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CuSithBell comments on Why Our Kind Can't Cooperate - Less Wrong
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In real life this is common, and the results are not always bad. It's incredibly common in mathematics. For example, Fermat's Last Theorem was a "last line" for a long time, until someone finally filled in the argument. It may also be worth mentioning that the experimental method is also "last line first". That is, at the start you state the hypothesis that you're about to test, and then you test the hypothesis - which test, depending on the result, may amount to an argument from evidence for the hypothesis.
Another case in point, this time from history: Darwin and natural selection. At some point in his research, natural selection occurred to him. It wasn't, at that point, something that he had very strong evidence for, which is why he spent a lot of time gathering evidence and building argument for it. So there's another "last line first" which turned out pretty well in the end.
I think the thing which is jumping out as strange to me is doing this after you've been convinced, seemingly to enhance your credence. Still, this is a good point.
The danger that Eliezer warns against is absolutely real. So what's special about math? In the case of math, I think that there is something special, and that is that it's really, really hard to make a bogus argument in math and pass it by somebody who's paying attention. In the case of experimental science, the experiment is deliberately constructed to take the result out of the hands of the experimenter. At least it should be. The experimenter only controls certain variables.
So why is there ever a danger? The problem seems to arise with the mode of argument that involves "the preponderance of evidence". That kind of argument is totally exposed to cherry-picking, allowing the cherry-picker to create whatever preponderance he wants. It is, unfortunately, maybe the most common argument that you'll find in the world.