) thestatement “All ravens are black.’’ But by elementary logic,“All ravens are black’’ is logically equivalent to “All nonblack objects are nonravens.’’ Since the statements areequivalent, any observation that confirms one confirms theother. But pink flamingos, orange shirts, and chartreuselampshades are all instances of nonblack objects, and thustend to confirm the statement “All nonblack objects arenonravens’’; therefore they must also confirm “All ravensare black.’’ Hence we arrive at the curious position of having pink flamingos, orange shirts, and chartreuse lamp-
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shades confirming the statement that all ravens are black!What is the problem? Well, it still is not clear to people. Two quick points should be made, however. One is thatmerely amassing instances of a statement is not enough toconfirm it. The second is that nonravens and nonblackobjects are much more numerous than ravens and blackobjects. Perhaps we could understand pink flamingos,orange shirts, and chartreuse lampshades as confirming,but only very slightly, the two equivalent statements above—not as much, in fact as a black raven would.Similarly, to confirm “All congressmen have problemswith grammar’’ one could go out, look for people who speakgrammatically, check to see that they’re not congressmen,and obtain some minuscule confirmation for the statement. More conclusive confirmation for it, though, wouldcome from attending congressional hearings.It is hard to say in general when an observation confirms (again, not necessarily conclusively verifies) a statement. Hempel considered some conditions that the notionof confirmation might be expected to satisfy. Two obviousones, it seemed, were (1) If an observation o confirms astatement h and h implies another statement k, then o confirms k; and (2) If an observation o confirms a statement kand another statement h implies k, then o confirms h.But from (1) and (2) we can derive the following: Let h= the theory of relativity, and k = the thermostat is above80°. Then its being hot in the room tends to confirm k (onany intuitive understanding of confirmation), and thustends by (2) to confirm the compound statement (h and k)
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(symbolically, hk), since hk implies k. Hence, by (1) h isconfirmed, since hk implies h. Thus we conclude that itsbeing hot in the room confirms the theory of relativity!Something is obviously wrong with (1) and (2) together. (2)seems especially suspect, but some (weakened) version of(2) is certainly often used, and is necessary in the everydaypractice of science. Even (1) is not always the case.
Let us consider one final oddity. Ever since Plato,knowledge has been taken by many philosophers to be justified true belief. A subject S is said to know a proposition Pif (a) P is true; (b) S believes that P is true; and (c) S is justified in believing that P is true. Edmund L. Gettier hasshown (1963) that these three ancient conditions are notsufficient to ensure knowledge.To show this, suppose that George and Martha are theonly applicants for a certain job at Lower Slobovia StateUniversity. Suppose further that George has strong evidence for the following compound proposition:(1) Martha is the person who will be hired, and Marthahas unruly hair this morning. George’s evidence for (1)might be that the chairman of the department informedhim that Martha’s specialty, unlike George’s, was the onethe department had been looking for. Further, he could seethat the department chairman and Martha had establishedan early rapport despite her unruly hair—while the chairman barely spoke to him, hurried him out of his office, and
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muttered something nasty-sounding to his secretary thatmade her laugh derisively.Proposition (1) implies:(2) The person who gets this job has unruly hair.George sees that (1) implies (2) and thus accepts (2) on thebasis of (1), for which, as we’ve seen, he has strong evidence.Clearly George is justified in believing that (2) is true.So far, so good. But suppose that, unknown to George,he, not Martha, will get the job. (The chairman, imagine,has a strange psychology.) Also unknown to George, his hairwas ruffled by the fan in the elevator, causing his cowlick tostand up. Proposition (2) is thus true even though (1) fromwhich it was inferred is false. Now all the following are true:(a) (2) is true; (b) George believes that (2) is true; and (c)George is justified in believing that (2) is true. But, ofcourse, it is quite clear that George does not know (2), since(2) is true in virtue of George’s own unruly hair, of whichhe is unaware. Thus justified, true belief does not constitute knowledge.If justified true belief does not constitute knowledge,the question remains: What does? One answer, due to thephilosophers Fred Dretske and Robert Nozick, involves thenotion of subjunctive or counterfactual conditionals. Theystate that S knows P if (a) P is true; (b) S believes P; (c) If Pweren’t true, S would not believe P; and (d) If P were true(but other minor things were different), S would stillbelieve P. For example, by these criteria George does notknow “The person who gets the job has unruly hair.’’ If theproposition in question weren’t true—say, George’s hair was
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not accidently ruffled by the fan in the elevator but he stillgot the job—George would still believe the proposition.Thus condition (c) is not met, and George cannot be saidto know the proposition. (Condition [d] handles otherrelated difficulties.)Since these four conditions, it can be argued, are sufficient for knowledge, and since such counterfactual andsubjunctive conditionals are also helpful in the analysis ofscientific law (and notions like “possible worlds’’), why notdeclare the war over and decree that these problems areresolved? The reason, for many philosophers, is that theanalysis of these conditionals is at least as problematic as thenotions they are meant to clarify.
Tr u th s , Ha l f -Tr u th s ,and StatisticsBenjamin Disraeli coined the phrase “Lies, damn lies, andstatistics,’’ and the phrase (as well as the sentiment) has lasted—though I like “Truths, half-truths, and statistics’’ better. In any case, even relatively simple applications of statistics can cause problems, not to mention the horrors associated with things like the often misinterpreted SPSS computer software (Statistical Programs for the SocialSciences).Probability and statistics, like geometry and mathematics in general, come in two flavors: pure and applied. Pureprobability theory is a formal calculus whose primitiveterms are uninterpreted and whose axioms are neither truenor false. These axioms originally arise from and are mademeaningful by real-life interpretations of terms like “probability,’’ “event,’’ and “random sample.’’ The problem withapplying probability and statistics is often not in the formalmathematical manipulations themselves, but in the appropriateness of the application, the validity of the interpretation, and indeed the “reasonableness’’ of the whole enterprise. This latter activity goes beyond mathematics into thesometimes murky realm of common sense and the philosophy of science (grue-bleen, ravens, etc.). Even though 1plus 1 equals 2, one glass of water plus one glass of popcorn
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does not equal two glasses of mixture. The mathematics isfine, the application is not.
martha: What did you get for the density of the block,George?george: Well, it weighed about 17 pounds and had a volume of about 29 cubic feet, so I guess the density is.58620689551 pounds per cubic foot. This calculator’sreally swell.
Babe Ruth and Lou Gehrig played baseball for the NewYork Yankees. Suppose Ruth had a higher batting averagethan Gehrig for the first half of the season. Suppose furtherthat during the second half of the season Ruth continued tohit for a higher batting average than Gehrig. Is it nevertheless possible for Gehrig’s batting average for the entire season to be higher than Ruth’s batting average for the entireseason? The fact that I’ve used up a paragraph asking thequestion indicates that the answer is yes, but how can it be?One way it can be is for Ruth during the first half of theseason to hit for an average of .344, getting 55 hits in 160times at bat; while Gehrig during this same time hits for anaverage of .342, getting 82 hits for 240 times at bat. Duringthe second half of the season Ruth’s average is .250, sincehe gets 60 hits in 240 times at bat; whereas Gehrig’s is
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.238, as he gets 38 hits in 160 times at bat. Nevertheless forthe season as a whole, Gehrig’s average of .300 is higherthan Ruth’s average of .287.Thus even the third-grade notion of an average can bemisused, not to mention (as I already have) things like complicated multidimensional analyses of variance.
If Waldo comes from country x, 30 percent of whosecitizens have a certain characteristic, then if we know nothing else about Waldo, it seems reasonable to assume thatthere is a 30 percent probability that Waldo shares thischaracteristic. If we later discover that Waldo belongs to acertain ethnic group 80 percent of whose members in theregion comprising countries x, y, and z have the characteristic in question, what now are Waldo’s chances of sharingthis characteristic? What if we subsequently determine thatWaldo belongs to a nation-x-wide organization only 15 percent of whose members have this characteristic? What nowcan we conclude, with all this information, about Waldo’schances of having the characteristic in question?
Wittgenstein writes about a man who, not being certainof an item he reads in the newspaper, buys a hundred copiesof the paper to reassure himself of its truth. Given theextent to which news-people and the media report each
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other’s reports, checking in several different papers orperiodicals is not much more intelligent.
News release: Abortions are becoming so popular insome countries that the waiting time to get one is lengthening rapidly. Experts predict that at this rate there will soonbe a one-year wait to get an abortion.The projection of “trends’’ linearly into the future isoften about as reliable as this “news release.’’
Most automobile accidents occur close to home, so wecan see that near one’s home is the most dangerous place todrive.Very few accidents occur when one is driving over 95mph, so it’s clear that driving this fast is actually quite safe.
It is a surprising, almost counterintuitive, fact that ifjust 23 people are chosen at random from a telephonedirectory, the probability is .5 (chances are about 50-50)that at least two of them will have the same birthday.Recently someone was trying to explain this oddity on atelevision talk show. The incredulous host thought the manmust be wrong, and so asked the studio audience how many
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people had the same birthday as he did, say March 19. Whennobody in the audience of about 150 people responded, thehost felt vindicated and the guest felt embarrassed. Actually,the question the host had raised was very different fromnthe one the guest had been discussing. It turns out that arandomly selected group of 253 people is required in orderfor the probability to be .5 that at least one member of thegroup has any specific birthdate (such as March 19); a groupof only 23 people is required for the probability to be .5that there is some birthdate in common.This is a specific instance of a very general phenomenon. Even though any particular event of a certain sort maybe quite rare, that some event of that sort will occur is notrare at all. The science writer Martin Gardner illustratesthis point with the story of a spinner that is equally likely tostop at any of the twenty-six letters of the alphabet. If thespinner is twirled 100 times and the results recorded, theprobability of any particular three-letter word, say “cat,’’appearing is quite small, whereas the probability of somethree-letter word appearing is very high.Columbus discovered the New World in 1492, while hisfellow Italian Enrico Fermi discovered the atomic world in1942. John Kennedy, elected president in 1960 and assassinated in office, had a Lincoln for a secretary, whileAbraham Lincoln, elected president in 1860 and assassinated in office, had a Kennedy for a secretary. As Gardnerhas noted, the acronym formed by the planets listed inorder—Mercury, Venus, Earth, Mars, Jupiter, Saturn,Uranus, Neptune, Pluto—is M V E M J S U N P, while that
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for the months is J F M A M J J A S O N D. In each case wehave an unlikely happening whose type, while almostimpossible to specify precisely, is quite likely to have someinstances. This has some relevance to evolution. That a particular branch should have evolved in a particular way isquite improbable. That some branch should have evolved insomething like the way in question is much less so.
The probability of getting at least one head on two flipsof a coin is .75. The chances of rain tomorrow are 75 percent. I think the odds of George’s marrying Martha are 3 to1. Does “probability = .75’’ mean the same thing in each ofthe above cases? The preceding discussion concerningunlikely coincidences might be summarized aphoristicallyas, “It’s very improbable that no improbable event willoccur.’’ Are both uses of “improbable’’ in this statementthe same?
Duhem, Poincaré, andthe Poconos-Catskill DietEven in the best of circumstances, applying the classicalcanons of scientific inference is not always straightforward.If a certain hypothesis H implies or makes more likely someevent I, then if that event I occurs, H is strengthened;whereas if the event I does not occur, H is weakened orrefuted. The raven and confirmation paradoxes demonstrated that I’s occurrence (the sighting of pink flamingoes,say) does not necessarily strengthen H (the statement that allravens are black). The French philosopher Pierre Duhem,on the other hand, showed that I’s nonoccurrence does notnecessarily refute or weaken H either.To see this, consider the Poconos-Catskill diet. Dr.Poconos, a Greek doctor from Pennsylvania, and Dr.Catskill, an Irish doctor from New York, prescribe twolarge servings of pastitsio (a meat, cheese, and noodle dish),three pieces of baklava, and four beers for every meal. Theyguarantee that after one week on this diet, a person will loseat least six pounds. George and Martha go on this diet andgain nine pounds in one week. Must Drs. Poconos andCatskill retract their hypothesis H, that the diet is effective,given that I, the loss of at least six pounds, did not occur?Of course not. They could maintain that a host of otherauxiliary, but tacit, hypotheses failed, and not their pethypothesis H, the Poconos-Catskill diet. Maybe the pastitsio
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was too heavily salted or not salted enough, maybe Georgeand Martha slept fourteen hours a day during the week, ormaybe the meals were not spaced correctly.Thus the nonoccurrence of I can never by itself refuteH, since there will always be scapegoats—auxiliary hypothesesto blame. What is being tested is never just “H implies I,’’but rather “H and H1 and H2 and H3 and . . . implies I,’’where the ellipsis indicates indefinitely many auxiliaryhypotheses. I’s nonoccurrence thus indicates only thateither H or (at least) one of the auxiliary hypotheses must befalse, not necessarily that H is false.Willard Van Orman Quine has gone even further thanDuhem in maintaining that experience never forces therejection of individual statements. He conceives of scienceas an integrated web of statements, procedures, and formalisms in contact with reality only at its edges. Any impactof the world on the web is distributed throughout the web,with no part (even logic) being absolutely immune and nopart having to bear the brunt of that impact alone.Adjustments can always be made in the whole web to accommodate the experience, but there is no unique way to makethese adjustments—simplicity, efficiency, and traditionbeing some of the criteria for a good web (science). Wecould thus accept that the Poconos-Catskill diet is an effective weight-loss program (as commercials often phrase it),but we would have to make fairly drastic changes in the restof our web.Similarly, in physics, if one wanted, for whatever reason, to accept some bizarre statement as true, one could do
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so by suitably and radically altering other statements, theinterpretation of certain physical terms, and so on. This isrelated to the traditional problem in the philosophy of science of where to draw the line between empirical physicsand a priori geometry. If one insists that geometry must beEuclidean, one’s physics (in certain astronomical contexts,say) becomes quite strange, with accelerations and forcesthat don’t make sense by traditional theoretical standards.An alternative is to switch to an appropriate non-Euclideangeometry that makes the physics simpler, but which itselfseems quite counterintuitive at first. Which geometryphysics combination to use depends on one’s purposes andis to some extent a matter of convention, as Henri Poincaréfirst noted (1913).
Of course, not just physics but any complicated phenomenon, especially one that involves human interaction,admits of many different and incompatible interpretations,each consistent with reality. The differing “theories’’ oftheir particular marriage entertained by a husband andwife, for example, often manifest this incomparability, asdoes the following Sufi story adapted from Masud Farzan’sAnother Way of Laughter:Since a Roman scholar was visiting Timur’s court, theemperor asked the Mulla to ready himself for a battle of witswith the scholar.
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The Mulla piled books with fictitious but impressivesounding titles on his donkey, books such as The Theory ofUniversal Bifurcants, Erosion and Civilization, A Critique of TolerantPurity, and Social Origins of Mental De-activation.On the day of the contest, the Mulla appeared in courtwith his donkey and books. His native wit and intelligenceoverwhelmed the Roman scholar, who finally decided to testthe Mulla’s knowledge of theoretical matters. The Romanscholar held up one finger.The Mulla answered with two fingers.The Roman held up three fingers.The Mulla responded with four.The scholar showed his whole palm, to which Mullaresponded with a closed fist.The scholar then opened his briefcase and took out anegg. The Mulla responded by digging an onion out of hispocket.The Roman said, “What’s your evidence?The Mulla answered, “The Theory of Universal Bifurcants, ACritique of Tolerant Purity,” etc.When the Roman sputtered that he’d never even heardof those titles, the Mulla responded: “Of course youhaven’t. Look and you will see hundreds of books you’venever read.”The Roman looked and was so impressed that he conceded defeat. Since no one had understood any of this,later, after refreshments had been served, the emperorleaned over and asked the Roman scholar the meaning of itall.
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“He is a brilliant man, this Mulla,” the Romanexplained. “When I held up one finger, meaning that therewas only one God, he held up two to say that He createdheaven and earth. I held up three fingers, meaning the conception-life-death cycle of man, to which the Mullaresponded by showing four fingers, indicating that the bodyis composed of four elements—earth, air, water, and fire.”“Well then, what about the egg and the onion?” theemperor pressed.“The egg was the symbol of the earth (the yolk) surrounded by the heavens. The Mulla produced an onion,indicating the layers of heavens about the earth. I asked himto support this claim to assign the same number of layers ofheavens as there are layers of onion skin, and he supportedhis claim by all those learned books of which I alas am ignorant. Your Mulla is a very learned man indeed.” The dejected Roman then departed.The emperor next asked the Mulla about the debate. Hereplied, “It was easy, Your Majesty. When he lifted a fingerof defiance to me, I held up two, meaning I’d poke both hiseyes out. When he held up three fingers indicating, I’msure, that he’d deliver three kicks, I returned his threat bythreatening four kicks. His whole palm, of course, meant aslap in my face, to which I responded with my clenched fist.Seeing I was serious, he began to be friendly and offered mean egg, so I offered him my onion.”Whether Mulla or Roman scholar was “right” is a matter of convention or tradition. As noted, conventionalism
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is the view that scientific laws are to a significant extent disguised conventions that reflect a decision, for one reason orother, to adopt one of various possible descriptions of aphenomenon. To return to physics, in some cases to ask ifone event precedes another makes as much sense as asking ifNew York is really to the right of Chicago.
George Carlin once listed six reasons for doing something or other: 1, b, III, four, E, vi. Notation, thoughclearly a matter of convention, is crucial. Imagine trying toexplain even some elementary mathematics, such as thequadratic formula, without good notation.
If the story of the Mulla and the Roman seems too farfetched, imagine a modern physicist trying to explain to anAustralian aborigine that neutrinos have no mass, or something about the properties of quarks and black holes.
Reduc tion i s m , Fa l l i bi l i s m ,and OpportunismThere are, of course, in addition to conventionalism manyother “isms” in the philosophy of science, some of whichshould at least be mentioned. The most common of these isopportunism, as illustrated by the following well-knownstory:martha: What are you doing out by the streetlight,George?george: I’m looking for my car keys.martha: But you dropped them on the grass near thebushes.george: I know, but the light is better out here.
The following story due to Leo Rosten (1968) makesthe same point, and also says something about the organization and contents of this book:A famous rabbi was asked by an admiring student “Howis it you always have a perfect parable for any topic?”The rabbi smiled and said “I’ll answer with a parable,”and told this story:Once there was a lieutenant in the tsar’s army who, riding through a small shtetl, noticed a hundred chalked cir-
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cles on the side of a barn, each one with a bullet hole in thecenter. The astonished lieutenant stopped the first passerby and inquired about all the bullseyes.The passerby sighed. “Oh that’s Shepsel, the shoemaker’s son. He’s a little peculiar.”“I don’t care. Anyone who’s that good a shot . . .”“You don’t understand,” interrupted the passerby.“You see, first Shepsel shoots and then he draws the chalkcircle.”The rabbi smiled. “That’s the way it is with me. I don’tlook for a parable to fit the subject. I introduce only subjects for which I have parables.”
Behaviorism is the philosophy of social science doctrineto the effect that psychological functioning is definable interms of overt behavioral manifestations. It, together with amindless respect for statistical formalism, has led to thepublishing of whole barrels full of trivial research of thefollowing form: If property x (humor, say) is operationallydefined in this way (number of chuckles elicited by a bookof cartoons), and property y (self-sufficiency, say) is operationally defined in this way (number of yes answers onsome “self-sufficiency questionnaire”), then the correlation coefficient between x and y is .621 (at least for studentsin Prof. George’s 8:30 psychology class).
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The existence of boredom implies the falsity of behaviorism, but the details of the derivation bore me.
Behaviorism is a type of reductionism, the latter beingany doctrine that claims to reduce what is apparently moresophisticated and complex to what is less so. Sometimes, ofcourse, this is possible; large parts of genetics can bereduced to molecular biology, and some parts of thermodynamics to statistical mechanics, for example. Sometimesthis reduction is not possible, at least not in any naturalmanner.Consider, for example, clocks or can openers. (I wonder if the preceding seven-word sentence has ever appearedin print before?) A clock is any device for keeping time, acan opener any device for opening cans. To characterizethese objects in purely physical terms, to reduce all talk ofclocks and can openers to the language of physics, would bea pointless (though theoretically possible) endeavor.Imagine what sort of physical description could encompassall and only sundials, grandfather clocks, wristwatches, digital clocks, etc. Any such purely physical description wouldbe a messy, unsystematic hodgepodge of ad hoc physicalstatements, and thus would offer no illumination on thepurposes or identification of clocks or can openers.Conversely, just as it is pointless though theoreticallypossible to describe clocks and can openers in purely physical terms, it is pointless though theoretically possible to
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eliminate theoretical physical terms like “neutrino,”“guard,” and “covalent bond” and replace them with anunsystematic hodgepodge of ordinary, everyday observational statements containing terms like “red,” “cold,” and“hard.” The predictive and organizational strength of scientific theories would, of course, be lost if this were done.Carl Hempel (1963) has shown explicitly how to rid scienceof theoretical terms (and thus cripple it) using Craig’s theorem, a well-known result from mathematical logic. Forsome reason, though, one never hears this referred to asreducing physics to the “theory of common sense.”
Richard Dawkins in his book The Selfish Gene seems toargue that our genes try to perpetuate themselves by fashioning our actions in such a way as to assure their own survival. Human behavior and culture is, on this sociobiological view, largely determined by our genes’ desire for selfpreservation. This reductionist account certainly needssupplementation. Note, for example, that different cultures with essentially the same genetic pool engender quitedifferent behaviors. Note, too, that we can carry the reduction even further. One might argue that it is not our genesthat are selfish, but a particular chemical bond in the genes.Our actions are not determined by our genes after all, butby a chemical bond in them that is trying to perpetuateitself. I am being simplistic, but so is the thesis as it is usually presented.
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x can be explained in terms of y. y does not have property P. So x does not have P.The greenness of grass, the blueness of the sky, the fleshtones of a human face can all be explained in terms of (theproperties of) atoms—frequencies and so on. But atomsthemselves are not colored. So grass is not green, the sky isnot blue, and faces are not flesh-colored.Similar arguments have been used to show that values,ethics, ideals, even intentions and beliefs, are illusions.
There is a story about a London man who spent his lifelooking around and recording everything he noticed in aseries of notebooks. He directed that after his death theseobservation notebooks be forwarded to the Royal Society, sothat scientists there could use them to fashion a new scientific theory.Though some people still think that science advances inthis way (consider some behaviorist research, e.g.), mostrealize that scientific research needs first a focus or problem; next, assumptions and hypotheses; and only then, newobservations. Scientific pronouncements should be supported (or at least supportable), and, as the Englishphilosopher Sir Karl Popper has stressed, they must becapable of being shown false. They must be falsifiable orfallible, at least in principle.
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When a plane crashes, many people remark that thesesorts of things always happen in threes. This belief is, asstated, completely unfalsifiable. Even if the next crash, twomonths later, occurs in Peru, and the third involves a smallprivate plane in an Arkansas cornfield, these people can stillmaintain that they have some special insight into aeronautical malfunctioning, or, more commonly, that “everyone”has this knowledge. Similar things can be said aboutremarks like “Whatever God wills, happens.”Popper has criticized Marxism and Freudianism forbeing too much like the above theory of plane crashes—thatis, for not being falsifiable and thus not being real sciences.He has in mind moves of the following sort: A Marxist predicts that the “ruling class” will respond in such and such anexploitive way to some crisis; when it doesn’t, this is attributed to some sneaky, co-opting policy of the ruling class.Similarly, an orthodox psychoanalyst might predict a certain kind of neurotic behavior; when his patient doesn’tbehave in this way, but in a quite contrary way, this is attributed to a “reaction-formation.” Popper does not quite sayso, but he hints that Freud is a fraud and that Karl Marxmakes less sense than Groucho.More generally, Popper is opposed to historicism, thedoctrine that there are immutable “laws of development”that govern the historical process and allow long-termsocial forecasts. One argument for Popper’s position is that
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scientific advances that obviously greatly affect social development are not predictable. How could someone predictDescartes’s discovery (or invention, if you like) of analyticgeometry, or Einstein’s discovery-invention of relativitytheory? Imagine a “futurologist” of the sixteenth (or nineteenth) century predicting the development of analyticgeometry (or relativity theory): if such a prediction had anydetailed content, it would in a sense already be fulfilled andthus not be a prediction at all.Less well known than Popper’s ideas on falsifiability ishis concern with the notion of randomness, the topic of thenext section.
Randomness and theB e r r y Ta s kgeorge: What’s Waldo’s phone number? I always forget.martha: Let me see. The Goodmans have 2 children, theFrankels have 3, the Passows have 7 dogs (though Myrtlemay be pregnant), the Youngs have 2 kids from his firstmarriage, 1 from hers, and 3 of their own, and theSturms have 9 pets in all. Waldo’s number is 237-2139.george: Thanks. It’s a good thing I know the area code.
(1) !!"!!"!!"!!"!!"!!"!!"!!"!!"!!"!!"!!"!!"!!"#$#$#$(2) "!""!"!"!""!""!""!"!"!""!"!"!"!""!"!""!"!"!#$#$#$(3) "!!!"!""!""!""!!!"!"!""!!"!""""!"!!"!"""!"!#$#$#$Why is it that the first sequence of !s and "s is likely tobe termed orderly or patterned, the last sequence randomor patternless, and the second somewhere in between?Trying to answer this simple question will lead to a definition of randomness, an important insight in the philosophyof science, and an alternative proof of Godel’s incompleteness theorem.Returning to the question, note that the first sequencehas an easily expressed pattern: two !s, then a ", repeatedindefinitely. The third sequence has no such pattern; while
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in the second sequence, !s always alternate with either oneor two "s, but the occurrence of one or two seems patternless.With examples like this in mind, the American computer scientist Gregory Chaitin and