Dan Schechtman, winner of the 2011 Nobel Prize in Chemistry for the discovery of quasi crystals, said: “The main lesson that I have learned over time is that a good scientist is a humble and listening scientist and not one that is sure 100 percent in what he reads in the textbooks.”
My research on groupoids and higher groupoids was started in the 1960s by a dissatisfaction with a van Kampen theorem that did not compute the fundamental group of the circle, a basic example: but groupoids were at the time regarded as "rubbish" by many senior mathematicians, and the idea of higher van Kampen theorems using higher groupoids was described by one such for 10 years as "ridiculous". (He gave in eventually!)
My worry is that people may be encouraged to follow high ups, rather than to analyse a programme on mathematical grounds, and so to develop their own feeling for mathematical structures.
January, 2015: One needs a variety of strategies, one of which is to look at what a theory does not do but somehow in principle should. This is the notion of anomaly. I have listed 5 anomalies in standard algebraic topology in this presentation Dec, 2014, Galway.
See also the advice given to me 1964 by S. Ulam, quoted in my web page discussing the issue of famous problems in category theory.
Alexander Grothendieck wrote to me that: "Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand -- and it always turned out that understanding was all that mattered."
So I always advocate writing and rewriting to make things clear to yourself, testing that by explaining to other people. At Bangor we explained to research students that a thesis must have a "thesis". So having decided on the latter, the first thing for the student to do is write up the background to that "thesis", which can always be expected to be a useful part of the final thesis. All sorts of things may turn up in that process.
