1344 NOTICES OF THE AMS VOLUME 52, NUMBER 11

Book Review

Science in the Looking Glass:What Do Scientists Really Know?Reviewed by Martin Gardner

Science in the Looking Glass: What Do ScientistsReally Know?E. Brian DaviesOxford University Press, 2003288 pages, US$29.95ISBN 0198525435

I’m not sure exactly what Brian Davies, a dis-tinguished mathematician at King’s College, Lon-don, intended the title of his fifth book to suggest.Reflect like a mirror the history and nature of sci-ence? Perhaps he also thought of his book as lead-ing readers into a dreamlike universe as fantasticas the world Alice first entered through a rabbit hole and later through a looking glass. Whateverthe intent, it is a brilliant work, beautifully written,and brimming with surprising information andstimulating philosophical speculations.

Before turning to my one caveat—unlike DaviesI’m an unabashed realist who believes that math-ematical objects and theorems are “out there” witha peculiar kind of reality that is independent ofminds and cultures—let me go over some of thebook’s highlights.

Davies begins with a discussion of the uncer-tainties of perception. Errors of seeing are demon-strated with two amazing optical illusions. One isa ring of slash strokes that seems to rotate as thepage is shifted forward and back. It is impossible,viewing the other illusion, not to be sure that one

Martin Gardner is the author of more than sixty books and wrote the “Mathematical Recreations” column forScientific American magazine for 25 years. He lives in Norman, Oklahoma.

square of a checker-board is darker thananother when bothare actually the sameshade. Language toocan be misleading.Davies does not buyNoam Chomsky’sclaim that there is agenetically transmit-ted deep “universalgrammar”. Interac-tion with environ-ment is sufficient toaccount for the abil-ity to speak. Apes

failed to speak because their throats lacked the apparatus necessary for producing a great varietyof sounds.

Descartes’s famous effort to separate mind frombody is thoroughly discredited. On the other hand,although Davies is convinced that consciousness—the awareness that one exists and has free will—isa function of a material brain, it is still a total mys-tery. He agrees with Roger Penrose, Oxford’s math-ematical physicist, that no computer made withwires and switches will ever become aware of whatit is doing. Assuming that we are no more than anenormously complex pattern of molecules, Daviesspeculates on the possibility that our pattern couldsomeday be scanned and transmitted to anotherplace like the up and down beaming of charactersin Star Trek. Can a simpler pattern, such as anapple, be so translocated?

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Physicists are hard at work trying to accomplishjust such a feat and have actually succeeded in tele-porting an atom. Transmitting a human, however,as Davies recognizes, raises profound questionsabout identity. After being “beamed down”, woulda translocated person be the same person or merelya replica? What if the technique produces two iden-tical persons? Philosophers, notably Locke, have agonized over just such thought experiments. Hundreds of science-fiction tales have consideredsuch possibilities. Penrose, by the way, has arguedthat if even an apple is transmitted, laws of quan-tum mechanics require total destruction of theoriginal. When the captain of the Enterprise isbeamed down to a planet, he cannot leave himselfbehind.

Davies’s chapters on pure mathematics cover awide range. He deals with imaginary and complexnumbers and the difficulties that arise with ratio-nal numbers when they are enormously large orsmall. “Hard” problems like the four-color-maptheorem have finally been proved, but by suchmonstrous computer printouts that the proof canbe checked only by another computer. Goldbach’sstill unsettled conjecture that every even numbergreater than four is the sum of two odd primes hasnow been confirmed for numbers up to 1014. This,Davies adds, “would be sufficient evidence for any-one except a mathematician.”

A page is devoted to the notorious Collantz con-jecture. Start with any number above 1. If even,halve it. If odd, replace it with 3n + 1. Continuedoing this. If the procedure ends with 1, stop. Theconjecture is that it will always stop. So far it hasstopped for all n up to 1012, but a proof remainselusive.

Davies reports the sensational discovery a fewyears ago by Manindra Agrawal and his two youngassistants in Kampur, India, of a simple rapidmethod of testing whether a huge number is or isn’tprime. The algorithm doesn’t generate factors, butmerely tests for primality, and does so in polyno-mial time! “Such discoveries,” Davies writes, “areamong the things which make it a joy to be a math-ematician.”

Several pages concern the innocent-seeming lit-tle problem of the three doors, which created sucha stir when Marilyn vos Savant published it in herweekly Parade column. Modeled with playing cardsit goes like this. Smith places three cards face downon a table. Only one card is an ace. You are askedto guess where the ace is by placing a finger on acard. Clearly the probability you guess right is 1/3.Smith, who knows where the ace is, now turns faceup a card that is not an ace. Two cards remain facedown. Does not the probability your finger is onthe ace go up to 1/2? It does not! It remains 1/3.If you now move your finger to the other card, theprobability it rests on the ace rises to 2/3! Savant

gave a correct solution, but thousands of mathe-maticians who should have known better wroteangry letters attacking the solution. The event evenmade the front page of the New York Times.

Davies’s chapters on the physical and biologi-cal sciences are as broad in scope and as illumi-nating as his chapters on mathematics. We learnabout the mind-bending paradoxes of relativityand quantum mechanics, about chaos theory, con-tinental drift, the ever-changing conjectures of cosmology, the anthropic principle, Thomas Kuhn’sshaky views about science revolutions and para-digm shifts, and a hundred other topics on thefrontiers of modern science.

A lengthy chapter on evolution rips apart the cur-rently fashionable claim by defenders of “intelligentdesign” that the “irreducible complexity” of eventhe simplest life forms could not have evolvedwithout the guidance of an intellgent designer,namely God. Davies ticks off a variety of facts thatsupport the randomness of mutations. Why shoulda competent designer, he asks, bother to producemillions of dinosaurs only to allow them to vanishexcept for some small ones that turned into birds?

The world’s vast amount of evil and sufferingis evidence, Davies is convinced, that there is notranscendent deity supervising evolution. As Mar-lene Dietrich once remarked (my quote), “If thereis a God, he must be crazy.” Davies reproduces alovely photograph of a snow crystal as evidence thatnatural laws combined with chance can produce in-tricate complexity.

I have touched on only a small fraction of themyriad of colorful accounts that Davies providesabout today’s science and mathematics. Let menow turn to my reasons for not accepting a basictheme of Davies’s book. I refer to his constantbashing of mathematical realism, especially thevigorous Platonism of Penrose and Kurt Gödel.

First of all, I prefer the term realism to Platon-ism. Why? Because it avoids all the dismal contro-versies over such universals as goodness, beauty,chairness, cowness, and so on, that so agitated theminds of the medieval scholastics. No modern re-alist believes for a moment that numbers and the-orems “exist” in the same way that stones and starsexist. Of course mathematical concepts are mentalconstructs and products of human culture. Every-thing persons think and do is part of culture. To saythat numbers are mental constructs is to say some-thing trivial—something no realist denies. Thedeeper question is whether these constructs havea peculiar, dimly understood kind of reality em-bedded in the universe in a way that is not mind-dependent. No human is needed to establish the factthat the geometrical shape of Aristotle’s vase is in-separable from the vase. A spiral is inseparablefrom a spiral galaxy. The four corners of a cube canno more be detached from a physical model of a

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Davies, like all anti-realists, slips into the languageof realism. He writes, “we can estimate how longit would take to find the first occurrence” (italicsmine) of a run of a thousand sevens. Again: The timeit would probably take “to find the sequence” wouldbe “vastly longer than the age of the universe”.The word “find” of course implies that the run already exists. Davies is usually careful to avoid theword “find” because it gives the game away. “A Platonic mathematician would say that either there exists [such a run] . . . or there does not. Thisis certainly psychologically comfortable, but it isnot necessary to accept it in order to be a mathe-matician.” So comfortable, in fact, that anti-realistsseldom hesitate to speak of “finding” (i.e., discov-ering) something when they really mean con-structing it.

William James somewhere speaks of digits as“sleeping” in π until some mathematician wakesthem up. It is a striking metaphor. A sleeping cat,however, has to sleep somewhere. To Davies andHersh the uncalculated digits of π sleep nowhere.They just pop into reality when a computer “con-structs” them.

Bertrand Russell, a firm realist, once wrote that2 + 2 = 4 even in the interior of the sun. As I haveoften said, if two dinosaurs met two other di-nosaurs in a clearing there would have been fourthere even if no humans were around to observethem. The equation 2 + 2 = 4 is a timeless truth,valid in all logically possible worlds because it iswhat philosophers since Kant have called analytic.Given the axioms of arithmetic 2 + 2 = 4 can betranslated into a string of symbols which, assum-ing the axiomatic system’s formation and trans-formation rules, arrive at A = A. Two plus two isfour for the same reason that there are three feetin a yard.

Like many anti-realists, Davies drifts close to akind of social solipsism in which even the externalworld fades into a hazy construction of our brains.He quotes favorably from Donald Hoffman’s bookVisual Intelligence: How We Create What We See.“Why,” Hoffman asks, “do we all see the samethings?” Why for instance, do we all see the samemoon? Everyone I know would at once answer, “Be-cause the moon doesn’t change.” Not Hoffman.His reason, so help me, is “because we all have thesame rules of construction.” We are not seeing amoon, out there, independent of us. We are seeingour constructions of the moon!

This is far more extreme than the opinion that2 + 2 = 4 because we all construct numbers thesame way. To suppose that people see the same cowbecause they have constructed the cow by the samerules boggles my mind. They see the same cow be-cause it is the same cow. “Realism,” I once heardRussell say in a lecture, “is not a dirty word.”

cube than from an ideal cube. The existence of op-tical illusions doesn’t prevent one from seeingeight corners. You can close your eyes and feel thecorners.

To a realist it is a misuse of language to say thatprimitive humans invented integers. What they didwas invent names, later symbols, for properties ofsets of discrete things such as fingers, pebbles, andelephants—things “out there”, independent ofhuman minds. Later they discovered the laws ofarithmetic because that was how pebbles behavedwhen manipulated. They didn’t invent thePythagorean theorem. They found it, out there,when they measured the sides of material right tri-angles.

If one is a theist, believing as Paul Dirac did thatGod is a great mathematician, or even in the pan-theistic deity of Spinoza and Einstein, then thelocus of mathematical reality moves to a tran-scendent realm outside Plato’s cave. The big debatebetween realism and constructivism evaporates.Paul Erdos liked to refer to God’s Book in which allthe most elegant proofs are recorded. From timeto time mathematicians are permitted briefglimpses into one of the Book’s infinity of pages.

In a curious way, numbers may be more real thanpebbles. Matter first dissolved into molecules, theninto atoms, then into particles, which are now dis-solving into vibrating loops of string or maybe intoPenrose’s twistors. And what are strings andtwistors made of? They are not made of anythingexcept numbers. If so, the numbers are as much“out there” as molecules. They could be the onlythings out there. As a friend once said, the universeseems to be made of nothing, yet somehow it man-ages to exist. As Ron Graham remarked, mathe-matical structure may be the fundamental reality.

No anti-realist such as Davies, and Reuben Hershwhom he admires, thinks the moon vanishes whenno one, not even a mouse, is observing it. If themoon is “out there”, why not admit that the moon’scircumference, divided by its diameter, is a closeapproximation of π even before mathematicianswere around to say it and will be true if humansbecame as extinct as dinosaurs?

To an anti-realist, π doesn’t really exist outsidethe minds of sentient creatures. A sequence in π ’sdecimal expansion, such as ten sevens in a row, isn’t“there” until a computer calculates it. Davies tellsan amusing story about how, in his book’s firstdraft, he wondered whether a computer wouldever find the sequence 0123456789 in π . ToDavies’s astonishment he later discovered that thissequence actually had been found. In the unlikelycase that readers would like to know, the run startsat π ’s 17,387,594,880th digit.

Davies takes up the question of whether one isallowed to say that somewhere in π is a run of athousand sevens. In talking about such things,

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The full complexity of reality is far be-yond our ability to grasp, but our lim-ited understanding has given us powerswhich we had no right to expect. Thereis no reason to believe that we are nearthe end of this road, and we may wellhardly be past the beginning. The jour-ney is what makes the enterprise fasci-nating. The fact that the full richness ofthe universe is beyond our limited com-prehension makes it no less so.

The conflict between realists and their criticsmay come down finally to the choice of a languagethat is the least confusing. As President Clinton fa-mously said, it all depends on what the meaningof is is.

Anti-realists are fond of claiming that mathe-matics, like science, is never certain. Morris Klineeven wrote a book titled Mathematics: The Loss ofCertainty. On the contrary, mathematics (includingformal logic) is the only place where there is no lossof certainty. In his book What is Mathematics, Re-ally?, Hersh argues that even laws of arithmetic areuncertain by considering a hotel that is missing athirteenth floor. Take an elevator up eight floors,then go five floors more, and you reach floor four-teen. Hersh apparently thinks this violates theequation 8 + 5 = 13. What he has done, of course,is jump from pure arithmetic to applied arithmetic,where applications are often uncertain.

Two beans plus two beans make four beans onlyif you assign to beans what Rudolf Carnap calleda correspondence rule. In this case the rule is thateach bean corresponds to 1. In the case of Hersh’selevator, if you assume that every floor corre-sponds to 1, then 8 floors plus 5 floors is sure tomake 13 floors. Without correspondence rules, ap-plications of mathematical truths are indeed un-certain. Two drops of water added to two drops canmake a single drop. Hersh and Philip J. Davis, intheir book The Mathematical Experience, give aneven funnier example. A cup of milk, they informus, added to a cup of popcorn doesn’t make twocups of the mixture.

Euclidean geometry is not rendered uncertain be-cause space-time is non-Euclidean. The PythagoreanTheorem is absolutely certain within the formal sys-tem of plane geometry. There is not the slightestdoubt that the angles of a Euclidean triangle addto 180 degrees. Science, on the other hand, is cor-rigible. Decades before Karl Popper, Charles Peircecoined the term fallibilism, and awareness that sci-ence is fallible goes back to the ancient Greek skep-tics. As Hume taught us, there is no logical reasonwhy the sun must rise tomorrow. For all we knowthere might be an unknown law of inertia thatwould suddenly stop the earth from rotating. Thisis in stark contrast with mathematics, where theuncertainty of science is incapable of inflicting injuries.

But enough about the tiresome, never-ending debate between the small minority of anti-realistsand the vast majority of mathematicians, includ-ing the greatest, who take realism for granted.They do their work without the slightest anxietyover the philosophical foundations of their craft.

What I admire most about Davies is his awe be-fore the terrible mystery of time and why the uni-verse, as Hawking recently wondered, “bothers toexist.” He is aware of how little we understand theworkings of Einstein’s Old One. To answer hisbook’s subtitle, scientists “really know” a greatdeal but what they don’t know is even vaster. Hereis how this book ends: