> if you have an empty bag, and add an apple, 1 is the cardinality of that bag. If you add another apple, then the cardinality is 2. The artificiality is high

> if you have an empty bag, and add an apple, 1 is the cardinality of that bag. If you add another apple, then the cardinality is 2. The artificiality is high here -- would you trade your bag of "2" apples for someone else's bag of two apples without asking about the size and quality of the apples? While counting numbers are wonderful things for examining things like "sets" it seem to me to be kind of silly to assign any "reality" to them. Discrete identical things are pretty much nonexistent at human scales. Continuous identical things, however, are numerous, where continuous here is not strict, but practical. This is not realistic as a foundation for mathematics. But I think the idea of using propositional logic and set theory was a mistake that we are only now starting to correct. Intuitionalists and category theorists (and finitists, frankly) are leading us in a better direction now.