For those with some time to spare, you might enjoy:
Clearing Up Mysteries -- The Original Goal
http://bayes.wustl.edu/etj/articles/cmystery.pdf
"Put most briefly, Einstein held that the QM formalism is incomplete and that it is the job of theoretical physics to supply the missing parts; Bohr claimed that there are no missing parts. To most, their positions seemed diametrically opposed; however, if we can understand better what Bohr was trying to say, it is possible to reconcile their positions and believe them both. Each had an important truth to tell us. But Bohr and Einstein could never understand each other because they were thinking on different levels. When Einstein says QM is incomplete, he means it in the ontological sense; when Bohr says QM is complete, he means it in the epistemological sense. Recognizing this, their statements are no longer contradictory. Indeed, Bohr's vague, puzzling sentences -- always groping for the right word, never finding it -- emerge from the fog and we see their underlying sense, if we keep in mind that Bohr's thinking is never on the ontological level traditional in physics. Always he is discussing not Nature, but our information about Nature. But physics did not have the vocabulary for expressing ideas on that level, hence the groping."
>we understand "infinity" just fine.
I enjoyed "Meta Math" by Gregory Chaitin:
http://arxiv.org/abs/math/0404335
"So the set of real numbers, while natural—indeed, immediately given— geometrically, nevertheless remains quite elusive: Why should I believe in a real number if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it? And each of these things happens with probability one! The real line from 0 to 1 looks more and more like a Swiss cheese, more and more like a stunningly black high-mountain sky studded with pin-pricks of light."