5 “Vocational Travelogue” is a very shorthand way of acknowledging that for a long time one reason people used to read fiction was for a kind of imaginative tourism to places and cultures they’d never get to really see; that modernity’s jetliners, TV, etc. have pretty well obsoleted this function; but that modern tech has also created such extreme vocational specialization that few people anymore are in a position to know much about any professional field but their own; and thus that a certain amount of fiction’s “touristic” function now consists in giving readers dramatized access to the nuts and bolts of different professional disciplines and specialties. It is not an accident that the first important Vocational Travelogues, novels like Hailey’s Airport and Hotel and Ed McBain’s “police procedurals,” began appearing in the late ’50s and early ’60s.
6 (Q.v. here WN’s marketing tag about GENIUS and MADNESS in FN4 supra, or UPGC’s flap copy’s heavy description of the novel as “about the search for truth at all costs, and the heavy price of finding it” [ sic ].)
7 In fairness to all concerned, this variability in readers’ mathematical backgrounds is a problem for pretty much anyone trying to write general-interest prose about math, a problem that Hardy refers to as “the restrictions under which I am writing. On the one hand my examples must be very simple, and intelligible to a reader who has no specialized mathematical knowledge…. And on the other hand my examples should be drawn from ‘pukka’ mathematics, the mathematics of the working professional mathematician.” Note that this sort of thing is a problem even for rather more “special-interest” writing like this book review itself. Is it, for example, necessary to inform or remind the average Science reader that Fermat’s Last Theorem (c. 1637) states that where n is an integer and n > 2, the equation x n + y n = z n has no nonzero integer solutions? or that Goldbach’s Conjecture (or rather the “strong” G.C. as reformulated by Leonhard Euler in 1742) is that every even integer > 4 can be expressed as the sum of two prime numbers, etc.? As it happens, this reviewer is not certain whether it’s necessary or not, and the fact that these lines have not been deleted by Science’s editors (i.e., that you are reading them at all) may indicate that the editors are not totally sure either.
8 Hardy, whose Apology talks about this better than anything else ever has, explains that “the mathematician’s patterns, like the painter’s or poet’s, must be beautiful; the ideas, liktheboue the colours or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics.”
9 (The assumption here will be that the typical Science reader already knows what “a priori,” “deductive truth,” and “logical proof” mean and is at least roughly familiar with the relationship between pure math and formal logic… if for no other reason than that to gloss tangential stuff like this would take up enormous amounts of space and time and might well also alienate the [presumably large] percentage of Science’s readership who already know the stuff and are apt to find such glosses not only otiose but annoying—this reviewer can actually imagine such readers looking increasingly aggrieved and impatient and saying to themselves, Whom does he think he’s talking to? All this is mentioned only to underscore once again the rhetorical diciness of the whole math-prose enterprise, a diciness that lies at the very center of this review’s criticisms of the actual novels to be discussed, which critical discussions are upcoming very, very shortly.)
10 It’s worth noting that as so much contemporary poetry, classical music, etc. becomes ever more abstract and involute and technically complex, their own audiences get ever smaller and more specialized. With very few exceptions, the people who truly “appreciate” a piece of language-poetry or an atonal fugue are people with extensive educations in the history and theory of these arts. And this increasing exclusivity in the U.S. arts has much less to do with good old “cultural elitism” than with our era’s tendency toward greater and greater specialization—it is not at all an accident that the majority of people who read contemporary poetry are themselves contemporary poets.
11 “Math Anxiety” is now a recognized term in educational psychology, and variants of the “I’m-back-in-high-school-and-sitting-in-my-AP-Calc-final-and-I’ve-forgotten-to-study-or-it-turns-out-all-my-pencils-have-pimento-in-them-instead-of-graphite” nightmare are so common they’re almost clichés.
12 “Average reader” is kind of a synecdoche for “people who read mainly for diversion or entertainment.” These people are American genre fiction’s basic audience. It is true that Hardy’s Apology, as well as novels from Don DeLillo’s Ratner’s Star and Thomas Pynchon’s Gravity’s Rainbow to Neal Stephenson’s Cryptonomicon have already deployed higher math in interesting and significant ways—but books like these are belle lettres, literature, for which the audience is, again, usually small and rather specialized. Genre books are mass books and are marketed accordingly.
13 The putative author of this problem, one “Anatole Millechamps de Beauregard” (b. 1791), is also fictitious, a kind of biographical hybrid of von Neumann and Galois, on whose florid life story—“Beauregard had a magnetic personality, and his appetite for wine, women and song was as great as for knowledge”; “One of Beauregard’s closest friends caught him in bed with his wife. Blind with rage, he strangled them both”—WN spends most of a chapter. The specular pun of Beauregard’s name, by the way, is not an accident: people in this novel are constantly saying stuff to each other like “Your findings lead directly into the high country of number theory. The view you offer is breathtaking.”
14 Like many of UPGC’s UPGiewsupporting characters, Ramanujan was a real number-theorist, an Indian savant discovered and mentored by Hardy. Robert Kanigal’s The Man Who Knew Infinity: A Life of the Genius Ramanujan is another of the post-Fermat math-bios now on the market.
15 The real source of this insight is Hardy, in his Apology’s famous “No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man’s game,” which UPGC’s narrator rips off without any attribution at all (p. 78: “Mathematics, you see, is a young man’s game. It is one of the few human endeavors where youth is a necessary requirement [ sic ] for greatness”). Actually, this FN is probably the place to point out that Doxiadis’s novel is filled with what appear to be little more than very slight rephrasings of stuff in Hardy’s Apology and/or C. P. Snow’s famous Foreword to it. Flipping through the two books at random, one might, e.g., compare UPGC’s “Anybody who claims that scientists—even the purest of the pure, the most abstract, high-flying mathematicians—are motivated exclusively by the Pursuit of Truth for the Good of Mankind, either has no idea what he’s talking about or is blatantly lying” with Hardy’s “So if a mathematician, or a chemist, or even a physiologist, were to tell me that the driving force in his work had been the desire to benefit humanity, then I should not believe him.” Or see Hardy’s “Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty,” and UPGC’s “Riemann had died at thirty-nine, Niels Henrik Abel at twenty-seven and Evariste Galois at a mere tragic twenty…”; or C. P. Snow’s description of the Hardy-Littlewood team as “the most famous collaboration in the history of mathematics” vs. Doxiadis’s narrator calling it “one of the most renowned partnerships in the history of mathematics.” On UPGC pages 129–30, Doxiadis even cribs nearly word for word a deathbed exchange between Hardy and Ramanujan and tags it with the footnote “Hardy also recounts the incident in his Mathematician’s Apology without, however, acknowledging my uncle’s presence,” which is not only intrusive and irritating but wrong, since it is not in the Apology but in Snow’s Foreword to it that the scene really appears.
It’s hard to know just how indictable UPGC is for its reliance on Hardy. It doesn’t seem like outright plagiarism, because plagiarism implies sneakiness, and Doxiadis has a fully attributed Hardy-quotation right up front as the novel’s epigraph. Plus it’s true that much commercial g
enre fiction has a long history of liberating stuff from established literary works. For the record, though, it’s still one of the more irksome things about UPGC.
16 The work Isaac’s doing for Arkanov is on “calibrator sets” and “K-reducibility,” two made-up terms that figure prominently in the plot’s math but are never specified or explained.
17 (Here the reviewer’s assumption is that if the T.P.P. is unfamiliar or the analogy unhelpful it can just be passed over with no hard feelings on either side.)
18 Rather than ever being specific about what all the complicated reasoning and complex equations are, WN employs the metaphor of mountain-climbing to try to evoke and describe what it feels like to do higher math. Actually, “employs” is the wrong word; the book repeats, exhausts, strip-mines the metaphor, pounding it again and again—“Every step I took, no matter how small, revealed new mountain- ne the wrongtops and unexpected canyons in this magnificent and bizarre region of mathematics”; “Another part of me had rushed ahead: it stood on the mountain pass, catching its breath as it watched the sun rising over a land that no human eyes had ever yet beheld”—until it becomes first grating—“Having completed the climb, we threw down our heavy backpacks and wiped the sweat from our brows. We were now standing together on the mountain pass, marvelling [ sic ] at the mathematical landscape”—and finally kind of funny—“Every time, I came tumbling back into base camp, dragging an avalanche of mistaken notions down with me.”
19 (Schogt’s original Dutch prose might, of course, be a thing of wonder)
20 The Wild Numbers’ American publisher seems equally culpable for the prose here. If Four Walls Eight Windows, Inc. is going to let an only semi-bilingual Philibert Schogt translate his own Dutch, why didn’t the FWEW editor bother to tell him that “television mast” should be “aerial” or “transmitter,” that “to pout” is intransitive and “to accommodate” takes a direct object, that phrases like “Shucks” and “city slicker” and “wine, women, and song” are now not idioms but ghastly clichés, or even that—no kidding—contemporary Americans do not bow to each other in formal greeting? Where was the editor? Was there an editor? Who did they think was going to read this stuff?
21 (from the original O Theios Petros kai i Eikasia tou Golbach)
22 And again: where was this book’s editor?
23 (And it’s just about as subtle w/r/t its thematics, with the narrator repeatedly and sans irony describing his uncle as an “Ideal Romantic Hero” [caps his] and saying stuff like “Think of the biblical Tree of Knowledge or the Prometheus of mythology. People like him have surpassed the common measure; they’ve come to know more than is necessary to man, and for this hubris they have to pay.”)
24 There’s a way more grievous example of this sort of thing involving Kurt Gödel and the plot’s first real crisis. Alan Turing (here a wide-eyed undergrad) accidentally exposes Petros to Gödel’s First Incompleteness Theorem in 1933, whereupon Petros freaks out because he fears that the Goldbach Conjecture may be one of the F.I. Theorem’s “formally unprovable” propositions. This is so implausible and reductive as to be almost offensive. As Science’s own readership is hereby presumed (q.v. FN9) to more or less know already, Gödel’s First Incompleteness Theorem is concerned with the abstract possibility of Completeness in axiomatic systems, and the formally unprovable propositions it succeeds in deriving are all very special self-reference-type cases—the mathematical equivalent of the “I am lying” paradox. To believe that the First Incompleteness Theorem 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 what the novel says is “the one and only, dizzying, terrifying question that had jumped into his mind the moment he’d heard of Gödel’s result…: what if the Incompleteness Theorem also applied to his problem? What if Goldbach’s Conjecture was unprovable?”
But then it gets even worse. Petros supposedly rushes off to Vienna and looks up Gödel
“a thin young man of average height, with small myopic eyes behind thick glasses
“I’ve spent my whole life trying to prove Goldbach’s Conjecture,” he told him in a low, intense voice, “and now you’re telling me it may be unprovable?”
Gödel’s pale face was now totally drained of color [ sic ].
“In theory, yes—”
“Damn theory, man!” Petros’ shout made the heads of the Sacher café’s distinguished clientèle [ sic ] turn in their direction. “I need to be certain, don’t you understand? I have a right to know whether I’m wasting my life!” He was squeezing his arm so hard that Gödel grimaced in pain….
Gödel was shaking. “I un-understand how you fe-feel, Professor,” he stammered, “but I-I’m afraid that for the time being there is no way to answer yo-your question.”
25 Some of these footnotes are so weird and U.S.-reader inappropriate that it’s worth giving a concrete example, such as let’s say p. 41’s FN to a line about the narrator enrolling in a U.S. college: “According to the American system, a student can go through the first two years of university without being obliged to declare an area of major concentration for his degree or, if he does so, is free to change his mind until the beginning of the Junior (third) year,” the very meaning of which is anyone’s guess.
26 N.B. here that the following main-text ¶ itself is geared to a very-strong-math-background audience; nobody else is going to get the ¶’s references, and this reviewer has neither the space nor the expertise to elucidate them. So feel free to skip it if you do not fit the ¶’s demographic.
27 Interested Science readers can find a discussion of Schnirelmann’s proof in W. Dunham’s Journey Through Genius: The Great Theorems of Mathematics (Wiley, 1990) but will probably have to don a miner’s helmet and go all the way back to Proceedings of the London Mathematical Society Series vol. 2 no. 44, 1938 for T. Estermann’s “On Goldbach’s Problem: Proof That Almost All Even Positive Integers Are Sums of Two Primes.”
28 Unless you are yourself a professional mathematician, the best place to find a nonlethal discussion of this proof (which is known in number theory as “Vinogradov’s Theorem”—that’s how famous this guy was) is in Section C of R. K. Guy’s Unsolved Problems in Number Theory (Springer-Verlag, 1994).
29 N.B.: End of audience-background-and-interest-restrictive main-text ¶.
30 You might further recall (from, e.g., Ovid’s Metamorphoses) that this bull ends up begetting on Minos’s queen the Minotaur, a hideous teratoid monster who has to be secreted in a special labyrinth and propitiated with human flesh, and who basically symbolizes the moral rot at the heart of Minos’s reign. That rot is, as Joseph Campbell describes it, a certain kind of alienated selfishness:
The return of the bull should have symbolized Minos’ selfless submission to the functions of his role. By the sacrilege of the refusal of the rite [of sacrifice], however,iceretur the individual cuts himself as a unit off from the larger whole of the community…. He is the hoarder of the general benefit. He is the monster avid for the greedy rights of “my and mine.”
31 Clearly, Petros’s real “sin” is not “Pride” so much as plain old selfishness, Greed. It’s not clear whether UPGC’s narrator truly fails to grasp this, or whether he is being presented as naive, or whether the whole thing’s just a translation problem.
32 Obvious though it is, Doxiadis appears to fear that his audience won’t get the compact irony here, so he has Hardy then rather sniffily advise Petros “that it might in the future be more profitable for him to stay in closer contact with his scientific colleagues.”
1 (N.B.: from The American Heritage Dictionary, Fourth Edition’s definition of “prosaic”: “consisting or characteristic of prose”; “lacking in imagination and spirit, dull.”)
2 (Numerals don’t count as words either, obviously.)
3 N.B. that this sort of problem is endemic to many of the trendy literary form
s that identify/congratulate themselves as transgressive. And it’s easy to see why. In regarding formal conventions primarily as “rules” to rebel against, the Professional Transgressor fails to see that conventions often become conventions precisely because of their power and utility, i.e., because of the paradoxical freedoms they permit the artist who understands how to use (not merely “obey”) them.
4 (Imagine offering a gymnast the chance to levitate and hang there unsupported, or an astronaut the prospect of a launch w/o rocket.)
5 Just in case these reasons [as well as the anthology’s real intended audience] are not yet obvious, q.v. the following announcement, variations of which appear in regular font on Best of The P.P.’s editorial page, in bold at the end of Johnson’s Intro, again in bold in an ad for The P.P. after the contributors’ bio-notes, and yet again, in a bold font so big it takes up the whole page, at the very end of the anthology: