Lately I've been reading about conceptions infinity, which is an important topic to natural philosophy and which Aristotle discusses in Book 3 of the Physics.
In any event, this is the reason I picked up David Foster Wallace's popular treatment of the mathematics of infinity (from Zeno up through Cantor). Wallace's writing is definitely mannered. He maintains a modern bias against Aristotle and in favor of the actuality of infinity, both of which points the book inadequately supports. (Sometime I'll have to do a full review.) Because of these flaws, the excellence of his explanation of the difference between zero and nothing is quite surprising:
It's a tricky difference [between the number 0 and the abstract word 'nothing'], but an important one. The Greeks' inability to see it was probably what kept them from being able to use 0 in their math, which cost them dearly. But 0 v. nothing is one of those abstract distinctions that's almost impossible to talk about directly; you more have to do it with examples. Imagine there's a certain math class, and in this class there's a fiendishly difficult 100-point midterm, and imagine that neither you nor I get even one point out of 100 on this exam. Except there's a difference: you are not in the class and didn't even take the exam, whereas I am and did. The fact that you received 0 points on the exam was thus irrelevant—your 0 means N/A, nothing—whereas my 0 is an actual zero. Or if you don't like that one, imagine that you and I are respectively female and male, both healthy 20-40 years of age, and we're both at the doctor's, and neither of us has had a menstrual period in the past ten weeks, in which case my total number of periods is nothing, whereas yours here is 0—and significant. End examples.
I suppose the difference can be summarized by noting that with zero, there is at least to start out with a possibility of having a something. Then of course the notion of possibility (vs. actuality) is critical to the whole notion of infinity....
David Foster Wallace, Everything and More: A Compact History of ∞ (New York: W.W. Norton and Company, 2003), 142.
Also of interest: Nothing Comes from Nothing