Dirac once said that he was mainly guided by mathematical beauty more than anything else in his discovery of the famous Dirac equation. Most of the deepest equations of physics are also the most beautiful ones e.g. Maxwell's equations of classical electrodynamics, Einstein's equations of general relativity. Beauty is always considered as an important guide in physics. My question is, can/should anyone trust mathematical aesthetics so much that even without experimental verification, one can be fairly confident of its validity? (Like Einstein once believed to have said - when asked what could have been his reaction if experiments showed GR was wrong - Then I would have felt sorry for the dear Lord)
Answer
Dear sb1, a good question. Well, beauty is a good guide in physics research but only for those whose sense of beauty is aligned with Nature's sense of beauty. ;-) Dirac was among them, at least when he was writing down his beautiful equation, but many others have a different sense of beauty that can easily lead them off the track.
The right sense of beauty is linked to the equations' rigidity and uniqueness. If a woman is beautiful, you may think that not a single thing could be improved about her. Every correction would damage this beauty. The same thing holds for the beautiful theories and equations in physics that simply "fit together". An important characteristic that can make a theory more rigid and constrained is symmetry - but it is not the only characteristic that can do so. For example, the nontrivial cancellation of various a priori conceivable theoretical problems - such as anomalies - also constrains theories and makes them "prettier" relatively to theories that haven't had to pass any similar theoretical tests.
Why are the equations and theories that "fit together" more likely to be the right description of Nature? Well, unless they're already falsified, they have many fewer parameters waiting to be adjusted than the competing - so far unfalsified - theories that are not so beautiful.
The "posterior" (after the comparison with the reality) probability that the "not so beautiful" candidate theory is valid is, by the Bayesian logic, multiplied by the probability $P(g=g_0)$ that the parameters $g$ take the right values to agree with the reality.
If the overall prior probability for the "beautiful" and "ugly" classes of theories are chosen to be equal, then the "ugly" theory is punished by the extra factor of $P(g=g_0)$, so it becomes less likely that it is the right theory that describes the observations. A more constrained point in the space of theories (constrained by symmetries and special consistency advantages) gets a "higher weight" because it's qualitatively different from the more "generic" or "uglier" points.
Nature has apparently chosen some repeatable laws that apply everything in the Universe (and maybe beyond) and that predict millions of phenomena from a very small amount of information about the laws that has to be known in advance. So it makes sense to extrapolate this observation and assume that the laws of physics are as constrained as possible, and in this sense, they must "fit together" and be "beautiful".
But again, one has to be very careful about this method to look for theories that becomes very unscientific unless the "beauty" of the mathematical structures may be justified by some technical arguments.
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