You may be familiar with the Heisenberg Uncertainty Principle, which states that the more precisely you know a particle's position, the less you can know about its momentum, and vice versa. The two values, position and momentum, are what mathematicians call conjugate variables. You can have a pretty good idea of each, or know a whole lot about one and very little about the other, or everything about one and have no clue about the other whatsoever. A full understanding of a particle requires both variables, but each variable precludes the other.
I've always suspected that there's some connection between this concept and the wave/particle duality, the strange quality of light (well, matter and energy generally, too) that it can display properties of a wave or a particle, but not both simultaneously. Similarly, the two leading theories of physics, General Relativity and the Standard Model of Quantum Mechanics, have both been spectacularly successful at predicting what they're supposed to predict, but they appear by all accounts to be incompatible with each other. Neither alone is sufficient to explain all of the universe, but each seems to contradict important elements of the other.
I have wondered on occasion if there might be a connection between these dualities and Gödel's Incompleteness Theorem. I first learned about this theorem, as many did, from Douglas R. Hofstadter's famous classic, Gödel, Escher, Bach: An Eternal Golden Braid, a book that has had an enormous influence on me. The upshot of it is this: a logical system will be either unsound (it will be able to prove falsehoods) or incomplete (it will be unable to prove some true statements); it cannot be both sound and complete. You can understand roughly how the proof works by considering this statement: "This statement cannot be proved." If you can prove that statement, then it's false and your system allows the proof of falsehoods, but if it's a true statement, then there exists at least one true statement that your system cannot prove.
Of course, I have no formal training in mathematics or physics (beyond a couple of first year university courses), and so I'm probably grossly oversimplifying these things or worse. I do have some background in philosophy, however, so I can talk about theories of ethics with a little more confidence, and I've always been struck by the way these exclusive choices between knowing momentum and position, or observing a wave or a particle, or explaining the universe at tiny or vast scales, have a parallel in the two main theories of ethics, deontology and consequentialism.
Consequentialism's best known incarnation is the Utilitarianism of John Stuart Mill, which evaluates the good or evil of an action by its consequences; an action is good if it leads to greater happiness than its alternatives, while an action that leads to greater unhappiness is evil. (An oversimplification, but it'll do.) Deontology, most associated with Immanuel Kant, explicitly denies that actual consequences have anything to do with the moral character of the choice. After all, we praise the moral character of someone who tries and fails to save someone from drowning, and condemn someone whose failed attempt at murder inadvertently saves a life.
I find both theories extremely persuasive, but of course, they are incompatible with one another: Mill says you must consider consequences, and Kant says you must not consider consequences. Yet it's my feeling that either system alone can only give a partial understanding of what morality is really all about; you need to understand both to have any hope of properly analyzing ethical problems. And unavoidably there will remain some questions that may not be answerable.