The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us
By Noson S. Yanofsky
(MIT Press, 424 pages, $29.95)
How rarely Reason guides the stubborn Choice / Rules the bold Hand, or prompts the suppliant Voice.” Dr. Johnson didn’t know the half of it. Not only does reason play a dismayingly small part in human affairs, but reason itself has built-in limitations that prevent our employing it in many cases where we should like to. Thanks to the work of 20th-century logicians, we now know that even mathematics, popularly thought to be the concentrated essence of reason, has zones of impenetrability.
Noson Yanofsky, a professor of computer science at New York’s City University, has written a survey of the border territories where reason gives way to the badlands of paradox, indeterminacy, undecidability, failure of intuition, and sheer physical impracticality. He covers a lot of ground, from simple word-conundrums like “This sentence is false” to deep issues in philosophy of science.
This is a dense and demanding book. There are seven full pages on Bell’s Theorem (which concerns quantum entanglement) and seven more on Turing’s Halting Problem (can we find out in advance whether any program will get stuck in a “hard loop”?). There are sixteen pages on P-NP (figuring whether the number of steps a problem requires for its solution is manageably reasonable, or preposterously un-reasonable), and twenty-two on the Anthropic Principle (which states, approximately, that the reason the universe appears to be fine-tuned for the existence of intelligent life is that if it weren’t, we wouldn’t be here to reason about it).
Hugh Everett’s many-universes interpretation of quantum theory gets a good airing, as do Galois Theory, Penrose tilings, and “strange loops.” Many fascinating minor oddities are covered: Goodstein’s astonishing theorem, for instance, and Paris and Kirby’s meta-theorem about it. Of the territory Yanofsky set out to cover, I don’t think he missed an inch.
Plainly this is not by any means light reading. I found myself asking the question publishers ask about a pop-science or pop-math manuscript: “Who’s it for?” The answer, I think, is that The Outer Limits of Reason will appeal most to intelligent teenagers or undergraduates keen to know something about the scope of our understanding. Readers at or past middle age who have any interest at all in these topics will find themselves frequently skipping passages on which their curiosity was sufficiently satisfied by Martin Gardner’s wonderful “Mathematical Games” column in Scientific American, 1956-81.
One of the great strengths of Yanofsky’s presentation is his very comprehensive bibliography. At the end of each chapter in his book are brief guides to further reading, keyed to that bibliography (which includes three of Martin Gardner’s sixty-odd books). Yanofsky credits himself with “the first popular exposition of the Kochen-Specker Theorem” in quantum ontology, but offers two different bibliographical references for more technical accounts. For many young newcomers to philosophy of science, this book could be an entry-level drug.
The book’s great weakness is the lack of any human dimension. This can be a difficult thing for an author to get right; not every reader wants to know the fine details about Galileo’s trials or Cantor’s depressive episodes. A dash of human interest never hurts, though, and Yanofsky is too sparing.
He also fails to engage with the uses and limits of reason in the social world. Some of the most elaborate exercises of reason occur in courts of law and in constitutional debates. Since legal cases are often wrongly decided and constitutions frequently fail, plainly there are limits here: it would have been good to hear something about them.
Yanofsky is also a bit too breezy about pseudoscience. “Why is it reasonable to check your blood pressure and not check your horoscope?” he asks. His answer: because astrology is a pseudoscience, since “its predictions have been shown to be different from observable facts.” Fair enough as regards horoscopes: but this understates the difficulty of marking off science in general from pseudoscience in general, a project so intractable that philosophers of science have wellnigh given up on it. Astrologers, Marxists, scientologists, Freudians, homeopathists, flat-earthists, and theologians of all religions make extensive use of reason. “Misuse,” a rigorous empiricist would say; but how does he know?
Out at the extreme, lunatics can be keenly rational, as G.K. Chesterton pointed out in Orthodoxy: “Imagination does not breed insanity. Exactly what does breed insanity is reason. Poets do not go mad; but chess-players do.” The exophthalmic barfly who buttonholes you to explain his theory about 9/11 being a Freemasons’ plot may be eminently reasonable. Anyone who writes for the public prints can tell you that the world is full of similar monomaniacs, many of them very skillful in the application of reason to a vast range of topics, including some of those covered by Yanofsky. We dwell among lunacy.
There is thus more to be said about the outer limits of reason than Yanofsky covers in his 350 pages. That’s really just an objection to the book’s title, though—and titles are not necessarily an author’s choice. What Yanofsky does cover, he covers very well, with clear explanations of some very difficult topics. The Outer Limits of Reason is a good, but strenuous, read.
Mr. Derbyshire will be writing "Shelf Life," a books column, every other week.
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