Dialetheism corrects a very long-standing error. Aristotle believed that it was important to have an axiom, the 'non-contradiciton' axiom because he thought it necessary to enable the law of the excluded middle - that is, something is either the case, or it is not the case.
When you see it, it's obvious - though, even today, some people fight against dialetheism. There clearly are some things that are both true and not true at the same time - not many of them, but certainly some of them.
Here's one: 'This sentence is false'. Well, if the sentence is true, then it is not true. Fair enough, it is true. Also, it is false. Where's the problem? The universe doesn't collapse because we notice that, and we still can tell that the sentence 'This sentence has seven words' is false, and 'This sentence has five words' is true.
What is remarkable, really, is that that Aristotle's error remained unnoticed for so very long. Many people, some extremely clever and thorough, took his mistake for granted. Even the great philosophers and mathematicians, Bertrand Russell and Alfred North Whitehead thought that their great work on mathematics had been floored by a simple contradiction, and died unhappy, not realising that it was only this tiny assumption that stood in the way of success. The moral of the story being that it's important to be hugely sceptical of even the most apparently reasonable assumption.
It's understandable that mathematicians don't like dialetheism - many mathematical proofs rely only on a contradiction, which doesn't mean that they're wrong, but does mean that they could be wrong, so they are 'unsafe' and somebody should work to find another way of proving them that is safe. Mathematicians should be pleased that there's all that extra work for people to get doctorates from, but optimistic mathematicians are not that thick on the ground.
Actually, I think it's very encouraging for the future. If such a simple error can lie unnoticed, under the noses of some of the cleverest people the world has known, then just how many more wonderful things are there just waiting to be revealed?
Never believe anything because somebody else, apparently wise and clever, says that it is the case - only accept things when you have clear evidence and have tested all the assumptions you've made, including unconscious ones, even if you have to work to ferret out just what those are.
