I have to interject here that this is where physicists and mathematicians diverge. Remember it was Hilbert who said "Every school boy in Gottingen knows more about four dimensional geometry than Einstein, yet it was he who created General Relativity not the mathematicians."
The history of physics is full of examples where physical intuition guided the choice of mathematical concepts and how they were to be interpreted.
When it comes to deciphering nature methinks one must keep a balance between one's intuition about the physical and rigor of the mathematicians. A noted mathematician, Rene Thom once quipped "The word rigor often preceeds the word mortis".