113

Dialogue on Scientific Theories and Fuzziness–Fuzzy-Philosophical Investigations

Rudolf Seising

113.1 Preliminary Remark

The historiography of modern science comprises some famous dialogues, startingwith thirtysix Socratic dialogues ascribed to Plato. Then there is Galileo’s famousDialogo sopra i due massimi sistemi del mondo that is the model for the fictitiousdebate on scientific theories and their relationships, that is presented in this chapter,and this subject of Philosophical Investigations reminds the reader of the posthu-mous edited book by Ludwig Wittgenstein.

Galileo Galilei (1564-1642) published his Dialogo – Dialogue Concerning theTwo Chief World Systems in 1632 [8]. It was the first book on science written inItalian (instead of Latin) and therein the author compared the new world systemof Nicolaus Copernicus (1473-1643) with the traditional one of Claudius Ptolemy(90-168).

Three protagonists debate on this subject, named Sagredo, Salviati, and Simplicio.While Simplicio argues for the Ptolomaic world system that was also the view ofAristotle, Salviati argues for the Copernican position, and Sagredo is an intelligentdisputant who starts with a neutral opinion. The debate takes place at the palace ofSagredo at the Canale Grande in Venice (see Figure 113.1, left).

About three and a half centuries later the three savants appeared again in the bookAre Quanta Real? [11] published by the Swiss physicist Josef Maria Jauch (1914-1974) and translated into English in 1989.1 In this book the three fight on scientificknowledge and nature perception against the background of the controversy on thefoundations of quantum mechanics.

Today, almost half a century later, in the times of coexistent real and virtualworlds, avatars for three scholars enter into a dialogue. These scholars are well-known for their work in the 20th century philosophy of science, and their avatarsmeet to debate the historical development of scientific theories and their fuzziness!Their Latin names are: Carolus, Thomas and Ludovicus.2

1 The German-written original Die Wirklichkeit der Quanten appeared already in 1973, [10].2 The word avatar means in Hinduism, a “manifestation” of a deity to earth. However, today

“avatar” denotes a computer-graphical representation in games or virtual worlds.

R. Seising et al. (Eds.): On Fuzziness: Volume 2, STUDFUZZ 299, pp. 813–829.DOI: 10.1007/978-3-642-35644-5_113 © Springer-Verlag Berlin Heidelberg 2013

814 113 Dialogue on Scientific Theories and Fuzziness

(a) (b)

Fig. 113.1. (a): Frontpage of Galilei’s Dialogo with copper engraving by Stefano della Bella(1610-1664); from left to right: Aristotle, Ptolemy and Copernicus; (b) The Campanile on thecampus of the University of California at Berkeley, CA

113.2 Introduction

Last year, Lotfi A. Zadeh gave me the permission to search, scan and photocopy thematerials in his office at the University of California at Berkeley and in his hometo continue my historical research on the genesis and development of the theory ofFuzzy Sets and Systems. One late evening, when I came home to the appartmentthat I have rented for my stay in Berkeley, I was terribly tired from work and I laiddown on the sofa. On TV there was an episode of the series Star Trek – The NextGeneration. I remember watching the English actor Patrick Stewart as the starshipEnterprise’s Captain Jean-Luc Picard. I know exactly that I gave some thought to thesimilarity of Steward and Lotfi Zadeh – may be because they both have bold heads. Itwas one of the Star Trek sequels where the actors used the so-called “holodeck”. StarTrek’s holodeck is a closed room that can replicate a wide variety of environmentsin which various people, objects, and places of past, present and future time can besimulated as a virtual reality. Usually the protagonists of the series use the holodecktechnology for their research and training, but also for recreation and entertaining.The crew members like to interact with the program and its characters. When I wasyoung, I was very interested in the episodes where the Star Trek actors met avatarsof historical notabilities. However, as all too often, this evening I fell asleep and thefollowing dialogue happened in my dream.

113.3 Dialogue 815

113.3 Dialogue

Carolus: Why are we here?

Ludovicus: Damned! – We agreed on not discussing such metaphysical questions:Why are we here? – What is there? – What is it like? – What is the meaning of life?– Why is there something? – Why isn’t there nothing? – Does God exist? – It wasmy deep hope that we conform with never debating senseless questions! – Whereofone cannot speak, thereof one must be silent.3

Carolus: Calm down! You are too sensitive to the sound of my words. I thought thatafter the many years you passed away you would not behave that Viennese! I justwanted to know: What place is this?

Thomas: This is the campus of Berkeley and we are sitting in front of the campanile,also called Sather Tower, UC Berkeley’s most recognizable symbol. It’s a bell andclock tower, completed in 1914. We are also close to the buildings of the philosophydepartment and the history department. Here, I became professor of the history ofscience in 1961. However, before that I studied in Harvard and later I got a profes-sorship of philosophy and history of science in Princeton. Did I ever told you that Iwas teaching in both of the departments ...

Ludovicus: Yes Thomas, you told us this a good many times! – Carolus, for a mo-ment I was thinking that we have arrived in Venice.

Carolus: In Venice is the Campanile di San Marco that is much older and this onehere has resemblance. Also when I was in England I saw the Joseph ChamberlainMemorial Clock Tower at the University of Birmingham, and there is also the Torredel Mangia in Siena, Italy. ... You’d think that all unversity campus have a tower butif you find one without tower, this hypothesis would be falsified ....

Ludovicus: Stop it, Carolus! You madden me. I was already looking for a poker! Itis not only the problem to know whether there is a tower or not. It is also a problemto decide what is a tower! – There exist so many buildings that could be consideredas towers in and we can imagine so many more towers ... I mean the Tower of Babel,the Tower of London, church spires, watch towers, siege towers, television towers,the Eiffel Tower, the Tower(s) of Hanoi and so on. What is common to them all?Don’t say: “There must be something common, or they would not be called ‘towers’” but look and see whether there is anything common to all. For if you look at themyou will not see something that is common to all, but similarities, relationships, anda whole series of them at that. To repeat: don’t think, but look! Look for exampleat look-outs, with their multifarious relationships. Now pass to church spires; hereyou find many correspondences with the first group, but many common features drop

3 This is a quotation by Wittgenstein; it is the last sentence in his Tractatus [16].

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out, and others appear. When we pass next to castle-towers, much that is commonis retained, but much is lost. Are they all “stonelike”? Compare television towerswith siege towers. Or is there always a viewing balcony, or a restaurant deck? Thinkof the Tower of Pisa. In watch towers there is a ladder; but for television towersthis feature has disappeared. Look at the wooden siege towers and a water towermade from concrete or stone. Think now of mine head towers; here is the elementof height, but how many other characteristic features have disappeared! sometimessimilarities of detail. And we can go through the many, many other groups of towersin the same way; can see how similarities crop up and disappear. And the result ofthis examination is: we see a complicated network of similarities overlapping andcrisscrossing: sometimes overall similarities.4

Carolus: How times flies, Ludovicus! I remember the days of neo-positivistic ideasin Vienna, in Berlin and later in Cambridge and Chicago. For me it is still one ofthe most remarkable developments in history of philosophy that you turned diamet-rically opposed your views on language. Nowadays it is obvious for us that not allphenomena could be expressed in terms of language. By the way, this was the reasonfor the break down of the Empiricism à la Carnap. Instead of analyis of the languageof science philosophers turned to the analysis of its theories and methodologies.

Ludovicus: Carolus, even your Logic of Scientific Discovery was published alreadyin 1934 in German it became not influential before the English edition appeared in1959.5 Because I passed away in 1951, I may be missed some information. Whatwas the big success of your book?

Carolus: My work is one of the milestones in History of Philosophy of Science andI became old enough to enjoy public famousness. The Logic of Scientific Discoveryheralded a shift in differentiating between science and non-science, metaphysics orpseudo-science. In the “pre-Popper-times” philosophers tried to fix this demarcationin scientific language ...

Thomas: Carolus, I hate it when you maintain this with such an arrogant manner.But in the merits you are right. However, Ludovicus’ Tractatus played the role of aBible for this view on philosophy of science, that was called “Logical Empiricism”or “Logical Positivism” or “Neopositivism” that was significantly involved by themembers of the Vienna Circle ...

Ludovicus: ... but that is a presentation of my previous philosophical view on lan-guage, logic and the world! As you mentioned already, I thought extremely dif-ferent when I came back to philosophy in the 1930s. Today philosophers distin-guish sharply between Wittgenstein I, the author of the Tractatus and Wittgenstein II,

4 As the reader will notice, this is a slightly variation of parts of §66 in Wittgenstein’s [17].5 See: [15]. Popper published the German book in 1934. He rewrote it in English, repub-

lished in 1959.

113.3 Dialogue 817

the author of the Philosophical Investigations, by the way, the latter was publishedin 1953, after my death, and I would have never published it this way! But that ’s lifeand that’s death!

Carolus: In my metatheory named “Critical Rationalism” the decision of what isscience and what is not science is related to theories and methods in these fields andnot in the precision of the terms of language. I created this alternative concept tothat of the Vienna Circle and the other Logical Empiricists who tried to analyze theconstitution or the structure of scientific theories by using modern logic, particularlyRudolf Carnap wrote in 1928 The Logical Structure of the World.6 For Carnap andmany others theories are sets of propositions and these propositions are built fromdata via induction.

I say: On the contrary! For Critical Rationalists scientific theories are not builtfrom data by induction! There is no logical way from data to theory! Theories arehypotheses or conjectures and scientists test these hypotheses in experiments withintend to refute them. Even a great number of positive test results can not confirm ascientific theory, but if there is only one outcome that is negative, this one counterex-ample shows that the theory is falsified. Look at this squirrel!

Carolus points to one of the many squirrels on the campus (see Figure 113.2,leftside).

There are so many squirrels on this campus and they all look different, they all liveon nuts and similar things but I know that some of them also consume meat and per-haps there are even “flying squirrels”. Well, that is a nice example for a little theoryor hypothesis. I say: There is no flying squirrel. It is not possibel to verify this theorybecause to this end we would have to investigate all existing squirrels and we haveto check wether they can fly or not, and may be they can but they don’t do duringthe time that we observe them. However, we can try to falsify this hypothesis andif we find one solely flying squirrel, then the hypothesis is refuted. To cut a longstory short: In Critical Rationalism the falsifiability is the criterion of demarcationbetween what is scientific and what is not.

Thomas: There is another reason why Logical Empirism has nowadays very fewsupporters. It seems very clear that we can not reduce all our knowledge to sensualdata. Therefore, we need so-called theoretical elements in addition to the empiricalones. These additional elements are being understood only in the context of a theory.They are more abstract, they are more distant from our perceptions than observa-tional terms. To factor these elements in Logical Empiricism Carnap and Carl G.Hempel introduced in the 1950s the so-called “double language model”.7 Whereasobservational and therefore non-theoretical terms are elements of the observationlanguage, theoretical terms are elements of the theoretical language.

6 Carnap published [4] in German in 1928, the English translation appeared in 1967.7 See [6, 9].

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Ludovicus: In former times we would have named this metaphysics!

Thomas: Later, also Willard van Orman Quine – we both were Fellows at Harvard– criticized the empiricist differentiation between “analytical” and “synthetical”. Inshort, the Logical Empiricism collapsed. If you want to name this metaphysics, thenI say: Without metaphysics no knowledge is possible!

Ludovicus: Let’s take Carolus’ example of “flying squirrels”: They exist! “Flyingsquirrels, scientifically known as Pteromyini or Petauristini, are a tribe of 44 speciesof squirrels (family Sciuridae).” Of course, the name is a theoretical term becauseWikipedia says: “Flying squirrels are not capable of powered flight like birds or bats;instead, they glide between trees. They are capable of obtaining lift within the courseof these flights, with flights recorded to 90 meters (295 ft).”8 Roughly spoken, wecan consider the term “flying” as an element of the theoretical language of the theoryof squirrels and its meaning is not our usual meaning of “flying”.

However, in Biology these flying squirrels belong to the family of squirrels, andthere are “tree squirrels, ground squirrels, chipmunks, marmots (including wood-chucks), flying squirrels, and prairie dogs.” 9 and I can think of no better expressionto characterize these similarities than “family resemblances”; for the various resem-blances between members of a family: build, features, colour of eyes, gait, tempera-ment, etc. etc. overlap and crisscross in the same way. And I shall say: “squirrels”form a family.10

(a) (b)

Fig. 113.2. (a): Squirrel on the Berkeley campus; (b) Structures of a theory T in the metas-tructuralist view of scientific theories

8 See: http://en.wikipedia.org/wiki/Flying_squirrel9 See for more details http://en.wikipedia.org/wiki/Squirrel

10 As the reader will notice, this is a slightly variation of parts of §67 in Wittgenstein’s [17].

113.3 Dialogue 819

Thomas: I am so sorry that we never met in lifetime, Ludovicus. Your philosophyof language games and familiy resemblance is exciting and I wonder why we don’ttry to combine it with my ideas on The Structure of Scientific Revolutions, that Ipublished here at Berkeley, 50 years ago.11

In this book I critisized Carolus’ view on theory dynamics in science. As I couldshow in many cases of my historical research work, no replacement of a theory byanother happened because of falsification.12 In my view theory change in science isnot a rational process and therefore we need assistance from sociology and psychol-ogy to explain the paths of science through history.

My historical research convinced me that there were periods of “prescience” thatlack any theory or paradigm, then there were periods of “normal science” withparadigm monopole and finally there were times of crisis that triggered “scientificrevolutions”. Most scientists in most periods have been “normal scientists”. Theyare involved with puzzle-solving. Only if there were many anomalies in oppositionto the current paradigm a crisis appeared and a scientific revolution could happen.However, we have discussed this very often in life, Carolus. Do you remember theconference organized by Imre Lakatos at Bedford College in London, 1965?13

Carolus: I conceded already in my lifetime: I did not concentrate my attention onthe periods of normal sciences. You can say that this was a failure! However, myinterests was the change of theories in the field of science. Yes, normal science existsand there are normal scientists but this is the bad thing! To my mind scientists haveto critically analyze their theories at any time and therefore I did not differentiatevarious periods. To make this clear: I still think that it is a scandalon that there isnormal science! In the early times, the adventure of science startet with the methodof trial and error and, as I said in my Logic of Scientific Discovery, from then untiltoday, the methodology of scientific progress is falsifiability.

To say a word to your “paradigm shifts”: I still think that the idea on rivalparadigms in times of a scientific crisis has an irrational element.

Thomas: In this point I agree with you, Carolus, it was straightforward my intentionto emphasize that these processes can not be explained in terms of pure rationality!In the way I see it, a paradigm shift can be activated by sociological, economical,political or other reasons and there is no logical determinism in theory dynamics.I have to emphasize that the theory in a new paradigm may use concepts that aretotaly different from the concept of the former theory in the old paradigm. Even ifthey have got the same or a similar name, the new concepts may have a very differentmeaning. It is also not necessary that a concept in the new paradigm has an analogonin the old paradigm!

11 See: [13]. Kuhn exemplified later that the idea to this book went back to 1947. In thatyear as a graduate student at Harvard University he was asked to teach a science class forhumanities undergraduates on historical case studies.

12 See for these examples Kuhn’s book [13, 12].13 For details see: [7].

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Carolus: This means that if two paradigms are incommensurable then we have nochance to compare them directly. Do you really want to join your irrational systemwith that “Wittgenstein II language philosophy” that is not strong rational as well?

Thomas: Yes, I think that it could be a very fruitful combination in philosophy andhistory of science!

Ludovicus: I appreciate very much this suggestion to step forward in our discussioninstead of backwards movements! May be you can explain what do you have in mindby this “combination”?

Thomas: Shortly after your death a second trend in obtaining systematic rational re-constructions of empirical theories was established, the so-called Suppes approach ...

Carolus: Patrick Suppes is a philosopher and statistician living in Stanford, veryclose to Berkeley.

Thomas: That’s him. – Well, to get a rational reconstruction of the theory in questionthe first step consists of an axiomatization that seeks to determine the mathematicalstructure of the theory. Now, the difference between the old view – we call it Car-nap approach – and the new Suppes approach can be found in the manner in whichthis task is performed. As you can imagine, Carnap was firmly convinced that onlyformal languages can provide suitable tools to achieve the desired precision. Con-sequently, the Carnap approach claims that a theory has to be axiomatized within aformal language.

Ludovicus: That’s right! – Could you please explain any more about this other view,the Suppes approach?

Thomas: In this approach one is able to axiomatize physical theories in a preciseway without recourse to formal languages.

Ludovicus: Eh? – I see, this turns toward my late philosophy ...

Thomas: No rush, Ludovicus! The Suppes approach – later the fully developedapproach was called “Metatheoretical Structuralims” – uses informal logic and settheory. Already in the 1950s Suppes proposed to include the axiomatization of em-pirical theories of science in the metamathematical program of the French group“Bourbaki” 14 and in the 1970s the physicist Joseph D. Sneed developed informalsemantics meant to consider not only mathematical aspects, but also application

14 “Nicolas Bourbaki” is the pseudonym of a group of mathematicians in France in the 20th-century. Since 1935 the group presented their mathematical research work in a series ofvolumes. It was the aim of the group to establish a founding of all mathematics on settheory. [3]

113.3 Dialogue 821

subjects of scientific theories in this framework, based on this method15 Sneed pre-sented this view as stating that all empirical claims of physical theories have the form“x is an S” where “is an S” is a set-theoretical predicate. Every physical system thatfulfills this predicate is called a model of the theory. Let’s take again our exampleof towers: “x is a tower ” means that every thing that fulfills the predicate “tower”is a model of our little tower theory. Thus, there is the set M(Towers) of all towermodels!

Carolus: What’s the use of it?

Ludovicus: I think I got the picture! The Carnap approach and also the other ap-proaches of Logical Empiricism regarded a theory as being a system of logical propo-sitions i.e. linguistic formations. In contrast, this metastructuralist view of Sneedconsiders a theory as an object that comprises mathematical structures.

Thomas: That’s it, Ludovicus. To make it clear, let’s get back to our little towerexample! Wikipedia explains: “A tower is a tall [architectural] structure, usuallytaller than it is wide, often by a significant margin.”16

Now, we follow the framework of the metastructuralist approach to reconstruct atheory of towers – however, this will be a very simple theory and we name it “Wiki-Theory of Towers”, WTT for short. First we have a set of all architectural structuresx that have a significant margin. We call this set Mp(WT T ) the potential models ofour “Wiki-Theory of Towers” WT T .

We recall the condition in Wikipedia’s sentence: the architectural structure has tobe taller that it is wide to be a tower. We can phrase this condition as an axiom ofour WT T : For this we have to indicate two magnitudes: the height H(x) and widthW (x). To formulate the W TT ’s axiom we say H(x)>>W (x).

Now, all potential models of WT T (all x ∈ Mp(W TT )) which in addition fulfillthis axiom are models of WT T . Therefore, they build the set M(W T T )

Carolus: That’s a nice set-theoretical finger exercise but can this approach pictureall wherewithal to present modern science studies? Can it model the relationshipsbetween scientific theories, eg. specialization, generalization, reduction, and of ut-most importance, the change and replacement of theories, the dynamics that was oursubject in the last half hour?

Thomas: This can be done by set theory. Metastructuralists use thereto the conceptsof set theoretical relations. – A specialization of a theory is represented by a morespecial axiom. That’s food for thought: From our Wikipedia theory of towers, WT T ,we could obtain a more special theory of towers by a special axiom, e.g. the theory

15 [14].16 Not to be confused with the structures of the metatheoretical structuralism we completed

Wikipedia’s definition by the adjective “architectural”; seehttp://en.wikipedia.org/wiki/Tower

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of wooden towers, the theory of university towers or the theory of towers that aretwo times hight as wide. Hence, we obtain a specialization of a theory by fulfillmentof this additional axiom. In my last example that would be H(x) = 2×W(x).

Therefore, the set of models of the more special theory, let’s name it WT TH=2×W ,are a subset of the models of the general theory WTT , i.e. we have the set rela-tion M(WT TH=2×W )⊆M(W T T ) for this specialization relationship between the twotheories.

Carolus: I wonder – and who’d have thunk that I would take the part of an Empirist– where is the connection to empirical phenomena in this view? Where are the sci-entists observation terms?

Thomas: You are right, Carolus, besides the layer of theoretical structures therehas to be another one that comprises the empirical data and phenomena. Thoughscientists establish laws and they introduce empirical theories that say that the lawshold for the data, they also observe real systems or phenomena and they measuredata. That is to say: To study systems or phenomena in reality, scientists connectthem with a theoretical structure. To this end they give the real systems a structurethemselves. Woflgang Balzer, one of the today’s metastructuralists wrote in thisregard: “How to do that is not clear! — This is one of the central problems in thephilosophy of science. [...] The problem is that we create a connection between realsystems and theoretical structures. We assume that this can be done. Without thisassumption it is senseless to talk about empirical science.”17

In Metastructuralism the connection between non-theoretical and theoretical struc-tures of a theory T , i.e. the potential models in Mp(T ) and the models in M(T ) arerepresented by another set theoretical relation, the so-called Theoretization. What isimportant is that in this view theoretical terms are theoretical relative to a theory T ,i.e. a concept is not theoretical at all but it is T -theoretical to the respective theory T .

Let me try to explain this in terms of our example, the theory of towers. Let’ssay hight H(x) and width W (x) are theoretical concepts relative to our theory WT T .That means that hight H(x) and width have got their meaning by the theory WT T ,they are WTT -theoretical concepts.

If we remove all theoretical terms of a theory T in its potential models Mp(T ), thenwe get structures in a T-non-theoretical layer; we call these structures the “partial po-tential models” of theory T and we name their set Mpp(T ). In our example, withoutthe WTT -theoretical concepts of hight H(x) and width W (x) we obtain Mpp(WT T ).These are obviously all architectural buildings.

Finally, every empirical theory T has a class I of intended application systemsthat is a subset of all partial potential models in Mpp(T ). E.g., for the tower exampleWikipedia says that all “architectural structures that have a significant margin”. Itry to draw the whole metastructuralist conceptualization on the ground. The setsMp(T ) and M(T ) and the sets Mpp(T ) and I are located in different “layers”:

17 The sentence that is quoted here is in the German book [1], the translation into English isby the author.

113.3 Dialogue 823

Thomas draws Figure 113.3 on the ground.

Fig. 113.3. Structures of the intended application systems of theory T , its theoretical structuresand of their Theoretization T ′

Mpp(T ) and I are structures in the layer of empirical concepts, whereas Mp(T ) andM(T ) are structures in a theoretical layer of this schema. And with a spotted linesI indicate the connection between the two layers, the “Theoretization” between thetheories T and T ′. Indeed, this is a set-theoretical relation for it holds: T ′ is a theo-retization of T if and only if Mpp(T ′)⊆ M(T ).

Carolus: Now, I am very interested in your answer to my next question: Have thesemetastructuralists also set-theoretical tools to model the change of theories, say, fromT old to T new, as the change from Ptolemy’s geocentric universe to Copernicus’ helio-centric model or from Newtonian Mechanics to Einstein’s Special Relativity Theory?Can they express these scientific revolutions by set theory?

Thomas: Well, Carolus, they try to do this. They defined an intertheoretical relationthat is called “reduction” to reconstruct these kind of theory change. Let me seewhether I can sketch this as well!

Thomas draws Figure 113.4 on the ground.

Thomas: If you have two theories, say Told and Tnew, they say that Told reduces Tnew

if the following conditions are fulfilled:

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1. ρ ⊆ Mp(Told)×Mp(Tnew)2. For all x and x′: if < x,x′ >∈ ρ and x′ ∈ M(Tnew), then x′ ∈ M(Told).

Ludovicus: Stop! Please stop! This discussion became to mathematical to my mind!– Thomas, it was your intention to find a combination of your ideas in history of sci-ence and my philosophical system of family resemblances. But what you discussedin the last minutes and what you showed in your drawings has nothing to do withmy philosophy. I doubt that this kind of set theoretical relation can represent the-ory changes like scientific revolutions. As you both agreed some time ago, thesescientific revolutions that you, Carolus, called “theory replacement” and that you,Thomas, explained as a paradigm shift, are not pure rational changes. Between theold and the new theory there is no one-to-one-relation, therefore we have to respectsome unsharpness in these transformations that can not be represented by hard math!

Fig. 113.4. The relation of reduction between theories T old and T new

Thomas: You are right, Ludovicus, and I wanted to add that to represent changingsof theories or paradigms instead of this reduction relation very often a so-called ap-proximative reduction is used.

Carolus: Oh, I see. With this “approximative reduction” they can try to reconstructparadigm shifts as the one from Newtonian Mechanics to Quantum Mechanics, or tothe Special Relativity Theory, isn’t it? How does this look like?

Thomas: Oh, there are different proposals. Some metastructuralists use the conceptof converging series of models of a theory, others favored to establish topologicalspaces of models ...

Ludovicus: Stop it, please! – Again I have to say that this sounds to high-mathematically in my mind! I doubt that all these approaches that base on our pre-cise system of mathematics will result in a success of our philosophical problems. Iam looking for a tool that is appropriate to model language games!

113.3 Dialogue 825

At this point in my dream I got in the middle of the dialogue. Hitherto I was just akind of a neutral bystander but now I could not keep still. – As is usually the casethe holodeck would not run a program without interaction of the user. Interestingly,in that dream I borrowed some phrases from Lotfi Zadeh.

I: Please accept my apologies. Excuse my interrupting. I overheared you interestingdebate. 50 years ago, in the year of the publication of your Structure of Scientific Rev-olutions, Thomas, a professor of Electrical Engineerig at this university in Berkeleywrote already on “the gap that reflects the fundamental inadequacy of the conven-tional mathematics — the mathematics of precisely-defined points, functions, sets,probability measures, etc. -– for coping with the analysis of biological systems.” Inthat paper he said that “we need a radically different kind of mathematics, the mathe-matics of fuzzy or cloudy quantities which are not describable in terms of probabilitydistributions.”18 Three years later he established a new mathematical theory19. Heintroduced “Fuzzy Sets” — as classes or sets that ‘“are not classes or sets in the usualsense of these terms, since they do not dichotomize all objects into those that belongto the class and those that do not”. In fuzzy sets ‘“there may be a continuous infinityof grades of membership, with the grade of membership of an object x in a fuzzyset A represented by a number fA(x) in the interval [0,1].”20. Please do not hesitateto disagree with me: I think that Fuzzy Sets are a suitable formalism to solve yourproblems.

Thomas: Who is this scientist at this university? It is a pity that I never met him!

I: The name is Zadeh, Lotfi Asker Zadeh, better known as Lotfi A. Zadeh! His of-fice on this campus is very close in Soda Hall. Having graduated with a Bachelorof Science in electrical engineering from the University of Tehran in 1942 and af-ter working for a year as a technical contractor with the United States army forcesin Iran, Zadeh came to the US in 1944. He applied to the Massachusetts Instituteof Technology (MIT) in Cambridge, Massachusetts, and was accepted to continuehis studies. For the thesis he completed with Robert M. Fano, he was awarded thedegree of Master of Science in 1946. Then, he did move to New York, where heobtained a position at Columbia University as an instructor. In 1959 he had becomea professor at Berkeley and 1963/64 he was chairman of the department of electri-cal engineering. Prior to the publication of his first paper on fuzzy sets in 1965,he was concerned with systems analysis, decision analysis and information systems.For his scientific work he received among many others the IEEE Richard W. Ham-ming Medal, the IEEE Medal of Honor, the ASME Rufus Oldenburger Medal, theB. Bolzano Medal, the Kampe de Feriet Medal, the Grigore Moisil Prize, the HondaPrize, the Okawa Prize, the IEEE Millennium Medal, the ACM 2001 Allen Newell

18 The reader will notice that this is a quotation of the paragraph in that Lotfi A. Zadeh usedthe word “fuzzy”in a journal article for the first time, see: [18, p. 857].

19 In 1965 Zadeh’s “Fuzzy Sets” appeared: [19].20 The reader will notice that these are quotations from [19]

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Award, the Nicolaus Copernicus Medal, the Franklin Institute Medal, the High StateAward ‘Friendship Order’, and 25 honorary doctorates.

Carolus: Well, that is very, very impressive! I do not know many names of thathighly honored scientists, but concerning his new mathematics, this “Fuzzy Set The-ory”: the fuzzy set’s “membership values between 0 and 1 sound very much to beprobabilities, that is nothing news!

I: Oh, scientists in the Fuzzy community had to listen to that very often in the last50 years, but Fuzzy Sets and probabilities are totally different! The theory representslinguistic uncertainties, you can use my fuzzy sets to label linguistic terms that arevalues of linguistic variables. If you would say, e.g., that this squirrel is brown orthat this tower is high, there is some impreciseness in these words ....

Ludovicus: That’s what I want to stress! Language is not precise!

I: Therefore you can use fuzzy sets to compute with these imprecise magnitudes andin Fuzzy Logic we established a tool for approximate reasoning. In the 1970s AbeMamdani and Sedrak Asilian used Zadeh’s concept of Fuzzy Algorithms to write asmall program that was able to control a steam engine and today there are so manyfuzzy application systems...

Carolus: Well, that sounds to be an example for the pragmatic way of Americanengineering and I can accept that engineers are comfortable with “good enough so-lutions”, but we are philosophers and our reasoning has to be exact.

Ludovicus: Come on, Carolus, your question reminds me the Gottlob Frege’s opin-ion! In my view it’s obvious that we need Fuzzy Logic and Approximate Reasoningin epistemology and philosophy of science! In our example of towers: One might saythat the concept “tower” is a concept with blurred edges. “But is a blurred concepta concept at all?” Is an indistinct photograph [of a tower] a picture of a tower atall? Is it even always an advantage to replace an indistinct picture by a sharp one?Isn’t the indistinct one often exactly what we need? Frege compares a concept to anarea and says that an area with vague boundaries cannot be called an area at all.This presumably means that we cannot do anything with it. But is it senseless to say:“Stand roughly there?”21

Carolus: So, Thomas, what do you have in mind? How would you use this theoryof fuzzy sets in philosophy and history of science?

Thomas: Why did nobody use fuzzy sets and fuzzy relations instead of usual setsand set relations to reconstruct all the structures in the elaborated Suppes approach?If I understand this man correctly then this “extension priciple” was very successful

21 As the reader will notice, this is a slightly variation of parts of §71 in Wittgenstein’s [17].

113.3 Dialogue 827

in engineering in the last decades in which there was a literally “Fuzzy Boom” inthe 1980s: after the steam engines they used fuzzy control in cameras, washing ma-chines, rice cookers, ...

Ludovicus: There! – Why did nobody try to establish a “Fuzzification” of thisMetastructuralism in philosophy and history of science?

I: Oh, I published some papers in conference proceedings and books to start this kindof research program a few years ago and I called it “Fuzzy Structuralism”. With thisview on philosophy of science I tried to find an approach to bridge the gap betweenscience and technology on the one hand and humanities and social sciences on theother hand!

Fig. 113.5. The holodeck-avatars debating at the Berkely campus

Ludovicus: We will get this out! – Carolus, because you said that our reasoning hasto be exact, I want to direct your attention once more to Wikipedia’s explanation:“A tower is a tall [architectural] structure, usually taller than it is wide, often by asignificant margin.” There are two words that are anything but exact, these words are“usually” and “often”. I would say that all so-called “definitions”, even in scienceinclude such imprecise concepts to a greater or lesser extent because they use lin-guistic terms. I think that Fuzzy Logic is suitable to represent the kind of reasoning

828 References

we need in philosophy. As I wrote already 50 years ago: The results of philosophyare the uncovering of one or another piece of plain nonsense and bumps that theunderstanding has got by running its head up against the limits of language.22.

Carolus: The sands are running out. We have to go.

Normally, Star Trek’s holodeck-stories close when the user of the holodeck programsays ‘Computer, end program!” but instead of that, I suddenly woke up very con-fused.

References

1. Balzer, W.: Empirische Theorien: Modelle –Strukturen – Beispiele. Die Grundzüge dermodernen Wissenschaftstheorie. Vieweg, Braunschweig (1982)

2. Balzer, W., Moulines, C.U., Sneed, J.D.: An Architectonic for Science. The StructuralistProgram. Reidel, Dordrecht (1987)

3. Bourbaki, N. (pseudo.): Elements of Mathematics: Theory of Sets. Addison-Wesley,Reading (1968)

4. Carnap, R.: Der Logische Aufbau der Welt. Felix Meiner Verlag, Leipzig (1928); English:The Logical Structure of the World. Pseudoproblems in Philosophy (translated by Rolf A.George). University of California Press, Berkeley (1967)

5. Carnap, R.: Logische Syntax der Sprache (= Schriften zur wissenschaftlichen Weltauffas-sung, Band 8). Julius Springer, Vienna (1934); English translation: The Logical Syntaxof Language. Kegan Paul, London (1937)

6. Carnap, R.: The Methodological Character of Theoretical Concepts. In: Feigl, H.,Michael, S. (eds.) Minnesota Studies in the Philosophy of Science. The Foundations ofScience and the Concepts of Psychology and Psychoanalysis, vol. I, pp. 38–76. Universityof Minnesota Press, Minneapolis (1956)

7. Fuller, S.: Kuhn vs. Popper: The Struggle for the Soul of Science. Columbia UniversityPress, New York (2005)

8. Galilei, G.: Dialogo sopra i due massimi sistemi del mondo. Landini, Florence (1632);English: Dialogue Concerning the Two Chief World System (translated by StillmanDrake). University of California Press Berkeley (1953)

9. Hempel, C.G.: The Theoretician’s Dilemma. In: Feigl, H., Scriven, M., Maxwell, G.(eds.) Minnesota Studies in the Philosophy of Science. Concepts, Theories and the Mind-body Problem, vol. II, pp. 37–98. University of Minnesota Press, Minneapolis (1958)

10. Jauch, J.M.: Die Wirklichkeit der Quanten. Ein zeitgenössischer galileischer Dialog.Hanser, Munich (1973)

11. Jauch, J.M.: Are Quanta Real? A Galilean Dialogue (with a new Foreword by DouglasR. Hofstadter). Indiana University Press, Bloomington (1989)

12. Kuhn, T.S.: The Copernican Revolution. Harvard University Press, Cambridge (1957)13. Kuhn, T.S.: The Structure of Scientific Revolutions, 1st edn. University of Chicago Press

(1962)14. Sneed, J.D.: The Logical Structure of Mathematical Physics. Reidel, Dordrecht (1971)

22 As the reader will notice, this is quote of a sentence in [17, §119].

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15. Popper, K.R.: Logik der Forschung. Mohr-Siebeck, Tübingen (1934, 1935); Englishtranslation: The Logic of Scientific Discovery. Routledge, London (1959)

16. Wittgenstein, L.: Logisch-philosophische Abhandlung, Wilhelm Ostwalds Annalen derNaturphilosophie (1921); also the Edition of the works of Wittgenstein:Werkausgabe,vol. 1, includes Tractatus. Suhrkamp, Frankfurt am Main (1980); English Translationprepared by C.K. Ogden with assistance from G.E. Moore, F.P. Ramsey, L. Wittgenstein:Tractatus Logico-Philosophicus. Routledge & Kegan Paul, London (1922)

17. Wittgenstein, L.: Philosophical Investigations. Blackwell Publishing, Oxford (1953)18. Zadeh, L.A.: From Circuit Theory to System Theory. Proceedings of the IRE 50, 856–865

(1962)19. Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965)