Math melodrama rings of reality.

Abstract

Cheers to D. F. Wallace for his spirited, witty, and informative review of the fictional “Math Melodrama” novels by Philibert Schogt ( The Wild Numbers ) and Apostolos Doxiadis ( Uncle Petros & Goldbach's Conjecture ) ( Science 's Compass, 22 Dec., p. [2263][1]). Cheers also to Science itself for conveying at some length and with considerable fidelity, through Wallace, some insights into the joys and anxieties associated with attempted sustained mathematical research. In his review, however, Wallace slightly misrepresents the thrust of Kurt Godel's First Incompleteness Theorem (FIT); further, Wallace uses unduly harsh language in note 17 when he criticizes Doxiadis for allowing his mathematically savvy protagonist Petros to fear that his chosen problem, the Goldbach Conjecture, may be “one of the [FIT's] formally unprovable propositions.” “This is so implausible and reductive as to be almost offensive,” writes Wallace. And later: “To believe that the [FIT] could apply to actual number-theoretic problems like the Goldbach Conjecture is so crude and confused that there is no way that a professional mathematician of Petros's attainments could possibly entertain [the thought].” FIT asserts that in any sufficiently rich, effectively axiomatizable first-order system ([1][2]), say first-order Peano Arithmetic, some first-order assertions will be undecidable, in the sense that they are true in some models of that system and false in others. In any given model, any (first-order) statement is true or false; it is the challenge of mathematics to determine in specific cases just which of the two it is, using any legitimate (not necessarily first-order) methods of proof. The character Petros is worried that the Goldbach Conjecture, a first-order statement in the system N of natural numbers (this is the so-called standard model of Peano Arithmetic), might be true in N but not provable by the (permissible, first-order) methods of Peano Arithmetic. We do not know, of course, whether the Conjecture is true in N or false in N ([2][3]). But Petros's worry strikes us (at the very least) as plausible or reasonable, not as crude or confused. Furthermore, contrary to Wallace's statement that “the formally unprovable propositions [that FIT] succeeds in deriving are all very special self-reference-type cases,” by no means is every statement known to be independent of Peano Arithmetic weird, contrived, or artificial. The combinatorial statement attributed to J. Paris ([3][4]), as well as the number-theoretic statement of Goodstein's Theorem and the graph/game-theoretic statement of “Hercules and the Hydra” ([4][5]), are natural mathematical statements that are easy to formulate (in a first-order way), and they are unprovable in Peano Arithmetic but nevertheless true in N. These and other non-self-referential propositions are discussed, for example, in ([5][6]) and ([6][7]). 1. [↵][8]Here, first-order refers to number-theoretic statements that are formulated in the logic involving the usual connectives such as “and,” “or,” and “not,” as well as quantifiers;the latter, however, may be applied only to numbers (as opposed to sets of numbers, as in second-order arithmetic). 2. [↵][9]If by chance the Goldbach Conjecture should turn out to be false (in N), it would be false in all models of Peano Arithmetic, and hence, by Godel's Completeness Theorem, refutable in Peano Arithmetic. Hence, if the Conjecture is independent of Peano Arithmetic, then it is true in N. 3. [↵][10]1. J. Barwise 1. J. Paris, 2. L. Harrington , in Handbook of Mathematical Logic, J. Barwise, Ed. (North-Holland, Amsterdam, 1977), pp. 1133-1142. 4. [↵][11]1. L. A. S. Kirby, 2. J. Paris , Bull. London Math. Soc 14, 285 (1982). [OpenUrl][12][FREE Full Text][13] 5. [↵][14]1. R. Kaye , Models of Peano Arithmetic, vol. 15 of Oxford Logic Guides (Clarendon Press, Oxford, 1991). 6. [↵][15]1. Z. Adamowicz, 2. P. Zbierski , Logic of Mathematics (Wiley, New York, 1997). # Response {#article-title-2} The reviewer hereby wincingly acknowledges that Comfort and Rothmaler make a good point. It's maybe possible to niggle with them about whether Doxiadis's Petros is actually freaked about the Goldbach Conjecture's undecidability per se or rather just about the possibility that it's “true but independent”—first, because there's no evidence that Petros knows anything about different models of first-order math (the book makes it clear that he is no logician), and second, because he's too rabidly ambitious to give one fig about the Conjecture's actual truth or falsity; he cares only whether he can prove it with first-order deductive tools. This bit of niggling doesn't affect the really winceworthy point of their letter, though, which is that language like “implausible and reductive” and “crude and confused” that I used to characterize Petros's reaction to the FIT is indeed “unduly harsh” and somewhat misleading. (Worse, my use of the terms “reductive” and “crude” appears itself to have been reductive/crude, so I can understand why it bothered smart readers.) Though I am grateful that Comfort and Rothmaler have corrected a misleading description of Petros's reaction to the FIT, I believe that what they've actually done here is catch me out in a writing-and-revision error rather than in a mathematical miscue. (This is the inevitable part of the Response where your reviewer tries to offer some kind of explanation/defense for his snafu, but I'll try to keep it maximally brief.) Note 17, which is where the discussion of Petros's horror about the FIT appeared in my book review, was originally longer than it was in Science , and the note included stuff about a scene in Doxiadis's novel right after Petros learns about the FIT and bites his wrist in horror. In this scene, Petros actually goes to Vienna and looks up poor little agoraphobic Kurt Godel and grabs him by the lapels and pretty much demands that K.G. tell him right there on the spot whether the Goldbach Conjecture is one of the Theorem's improvable propositions, Petros saying stuff in the scene like “Damn theory, man!…I have a right to know whether I'm wasting my life!” ([1][16]). It is one of the worst scenes in the book—incongruous, soap-operaish, unintentionally funny—and in retrospect I see now that it's really more the Petros-Godel exchange that is “implausible…offensive,” or maybe rather that I let my strong readerly dislike of that scene color the way I saw Petros's whole reaction to the FIT. The problems here were intensified when the account in note 17 of the Petros-Godel scene got cut by the editor ([2][17]), whereupon harsh language evoked by that scene and (yes, unduly) applied to the FIT itself lost not only its proper referent but any possible indication of its real (if, yes, confused) motivation. All that said, I still contend that the overwhelming majority of things in the book I said were silly and/or confused really are silly and/or confused. Quandoque bonus dormitat Homerus. 1. [↵][18]Uncle Petros & Goldbach's Conjecture (Bloomsbury USA, New York, 2000)This scene is on pp. 140–142 of Doxiadis's. 2. [↵][19]This does not mean that errors/misrepresentations were the editor's fault or just the result of cutting. If the reviewer acquiesces to a cut, he is responsible for cleaning up any errors or incongruities that are created by the cut, and this I clearly failed to do here. 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