I am now at the point in my academic career where some people see fit to ask me for advice. As shocking as this change is to me, I have found that when people ask me for advice, they mean it sincerely and take my advice seriously. It is thus incumbent upon me to be just as serious and sincere in what advice I give. Tonight, I'd like to ponder one particular kind of advice I have found myself giving in recent weeks.
In physics, we use mathematics. A lot. Math is the language of physics and the method by which our science is performed. How, then, can physics ever advance in directions unanticipated by math? Perhaps more to the point, since mathematical thinking is not inherent but a skill, how can we check ourselves against mistakes in our mathematics? In physics, we often find that intuition is the answer to these problems. A good physicist will have developed a keen intuition that guides them to new and novel discoveries, checks them against mistakes and that helps with the interpretation of very abstract concepts.
This answer, as correct as it may be, is nonetheless incomplete, however. Our intuitions are ill-equipped to deal with phenomena outside the "middle world" in which nothing is too big, too small or too fast. Thus, when we first encounter quantum mechanics, for instance, our intuitions often betray us, leading us to reject such beautiful facets of the theory as entanglement. As we learn, we must fight our intuitions as much as we utilize them.
It is common to paint this quandary as being about intuition versus mathematical truth, but herein lies my perhaps not-so-humble advice: trust your intuition, but trust the mathematics more. Our intuitions are wonderful tools for doing science, but at the end of the day, it is mathematics upon which our theories must be founded. When we encounter something unintuitive, it does indeed behoove us to exercise extra caution and skepticism, but these must ultimately give way to experiment and to theory.
As we learn to put our trust (not faith) in the explanatory power of mathematics and the veracity of our experiments, then we can build up a new intuition that serves us more faithfully. In short, by trusting in mathematics, we turn that most admirable of human qualities to our intuitions: the capacity for self-improvement. Rather than thinking of intuition and mathematical truth as being at odds, we can see conflicts between these as opportunities to learn and to improve. The two tools, when used to their fullest, work together in a virtuous cycle that results in the expansion of knowledge.