The Bactra Review: Occasional and eclectic book reviews by Cosma Shalizi   60

Labyrinths of Reason

Puzzles, Paradoxes and the Frailty of Knowledge

by William Poundstone

New York: Anchor Press/Doubleday, 1988

Through the Twilight Zone, or, Que sais-je?

Brains in vats! Disturbing bets with mind-reading magicians! Endless, sinister mazes! Bizarre manuscripts in unknown tongues! Lightning-spawned swamp-men! Condemned criminals who cannot be hanged! Waking up in a world where everything you know is wrong!

These sound like plot summaries from the old Twilight Zone, or side-shows in the Circus of Dr. Lao, and certainly not serious topics of debate among earnest philosophers who have set themselves the task of drawing their subject-matter and their methods from science and mathematical logic. But all of these have been, and most still are, subjects of active debate precisely among the analytical philosophers, the spawn of Frege, Russell, Moore, Wittgenstein and the rest of that mob. I even left out some weirder things, like grue and bleen, or Quine's hordes of possible people filling empty doorways, or holding conversations with books. The fact is that such philosophy is two-faced, like a coin. One side is decorated with symbols from mathematical logic, to show its severe, detailed nature, its disdain for such fripperies as politics or aesthetics or even (for the most part) ethics in favor of the really fundamental problems of knowledge. (What does it matter whether an act is good or bad if you can't even tell what it is?) This side is formal in several senses, and just a bit (or more than just a bit) pedantic, aspiring to be like Principia Mathematica or one of J. L. Austin's hours-long discourses on such words as "if." On the reverse is a portrait of Epimenides the Cretan: from the first analytical philosophy has been swamped in paradoxes and puzzles. Sometimes these are formal themselves (like several versions of Russell's paradox, or the Burali-Forti paradox), but more often they are verbal, indeed taking the form of stories. The point of the formality is to resolve the paradoxes and solve the puzzles, so that we can have a better idea of what constitutes knowledge, of what we do know and what we can know; they proliferate because people keep coming up with what look like ways of (so to speak) breaking those resolutions. One comes to the formalities with (as Russell advised) a mind well-stocked with puzzling and paradoxical stories, or the formalities seem pointless. (This is not to say that some of the formalities don't seem pointless even if you do know the stories.)

Now, for a popularization to attempt to plunge in to the formalism would be fatal. Not only would the number of people able to wrap their minds around it be depressingly small (owing to the dismal failures of our educational institutions), but it makes very little sense without the stories, without (as Dennett likes to put it) the "intuition pumps." But while the stories can be grasped (not even American schools can prevent students from understanding stories, at least not until they start graduate studies in literature), they need to be seasoned with at least a taste of the solutions and the arguments around them, or they're merely perplexing, like a bunch of mediocre koans. Poundstone, in what is clearly the best popular book on analytical philosophy (so good it never admits to being one), steers a middle course, using the stories for anchorage, and avoiding almost all technical details, but providing plenty of argumentation and (purely verbal) reasoning. Many of the stories are from Borges: fair enough, since the venerable Jorge took some of his best-known conceits from Russell.

In this amiable, discursive manner Poundstone reduces the reader to a quivering bundle of doubts. The book itself is in some manner a paradox: it uses our "knowledge painfully acquired" of such subjects as logic, neuropsychology and the theory of computational complexity to undermine our belief in that self-same knowledge; it is one vast undercutting maneuver (p. 140). For instance: surely deductive logic is a secure citadel if anything is. But this is only so if the premises from which it proceeds are mutually consistent; otherwise one can deduce anything one likes (the classic example: prove that if 1+1=3, then Bertrand Russell was the Pope). Fine: establishing consistency is just another problem in deductive logic, called the "satisfiability problem" or just "satisfiability"; how hard could it be? Well, it turns to be exceedingly hard indeed; the time needed to solve it grows exponentially with the number of premises for every known method of finding solutions, so that even modest-seeming collections of premises can not be proved consistent within a human lifetime. But the demonstration that satisfiability is an exponential-time problem is itself a mathematical deduction...

"What, then, is time? If no one asks me, I know what it is. If I wish to explain it to him who asks me, I do not know": thus, famously, St. Augustine (Confessions, Bk. XI, ch. xiv, 17); less famously, it is followed by Augustine's own theory of time. We are in a like case today when it comes to knowledge. We have figured out how to learn about what makes the world go round, and to trace out the path of its gyrations. It's when someone asks us why we are able to learn that we're lost, and even worry that we can't actually know anything in the first place. By pursuing the ideal of perfect knowledge, we learn that the pursuit cannot end in success, not for us thinking radishes, probably not for anything at all. What is to be done?

As the proverb advises, "If at first you don't succeed, change the rules." We must accept that our knowledge is, in the nature of things, frail and can break, that we cannot attain real certainty. If we compare it to an edifice --- Barzun's House of Intellect, say --- it is not a solid and unshakable edifice of marble, but a mess of bamboo scaffolding. We can shore up pieces of it, and we've found that when we make those pieces with certain tools, what we make isn't likely to give way beneath us; but that's as far as it goes. The whole thing is rickety and chancy and founded on nothing so nearly solid as sand, but it's the home we've built for ourselves. Poundstone conducts a nearly ideal tour through this structure, showing it off the way it should be shown off: clearly, engagingly, with a light touch, an affectionate familiarity with its dustier corners, and a perverse pride in its failings. Anyone who spent a gloomy afternoon as a teenager wondering whether they were, in fact, a sort of low-class nightmare will appreciate his tour; and I suspect even the most hardened of analytical philosophers will come away having learned something.

ix + 274 pp., 10 black and white diagrams, black and white reproduction of a page from the Voynich manuscript, bibliography of works cited, index of names and subjects
Computers and Computing / Logic / Mind, Consciousness, etc. / Philosophy / Philosophy of Science / Popular Science
Out of print as a hardback (ISBN 0385242611); in print as a paperback, ISBN 0385242719 [buy from Powell's], LoC Q175 P8816
18 September 1998; thanks to Don Smith for pointing out an goof about NP-completeness.