Well, I'm not entirely convinced the phrase "order academic fields by rigor and exactness" is a completely meaningful one. It implies a level of direct comparability that I'm not confident exists. I certainly agree that humanities makes for very bad science, but then, so does basket weaving. The flip side is that science has not developed a particularly useful vocabulary for dealing with nuance, ambiguity, or irony.
I'm also not sure a philosopher/cultural critic without significant scientific training is bound to write nonsense, so long as they actually stick to their field of expertise. Now, it may well be true that humanities sorts are more prone to straying from their actual areas of expertise - certainly a study demonstrating that would not surprise me. But I think that one can write for a very, very long time about sexual politics in Victorian literature without ever running into a situation where lack of knowledge of science beyond a high school level is going to be a problem. It's certainly difficult to imagine it resulting in nonsense production that goes beyond a stray sentence here or there.
And yes, the Sokal Affair clearly reflects badly on the editors of Social Text. But that's why I compared it to getting nonsense published in a newspaper - which has been proven possible from the local level up to major international papers. I also would not describe Social Text as "eager" to publish Sokal's essay. They rejected it initially, and only dusted it off because it was directly relevant to a special issue they were publishing and, probably more importantly for a paper journal, it was very short. But more to the point, Social Text is not a scholarly journal. I wasn't in the field at the time of the Sokal Affair, so any sense I have of its reputation is second hand, but if an un-peer reviewed journal was being treated as equivalent to PMLA or Critical Inquiry or something, that, much more than the Sokal Affair, is damning evidence against the humanities.
I should note that I am also more hostile towards Sokal for his completely and irredeemably awful book Fashionable Nonsense than I am for the Sokal Affair. He, at least, does not overstate the significance of his own hoax. Whereas Fashionable Nonsense is a sufficiently wretched book that I would generally advise people that their time would be better spent reading up on the intricacies of crystal energy healing.
To be perfectly honest, I know far more people in hard sciences who look down on postmodernist scholars as wooly nonsense-peddlers than I do postmodernists who reject the sciences or rationalism. This is, admittedly, anecdotal evidence, but I can honestly say that I have never seen a piece of anti-science writing out of the humanities half as perniciously irresponsible as Alan Sokal's "work." Certainly nothing that is as reflexively cited in discussions. To be honest, I find an exasperating tendency among math/science people to simply stop their reading on postmodernism with the Sokal Affair and decide they've got the matter nailed, despite the fact that almost nobody bothers to mention that Social Text isn't a peer-reviewed journal, and thus Sokal accomplished something about as hard as publishing a fraudulent piece in a local paper for a semi-major city.
So I'm inclined to be skeptical about which side of that debate is more prone to being infected with bad thinking. The biggest problem postmodernists usually have with math, science, and other more rational fields is that those are not their field, and they're not experts in them. Sit one of them down with some Feynman lectures, though, and they'll generally be fine, because they're generally speaking really smart people who just aren't specialized in that stuff. But the thing is, most of them know that they don't understand what goes on past freshman year in any math or science courses. Puzzlingly, science people seem to freely assume they understand graduate humanities work with alarming regularity. Of course, this failing on their part in no way invalidates their field - any more than the fact that someone who majored in English hasn't taken many math courses invalidates their work. :)
I'd delight in telling you you're wrong, but you're mostly not.
I would say that I don't think that postmodernism is lacking in rigor. Certainly, having been on both ends of peer review in the humanities, it does not seem to me that the process lets through a lot of flamingly inaccurate crap, beyond the sort of expected problems you get in the margins of well-studied ground. Frankly, in my own research, I'd have an easier time sailing a howler about the history of video games past peer review than I would a howler about the applications of Derrida.
I'm also not sure it does as badly as you say on Vladimir M's heuristics. Looking quickly, for my own field, there's still a ton of low-hanging fruit. Yeah, the major canonical works of literature assigned to undergraduates are pretty well-covered in the literature, but if you're working in popular culture of any era, you have basically no excuse for running out of things to say. The ideology test is a little trickier, since there are areas of literary criticism - feminist, queer, and racial studies, most obviously - that are explicitly ideological. But, of course, we have to be careful with ideology as a warning sign, because arguably at this point climate science and biology are ideologically poisoned. Ideology takes place heavily outside of the academy, and I'm loathe to say that just because something has become a political hot potato the academics of it are prima facia problematic.
But I think that postmodernism, broadly speaking, also hedges well against ideological bias. The thing that it is most easy to completely miss about postmodernism is that it is, by and large, a study of epistemologies. Yes, postmodernist literary criticism makes massive use of Freudian psychoanalysis despite the complete discrediting of Freudianism as an empirical practice. But we don't psychoanalyze real people. We psychoanalyze fictional people. The question is not "does there exist a real person with an Oedipus complex," it's "is the Oedipus complex a sufficiently well-known concept that it would be either included in a work of fiction by an author or read into a work of fiction by a reader?" In other words, since we're generally dealing with man-made structures, the question is less "Is foo true" and more "Is foo part of what is talked about and thought about regarding this subject?"
This helps a lot with the ideological bias issue, because it means that postmodernism can actually be ideologically neutral on the political issues it seemingly invests in. This is something I tell my students whenever one of them gets fussy about the fact that I assign literature with gay people in it. You don't actually have to accept the premise "homosexuality is not a valid basis for discrimination" in order to understand how gay writers articulate their own experience in literature.
Now, in practice, of course, literary scholars are generally liberals who are supportive of gay rights, feminism, and further work towards racial and economic equality. But then again, the ideological bias in the academy extends far beyond humanities departments - it's increasingly hard for an American climate scientist to be a Republican too. At the end of the day, this is somewhat irreducible - after all, reality probably does, in the long run, lend more support to some political viewpoints than others.
In any case, I tend to think that postmodernism is something that just about everybody should be broadly familiar with. I think there are some serious problems with how we teach it, and some serious problems with how it's attacked (Alan Sokal should never, ever be referenced by anyone who is trying to make a serious point about humanities scholarship. His "research" on postmodernism makes his hoax article look like sound scholarship by comparison), but the basic elevator summary of postmodernism is "It's the process of learning to account for epistemological differences when dealing with communications." Which is something that is probably useful if, you know, you ever intend to talk to someone you disagree with in your life.
Also, can I just say that the Stanford Encyclopedia entry you linked to is absolutely terribly written? Clue for writers explaining concepts that use algebraic or pseudo-algebraic notation - it's really nice if you bother to define what S and T mean before you start throwing them around.
I would agree if one is writing about Victorian sexual politics straight-up, however I was careful to specify the sexual politics of Victorian literature. For which Freudianism, notoriously wrong as it is, is highly relevant because it was enormously popular for a chunk of the time period, and did directly influence writers (more particularly in the early 20th century than the late 19th, but still). Certainly it had much more influence than post-Victorian science that the authors could not possibly have been aware of.
Which seems to me one of the hedges that postmodernism usefully offers. The decision to approach Victorian literary sexual politics in terms of the thinking of the time and to treat it as a phenomenon of that culture is, to my mind, quintessentially postmodernist.