As you say, the way string theory requires all these extra dimensions and this comes from certain consistency requirements about how string should behave and so on.
Roger PenrosePeople think of these eureka moments and my feeling is that they tend to be little things, a little realisation and then a little realisation built on that.
Roger PenroseWell, gauge theory is very fundamental to our understanding of physical forces these days. But they are also dependent on a mathematical idea, which has been around for longer than gauge theory has.
Roger PenroseThere are considerable mysteries surrounding the strange values that Nature's actual particles have for their mass and charge. For example, there is the unexplained 'fine structure constant' ... governing the strength of electromagnetic interactions.
Roger PenroseNo doubt there are some who, when confronted with a line of mathematical symbols, however simply presented, can only see the face of a stern parent or teacher who tried to force into them a non-comprehending parrot-like apparent competence--a duty and a duty alone--and no hint of magic or beauty of the subject might be allowed to come through.
Roger Penrose