when awareness comes into play, recognition is possible. First comes the very basic beginnings of self-awareness and awareness of subjective consciousness. This is the root of the ego, the personal identity. As that awareness deepens, it becomes possible to recognize awareness in others, and eventually the more subtle "consciousness" behind other aware beings, and then even further, the "consciousness" behind *ALL* beings, whether aware or not, animate or inert.
Of course, the ramblings and clutter of the ego-mind often interfere with discernment and make it hard to perceive consciousness & awareness in others, but when the mind is pure, seeing these things is just like looking in a mirror. There's a sense of familiarity... recognition. To a pure mind it's obvious. Everywhere you look you see yourself reflected back at you, even in inert matter. But even in an average mind, on a very deep, "subconscious" level, that familiarity is there also, even if the mind doesn't realize or understand it. And that's how you can tell. It just takes discernment, discrimination, awareness, keen observation, a deep sensitivity, "attention to detail", the ability to see "beneath the surface". Everyone is capable of these qualities if they devote the necessary effort and attention in order to "cultivate" them.
But, even though the existence of consciousness and awareness can be "proven" (recognized) in others, it is still first and foremost dependant on your SUBJECTIVE awareness. The "proof" of awareness in another is only an extension of the "proof" of awareness in yourself, neither of which can be proven in an objective way.
Like you said though, there are subjective sciences, and infact this is the basis of all religion. These subjective sciences say "Do X technique, and realize Y result". They aren't concerned with proving anything to you, they simply give you the method in which you can find the proof yourself. All meditation is science. Subjective science. Do the method, and you will become both the result and the proof simultaneously.
The problem with religion though, is that all these techniques, methods, abstract concepts, etc. are often conveyed through a great deal of symbolism, and that symbolism gets grossly misunderstood by the "blind masses". That "misunderstood symbolism" gets taken literally, and people latch onto those things to fuel their need for identity. And of course there are also those who really have a deep "urge" to understand these things, but simply aren't capable of seeing past the symbolism yet. And that's fine, that's just "where they're at".
But even concepts by great scientific minds are often misunderstood by the masses, and even those ideas become distorted and impure, and the followers of those ideas are no different than the blind worshippers of X religion or cult. The work of great minds, whether scientific or religious, will always be misunderstood and made impure by the masses. It's just how it is right now. Humankind is on the verge of one of those "awareness leaps" I mentioned earlier, and a few individuals here and there just happen to get a little eager and leap a head before the rest. ;) -- or in other words, the "awareness leap" is like a magnet, and "humankind" is a clump of metal shavings. As you hold the magnet over the shavings and move it closer, very slowly, at first only a few shavings will jump up to the magnet (these are the "leading edge" great minds of both
Science AND religion). The rest of the shavings still remain "down there", on a lower level. But as you move the magnet closer, the amount of shavings being lifted increases at an exponential rate, until it appears like a large "clump" just gets lifted all at once. This is how leaps in "collective awareness" happen also, and it is also very closely related to the "hundredth monkey" concept (read about this on the internet if you don't know about it)
So anyway. Easy, right?
--James