Don't be a puss! (A rejoinder to Skeptic Cat)
On a personal level: Hmm, guilty until proven innocent or innocent until proven guilty? Demanding 'proof of innocence' frequently in a relationship is incredibly corrosive, not to mention time-consuming and emotionally draining. In fact, I would argue that without trust there is no real relationship. So, proof of what and by whom? And what constitutes a proof (as Alex notes)--news reports, gossip?
In mathematics: How often does a mathematician confirm the truth of theorems he uses? Don't most wait for apparent inconsistencies or other inadequacies to appear before closely examining a mathematical edifice skeptically? And even then, don't most mathematicians trust the 'authorities' in the field to resolve the problem?
In some sense, a belief that a good, pragmatic approximation to 'the truth' will out is implicit in people's approaches to most endeavors. (E.g., see Michael Atiyah's comments on Proof in Advice to a Young Mathematician.) So, in the spirit of Euler, Fourier, Riemann, Heaviside, and Ramanujan (some of the most productive personalities in mathematics), don't be a puss, have the courage to forge ahead (albeit not blindly) until your belief is no longer tenable.