Please forgive my non-descriptive title.
I'm an aspiring physicist who feels no physicist can ever achieve in his subject without Mathematics whether he acquires it purely intuitively (like Faraday) or rigorously (like Cauchy).
My question is how am I to approach mathematics? I'm currently working on Spivak's Calculus and also learning a rigorous foundation of number systems with another book. Some people have gone to the extent of telling me that I'm wasting my time with such rigour and my way is more suited for pure Mathematics.
I'm confused as I feel the rigour is essential if I'm to ever to progress to topics such as differential geometry and the calculus of variations. On the other hand those topics were long established before the rigour of today played a major role.
Any guidance would be much appreciated.