the Russian mathematician A. N. Kolmogorov defined the complexity of asequence of !s and "s to be the length of the shortest computer program that will generate the sequence in question.For the sake of uniformity, assume that the language of theprogram is coded into a sequence of !s and "s so that theprogram itself can be considered a sequence of !s and "s.Thus we can take the length of a program to be simply thenumber of !s and "s in it. If the program is the shortest oneto generate a given sequence, the program’s length in bits(!s or "s) is the complexity of that given sequence.A program that generates the first sequence above willjust be a translation into machine language of the followingrecipe: two !s, then a ", repeated x times. This programshould be quite short compared to the length of thesequence, which, let us assume, is 10 trillion bits; and thusthe sequence has, despite its length, a complexity of only 1million bits, say.A program that generates the second sequence would bea translation of the following: one ! alternating with eitherone or two "s, the pattern of 1s being one-two-one-onetwo-two-two-one-one-two-one-one-one-two-one-twoone-one . . . If the second sequence were very long, say 10trillion bits again, and this pattern continued, any programthat generated it would have to be quite long to give the pat-
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tern of the intervening "s. Still, the shortest such program,considered as a sequence of !s and "s itself, would be considerably shorter than the 10-trillion-bit sequence it wasgenerating. The complexity of this sequence thus might beonly 5 trillion, for example.With the third sequence the situation is different. Thesequence, let us assume, is so disorderly throughout its 10trillion-bit length that no program we might use to generate it would be any shorter than the sequence itself. Since allthe program can do in this case is just list the bits in thesequence, there is no way it can be shortened. Such a program, itself expressed in bits, is at least as long as thesequence it is supposed to generate, whose complexity thusis at least 10 trillion. The sequence is random.More formally, we can define a sequence to be random ifits complexity is (roughly) equal to its length; that is, if theshortest program capable of generating it has (roughly) thesame length as the sequence itself. A sequence is then notrandom if its complexity is less than its length. Sequence (3)is random on this account, while sequences (1) and (2) arenot. Two important consequences of these definitions are(a) If two sequences of different lengths are random, thelonger one is more complex; and (b) For any given wholenumber x, the vast majority of sequences of bits of length xare random; there are only a relatively few low-complexityones of any given length.
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Intelligibility or precision: to combine the two is impossible.Bertrand Russell
Attempts have been made to apply these formal notionsof complexity and randomness more generally. A.Solomonoff theorized, for example, that a scientist’s observations could be coded up into a sequence of !s and "s. Thegoal of science then would be to find short programs (algorithms, recipes) capable of generating (deriving, predicting) these observations. Such a program, so the story goes,would be a scientific theory, and the shorter it was, relativeto the phenomena it predicted, the more powerful it wouldbe. Random events would not be predictable, except in avery fractured sense by a program that simply listed them.There are, of course, serious problems in trying toextend the use of these technical notions to more generalcontexts. Where do these sequences of !s and "s come fromin the first place? Exactly how are observations to be codedinto a sequence of bits? Or prediction sequences decoded?What relation do they have to other significant categories?(Recall the sportscaster in a hurry: “And now for the baseball scores—6 to 2, 4 to 1, 8 to 5, 7 to 3, 5 to 0, and in a realslug-fest, 14 to 12.”) It is impossible to divorce thesesequences from the way they are obtained and employed,and from the human interests and values that led to theirdiscovery and explanatory significance. Without this supporting scientific and cultural background, the sequencesare meaningless.Wittgenstein once remarked, “That Newtonianmechanics can be used to describe the world tells nothing
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about the world. But this does tell us something—that it canbe used to describe the world in the way in which we do in fact useit.” The same thing can be said about scientific theoriesconceived of as programs that generate predictions: the wayin which we do in fact use them is their scientific content.The reductionist tendency to be seduced by the simplicityand exactitude of this account of scientific theories as prediction-generating programs, and the consequent desire todeclare that science is nothing but the study of such programs, should be resisted.
Ah but a man’s reach should exceed his grasp,Or what’s a heaven for.—Robert Browning
The formal notions of complexity and randomness,though they suffer from the limitations discussed above, arenevertheless often suggestive and useful. Chaitin hasemployed them and a detoxified version of the Berry paradox to give an alternative proof of Godel’s first incompleteness theorem. Since the proof sheds a somewhat differentlight on this famous result, a sketch of it follows.The Berry paradox, first published in 1908 by BertrandRussell and attributed to a Mr. Berry, asks one to considerthe following task: “Find the smallest whole number thatrequires in order to be specified more words than there arein this sentence.” Examples such as the number of hairs onmy head, the number of different states of a Rubik cube,and the speed of light in centimeters per decade, each spec-
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ify, using fewer than the number of words in the given sentence, some particular whole number. The paradoxicalnature of the task becomes clear when we realize that theBerry sentence specifies a particular whole number that, bydefinition, it contains too few words to specify.What yields a paradox in English can be modified(detoxified) to yield a statement in a formal system that cannot be proved, and whose negation also cannot be proved.Consider a formal axiomatic system of arithmetic expressedin some formal language that contains symbols for addition, multiplication, and so on. This system—formulas,axioms of arithmetic, rules of inference—can be encodedinto a sequence of 0s and 1s, a binary program P whoselength in bits is L(P). We can then conceive of a computerexecuting this program and over time generating from it thetheorems of the system (encoded, of course, in bits). Stateda little differently, the program P generates sequences ofbits that we interpret as the translations of statements inarithmetic, statements that the formal system has proven.Now we ask whether the system is complete. For everyarithmetic statement A, is it the case that either A or itsnegation, ⬃A, will always be a theorem? (Is the sequence ofbits associated with either A or ⬃A eventually generated bythe computer?)To see that the answer to this question is “No,” we makea crucial alteration of the Berry sentence: we replace “thatrequires,” a metalevel notion expressed in English, by “thatcan be proved to require,” an object-level notion expressible in terms of !s and "s. (We assume that enough arith-
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metic is included in the original formal system to permit“talk,” via some reasonable code involving 0s and 1s, ofnotions like provability, complexity, etc. within the system.)Recalling that the complexity of a sequence of bits is thelength of the shortest program needed to generate thesequence, we find that even the altered Berry task is impossible, though not paradox-inducing: “Find—that is, generate—a sequence of bits that can be proved to be of complexity greater than the number of bits in this program.”(Again, enough arithmetic is assumed to be included in theoriginal formal system to allow, via some sort of coding, forthe self-reference of “in this program.”) The program cannot generate such a sequence, since any sequence that thearithmetic program P generates must, by definition of complexity, be of complexity less than P itself is. Stated alternatively, there is a limit to the complexity of the sequences(translations of arithmetic statements) generated (proved)by P. That limit is, by the definition of complexity, thelength, L(P′), of the shortest program encoding P.Since a sequence is random if the length of any program generating it is at least as long as the sequence itself,we can further conclude that a formal system can generate arandom sequence only if the sequence is less complex thanthe sequence encoding the system. That is, the only randomsequences that the arithmetic system P can generate are necessarily less complex than P.Hence we finally get to the undecidable statements thatGodel promises us. Take g to be a random sequence of bitsof complexity greater than that of P encoded into bits. Thisis always possible, since for any whole number x, most
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sequences of length x are random. Then the statement G: “gis random,” properly encoded into bits, is unprovable bythe system P. (Remember, “random” is defined in terms of“complexity,” which is defined in terms of lengths of programs, all of which talk can be translated into arithmeticand then via code into a sequence of bits. Thus the statement “g is random” can be translated into a sequence ofbits.) P cannot generate the sequence that is the translationof “g is random” for the reason discussed above; thesequence is too complex. Neither can P generate thesequence that represents the negation of “g is random,”because g is random; since the axioms of P are true and therules of inference preserve truth, only true statements canbe proved. Therefore G can be neither proved nor disproved.Godel’s theorem can thus be interpreted as a consequence of the limited complexity of any formal arithmeticsystem, a limitation affecting human minds as well asmachine programs. There is not the temptation, on thisinterpretation of the theorem, to dismiss many of the goalsof artificial intelligence as impossible in principle, sincemachines cannot “step outside themselves” to the metalevel. This latter attempted refutation of mind-as-machine isrendered superficially plausible by more-standard proofs ofGodel’s theorem. Lastly, Chaitin’s proof suggests thatprogress in mathematics is at times not so different fromprogress in other sciences. Instead of looking for new facts,one looks for new, true, independent axioms whose addition will make the relevant formal systems (or rather theirtranslations into bits) more complex.
Determinism andSmart ComputersAlthough falsifiability and verifiability (testability for short)are important properties, it is unwise to be too rigid ininterpreting these terms or too hasty in rejecting any pronouncements that are not testable (under any interpretation). Metaphysics is untestable and, as Friedrich Waissmanresponded in countering some zealous logical positivists,“To believe that metaphysics is nonsense is nonsense.”The metaphysical doctrines of determinism and indeterminism are certainly not testable in any strong sense.How is one to test that every event is determined, or that atleast one event is not, whatever is meant by “determined”?Certain stronger, more precise theses can be formulated andrefuted, though. Consider, for example, provable determinism, which maintains that every question in an appropriate language of science can be decided one way or theother on the basis of certain physical and mathematicallaws. This is simply false by Godel’s theorem, since incompleteness infects even formalized physical theories thatinclude arithmetic in their formalization. A doctrinairedeterminist can still maintain that the answer to every question is “determined” by “states of affairs,” but not that everyanswer can be proved to follow from the given physical lawsand mathematical theories.
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Consider a computer, the “IBM-Cyber-007-know-itall-smarty-pants,” into which has been programmed (insome suitable language) the most complete scientificknowledge of the day, the initial conditions of all particles,and elaborate mathematical techniques and formulas.Imagine further that “know-it-all” answers only Yes or Noquestions, and that its output device is constructed in sucha way that a Yes answer turns off an attached lightbulb and aNo answer turns it on. If one asks this impressive machinesomething about the external world, the machine willrespond, let us assume, flawlessly. However, if one asks it ifits lightbulb will be on in one hour, “know-it-all” isstumped and cannot answer either way. This question, atleast, is “undetermined” by the laws and axioms of its program (although an onlooking computer might be able toanswer it).
Related to the “know-it-all” computer is the followingphenomenon: In predicting what a person will decide, it isoften very important to keep this predictive “information”secret from the person deciding. The scare quotes around“information” are meant to indicate that this peculiar typeof information loses its value, becomes obsolete, if given tothe person involved; it changes the person. The information, while it may be correct and true, is not universal. The
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onlooker and the deciding agent (compare “know-it-all”)have complementary viewpoints. As D. M. MacKay has written, “To us, our choice is logically indeterminate until wemake it. For us, choosing is not something to be observedor predicted, but to be done.”It is suggestive that in quantum mechanics, where indeterminism and subject-object interaction also play a crucialrole, we also have the phenomenon of complementary viewpoints. A scientist might know the position of a particle, butthat knowledge is incompatible with knowledge of the particle’s momentum since, as the Heisenberg uncertaintyprinciple states, the process of determining one necessarilyaffects the other.
Statistical laws have been around for a long time andmay reflect either our ignorance of things or the nature ofthings. In philosophical parlance, they may reflect either anepistemic or a metaphysical condition. As far as microphysical phenomena go, quantum mechanics points to the latterconclusion. There is a fundamentally and irreducibly probabilistic aspect to atomic and subatomic happenings; somephenomena simply are random. Whether this microphysical indeterminism somehow percolates “up” through thestructured complexity of the human (or inhuman) brain,resulting in free will, is an open question.In any case, just as the mathematical existence of longrandom sequences of bits was used in the last section to
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establish Godel’s theorem, and thus the falsity of provabledeterminism, the physical existence of random subatomichappenings demonstrates the untenability of determinismproper.Some have been unwilling to accept this and haveobjected, as Einstein did, that “God does not throw dice,”that there must be some “hidden variables” whose values,were we to know them, would lead to completely deterministic predictions. That this is logically and physically impossible is demonstrated by self-implicating questions andobservations and the quantum mechanical facts. Just howextraordinary one such fact is, is clarified by an inequalitydue to J. S. Bell.
Bell’s Inequalityand Weirdness
“If any of them can explain it,” said Alice, “I’ll give him sixpence. I don’t believe there’s an atom of meaning in it.”“If there’s no meaning in it,” said the King, “that saves a worldof trouble, you know, as we needn’t try to find any.”
This section will demonstrate, I hope, that one of the leastproblematical aspects of quantum mechanics is its probabilistic nature. More serious is the fact that it almost makesno sense. Almost. Imagine a device having three unconnected parts, as in figures 1 and 2 below. The part C in themiddle has a button that, when pressed, sends out a particleto each of the detecting devices A and B. The dials on A andB can be set before the button is pressed, or even while theparticles are in flight. (The distance between A and C neednot equal the distance between B and C, and the dial for Bcan even be set after a particle arrives at A but before onearrives at B.) When the particles do arrive at A and B, a lightflashes. The light, which may be red or green, flashes nomatter how the dials on A and B are set, though whether it isthe red or green light that flashes may depend on the dialsettings. The dials are set independently of one another.
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Figure 1. The complete device. A and B are the two detectors.C is the box from which the two particles emerge.
Figure 2. A detector. Particles enter on the right. The red (R) andgreen (G) lights are on the left. The switch is set to 1.
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The nature of the particles, the construction of thedevice, in fact all the technical details are irrelevant. All thatis important is that such a device can be built with absolutely no connections between the three parts A, B, and C. Whenthe button is pressed, the particles are ejected and the colors of the lights on A and B are recorded. If this is repeatedmany times we can discern some pattern to the flashings. Weindicate the outcome of any single episode by using a simple notation (due to N. David Mermin [1981], whose general approach I am following): 21RG, for example, meansthat A, whose dial was set at 2, flashed R while B, whose dialwas set at 1, flashed G. The record of many repetitions thuswould look like the following: 32RG, 21GR, 33RR, 22RR,13RG, 32GR, 23GG, 12RG, 12GG, 11GG, 21GR, 22GG . . .(An analogy might be helpful. Imagine two psychologists A and B in different cities, say Albuquerque andBuffalo. Married couples in Chicago separate and travel tothese cities, one spouse to Albuquerque, the other toBuffalo, in order to be asked exactly one of three Yes or Noquestions. The psychologists’ records after examining manysuch couples would look like the following: 31YY, 21YN,22NN, 13NY, 11NN, 21YY, 21NY, 33YY, 21NY . . .21YN would mean, as in the particle case, that the psychologist in Albuquerque asked question 2 and received aYes response while the psychologist in Buffalo asked question 1 and received a No response.)On examining this record we would notice severalthings: (a) When the dials on A and B have the same setting(11, 22, 33), the detectors always flash lights of the same
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color, either both red or both green. (b) For these settings,red and green pairs of flashes appear randomly with equalfrequencies. (c) When the dials on A and B have differentsettings (12, 13, 21, 23, 31, 32), we note that the detectorsflash lights of the same color only one-fourth of the time,red and green pairs of flashes appearing randomly withequal frequencies. (d) Three-quarters of the time that thedials are set differently, the lights that flash are of differentcolors, RG and GR appearing randomly with equal frequencies.It is important to remember observation (c), that whenthe dials on A and B are set differently (12, 13, 21, 23, 31,32), these detectors flash the same color only one-fourth ofthe time. Millions of trials can be run to be certain of thisproportion.Given all this, so what? Well, some strange conclusionsseem forced on us. We conjecture that properties of eachparticle determine the color its detector will flash for eachof the three dial settings. This seems to be the only way toexplain why the detectors always flash the same color whentheir settings are the same: the detectors A and B must beresponding to some (shared) property of the two particles—size, speed, spin, whatever.(If, unlikely as it may seem, the members of each andevery married couple were always to answer the same way,either both Yes or both No, whenever the psychologists inAlbuquerque and Buffalo asked the same-numbered question, it would seem natural to conclude that the psychologists were measuring some real property of these couples.)
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It would thus appear that there are eight types of particles—RRR, RRG, RGR, GRR, GGR, GRG, RGG, GGG—and thatwhen the dial settings are the same, the flashing of the samecolor on both detectors indicates the same property presentfor both particles. Thus if RRG particles are ejected by C andif the dials of both detectors are set at 2, the red light willflash at both A and B. If the dials are both set (independently, remember) at 3, green lights will flash at both. Onthe other hand, if A’s dial is set at 1 or 2 and B’s at 3, A willflash red and B green. Similarly, if GGG particles are ejected, both detectors will flash green no matter what their settings.(To continue with the married couples analogy, weconclude that there are eight types of couples—YYY, YYN,YNY, NYY, YNN, NYN, NNY, NNN. Consider a couple of typeYNY. If the members of this couple leave Chicago, arrive atAlbuquerque and Buffalo, and are both asked question 3,they will both answer Yes. If only the Albuquerque individual is asked question 3, while the Buffalo member is askedquestion 2, the latter will answer No while the former willstill answer Yes. Similarly, if a couple of type NYY are askedquestions 1 and 2 respectively, they will answer No and Yesrespectively.)Although he didn’t consider the (analogue of the) casewhere the dials have different settings, Einstein concludedfrom a similar situation (the Einstein, Podolsky, Rosenexperiment—EPR for short) that there must be “hiddenvariables” (the RRG, GRG, etc.) that determine whether thedetectors will flash red or green, and that will explain why
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both detectors always flash the same color when their settingsare the same. Note that in the case of the married couplesthis conclusion certainly seems reasonable enough.So far, this whole discussion may seem a bit out of placeand not worth pursuing (or even beginning)—somethinglike a long shaggy-dog story without a punch line. Well, thepunch line is coming. I quote from N. David Mermin’sarticle “Quantum Mysteries for Everyone,” where he writes:“The apparent inevitability of this explanation (above) forthe perfect correlations when the dial settings are the sameforms the basis for the conundrum posed by the device. Forthe explanation is quite incompatible with what happenswhen the dial settings are different” (1981).Consider a particle of type GRG. Out of the six possibledial settings that differ (12, 21, 13, 31, 23, 32), only 13 and31 will result in the same color flashing (green in this case).Thus for this type of particle, GRG, the same color shouldflash one-third of the time. Similarly, for particles of typesGRR, RGR, RRG, RGG, and GGR the same color should flash onthe detectors at A and B one-third of the time. Particles oftype RRR or GGG should always flash the same color when thedial settings differ. Thus we can conclude that the only reasonable account of why the colors are always the same whenthe dial settings are the same implies that when the dial settings are different, the colors flashed by the detectorsshould be the same (at least) one-third of the time—more ifthere are RRR or GGG particles. But empirically this justdoesn’t happen: when the settings differ, the colors flashedby the detectors are the same only one-fourth of the time.
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(This contrasts with what happens in the case of themarried couples. Consider one such couple, say of typeYNY. Out of the six possible question settings that differ(12, 21, 13, 31, 23, 32), only 13 and 31 will result in thesame answer being given (Yes for this couple). Thus for acouple of type YNY, the same answer should be given onethird of the time. Similarly, for couples of types YNN, NYN,NNY, NYY, and YYN the same answer should be given inAlbuquerque and Buffalo one-third of the time. Sincethere are also some couples of type YYY or NNN, the sameanswer should actually be given more than one-third of thetime.)Weirdly enough, the fraction one-fourth is what quantum mechanics predicts, contrary to the above “commonsense” estimate of at least one-third developed in a slightlydifferent context by the physicist J. S. Bell in 1964. (In thecase of the married couples, given the quite unlikelyassumption that couple members always do give the sameanswer when asked the same question, the estimate that atleast one-third of the answers will be the same when thequestions differ is, as we’ve seen, correct.)
Realism is the commonsense philosophical view thatphysical objects exist independently of being perceived. Itwas only considered to be a philosophy of science because acase against it had been made by idealists (e.g., Berkeley inthe eighteenth century), who maintained that physical
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objects are mind-dependent, and their being consists inbeing perceived. Most red-blooded scientists and philosophers are still realists, but the quantum effects exhibited inmicrophysical (and macrophysical) phenomena put a gooddeal of new stress on this position.There is no intelligible model or mechanism thatexplains these phenomena, there are just certain rules forcalculating and predicting probabilities (as above, 1⁄4 insteadof ≥ 1⁄3). As David Mermin writes, “The device behaves as itbehaves. . . . It is not the Copenhagen interpretation ofquantum mechanics that is strange, but the world itself.”Physicists either become slightly mystical, postulating somequality of “wholeness” to nature (in particular to the particle pairs being measured, or to the unconnected parts A, B,and C of the device itself), or they become hardheadedlypositivistic, taking refuge in their rules and formulas andabjuring intelligibility or explanatory models altogether.Some nonphysicists (and even a few physicists) haveinterpreted the above experiment as indicating the existence of telepathy or instantaneous communication. Thestandard so-called Copenhagen interpretation of quantummechanics rules this out, but the experimental results cannevertheless be interpreted in this way. (Even if they are,however, the “telepathy” or “instantaneous communication” involved is not of a sort that should be very comforting to these people; absolutely no message can be sent via it.The original Einstein–Podolsky–Rosen thought experiment is a simpler version of the one I’ve sketched here.Thus, to put it acronymically, EPR does not imply ESP.)
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I wrote in the first chapter that if one understands therelevant philosophical point, one gets the joke. Not fullyunderstanding quantum mechanics, we can’t laugh quite yet(at least not heartily) at the above or other quantummechanical puzzles. Still, I think someday we will—though,contrary to Einstein, I think that God or nature or theGreat Tortoise really does throw dice.
A jest unseen, inscrutable, invisible, as a nose on—William Shakespearea man’s face.
We dance round in a ring and suppose,But the Secret sits in the middle and knows.
—Robert Frost
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atomic piles. Only $675. Cheap when you considerhow proud you’ll be to hear your son say “MyDaddy’s a QUANTUM MECHANIC!”
—anonymous, magazine parody
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On AssumptionsRisking cerebral whiplash, let me momentarily flash back2500 years to Zeno of Elea. Zeno wrote that Achilles couldnever catch the turtle, since by the time Achilles hadreached the place P! from which the turtle had started, theturtle would have advanced to P1. When Achilles reached P1,the turtle would have advanced to P2; by the time Achilleshad reached P2, the turtle would have advanced to P3; and soon. Achilles would have to cover an infinite number of suchintervals before he could catch and surpass the turtle. Weknow now that the sum of an infinite number of (interval)lengths can be finite (e.g., 1⁄2 + 1⁄4 + 1⁄8 + 1⁄16 + 1⁄32 +. . . = 1).Along with some theorems on infinite sets due to GeorgCantor, this knowledge resolves Zeno’s paradox.It’s a reasonable conjecture that some comparable sortof conceptual advance will be necessary if we are to understand (and not just predict) quantum phenomena. Thepresuppositions of classical physics that will probably haveto be modified concern the notion of “thinghood.” What isit to be a thing? How is it that things persist through time?Are things distinct, or are they fuzzy? Are they independent, or somehow interconnected? It may very well turn outthat we do not know very much about things.The American philosopher Hilary Putnam has writtenthat just as the development of non-Euclidean geometrymade possible the easy expression of Einstein’s relativity
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theory, so might the development of a quantum nonclassical logic make easier and more natural the expression ofquantum theoretical insights about “things.” The law of theexcluded middle, for example, might be modified (or itmight not).
Imagine a list, in some order, of all the infinitely manynumbers between 0 and 10: 7.1, 2 33⁄49, π, 9 112⁄219, 52,319, 2.86312, . . . , e2, 5 11/103, . . . . Around the first number, place an interval of length 1/2; around the second number, an interval of length 1/4; around the third number, aninterval of 1/8; around the fourth, an interval of 1⁄16, and soon. The sum of these intervals is 1⁄2 + 1⁄4 + 1⁄8 + 1⁄16 + 1⁄32 +. . . =1, yet to most people it would seem that these intervals coverall the points on the number line between 0 and 10. Whatassumption are they (or you) making that Georg Cantordidn’t make?
George is running up the aisle of a train traveling at 20mph. If he’s running 10 mph with respect to the train, he’smoving at 30 mph with respect to the ground. It’s naturalto assume that velocities always add in this way—but the theory of special relativity states that this is not the case for verylarge velocities. Likewise, it’s natural to assume that of anytwo space-time events, one must necessarily precede theother for all observers—but again, it isn’t so.
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Martha and a big dog are standing at a bus stop. Waldoapproaches them and asks if her dog bites. She assures himthat her dog is very friendly and doesn’t bite, whereuponWaldo pets the dog. The dog bites his arms and legs andthoroughly mauls Waldo, who screams at Martha, “I thoughyou said your dog doesn’t bite!” Martha responds quiteinnocently, “Oh, that’s not my dog.”
Waldo asked his doctor how to improve his relationshipwith his wife. The doctor advised him to take a ten-milewalk each night so he wouldn’t be so irritable, and to callhim in a