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Vladimir_M comments on Costs and Benefits of Scholarship - Less Wrong
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In my opinion, the main danger of reading things like postmodernism is that one might get infected by the smug attitude of superiority that looks upon rigorous, precise, down-to-earth rational thinking as blinkered and nerdy, without having anything better to offer instead.
Now of course, there are subjects that nobody yet knows how to approach with rigorous and precise thinking, and all attempts to do so regularly end up in blinkered and nerdy discussions without much connection to reality. In these subjects, a more fuzzy approach is indeed the only viable alternative. The trouble is, those humanists who, as you say, can barely count without using their fingers tend to greatly overestimate the extent of these subjects, and their self-satisfied and smug attitude can be very infections for a lot of people.
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. :)
Oh, I certainly don't think that the average hard scientist's view and knowledge of humanities are much better. However, when you order academic fields by rigor and exactness, with pure mathematics on one end and humanities on the other, then as a very general rule, in order to avoid writing nonsense in your own field, you must not have misconceptions about fields that are more exact than yours, whereas knowledge of less exact fields is normally not important for scholarly work.
Thus, for example, a physicist can be completely ignorant about philosophy and humanities and nevertheless consistently produce top-quality physics, whereas a philosopher or a cultural critic who is completely ignorant of natural sciences will inevitably end up writing nonsense at least occasionally. So while both of them may have an equally distorted and ignorant view of each other's field, the latter's work will likely suffer far more as a consequence.
Regarding the Sokal affair, I agree that it's usually overblown far beyond its real significance (and physicists should also be more humble in light of the more recent Bogdanoff affair). However, I think your minimization of it is also exaggerated. Regardless of whether a journal is peer-reviewed, editors should be held responsible for what they decide to publish. I don't think the editors of Social Text would have been so eager to publish Sokal's essay if it hadn't pandered so consistently to their ideology.
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.
I strongly disagree here. To write meaningfully about sexual politics, you must have a model of sexual and other related aspects of human thought and behavior, and modern science has a whole lot to say about that. (Of course, the relevant science is still very incomplete and far from settled, but that makes it even more important to be knowledgeable about it, in order to separate solid insight from speculation.) If you lack that knowledge, your model is likely to be wrong in at least some ways that could be corrected by familiarizing yourself with the relevant science, and this is likely to show in your writing. Moreover, there is a whole lot of spurious pseudo-insight in this area (Freudianism and its offshoots being the most notorious example), and if you're not familiar with science beyond a high school level, you may well end up swallowing a lot of such nonsense believing it to be solid insight and incorporating it into your work.
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.