Is it wise to develop skills that do not immediately benefit short-term research productivity? I have a master in physics and philosophy and have a keen interest in mathematics. Currently I am doing a PhD in philosophy. On a personal level, however, I cherish the interdisciplinary ideal of combining mathematics, physics and philosophy. I am well aware that realizing this ideal requires an enormous amount of work and talent – and that very likely I cannot 'completely' realize it. 

On a structural level, I suppose this is an unfortunate side-effect of the level of specialization in academia today. On a subjective level, however, I feel the need to “revolt” against such over-specialization. It a shame that some "philosophers of science" in my branch haven't solved one differential equation in their life; yet they intend to reflect on what physicists do. I am convinced that combining different disciplines is a worthy thing to do. I suspect many people would agree with me, but the competitive reality of academia makes it very hard to realize.

The time one invests in a secondary area of interest (i.e. mathematics, physics) takes away time to do research in one's main area (i.e. philosophy). The reduced time would (on the short-term) make me less productive and successful. However, in the long-term, it might allow me to produce work of more value and depth.


(1) Is it wise for a PhD student, somewhere below on the academic ladder, to gamble already on long-term investments, or should one give in to the short-term safety and (over)specialize in one area? 
(2) Is there some middle-way between the two extremes? 
(3) Is it more useful in later stages of one's career to have a broad skill-set or have more publications in one (limited) area?
I think this is a good question, but I have only a brief, partial answer: you may have to accept that, in pursuing the interdisciplinary topics that interest you, the time and energy you spend will not be repaid with any kind of academic currency, but only in less tangible forms.  In other words, you may never get a better job, get an extra paper published, or even get an extra pat on the back from a colleague; but you may nonetheless, in a subtle and unsung way, advance the anti-specialization principles you espouse simply by doggedly producing quality work that embodies them.  This may sound cynical, but I mean it seriously.  Of course, you need enough tangible reward to keep your career going, but it sounds like you're asking about how or whether to invest the energy to go the extra mile above and beyond that.  It's possible to derive internal comfort from believing that you're doing your best to produce the kind of work you think needs to be produced, even if doing so doesn't earn you any praise or advance your career in any concrete way.  In the end, whether you continue to do this work will depend on whether you can be happy and sane doing it, and drawing strength from your inner conviction, rather than from external validation, can help with that.