Keys to Infinity
Reviewed by Stan Kelly-Bootle.
I am delighted to add Clifford Pickover's latest recreational romp to my endless collection of books with at least one occurrence of the string "infinity" on their spines [Note 1]. By definition, "popular" mathematics books are aimed at a wide, non- and semi-academic, largely point-of-sale audience. Publishers, therefore, employ the keenest semioticians (the bigger houses can afford full oticians) to create mind-grabbing titles and eye-catching covers. Certain colors, themes and keywords provably attract the casual browser [Note 2]. Thus crimson "Catastrophe" outsells text-brown "Morphogenetic Instability," and fractal-rainbow "Chaos" tickles a wider fancy than Springer-off-yellow "Feigenbaum Transformations." (And, yes, there's also price as a key marketing factor: $24.95 [Note 3] is the canonical figure for optional, spur-of-the-moment purchases, compared with $69.95 for mandatory course material.)
But beyond these fashionably hot-pants topics, there's clearly an enduring, insatiable demand for books on "Infinity," and especially for those that offer "keys" to that least effable of all transcendentalities [Note 4]. Hilbert's oft-quoted
The infinite has always stirred the emotions of mankind more deeply than any other question; the infinite has stimulated and fertilized reason (my italics) as few other ideas have; but also the infinite, more than any other notion, is in need of clarification. [Note 5]
was echoed by the cashier at Borders who told me, "People just can't get enough of this stuff." The shelves at Borders reveal that publishers and prophets have not been shy in trying to quench our metaphysical thirst (cf. John 4:13-14). Sadly, Pickover's book finds itself competing for store-space and mind-share with steaming heaps of irrational fertilizer: Nostradamus on Who-Killed-JFK, Oxfordian Authorship [Note 6], Crystal Faith Therapy, Alien Chariot Abductions, Arks Lost & Found, Pyramidiot Numerology, Martian Portraiture, Gellerian Telekinesis, Monastic Levitation, and Hidden End-Time Bible Codes. (Dare I add the glut of Java books, a sort of computer-aided tulipomania?) These topics seem to guarantee a place in the best-selling charts, stretching the usual fiction and non-fiction categories. And this in spite of the fact that the works of Boas, Randi, Allen, Casti, Gardner, Stewart, Flew, Pickover, et al., each in their separate skeptical ways, offer more pure fun and entertainment than the entire pseudoscience "industry."
Have I missed your favorite paranormality? The latest, most-depressing, is a tape-recording from Hell itself: endless-loop screams of the eternally damned picked up on a (presumably) lake-of-fire-resistant microphone lowered into a sulfurous, nine-mile deep hole somewhere in Alaska. Or was it a bottomless pit in Siberia? [Note 7] The tape has been broadcast on those X-file, "Unsolved Mysteries" programs that always cop out with a concluding "Perhaps we may never know."
Let's hope that browsers looking for an occult fix are attracted by the title and rouge-et-noir cover of Keys to Infinity. Flipping the pages, will they be tempted by chapters called "Ladders to Heaven" and "Slides in Hell" illustrated by Gustave Dor*'s sublime engravings of Dante's Divina commedia? There's Paradiso on page xii with "the numbers of angels increasing to infinity the higher one ascends." Note, though, how the wing-spans and halos seem to get smaller and smaller: the equally baffling downside to the infinite known as the infinitesimal. Is heavenly rank inversely proportional to self-magnification? And there's a strange Inferno on page 84: a sea of sweating, full-frontal nudes-hardly a torment for any self-respecting sadomasochist among the damned. Pickover uses these images of eternal bliss and punishment to illustrate how our religious concepts of infinity are deeply embedded from childhood. The Preface traces his own fascination with infinity and large numbers from early encounters with Kao-Lung (the highest purpose to which mortal man may aspire is "to touch the hem of Eternity" in a blade of grass) and with the Augustinian God who thinks infinite thoughts and "knows all numbers." [Note 8]
But, as you work through Keys to Infinity, you move from this awe-struck religiosity to probe what I see as the more fruitful, human-grounded aspects of the intangible: from godspeak to mathtalk; from Jerusalem to Athens [Note 9]. The journey reflects the fitful emergence of science, from metaphysical (idle?) speculation to Baconian experiment; from dogma to maybe-what-if; from the literal to the metaphorical (but no less miraculous) heavens and hells lurking beneath our skulls.
Each of Pickover's thirty-one chapters challenges the reader with a problem or paradox related to the infinite (or to horrendously large numbers headed in that direction). Although many of the topics are based on his diverse journal contributions, this is not a quick cut'n'paste job. Even if you've followed his regular "Brain Boggler" columns in Discover magazine or his occasional articles in Computers and Graphics, The Visual Computer, and Skeptical Inquirer, you'll welcome this stoutly-bound collection with previously unpublished illustrations and reader feedback. The text is newly spiced in abundance with wonderfully apt quotations, a good mix of familiar sources (Job, Socrates, Hamlet, Pascal, Sagan) and (to me) unknown, intriguing authors (Poundstone, Frankowski, Zebrowski).
In keeping with Pickover's computer-science affiliation (the IBM TJ. Watson Research Center), many of the problems are presented with exploratory source code in a pleasant diversity of programming languages: not just your boring BASIC and predictable C, but IBM's regal REXX, Mark Jones's "lazy Yale" Gofer, and Ken Iverson's J (for those who think APL is over-friendly). There's no attached diskette (which would boost the canonical price from $24.95 to $34.95), so if you want to run the programs, you must key in the source and apply the appropriate compiler or interpreter. And a good thing too! Typing and debugging code (yours or another's) is the essential road to syntactical understanding. (Semantics comes later, all in good time!) This is modem, interactive learning, not your medieval, rote catechism! Dore's black-and-white visions give way to Pickover's many-colored fractals [Note. Are we touching the hem of Joseph's coat (Gen 37:3) or simply fiddling with the initial, divinely sensitive values of a recursive graphics algorithm? Whether God plays dice or not (some think she has better things to do), Pickover lets us play God on his "Infinite Keyboard." In Chapter 3 (Infinity Machines), we encounter the Nikola Tesla "paradox." Tesla, we read, would not eat "countable" food items unless served in multiples of three. Was he merely eccentric (aren't we all, more or less?) or stark-raving Trinitarian bonkers? Tesla was saved by being so madly prolific that a fair number of his inventions actually worked, shocking Edison and the scientific establishment. (As Kuhn's bumper sticker reminds us: SHIFTS HAPPEN.) Pickover explores the Tesla Society's obsession with perpetual motion devices and related inventions, such as the lABS (Infinity Antilock Brake System), that rely on the convergence of Pickover's "Infinity Program" will run while (sum
Reviewers love to act the doryphore ("one who takes excessive delight in finding small errors"). The best I can do is to report a few minor glitches. The reference to page ix on page xv should be to page xi. The program on page 21 should initialize the variable result to 0 before the while loop starts. In the final program on page 200, Fibonacci (X) should be Fibonacci (N). There's an index entry to "Erds, Paul 170," whereas on page 170, and elsewhere, we encounter "Paul Erdos" with the wrong, non-Hungarian umlaut!
Finally, Pickover provides many pointers to Internet newsgroups and web sites together with a generous "Further Reading" list for those of us who "just can't get enough of this stuff." Missing, no doubt due to paginational finitude, is one of my favorite philosphicomathematico-theological works: The Infinite, A.W. Moore, Routledge, London and New York, 1990.