Mathematical Recreations; January 2000; Scientific American Magazine; by Stewart; 2 Page(s)
In ordinary discourse when we say something is impossible, we often don't mean that. What we mean is that we can't see any way to achieve it. Many people once thought it was impossible for machines heavier than air to fly, and before that a lot of people thought it was impossible for machines heavier than water to float. Human ingenuity frequently overcomes apparent impossibilities.
In mathematics, though, impossibility is something you can often prove. For instance, 3 is not an integer power of 2. One way to prove this is to ask what power of 2 could equal 3 and observe that 1 is too small (21=2) and that 2 is too big (22=4). Impossibility proofs, however, function only within the world of mathematics as it is currently set up: if you change the rules of the game, different things may happen. For example, in the set of integers "modulo 5," any multiple of 5 is considered to be 0 and any number above 5 is converted to the remainder after division by 5. Under these rules, 3=23. This doesn't mean that my original impossibility statement is wrong, because the context has been changed. It just means I have to be careful to define what I'm talking about.