Fields that can and do test their theories against reality in practical, empirical tests, that discard falsified theories, and that have a corpus of supporting evidence that forces all similarly equipped observers to the same conclusion, are sciences. The rest are pseudosciences whose status is clearly shown by innumerable articles whose titles end in a question mark.
Having a corpus of poor quality articles does not exclude you from being a science. Having dissenting groups, and areas where existing evidence permits more than one conclusion, does not exclude you from being a science. Quite the opposite: if there was no dissent, there would be no science.
That's true, but if all that exist are poor-quality articles, then it's not science -- there must be a corpus of testable, falsifiable theories beyond the chaff. This is why psychology isn't a science -- there are lots of poor-quality articles, and there's nothing else.
> if there was no dissent, there would be no science.
Nonsense. A scientific dissent must be accompanied by evidence, as does the thing being dissented against. Dissent per se means nothing in and of itself.
No, they aren't. A mathematical proof is by definition unfalsifiable. If an error is found in a proof (as with Wiles' proof of Fermat's Last Theorem), that it isn't a proof ... yet. And mathematical proofs aren't empirical, another requirement for science.
Scientific theories are falsifiable in perpetuity because the possibility always exists for new empirical evidence to show up that falsifies an existing theory. This possibility doesn't exist for mathematical proofs.