Pi in the Bible?

Steven Dutch, Natural and Applied Sciences,University of Wisconsin - Green Bay

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Created 3 February 1998, Last Update 3 February 1998

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Many of the philosophical questions about the nature of scientific proof revolve around theimpossibility of devising a completely rigorous description of scientific methods. Formal logic isvery powerful, but science uses informal logic as well. An example of a formal logical argument isthe following: (1) Aluminum atoms have 13 protons in their nuclei (2) This atom has 12 protons in its nucleus (3) Therefore, the atom is not aluminum (actually it is magnesium.)This argument is an example of deductive logic. It extracts latent informationfrom existing facts but does not generate any new facts. To a scientist, the interesting item is thefirst proposition. To obtain this fact, science first had to determine that there was such a materialasaluminum, that there were atoms, that atoms had nuclei, and that each atom had a specific numberof protons in its nucleus. Just because all aluminum atoms tested to date have had 13 protons,howdo we know this rule will always hold? The process of using facts to generate new rules is calledinduction, and so far it has defied all attempts to define a rigorous set of rules for it. For onething, induction deals with approximation. The experiments that first showed that elementscombine in fixed proportions, and thereby supported the concept of matter being made up ofdiscrete atoms, did no tactually yield exact results. An experiment to determine the ratio ofhydrogen atoms to oxygen atoms in water might not yield an exact value of 2.00, but 1.95 or2.03.There are no hard and fast rules for deciding how "approximate" a result can be and still count asevidence for a theory because the rules will depend on the difficulty of the experiment and all thefactors that might cause experimental error. In some cases, the rules for inductive logic are quite different from the rules for deductivelogic. For example, consider one of the most famous scientific proofs in history. If the theory thatplanets orbit the Sun is correct, the planets that are closer to the Sun than earth should appear ascrescents under certain circumstances. Galileo observed Venus as a crescent and therefore provedthat planets orbit the Sun. Correct? Actually, no! All Galileo showed was that Venus comes between the earth and the Sun. Thereare geocentric models of the Solar System that also allow Venus to show phases.More importantly, in formal logic, the reasoning is a fallacy called "confirming the consequent".To see why the argument can be dangerous, consider this all-too-real example: (1) If people are lazy, they will live in run-down houses. (2) Many poor people live in run-down houses. (3) Therefore poor people are lazyThe major fallacy in the argument, of course, is that there are many reasons other than lazinesswhy poor people might live in run-down housing, such as inability to afford better housing. Agood deal of the controversy that accompanied the birth of modern science revolved aroundshowing that methods that were logical fallacies in formal logic nevetheless worked in real life.Thanks to medieval theology and the recovery of classical learning, the Renaissance world had agood grasp of deductive logic. Hammering out some of the rules of inductive logic was oftenanother matter. Confirming the consequent may be a fallacy in strict formal logic, but it does workin science. In order to use it,though, we have to eliminate all the other rival explanations of theconsequence. Even deciding whether a piece of evidence supports a theorycan lead to interesting paradoxes.It is possible for a piece ofevidence to confirm a theory but at the same time undermineit!Consider the hypothesis "all humans are less than 100 feet tall".A 99-foot human "confirms" thehypothesis, but also makes it seema good deal less likely. The paradox in this case lies inthehidden assumption we all make that all humans are more or lessthe same height and none areanywhere near 100 feet tall. Findinga 99-foot tall person "confirms" the hypothesis butdemolishesthe hidden assumption that went along with it. Galileo'stelescope, similarly, did not somuch confirm heliocentricastronomy, as wreck a lot of the unwritten assumptions that wentalongwith the old astronomy. Another interesting paradox involves deciding what is and isnot evidence. "All crows areblack" is logically the same assaying "anything that is not black is not a crow". Thus agreenVolkswagen "confirms" the idea that all crows are black?! Thelogic certainly seems strange.When we are confronted with such aproblem we tend unconsciously to select the simplestapproach.The set of crows is smaller and more uniform than the set of allobjects that are notblack, so it makes intuitive sense toapproach the hypothesis by examining the set of crows,andlooking at the problem from the reverse perspective strikes us asstrange. Nevertheless, wecould go about testing the hypothesisby looking at all the non-black objects in the world andseeingif, say, a white crow turned up (It is entirely likely that analbino crow would be found oncein a while). But if we arelooking for crows in the woods, do we look for crows, or do welook atand discard all the non-crow objects without eventhinking about it, or do we actually do both atthe same time?Suppose instead of looking for crows, we are sorting steel andaluminum cans forrecycling. Does a magnetic separator selectsteel cans, or does it reject aluminum, or does it in factdoboth at the same time? Part of the apparent paradox is that weare imposing a false dichotomyon what is, in reality, a singleunified process. This paradox relates directly to one of the majorcontroversies in the philosophy of science. Arescientifictheories tested by attempting to confirm them, or by attemptingto refute them? Pastexperience suggests that we cannot beabsolutely confident that even the best-established theorywillnot need to be revised in the future, and so some philosophers ofscience, notably Karl Popper,have focused on the criterion of"refutability". In this view, a theory can be consideredscientificallytestable if there is some conceivable way thetheory might be refutable. Theories that surviverepeatedattempts to refute them can be considered valid, at least for thetime being. However, justas we cannot be sure a theory has beenproven for all time, we cannot be certain that a theory hasbeenrefuted for all time; there may just be some undetected flaw inthe refutation, or someunforeseen way the theory may turn out tobe true. For example, some scientists were once quiteconfidentthat they had thoroughly refuted the theory of continental drift.Refutation is alsounsatisfactory as a universal criterion foranother reason; there are many ideas in science such astheprinciple of uniformitarianism or the idea that the laws ofnature are the same throughout theUniverse that are not entirelyrefutable but which we nevertheless have every reason to believearevalid. When we deal with real-life scientific confirmation, we usea combination of confirmation andrefutation. Suppose we test theidea that two chemicals will react to give a certain product. Insucha case there is only one possible way an experiment can beconsistent with a theory, and everyother outcome refutes thetheory. We do the experiment and in effect test all thepossiblerefutations at once; if the theory passes the test, we arejustified in saying the theory hasbeen confirmed. In general, ifa theory makes a limited and highly specific claim, and the claimissubstantiated, it makes sense to view the testing process fromthe standpoint of confirmation. In other cases it may not be possible to confirm a theoryto the exclusion of all other competingtheories, but the theorymay have vulnerable points that can be tested, and if the theoryfails atthese points the theory as a whole is refuted. In such acase it makes more sense to view thetesting process from thestandpoint of refutation. Most pseudoscience is vulnerable torefutation.We may not be able to test Velikovsky's hypothesis asa whole, but we can certainly show thatcritical elements of thetheory do not work. It is interesting to note that mostpseudoscientists areinterested only in the tests that confirmtheir own theories and shrug off refutations as onlyaffectingpart of their theories and hence unimportant. However, just as amissile need not destroyevery part of a plane to destroy theplane itself, testing need not disprove every element of atheoryto demolish the theory as a whole. In many cases the testing of a scientific theory involvesattacking the theory from bothdirections at once; seekingevidence that supports the theory while at the same time lookingfortests that might refute the theory. We cannot divide theseprocesses neatly into the categories ofconfirmation orrefutation. A negative search for confirming evidence counts onthe side ofrefutation, whereas if the theory survives repeatedattempts to refute it we would count thatevidence asconfirmation. A good example of such a multi-faceted approach wasthe testing of thepolywater theory (Chapter 4). Some scientiststried to "confirm" the theory, by attempting tosynthesizepolywater in the laboratory, others tried to "refute" the theoryby finding experimentalflaws or alternative explanations for theevidence. The failure by the "confirming" scientists toobtainthe evidence they sought was actually the strongest evidence infavor of refutation. In thiscase refutation won out. Thereprobably is no single criterion for scientific confirmation, andwewould do better to think in terms of a battery of criteriathat all have their utility under differentcircumstances.These criteria include supporting evidence, failure of attemptsto refute, logicalconsistency, economy and simplicity, lack ofad hoc postulates, consistency with known facts,and others. There are a number of interesting paradoxes that seem tocall objective reality itself, or at leastour conceptions of it,into question. One of the most famous paradoxes is NelsonGoodman's"grue" paradox. An object is "grue" if it is greenbefore January 1, 2000 and blue afterward, and"bleen" if it isblue before January 1, 2000 and green afterward. The paradox isthis; the categoriesseem thoroughly artificial, and yet we couldjust as logically say a green object is "grue" beforeJanuary 1,2000 and "bleen" afterward, or a blue object is "bleen" beforeand "grue" afterward.What criterion do we use to separate theconcepts green and blue from "bleen" and "grue"? We donot havemuch time left to resolve this problem! We can find an identical version of this paradox a lotcloser to home than the year 2000. Atraffic light changes fromgreen to yellow to red and back to green. Could we not say thelight is"grellow" if it changes from green to yellow in aspecified interval, and so on? We could then justas easily saythe traffic light changes from "grellow" to "yed" to "reen" andthen back to "grellow". So long as we persist in believing that our conceptualcategories are merely abstract constructswith no intrinsicreality of their own, the "grue" paradox and similar paradoxeswill remain puzzles.Similarly, there will be no logical way ofdistinguishing "all crows are black" from "anything that isnotblack is not a crow". But conceptual categories are rooted inreality. In a very real way,paradoxes of this sort are a goodexample of a method of testing called reductio ad absurdum--reduction to the absurd. We assume for purposes of argument thatan idea is true, in this case theidea that conceptual categorieslike green and blue are purely arbitrary, and show thattheassumption leads to a contradiction and hence must be false.Objects do not spontaneouslychange color suddenly, so that it it is objectively more sensible to treat objects as having constantcolor, and refer to the exceptions as "changing color", rather than build the change into thedefinition of color. For cases where variability is as important as color, we do have conceptualcategories to reflect that fact: "striped","spotted", "variegated", "blinking", "alternating", and thelike. Similarly, it is an objective fact that there are far fewer crows than non-black objects, andthat crows are more similar to one another than non-black objects are. It therefore makesobjective sense to approach the confirmation problem from the vantage point"all crows are black"rather than the reverse. There are other cases where an approach from the other angle is better.Given the two propositions "any working used car is worth buying" and "any used car that doesnot work is not worth buying", most buyers will find it convenient to weed out the lemons ratherthan test all the working cars on the lot. There are many examples in science of problems that are best approached from one viewpointat some times and the other at other times. In statistics, if we want to find the probability of asingle event taking place in a single trial, the simplest approach is to calculate the probability ofthe event. But if wewant to know what the chances are that the event will take place at least onceduring a certain number of trials, the simplest method is to calculate the probability that the eventwill notoccur at all, and then subtract the result from 1. This approach is a lot simpler thanenumerating all the possible ways the event might occur. Generally the best approach is to attackthe problem from the position that involves the fewest complexities.