I think you misunderstand the nature of argument. All arguments rest on a set of premises, and those premises themselves rest on premises, recursively so until

I think you misunderstand the nature of argument. All arguments rest on a set of premises, and those premises themselves rest on premises, recursively so until we reach axioms which we simply accept to be true without further justification - like that the external world exists in some sense. Feel free to doubt that, by all means. Proof is not accessible to us - not a point the existentialists made (as you seem to believe), but rather something they took as self-evident and were responding to. Analytic philosophy basically contends that if we assume axioms hold (correspond to the world in some sense) and we construct arguments from those premises which conform to logical rules, then the conclusions of those arguments are also true of the world in some sense. This is, of course, doubtable in various ways - but you pretty much have to abandon reason itself as the alternative.