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Delta: A Paradox Logic (Series on Knots and Everything , Vol 16) By Nathaniel Hellerstein

I usually enjoy mathematics books, and it is rare that a book on the subject is incompetent. Hellerstein's book is, unfortunately, one such. It begins by citing, as though they were true conundrums, many famous and lesser known "paradoxes". Most have long been known to harbor no real paradox. Here are some examples:

He asks (p 10) whether aleph null (the cardinality of the integers) is even or odd. It is the question that is odd, as evenness and oddness are defined for integers and not for aleph null. He says that it "is a counting number, it is presumably an integer..." however, it is most definitely not a counting number, and not an integer. In defining the counting numbers we use a concept called the successor of a number, define zero, and build from there. Aleph null, not being the successor of a counting number is simply not one of them. His word "presumably" is a dead giveaway; on what basis does he presume? This is mathematics, we conjeccture, prove, or disprove. We do not presume.

Another class of what he thinks are paradoxes are merely cases where he has used undefined terms. For example, his "Paradox of the Boundary" where he asks, "Is noon A.M. or P.M.?" I would ask: Define what you mean by A.M. and P.M.?If A.M.'s definition ends, "up to and including 12:00, then noon is A.M." If A.M.'s definition ends "up to but not including 12:00", then noon is not A.M. As soon as you define noon, midnight, A.M., and P.M., the problem disappears. Similarly for his other questions of this type, for example, "Which country owns the border?" or "Is zero plus or minus?" (a better mathematician would have asked, "Is zero positive or negative?"). It is as if he is ignorant of the properties of open and closed sets.

His "Size Paradoxes" also vanish when we define our terms with precision. "Surely," he says, "one grain of sand does not constitute a heap of sand. Surely adding another grain will not make it a heap. Nor will adding another, or another, or another. In fact, it seems absurd to say that adding one single gran of sand will turn a non-heap into a heap." The problem is that we have no definition of a heap of sand. We can, however, create one, and as soon as we do, the paradox disappears. A definition in terms of number of grains of sand will do, or -- if that seems overly rigid -- then define a heap of sand empirically, by asking, say, 41 randomly chosen people to decide, and going with the majority. But best is to simply recognize that there is no fixed definition, that we cannot determine whether some small piles of sand constitute a heap or not, that there is no logical contradiction, and that therefore there is no paradox.

The paradox of the smallest uninteresting number begins by asking if there are any uninteresting whole numbers. There cannot be, because the set of uninteresting whole numbers (UHNs) has a smallest member. The smallest UHN is, obviously, interesting, so we remove it from the set. But the new set has a smallest, and we remove number after number until there is nothing left. However, "uninteresting" is not well-defined. Given a number, there is no test we can apply to determine whether it is an UHN. In fact, a number that is uninteresting one day may prove interesting the next to the same observer. Again, there is no paradox, just his confusion.

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