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. Consider this argument of Hellerstein's: "Let us attempt to evaluate finiteness. Let F = 'finitude', or 'finity'; the generic finite expression. You may replace it with any finite expression." He does not define 'finitude' or 'finity' and I have no clear idea of what they mean. But he seems to say that we can replace it (that is F) with any finite expression. If I can replace F with any finite expression, I will choose "3". He may mean "Let F be an arithmetic expression whose result is a whole number." but he seems to want to leave things unclear. He goes on, "Is Finity finite?" If Finity is F (which he said) and F = 3 (which he permits), then the answer is obviously 'yes'. But he ignores the obvious answer to go on and state, "If F is finite, then you can replace it by F+1, and thus by F + 2, F + 3, etc. But such a substitution, indefinitely prolonged, yields an infinity." He is simply wrong because, for all real numbers k, F + k is finite. Let us restate his sentence more precisely: He means that if the assertion "F is finite" is true, then the assertion "F + 1 is finite" is true, and every assertion of the form "F + k is finite" is true for all real numbers k. This is correct, and it is correct for the integers and the whole numbers, too. We never get any infinities, just larger and larger finite numbers. Another class of paradoxes are the liar and barber paradoxes. To explode them, we need only look at the method of proof by contradiction, often used in mathematics. An elementary exercise in formal logic proves that proof by contradiction is valid. The most famous such proof is that the square root of 2 is not rational. I won't repeat the proof here, but the method is this: you assume that the square root of 2 is rational, in particular it is the ratio of two integers p and q, and then you show that this leads to a contradiction, whatever integers you choose for p and q. You therefore reject the supposition that the square root of 2 is rational. In the barber paradox we suppose a village where all men are clean shaven (a man is, for the sake of this puzzle, assumed to be someone who needs to be shaved in order to be clean shaven, though this assumption is rarely stated explicitly), and we know that the barber shaves every man who does not shave himself and only such men. The paradox comes about when we ask: Who shaves the barber? If it is the barber, then he shaves himself and therefore is not shaved by the barber, that is, himself. If the barber does not shave himself, then by our assumption, he is shaved byh the barber, that is himself. This self-contradictory conclusion arises only because we made an assumption that there can be such a village. This so-called paradox is simply a proof that a village so defined cannot exist. There is no paradox, there is no village. The book's front matter states that it is printed on acid-free paper, so it won't fall apart. However, books can fall apart in other ways. This one was not only printed on acid-free paper, it is also on mathematics-free paper. |