>Everyone's familiar with Descartes' "I think, therefor I am", but how can I be sure anyone else exists? Is there a empirical method towards confirming someone's existence?
Yes, everyone exists, because we live in a world of atoms, and atoms arranged in a certain little way makes a thing we call "people." If you went blind, mute, lost your sense of touch and ability to hear, they would still be there, watching you flail about.
If you don't buy the idea that reality exists outside observation, then nevermind. You don't have a reference point for anything at-all, in fact, but now you have a new question to ask: why do you care if anyone else exists? You may be entirely alone, you may not know for certain whether anyone else or anything exists, but why do you care?
Even if you proved for certain that no one else existed at-all, you're still experiencing their presence, you don't fly without wings, you haven't won four lotteries in five minutes, and you're not eighty feet tall. If you think you can change those facts -- well, do it. If your bet is wrong, reality exists and your brain is fooling you, you'll wind up some shade of mentally ill, but such fine control over the senses is rare and worth pursuing.
If you can't do those things, then your behavior shouldn't change at all. You've been dropped into a very protracted nightmare, and you have lots of imaginary friends and foes along for the ride. Be nice, and be thankful there's anyone else with you at all, even shitty people, because nothingness is worse.
>But that's a purely rational method. You can look up Husserl's argument against solipsism, it should be in "Formal and Transcendental Logic" and "Cartesian Meditations" somewhere.
Husserl was implicitly solipsistic, even permitting his objections, because he hinged his defense on intersubjectivity. No matter how empathic and "a-priori" you claim the community your transcendental lens participates in is, it must still be observed to become certain, since transcendentalism demands that observation be the root of objectivity.
Yet observations are of fault, and every step of the twinning with the transcendental community is guilty of assuming the reality of the community in the first place. Assuming there is a part of one's self one can not observe guiding them, some principle that generates falsity, or a third party intervening, it is easy to construct a scenario in which the entire experience is falsified and returned as though it is real without ever truly connecting. /x/ is intimately familiar with at least one real-world analogy in schizophrenic tulpafags, who commit every step of Husserl's proof of community- with themself, and no one else.
There's just no way around it, really, as a "non-solipsistic" reality which has a purely experiential component can only ever be a matter of theology. That's not a problem, of course, but it must be understood as such to move forward in a productive direction.
>and you are willing to trust your sensory experience
You can't, that's the entire point. Solipsism isn't a real position for anyone but loons, it's a thought experiment about the implications of the inadequacies of qualia. That's also why it's usually used to object to theories, so as to say, 'if your assumptions lead to solipsism, somewhere you must have a logical fault.'
That's jumping the wagon a bit, but it's still a valuable divisor in theories. Either your interpretation of reality is tailored to allow a single ego explanation, or refuses to allow a single ego explanation.