My previous reply wasn't very helpful, sorry. Let me reiterate what I said above: making assumptions isn't so much rational as unavoidable. And so you ask "then, should we believe in the external world?"
Well, this question has two answers. The first is that there is no argument that will convince an agent who didn't make any assumptions that they should believe in an external world. In fact, there is no truth so self-evident it can convince any reasoner. For an illustration of this, see What the Tortoise Said to Achilles. Thus, from a perspective that makes no assumptions, no assumption is particularly better than another.
There is a problem with the first answer, though. This is that "the perspective that makes no assumptions" is the epistemological equivalent of someone with a rock in their head. It's even worse than the tortoise - it can't talk, it can't reason, because it doesn't assume even provisionally that the external world exists or that (A and A->B) -> B. You can't convince it of anything not because all positions are unworthy, but because there's no point trying to convince a rock.
The second answer is that of course you should believe in the external world, and common sense, and all that good stuff. Now, you may say "but you're using your admittedly biased brain to say that, so it's no good," but, I ask you, what else should I use? My kidneys?
If you prefer a slightly more sophisticated treatment, consider different agents interpreting "should we believe in the external world" with different meanings of the word "should". We can call ours human_should, and yes, you human_should believe in the external world. But the word no_assumptions_should does not, in fact, have a definition, because the agent with no assumptions, the guy with a rock in his head, does not assume up any standards to judge actions with. Lacking this alternative, the human_reasonable course of action is to interpret your question as ""human_should we believe in the external world," to which the answer is yes.
The second answer is that of course you should believe in the external world, and common sense, and all that good stuff. Now, you may say "but you're using your admittedly biased brain to say that, so it's no good," but, I ask you, what else should I use? My kidneys?
This is the place to whip out the farmer/directions joke. The one that ends, "you just can't get there from here."
I have naturally read the material here, but am still not sure how to act on two questions.
1: I've been arguing out the question of Foundationalism v.s Coherentism v.s other similiarly basic methods of justifying knowledge (e.g. infinitism, pragmatism). The discussion left off with two problems for Foundationalism.
a: The Evil Demon argument, particularly the problem of memory. When following any piece of reason, an Evil Demon could theoretically fool my reason into thinking that it had reasoned correctly when it hadn't, or fool my memory into thinking I'd reasoned properly before with reasoning I'd never done. Since a Foundationalist either is a weak Foundationalist (and runs into severe problems) or must discard all but self-evident and incorrigible assumptions (of which memory is not one), I'm stuffed.
(Then again, it has been argued, if a Coherentist were decieved by an evil demon they could be decieved into thinking data coheres when it doesn't. Since their belief rests upon the assumption that their beliefs cohere, should they not discard if they can't know if it coheres or not? The seems to cohere formulation has it's own problem)
b: Even if that's discarded, there is still the problem of how Strong Foundationalist beliefs are justified within a Strong Foundationalist system. Strong Foundationalism is neither self-evident nor incorrigible, after all.
I know myself well enough to know I have an unusually strong (even for a non-rationalist) irrational emotive bias in favour of Foundationalism, and even I begin to suspect I've lost the argument (though some people arguing on my side would disagree). Just to confirm, though- have I lost? What should I do now, either way?
2: What to say on the question of skepticism (on which so far I've technically said nothing)? If I remember correctly Elizier has spoken of philosophy as how to act in the world, but I'm arguing with somebody who maintains as an axiom that the purpose of Philosophy is to find truth, whether useful or useless, in whatever area is under discussion.
3: Finally, how do I speak intelligently on the Contextualist v.s Invariantist problem? I can see in basic that it is an empirical problem and therefore not part of abstract philosophy, but that isn't the same thing as having an answer. It would be good to know where to look up enough neuroscience to at least make an intelligent contribution to the discussion.