When we write down a complex logical argument, it seems there is a pretty good chance that while writing down the proof, we will write down sentences that express propositions which we do not believe. Some of these sentences will be as part of a conditional proof, and those are not puzzling. But some of the sentences that express propositions which it seems we do not believe will be simply asserted. For instance, in the middle of our argument, we might make a claim that involves some complex logical or mathematical formula, which we then expand out using appropriate manipulation rules. But the expanded out claim may well exceed our mental capacities: we can handle it on paper, but it is just too complex for us to believe, it seems.
If this is right, then sincerity does not require that I believe what I say. (I assume the rules for sincerity do not depend on whether I am writing or speaking.) All that is required is that I believe that what I am saying is true. (What should I say about the speaker meaning in such a case?)
Or so it seems. But here is a curious test case. Politician reads a speech that her speechwriter wrote. She trusts her speechwriter to write only truths. The politician did not read over the speech ahead of time. She enunciates the sentences carefully, but she is distracted and pays no attention to what the text says. I think we would say that she is not being fully sincere. Maybe, though, our standards for sincerity are unfairly high in the case of politicians. Or maybe there are different kinds of sincerity—there is the bare sincerity which is one's duty in speech, and there is something that one might call "real sincerity" which entails conviction (where conviction is belief and more).
On the other hand, there is a different way of looking at the case of the complex sentence in an argument (this is inspired by some things that David Manley said based on his book with John Hawthorne). Maybe we can simply gain access to the proposition by means of the sentence, without having ourselves to understand or even parse the sentence.
Or maybe the things in the middle of proofs should not count as assertions. Perhaps making steps in proof is a mechanical procedure, akin to punching buttons on a calculator and likewise intrinsically devoid of propositional content, aimed at producing empirical evidence of the truth of some entailment. (That the evidence produced by a complex proof is empirical in nature is clear to me, weird as it may sound. One reason is that memory is intricately involved.)