When we first meet Bernard, he's on an undercover mission: to get the information he needs out of Hannah without her figuring out that he wrote a particularly mean review trashing her book. Even under the name of Peacock, however, he doesn't manage to hide his obnoxious opinions for long enough to get on Hannah's good side. In fact, he puts his foot in his mouth, big time, by sharing his views about where the accent should fall in the word "ha-ha" (not the laugh; in this case, it's a trench used instead of a fence to keep out livestock without blocking the view in a garden).
BERNARD: A theory of mine. Ha-hah, not ha-ha. If you were strolling down the garden and all of a sudden the ground gave way at your feet, you're not going to go 'ha-ha', you're going to jump back and go 'ha-hah!', or more probably, 'Bloody 'ell!' [...] In France, you know, 'ha-ha' is used to denote a strikingly ugly woman, a much more likely bet for something that keeps the cows off the lawn. (This is not going well for Bernard but he seems blithely unaware.) (1.2)
A tip: if you're talking to someone who just wrote a book arguing that a previously dismissed woman author should be taken seriously, making sexist comments is probably not going to earn you any points. The fact that Bernard 1) goes off on this tangent in the first place, and 2) is totally unaware of the bad effect he's having on Hannah, shows his complete self-absorption. He's pretty much incapable of taking other people's points of view into consideration. While there may be something positive to be said about single-minded determination (which seems to be a factor in Bernard's initial success in researching Byron), it also almost earns him a kick in the balls from Hannah. Perhaps being stubborn and blind to what's going on around you is not the safest way of moving through the world.
Bernard searches for facts just because other and, in his opinion, inferior people need to be persuaded of something that is obviously true. He's completely certain that he's right before he even starts his research. As he puts it to Hannah:
BERNARD: I'll tell you your problem. No guts. [...] By which I mean a visceral belief in yourself. Gut instinct. The part of you which doesn't reason. The certainty for which there is no back-reference. Because time is reversed. Tock, tick goes the universe and then recovers itself, but it was enough, you were in there and you bloody know. (3.4)
The fact that this debate is with Hannah rather than with Valentine puts an interesting spin on it: while it might be tempting to think Bernard = literature = instinct, and Valentine = science = reason, it's not quite so simple. Hannah says that literary arguments also require reason and evidence (remember that when you're writing your English paper), while Thomasina's ideas suggest that scientific discovery can come, like artistic inspiration, in a flash of intuition (as Valentine says, "She saw what things meant, way ahead, like seeing a picture" [II.7]).
If the divide between intuition and reason isn't literature vs. science, or bad thinking vs. good thinking, then what is it? What do Bernard's successes and failures, as well as those of the other characters, suggest about how these two kinds of thinking can and should work together to create knowledge?
While the ruin of all your plans (in a particularly painful and public way, no less) is usually the grounds for some rethinking of your actions, Bernard misses the boat entirely on the idea of positive character development. He does have one glimmering of a realization that he could have acted better:
BERNARD: I've proved Byron was here and as far as I'm concerned he wrote those lines as sure as he shot that hare. If only I hadn't somehow ... made it all about killing Chater. Why didn't you stop me?! (2.7)
Self-knowledge: FAIL. While Bernard does notice, briefly, that he would have been better off if he hadn't insisted on publicizing the story despite his shaky evidence, his last sentence demonstrates that he feels no sense of personal responsibility whatsoever. It also suggests that Bernard's tendency to rewrite history includes his own life experiences: as you may remember, Hannah did everything short of tying Bernard up to try to keep him from going forward with his hare-brained scheme. Bernard's self-centered tunnel vision not only causes him to fail in the first place, it also prevents him from learning from that failure.
Just in case any of us holds out any hope that Bernard might still become a better person, his parting shot puts down that thought once and for all:
HANNAH: Actually, I've got a good idea who he was, but I can't prove it.
BERNARD: (With a carefree expansive gesture) Publish! (2.7)
While self-confidence can be a great thing to have, Bernard's failure to learn from his experience suggests that a little goes a long way. His final cry of "Publish!" echoes his ongoing obsession with publicity, but also invokes the play's concern with writing as record – if Hannah doesn't put her ideas, right or wrong, into text, how will anyone know that they even exist? Bernard gets proven wrong on the Byron question, but does that mean he was wrong to share his ideas? Is there something salvageable in Bernard's methods?