Its fun thinking: "uhm, what can I say about this word..." and then start having ideas bubble in ones brain! :D Mathematics has a way of molding your world view, and apparently, for me, this word has been completely taken over! hehe...
Contradiction: Self-negating; An argument or expression which concludes falsifying itself.
Contradictions are invaluable in the discovery of new truths; many famous contradictions are a critical part in the growth of knowledge and our understanding. In logic (mathematics) demonstration through contradiction is indeed one of the most powerful forms of proof! For example; a fables narates how a Greek named Hippasus drowned over a "proof through contradiction": most Pythagoreans thought there was no such thing as irrational numbers (as it went against their beliefs on the beauty of nature); so like all other numbers must indead be rational... except that this leads to a contradiction. As the story states, Hippasus must have noted this, apparently sharing the demonstration just before being told to go drown somewhere.
Contradictions are not paradoxes, and are neither undecidable nor incomplete; this is why philosopher's and mathematitians have found salvation and beauty in carrying out proofs through contradiction. The weight of this perspective can be condensed in the following quote by Ayn Rand: "Contradictions do not exist. Whenever you think you are facing a contradiction, check your premises. You will find that one of them is wrong." Thats exactly how we can demonstrate irrational numbers exist. Because of this; it was once assumed that mathematics was a self healing body of absolute knowledge we simply pearing into; Galileo stated: "Mathematics is the language through which God has written the universe".
So: contradictory as it seems, contradictions have been the source of truth, new realizations and a more secure untainted view of our reality.