What should we ask of a theory? That's a somewhat broad question, but one vitally worth answering. Specifically, what should physicists ask of a physical theory? Already the classic divide between theoreticians and experimentalists springs up. The theoreticians would likely extol the virtues of beauty and logic in a physical theory, while their experimental counterparts would be more practical, demanding simply that a theory work. Much as I would love to smooth over the division, I must here side with the experimentalists. All we can ask of a theory is that it work.
What's wrong with beauty? Nothing. I love beautiful theories, Maxwell's breathtaking equations being the first ones that spring to mind. But how did Maxwell discover these equations? Not by looking for what was the prettiest, that's for sure. He looked for an explanation that worked, and found that the explanation that worked was beautiful. This happens so often, I believe, because the world is beautiful and we were created to appreciate and seek the beauty in the world. Nevertheless, beauty in a theory is a “bonus,” as it were. I can think of half a dozen ways of putting together the quarks, or of making relativity, that would be just as beautiful as the contributions of Gell-Mann and of Einstein to those subjects. So why are Gell-Mann and Einstein famous physicists while I'm a mere amateur? Because their theories are not only beautiful, they work – and they were found by searching for a theory that works. Let us not look for beautiful theories and hope they will work; let us look for working theories and know they will be beautiful.
What does it mean that a theory work? Obviously, we can't just say that a theory works in the same way as a car or a tool works. However, the tool analogy is closer than it seems, because a theory has a specific job to do, also – namely, to describe some aspect of the observed physical world. So a theory that works is a theory that describes the world, and it seems reasonable to add that we must be able to make predictions with it, and that the predictions must be accurate within their own sphere. (This last prepositional phrase allows theories like Newton's mechanics to make the definition of a working theory, even after the advent of relativity and quantum mechanics.)
What about replacement theories? When the old theory really fails to properly describe the world, a new theory is always to be commended, assuming it works (i.e. describes the world) better than the old theory. Sometimes, however, the new theory is put forth on purely hypothetical considerations, or considerations of beauty. What then? My advice to these theorists would be: Play with your theory a little bit. See whether it makes the same predictions as the old theory. If it does, is it easier to use? If so, you've found an alternative formulation or method that could be worth a shot. This is where beauty can become a “tiebreaker question” between two otherwise identical-in-practice theories. If it makes different predictions than the old theory, have they been tested already? Are they reasonable? If they are reasonable but untested, it has the potential to be an entirely new theory, and my advice is to ask the nearest experimentalist about testing it.
How do we judge a theory? What should the international scientific community do when some crazy-haired guy in a patent office says he's improved on Newton? I suggest that any new theory be judged based on its adherence to three simple standards:
Does it explain everything its predecessor explains? (If not, it's falsified.)
Does it explain something its predecessor doesn't explain? (If not, it's unnecessary.)
Does it explain anomalous observations other than those that inspired it? In other words, does the idea designed to explain unusual occurrences A and B also explain unusual occurrences C and D? (If not, it's undesirably ad hoc.)
Although requirement (3) appears the most rigorous, right now in physics requirement (2) is the hardest to meet. Quantum theory explains everything observed on the small scale, relativity explains everything observed on the large scale, the point where they might come into contradiction is far beyond the limits of current observation. So, unfortunately for the physicists, most puzzles in physics are those of beauty, logic, and order – not ones of “I saw this particle do that, and it shouldn't have, according to our theory's predictions.” I don't believe that beauty is a characteristic we should make our primary goal in physics, but I do believe that the most effective theories have always been the most beautiful ones, so it is likely that the anomalies from what the Standard Model “ought” to look like really do indicate anomalies from what actually happens, which we simply haven't been able to observe yet. But can a theory based, like string theory, on what logical considerations would indicate really be a theory that ends up describing the real world? Only time and experiments will tell. ‡