1Gdel, Kuhn, Popper, and Feyerabend*

Jonathan P. SeldinDepartment of Mathematics and Computer Science

University of Lethbridge4401 University Drive

Lethbridge, Alberta, Canadajonathan.seldin@uleth.ca

http: //www.cs.uleth.ca/~seldinJanuary 5, 2004

AbstractThomas Kuhn (Kuhn, 1962) has presented a view of

the history of science as a succession of paradigmswhich are not completely comparable with each other.Karl Popper (Popper, 1970) has attacked this view asbeing relativistic and denying that there is objectivescientific truth. Popper seems to be saying that in orderfor science to be objectively true, every two scientifictheories must be completely comparable. In taking thisposition, Popper is making a claim for a kind ofcompleteness that is ruled out for theories strong enoughto be interesting by Gdels Incompleteness Theoremand related results. In this paper, GdelsIncompleteness Theorem and related results will be usedto draw conclusions about this conflict between Kuhnand Popper. In particular, it will be argued that if Popperhad taken Gdels Incompleteness Theorem into account,he would have wound up with a position consistent withthat of Kuhn. Gdels Incompleteness Theorem will also

* This work was supported in part by a grant from the Natural Sciences and EngineeringResearch Council of Canada. A preliminary version of this paper was presented at a forum ofSigma Xi held at McGill University in Montreal on April 18, 1997. A later preliminaryversion was presented at a colloquium of the Department of Philosophy at the University ofLethbridge in January, 1999. I would like to thank the anonymous referees for their helpfulsuggestions.

2be used to argue that Feyerabend (Feyerabend, 1975)does not really have an argument that there is noobjective scientific method. The paper will close withsome remarks on what might really count as objectivescientific truth.

1. The dispute between Kuhn and Popper

Thomas Kuhn (Kuhn, 1962) has presented a view of the history of science thatdiffers from the previously commonly held view of science as a gradualaccumulation of knowledge. He presents the history of science as a successionof periods of normal science, each of which is determined by a paradigm.When for some reason the paradigm is no longer adequate for normal scienceto proceed in the usual way, there is a scientific revolution, and normalscience only resumes when a new paradigm is accepted by the scientificcommunity involved. Usually, this is a group of specialists, but sometimes itmay be a larger group than that.

One of the most controversial aspects of Kuhns ideas is his view thatsuccessive paradigms are not completely compatible. He says, for example,Like the choice between competing political institutions, that betweencompeting paradigms proves to be a choice between incompatible modes ofcommunity life. (Kuhn, 1962, p. 94, 1970 edition; all citations from Kuhn,1962, will be from the 1970 edition.) Later, he says, In so far as their onlyrecourse to that world is through what they see and do, we may want to saythat after a revolution scientists are responding to a different world. (Ibid., p.111)

As readers of this volume will be aware, Karl Popper is known for theview that science consists of the bold formation of theories as hypotheseswhich are then subjected to attempts to falsify them. This seems in manyways close to the ideas that Thomas Kuhn proposed. However, Karl Popperobjected to Kuhns views on the grounds that they represented relativism. In(Popper, 1970, p. 56), Popper says of Kuhns views:

...Kuhn suggests that the rationality of science presupposes theacceptance of a common framework. He suggests that rationalitydepends upon something like a common language and a common setof assumptions. He suggests that rational discussion, and rationalcriticism, is only possible if we have agreed on the fundamentals.

3This is a widely accepted and indeed a fashionable thesis: thethesis of relativism. And it is a logical thesis.

I regard this thesis as mistaken. (Emphasis in the original)

And he goes on to say later on the same page:

I should like just to indicate briefly why I am not a relativist: I dobelieve in absolute or objective truth, in Tarskis sense (althoughI am, of course, not an absolutist in the sense of thinking that I, oranybody else, has the truth in his pocket.) I do not doubt that this isone of the points on which we are most deeply divided; and it is alogical point. (Authors footnote omitted)

And again later on the same page:

The central point is that a critical discussion and a comparison ofthe various frameworks is always possible. It is just a dogmaadangerous dogmathat the different frameworks are like mutuallyuntranslatable languages.

Popper was not the only one who thought that Kuhn was relativist: Thereis nobody else than Thomas Kuhn who contributed more to the widespreadacceptance of cognitive relativism in the recent years. (Watanabe, 1991, p.25)

Popper did eventually accept that he had misinterpreted Kuhns views. Hesays of the view that comparison of different scientific theories requires anagreement on the general framework, a view with which he disagrees:

... I originally had in mind Thomas Kuhn ... However, as Kuhnpoints out, this interpretation was based on a misunderstanding ofhis views (see his Kuhn, 1970, and his Postscript 1969 to Kuhn,1970), and I am very ready to accept his correction. Nevertheless, Iregard the view here discussed as influential. (Popper, 1994,Footnote 19, p. 63)

There seem to be other disagreements between Kuhn and Popper. Kuhnconcludes that it may not be possible to say that as science progresses it isbringing scientists closer to the truth:

... scientific progress is not quite what we had taken it to be. ... Inthe sciences there need not be progress of another sort. We may, tobe more precise, have to relinquish the notion, explicit or implicit,

4that changes of paradigm carry scientists and those who learn fromthem closer and closer to the truth. (Kuhn, 1962, p. 170)

On the other hand, Popper says about scientific theories: The aim is to findtheories which, in the light of critical discussion, get nearer to the truth.(Popper, 1970, p. 57) An indication of what Popper means by this is asfollows:

I shall give here a somewhat unsystematic list of six types of case inwhich we should be inclined to say of a theory t1 that it issuperseded by t2 in the sense that t2 seemsas far as we knowtocorrespond better to the facts than t1 in some sense or other.

(1) t2 makes more precise assertions than t1, and these moreprecise assertions stand up to more precise tests.

(2) t2 takes account of, and explains, more facts than t1 (whichwill include, for example the above case that, other things beingequal, t2s assertions are more precise).

(3) t2 describes, or explains, the facts in more detail than t1.(4) t2 has passed tests which t1 has failed to pass.(5) t2 has suggested new experimental tests, not considered

before t2 was designed (and not suggested by t1, and perhaps noteven applicable to t1); and t2 has passed these tests.

(6) t2 has unified or connected various hitherto unrelated problems.If we reflect upon this list, then we can see that the contents of

the theories t1 and t2 play an important role in it. (It will beremembered that the logical content of a statement or theory a is theclass of all statements which follow logically from a, while I havedefined the empirical content of a as the class of all basic statementswhich contradict a.) For in our list of six cases, the empiricalcontent of theory t2 exceeds that of theory t1. (Popper, 1963, 4

th

edition 1972, Chapter 10, p. 232, authors footnote omitted)

This suggests that Popper is aiming to give a set of deductive rules to expressthe idea that one theory is nearer to the truth than another. The ideapresupposes the kind of relationship between theories that Kuhn claims doesnot always exist between paradigms, but is rather excluded by theincommensurability. Some of Poppers remarks seem to indicate that hethought that Kuhns notion of incommensurability implied the impossibility of

5comparison between paradigms. However, Kuhn, in!(Kuhn, 1983) explicitlydenied this, claiming that his idea of incommensurability was really localincommensurability:

Applied to the conceptual vocabulary deployed in and around ascientific theory, the term incommensurability functionsmetaphorically. The phrase no common measure becomes nocommon language. The claim that two theories areincommensurable is then the claim that there is no language, neutralor otherwise, into which both theories, conceived as sets ofsentences, can be translated without residue or loss. No more in itsmetaphorical than its literal form does incommensurability implyincomparability, and for much the same reason. Most of the termscommon to the two theories function the same way in both; theirmeanings, whatever those may be, are preserved; their translation issimply homophonic. Only for a small subgroup of (usuallyinterdefined) terms and for sentences containing them do problemsof translatability arise. The claim that two theories areincommensurable is more modest than many of its critics havesupposed.

I shall call this modest version of incommensurability localincommensurability. Insofar as incommensurability was a claimabout language, about meaning change, its local form is my originalversion. (Kuhn, 1983, pp. 670671)

I believe that if Popper had taken more account of Gdels IncompletenessTheorem, he would have found his position much closer to that of Kuhn thanhe realized. (Popper was certainly aware of Gdels Incompleteness Theorem;he refers to it, for example, in!(Popper, 1963, 4th edition 1972, Chapter 11, pp.269ff and Chapter 14, p. 310). But he seems to have missed this particularapplication of it.)

2. Gdels Incompleteness Theorem and the Kuhn-Popper Dispute

Gdels Incompleteness Theorem says that any consistent formal system thatincludes the elementary theory of natural numbers is necessarily incomplete.It is proved by mapping the formal objects of the formal system to the natural

6numbers using a one-to-one mapping so that some sentences about naturalnumbers become, via this mapping, sentences about the formal system itselfand then constructing a sentence of the formal system which, via this mapping,asserts its own unprovability. The condition that the formal system includesthe elementary theory of natural numbers is often expressed by saying that thesystem is strong enough to be interesting.

Gdels Incompleteness Theorem applies directly only to formal systems,that is to deductive theories that are so strictly formalized that it is possible todetermine mechanically whether or not a sequence of sentences constitutes avalid proof. But indirectly it applies to theories that are capable of thiscomplete formalization, even if they are not presented in this fully formalizedform. If one considers the process by which large parts of mathematics wereformalized in the late nineteenth and early twentieth centuries, one can see thatthe process which led to these formal systems was a matter of bringing hiddenassumptions into the open, a process which required the analysis of someconcepts that had not been previously analyzed. It would thus appear thatmost deductive theories could be formalized if somebody were willing to dothe necessary work.

Furthermore, there are a number of results related to GdelsIncompleteness Theorem. As an example, there is the undecidability of thehalting problem, which says that for an idealized computer with unlimitedmemory and time, there is no program which will take as input anotherprogram and input data and return the answer to whether or not the computerrunning second program with the given input data will stop after a finiteamount of time or continue forever in an infinite loop. There are a number ofother such examples. A consideration of all of these related results suggestsstrongly that even where Gdels Incompleteness Theorem itself does notapply directly, it would be a mistake to assume completeness.

To see how these results might apply to the dispute between Popper andKuhn, suppose that T1 and T2 are two scientific theories to be compared. Forexample, T2 might be a theory which has replaced T1 in what Kuhn calls ascientific revolution. Any comparison would have to be carried out in sometheory, T (which might be one of T1 or T2). The theory T will have to bestrong enough to include references to T1 and T2 and to draw conclusions fromthose references. Assuming that T1 and T 2 are strong enough to beinteresting, T will also have to be strong enough to be interesting. Thismeans that assuming completeness would be a mistake. (Even if we arethinking in terms of semantics [model theory], where the basic definitions doseem to imply completeness, the comparison is likely to involve deductions,and the definitions used in model theory are insufficient to allow a

7determination of the validity of all possible deductions. It follows that therecannot be a guarantee that it will be possible in T to obtain a completecomparison of all aspects of T1 and T2. This is why I claim that Kuhns notionof incommensurability, especially in the version of local incommensurabilityof!(Kuhn, 1983) should be considered an expected consequence of GdelsIncompleteness Theorem.

(It should be noted that Kuhns claims for [local] incommensurability werebased on historical considerations.)

A similar argument will, I believe, show why we cannot talk about scienceprogressing to theories that come ever closer to the truth about the world. Tosay that one theory, say T1, is closer to the truth than another, say T2, requiresa theory, T, in which to carry out this reasoning. Since we are dealing withscientific theories, it is reasonable to suppose that T would also be strongenough to be interesting, and so it would be a mistake to assumecompleteness for it. This would imply that it would be a mistake to assumethat there is a complete definition of truth, and therefore that it is not alwayspossible to say that newer theories are closer to the truth than older ones.Thus, we should consider it a consequence of Gdels IncompletenessTheorem that we cannot always talk about scientific progress getting closer tothe truth.

It thus appears that Popper was originally more mistaken about the Kuhnsviews being relativist and subjective than he later realized. This seemsespecially likely since Kuhn implies in (Kuhn, 1962) that every paradigm willeventually be found inadequate: surely this implies that there is some sort ofobjective reality against which these paradigms are being tested. Theimplication of the incommensurability of paradigms is that our ability to knowand talk about this objective reality is limited, not that this objective realitydoes not exist. In fact, Kuhn himself points out (Kuhn, 1970) that except forthis one difference, (i.e., Poppers claim that the incomparability of paradigmsimplies relativity) his views are very similar to those of Karl Popper, and whenhe and Popper are analyzing the same historical episode in science theiranalyses are usually very much alike. Perhaps we can say that, roughly,

Popper + Gdel = Kuhn.

3. Gdels Incompleteness Theorem and PaulFeyerabend

8Gdels Incompleteness Theorem also has something to tell us about the viewsexpressed by Paul Feyerabend in (Feyerabend, 1975). Feyerabend thinks thathe is arguing that there is no such thing as a scientific method. But when hisarguments are stripped of rhetoric, what is left is a catalog of instances inwhich various descriptions of scientific methodology fail to correspond withthe way science has actually progressed. Feyerabend argues correctly that thisshows that the descriptions of scientific methodology of which he is speakingdo not fit science as it has actually occurred, and he goes on to suggest that nocomplete and consistent description of scientific methodology is possible.

Feyerabend then argues that the absence of a sound and completedescription of the scientific method means that there is no such method. But,again, this argument is based on the same kind of claim of completeness asPoppers. In fact, Gdels Incompleteness Theorem and related results shouldlead us to expect that no description of the scientific method can be both soundand complete. Feyerabend may have produced some evidence that we cannever soundly and completely describe the scientific method, but he has noevidence whatever that there is no such method. (If there can be no sound andcomplete description of a scientific method, perhaps method is the wrongword: perhaps we should be speaking of a scientific approach.) The evidenceFeyerabend presents in!(Feyerabend, 1975) simply does not support his thesisthat anything goes in science.

(This argument applies only to what Feyerabend says in!(Feyerabend,1975), and not to any of his other works.)

4. The objective truth of science

At this point some readers may wonder what we can say to those who reallydeny the value or truth of science. I think there is an answer to them, but it isnot the same kind of answer that Popper seems to have been seeking. Theanswer I have in mind does not try to prove the truth of every assertion of theaccepted scientific paradigms. Instead, it concentrates on those questions thatare of interest to legislative bodies and courts involving matters of publicpolicy or the value of scientific testimony. Legislative bodies and courts areinterested in evidence for their conclusions on these issues. And I think thatthe refereeing procedure that is standard in the best scientific journals is theprocedure that generates the best evidence for or against the assertionsinvolved. (Ayala and Black, 1993) argue from the position of Popper that thecourts should pay attention to refereed scientific journals. This idea gives us abasis for saying that there is an objective basis for these assertions. Of course,at a later time, the evidence may change with regard to any particular question,

9but at the time the question is being considered, a consensus of the literatureby specialists on the matter in question in refereed scientific journals, if such aconsensus exists, is the best evidence available on that question. It is possiblethat at a given time there is no consensus of the literature by specialists on thematter in refereed scientific journals; in this case, the legislatures and courtswould have to conclude that there is insufficient scientific evidence to settlethe matter at hand scientifically. This is a way we can refer to science asembodying objective truth.

This idea seems to have something in common with the Poppersdiscussion of the establishment of basic statements in (Popper, 1959, 8thimpression 1975, Chapter 5), especially the last two pages on jury verdicts.What is common is the idea that the determination of truth involved is notabsolute and unchanging, but depends on the (scientific or legal) context.However, unlike Popper, I am not assuming any characterization of the logicalforms of questions which come before legislatures and juries, as Popper seemsto do in (Popper, 1959, 8th impression 1975, Chapter 5).

There is one important qualification we must make here: it is importantthat this refereeing process be conducted in a way consistent with the practicesof the scientific specialty in question and is free from external influences, suchas economic interests. I think most of us know from our own personalexperience that if we have a reason to want something to be the case, it is easyto convince ourselves that it is the case even if the evidence does not supportthat conclusion. This means that each of us must constantly be on guardagainst making this kind of mistake in our own individual work. Economicinterest can cause a very powerful desire for something to be the case even if itis not. This suggests that in areas of science in which there are importanteconomic interests that are affected by the results of scientific research, specialefforts are needed to protect the process of scientific research and refereeingfrom the effects of those economic interests. For a troubling example showinghow medical research financed by a pharmaceutical company may be affectedby economic interests, see (Canadian Association of University Teachers,2001).

Of course, there are questions that arise in science that are neverconsidered by courts or legislative bodies: an example is the question thatarose in the early days of the special theory of relativity, when there was analternative theory due to H. A. Lorentz based on classical physics which madeexactly the same predictions as Einsteins theory. Einsteins arguments for hisapproach did not constitute the same kind of evidence from the point of viewof courts and legislative bodies as, say, the scientific evidence that AIDS iscaused by the HIV virus. But then these arguments of Einstein were on asubject unlikely to get before a court or legislative body. It is at the point at

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which science touches public policy or trials through courts and legislativebodies that we are best able to talk about the objective reality of science. (I amthus dividing science in something of the way Hilbert divided mathematicsinto its real and ideal parts.)

References

Francisco!J. Ayala and Bert Black, 1993, Science and the courts, AmericanScientist, Vol 81, 230239.

Canadian Association of University Teachers, 2001, Report of the Committeeof Inquiry on the Case Involving Dr. Nancy Olivieri, the Hospital for SickChildren, the University of Toronto, and Apotex Inc, available from theirw e b s i t e a thttp://www.caut.ca/english/issues/Acadfreedom/olivieri.asp.

Paul Feyerabend. 1975, Against Method: Outline of an Anarchistic Theory ofKnowledge, London and Atlantic Highlands: NLB and Humanities Press.

Thomas!S. Kuhn, 1962, The Structure of Scientific Revolutions, Secondedition, Chicago: University of Chicago Press, 1970.

Thomas!S. Kuhn, 1970, Reflections on my critics, in Criticism and theGrowth of Knowledge: Proceedings of the International Colloquium inthe Philosophy of Science, London, 1965, volume!4, Cambridge:Cambridge University Press, 231278, Cambridge.

Thomas!S. Kuhn, 1983, Commensurability, comparability, communicability,in Peter!D. Asquith and Thomas Nickles, editors, PSA 1982: Proceedingsof the 1982 biennial meeting of the Philosophy of Science Association,volume!2, East Lansing, MI: Philosophy of Science Association,669688

Karl!R. Popper, 1934 , Logik der Forschung , Vienna: Julius Springer.English translation in (Popper, 1959).

Karl!R. Popper, 1959, The Logic of Scientific Discovery, 8th Impression,London: Hutchinson 1975.

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Karl!R. Popper, 1963, Conjectures and Refutations: The Growth of ScientificKnowledge, Fourth edition (revised), London and Henley: Routledge andKegan Paul 1972.

Karl!R. Popper, 1970, Normal science and its dangers, in Criticism and theGrowth of Knowledge: Proceedings of the International Colloquium inthe Philosophy of Science, London, 1965, volume!4, Cambridge:Cambridge University Press, 5158.

Karl!R. Popper, 1994, The myth of the framework, in Karl R. Popper (M.!A.Notturno, editor), The Myth of the Framework: In Defense of Science andRationality, London and New York: Routledge, Chapter 2, 3364.

Satosi Watanabe, 1991, The foundations of cognitive relativity, Annals of theJapan Association for the Philosophy of Science, Vol 8, 2348.