Someone who knows much more about food science than I suggested that alum and/or lime may interfere with the dissolution of pectin. Pectin is what glues the cellulose in plant tissues together. It's water soluble--which is why fruits and vegetables soften as they cook (and why the liquid part of jams thicken as the fruit cooks--the pectin migrates from the fruit to the sugar syrup). If alum or lime "harden" the pectin (that is, make it more difficult to dissolve) before the cooking /processing that would account for the crispness of the pickles. It would also explain why the alum is used a rinse BEFORE the pickling begions. It's only a hypothesis, of course, but it sounds pretty good to me...