Yesterday I attended a talk here at UoP by an academic from Swansea. He was a nice chap who gave a brilliant talk on a subject he passionately believes in: string theory and the AdS/CFT (Anti-deSitter Space/Conformal Field Theory) duality. So while being tremendously impressed by him, I was unimpressed by string theory.
The summary is that a unbelievably complex string theory in 11 dimensions looks rather like a 4 dimensional Conformal Field Theory without gravity. At the end of his talk I asked him whether "it might be just a meaningless coincidence that 11-d string theory looks a bit like 4-d CFT?' and I told him that an experimental test was needed somewhere. He said that when you actually do the calculation the similarity is so miraculous, even from such a complicated theory, that it must be right. Not necessarily! Maybe it is just coincidentally how the maths works out in the collapse from 11 dimensions to 4. Maybe theorists have now played around with so many 11-d string models that they have finally found one that by accident looks like the 4-d CFT model? Ptolemy's epicycles were very complex and reproduced the observed behaviour, but were wrong. To do physics, string theorists have to suggest an experimental test whose behaviour would be different depending on whether string theory is true or not, otherwise it is mathematics, not physics.
Mathematics is essential for setting up predictive models, but is just a human invention, a derived thing, and I don't believe you can get paradigm shifts in physical theory by just "mixing up" old mathematical models. In the same way, there are probably an infinite number of concepts whose description lies outside the words we have, and you'll never get to them using existing words. Something non-verbal from outside is needed as a initial guide: from observation & intuition and then words (or maths) can be invented to describe them.
Unfortunately, the uber-mathematical approach is the fashion in theoretical physics right now, and it always reminds me of a quote about the death of ancient Greek science: "speculation went way beyond the testable, and into metaphysics" (words or maths with no input from nature) , so to avoid this I always try to think about anomalous observations, which is very difficult, messy and even misleading sometimes, but at least makes me feel I am doing something real and new.