I hope you like this brief answer to what is actually an incredibly complex question. The simple answer is that, in everyday mathematics, in number base systems higher than ?4,? 2 + 2 = 4 because it is defined as such. Based on the definitions of the number ?2?, the number ?4?, and the mathematical operation of addition, the answer is always the same. It is at the basis of all number theory and other branches of mathematics. Simply put, if you take a pile of objects that we designate as consisting of ?2? objects, and place it with another identical pile, the count of the resulting pile of objects is equal to what we label as the number ?4.? Of course, there are other number systems and other ways of doing math where that definition is not used, but they don?t generally produce very useful results for everyday applications.. It may surprise you to learn that an entire book, Principia Mathematica, (the one by Whitehead and Russell, 1910-1913, not the one by Sir Isaac Newton in 1687) devotes several hundred pages to deriving an explanation of just why 2 + 2 = 4 (actually, as I recall, it was 1 + 1 = 2) A complete proof, such as the one presented in Principia Mathematica, involves 1,789 sub-theorems consisting of 19,731 individual steps. As some others have already commented, the question about quantum mechanics only applies to extremely small events at and below the atomic level. The mathematics involved in solving questions in quantum mechanics seldom even involves addition. In the simplest sense, as I once heard an old friend, John Van Vleck, Nobel Laureate Physics, 1977, say, you should always think of any question involving quantum mechanics as being a question of probabilities. In that sense, in quantum mechanics 2 + 2 never = 4 with any certainty. One thing to keep in mind is that when you start getting into the details of advanced physics then you must switch languages. The reason things don?t seem to make sense at the atomic and sub-atomic levels is that non-physicists are forced to discuss things in English or some other language, which leads to what appear to be contradictions. In reality, you can only discuss such things in mathematics, which is why you will find most scientists very reluctant to talk about their work with non-scientists. Think about it this way. You would never use chemistry notation to show someone how to bake a cake. In the same way, you simply can?t explain most advanced physics concepts in English or any spoken language; the best you can do is summarize, simplify, and generalize ? the language simply won?t support detailed explanations without resulting in apparent paradoxes or contradictions.
