"Are Numbers Real?" by Brian Clegg

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"Are Numbers Real?" by Brian Clegg

Post by Upton_O_Goode » Sat Jul 08, 2017 3:41 pm

I just checked this book out of the public library. Couldn't resist it, numbers geek that I am. The author is a British physicist who writes a lot of popular books. He's a very engaging writer, and I did enjoy this book. The chapter on James Clerk Maxwell taught me many things about Maxwell and Maxwell's Equations that I didn't know. Clearly, he's much better versed in this area than I am.

The chapter on infinity is also generally good, but here I wish to point out a couple of mathematical mistakes, in one of which he "proves" a false theorem. This "theorem" is on page 182. He constructs what he believes is a covering of the positive real numbers by a countable set of intervals of total length 1. Now, it's apparent to anyone who knows measure theory that this is impossible. Lebesgue measure is countably subadditive, and that means the measure of a union of intervals is never greater than the sum of the lengths of the intervals. You can't cover any interval of length greater than 1 by his construction. His idea is to enumerate the positive rational numbers as r1, r2, r3,.... then consider the intervals (r1-1/4, r1+1/4), (r2-1/8, r2+1/8), (r3-1/16, r3+1/16).... He asserts without proof that these intervals cover all of the positive real numbers. Intuitively, it seems that they might, but in fact they can't. Indeed, if he means OPEN intervals, as I have assumed here, you can't even cover the closed interval [0,1] this way, because the Heine–Borel theorem would imply that some finite set of these intervals already covers it, and thus you'd have a finite set of intervals of total length less than 1 covering an interval of length 1. If you allowed them to include their endpoints, you could do this, just barely, since [0,1] is the union of the sets [1/2,1], [0,1/4],[1/4,3/8],[3/8,7/16].... But you couldn't cover any interval of length larger than 1 that way.

Later edit: To be fair, Clegg says only that "it seems" that the whole line is covered. On the other hand, he never comes out and says that it isn't covered. He says only that this is a "mind-bending" paradox. Well, it is a bit strange that one can cover a dense subset of the line with a collection of intervals of arbitrarily small total length. But I wouldn't call it a paradox. Let's be generous here and say he had a slight lapse in drafting this page and left it incomplete.

On the following page, he gets the history slightly wrong, claiming that Gödel's incompleteness theorem implies that the continuum hypothesis is undecidable. It implies nothing of the kind. Gödel's undecidable proposition is provable in the metalanguage, but could be deduced within the object language only if the axioms for that language were inconsistent. Clegg has conflated Gödel's 1931 incompleteness theorem with his 1938 paper showing that if Zermelo-Fraenkel set theory is consistent, it remains consistent when the axioms are augmented simultaneously by (1) the axiom of choice, (2) the continuum hypothesis, (3) the hypothesis that there exists a co-analytic set that isn't analytic, and (4) a proposition so obscure that I can't remember what it is and am too lazy to look up. Thus, you can't DISPROVE the continuum hypothesis. That you also can't prove it was shown in 1963 by Paul Cohen. It is independent, which is not the same thing as being undecidable.

He also gives, on p. 186, a list of the ZFC axioms, from which he omits the important axiom of regularity, needed to avoid Russell's paradox in most versions of the theory. His account of Russell's paradox is also condensed and leaves out the very large hint that Russell got, when he proposed "the set of all sets" as a refutation of Cantor's proof that there is no largest cardinal number. (Try working through Cantor's proof in application to "the set of all sets" and you'll immediately arrive at Russell's paradox.)

But these are minor quibbles. The man is a physicist, and ventures to write about mathematics. Good for him. I would hope to be treated kindly in connection with anything I wrote about physics, so I'm giving him a pass. As I say, he's a very engaging writer.
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Re: "Are Numbers Real?" by Brian Clegg

Post by Poodle » Sat Jul 08, 2017 9:02 pm

To be fair to Mr. Clegg, he's actually a science writer rather than a physicist and he's done a lot to popularise science for the Great Unwashed. Even so, he shouldn't be coming up with whoppers - but it's not my field so I can't really judge.