thermodynamics
.. became interesting for me once entropy was introduced .. I remember reading about a paper by Wheeler & Feynman from the late 40s (I think), which addressed the fascinating question: Maxwell's equations are time-symmetric, and there are even solutions which suggest the existence of 'advance waves', meaning EM waves travelling backwards in time. So the question is, why don't we see them? - they obviously don't exist in the real world .. W & F showed mathematically, using entropy considerations, that the advance waves are cancelled out by interference with other, 'retardant' waves, and so we never actually 'see' them. Great stuff - Yankee ingenuity applied to theoretical physics.