Taboo "exists". Does the physical world contain things you don't see? Also, lack of absolute certainty doesn't imply confidence in absence, one shouldn't demand unavailable kind of proof and take its absence as evidence.

What Nesov and ArisKatsaris said.

Also, there's no crisp line saying images on the retina can constitute "proof" but memories cannot. Both are indirect evidence, the difference is in degree not in kind. Especially if you know yourself to have poor eyesight but reliable memory.

Your friend is an amateur. I don't have proof that people exist even when I can observe them directly in front of me.

(I do have really really really strong evidence, though. And when they leave and I'm left with mere really really strong evidence, that's still good enough for me too.)

How does his belief pay rent (http://lesswrong.com/lw/i3/making_beliefs_pay_rent_in_anticipated_experiences/)? Try to figure out how his predictions about the world differ from yours. If they don't, his extra belief isn't paying any rent; if they do, then you'll have an obvious test to see who's correct.

How could you come into contact with them again if you were in different universes? If your universe constantly communicates with theirs in order to know when to combine (when you begin to occupy adjacent corresponding space), in what sense do you occupy 'different universes'? How do you know there isn't just a wall that appears whenever you can't measure it - and how do you know it's there if you can't by definition measure it? (Note: only ask him these questions if you want to get back at him for confusing you.)

Also The Simple Truth, that one's good too.

Edit: I'm pretty sure the relevant bit is

Inspector Darwin looks at the two arguers, both apparently unwilling to give up their positions. “Listen,” Darwin says, more kindly now, “I have a simple notion for resolving your dispute. You say,” says Darwin, pointing to Mark, “that people’s beliefs alter their personal realities. And you fervently believe,” his finger swivels to point at Autrey, “that Mark’s beliefs can’t alter reality. So let Mark believe really hard that he can fly, and then step off a cliff. Mark shall see himself fly away like a bird, and Autrey shall see him plummet down and go splat, and you shall both be happy.”

We all pause, considering this.

“It sounds reasonable…” Mark says finally.

“There’s a cliff right there,” observes Inspector Darwin.

Autrey is wearing a look of intense concentration. Finally he shouts: “Wait! If that were true, we would all have long since departed into our own private universes, in which case the other people here are only figments of your imagination – there’s no point in trying to prove anything to us -”

A long dwindling scream comes from the nearby cliff, followed by a dull and lonely splat. Inspector Darwin flips his clipboard to the page that shows the current gene pool and pencils in a slightly lower frequency for Mark’s alleles.

(Emphasis mine.)

The presence of memories in your mind is evidence about things you can no longer see. When someone leaves your presence, you are left with a memory of them having existed. This should be very convincing evidence to you; it is quite rare to have clear memories of people that don't exist. On the other hand, if you try to communicate this evidence to other people, it's less convincing to them, because you might miscommunicate, or lie, and these are much more common than false memories.

Of course, it is possible that the person you remember having existed, stepped into an incinerator or otherwise stopped existing while you weren't looking. But focusing on sufficiently-rare exceptions is a mistake; you can't ever have 100-point-0 certainty about anything (that would require a literally-infinite amount of evidence), but 99-and-some-more-nines percent certainty really is sufficient. Emphasizing the difference between these two levels of certainty is usually an instance of the fallacy of gray.

As for the "personal worlds" stuff - we do in fact all occupy the same world, and talking about peoples' individual worlds is just a confused way of talking about the contents of different peoples' heads.

Others have linked some relevant articles, but I think the one that most directly addresses your question is Belief in the Implied Invisible. (I tried to expand it in the context of Tree Falls in a Forest ...)

Long story short, in order to have a model that says the person stops existing when unobserved, you immensely complicate your description of the world, because you would have to throw out all kinds of regularities you rely on, like conservation of mass.

You can certainly get away with simply positing the concept of "my reality" like your friend does, and say someone stops existing in that sense, but it's unclear what would be the point of doing so, since the concept is (as far as I can tell) precisely identical to "I don't detect that person", so you already have an expression for it.

gravity / existence of neptune

I mean, he challenged me to prove that my mother existed, without seeing her. Obviously I couldn't.

Beliefs pay rent in anticipated experience. Your belief that your mother exists means that you would expect to see her if you went to wherever she lives or whatever. So he wants proof? Bring him to her. Can't be done in class though.

Of course he might well have meant something entirely different by "exist" than I assumed above, but the ordinary interpretation of "my mother exists" (when she's not in the speaker's presence) is something like, "I have a model in my head saying that we'll see my mother if we go to [location]".

If he means something else though, then you'll simply have to ask what that something else is.

I mean, he challenged me to prove that my mother existed, without seeing her. Obviously I couldn't.

You could, if you have sufficient sharp ears. Or smell sense. The point is, if you accept current physical law, even if your mother is space-like separated from you, there'll come a time when your future cones will interact, and in that case, provided you have sufficient sharp instruments, you could prove her existence. You could even entangle your matter with her, thereby reaching non-locality (or creating another universe, if you prefer realism). The only way to physically separate yourself from your mother is to make her cross an event horizon (cosmological or otherwise): in that case, no more physical interaction can occur and she will be gone for good.

Otherwise, as others have noted, the only way left is solipsism.

Everything is connected. My mother provided my existence in the first place. When she walks out of the room, she still breathes in oxygen and breathes out carbon dioxide, she still consumes food and water, releases methane, sheds skin as dust, absorbs sunlight, and consumes products. This book I hold in my hand, it required the existence of a bookbinder, of a writer, of an editor, an accountant, a tax man, a delivery man, a construction worker, a copyright official, and on and on. These people poured their existence into that object, just as all of my ancestors going back a billion years poured their existence into me. Right now, me typing this on a screen, requires all of them, everything, all connected, all pouring into this moment. Without those people, without those worlds and physical laws, without even you, without your questions and thoughts, I wouldn't be here, doing this; I would be someone else, or nothing at all.

I am, therefore you are.

He does not understand the difference between map and territory. He thinks that he thinks that existence in the territory depends on existence in his map.