Approach to Science

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Leszek Nowak Department of Philosophy, Adam Mickiewicz University

THE IDEALIZATIONAL APPROACH TO SCIENCE:

A NEW SURVEY(*)

(I ) Five paradigms of idealization

Various approaches to idealization differ, first and foremost, as to what the paradigmatic case of this procedure is. One may distinguish at least fiveapproaches to idealization. Each of them localizes this procedure in a differentelement of the theory construction: at the level of the construction of scientific

facts, of theoretical notions, of laws, etc.(1) The neo-Duhemian paradigm. Idealization is basically a method of transforming raw data. F or instance, systematic errors that are generated bymeasuring devices are corrected and due to that scientific facts can serve thegoals of testing, explaining, etc. It is Suppe's semantic theory of science (e.g.Suppe 1972) that is an explication and a developme nt of this kind of approachthus deserving the name of neo-Duhemian paradigm 1.

(2) The neo-Weberian paradigm. Idealization is basically a method of constructing scientific notions. Having a certain typology in mind, one mayidentify its extreme member. If the member is an empty set, it is termed an ideal

type and the notion attached to it is labelled idealization. It is particular notions,or their definitions, that exemplify idealizations in science. The source of thisapproach lies in Max Weber's methodology. In modern philosophy of science it isHempel's conception that is an explication of Weberian ideas(Hempel/Oppenheim 1936, Hempel 1961).

(3) The neo-Leibnizian paradigm. Idealization is a deliberate falsification whichnever attempts to be more than truthlike. An idealizational statement is a specialtype of a counterfactual which has to do with what goes on at possible worldsgiven by the antecedent of that statement. The smallest is the distance betweenthe intended possible world of the kind and the actual world, the truer thecounterfactual is. That conception has been developed by Lewis (1973, 1986).

(*) By Leszek Nowak. A significantly expanded version of the paper published in: J.Brzezi ski, L. Nowak, Idealization III: Approximation and Truth , Amsterdam/Atlanta1992, pp.9-63.

1 Another, and formally elaborated, approach of the neo-Duhemian type isformulated by Wjcicki (1974, 1979), cf. also below Chap. 19 .


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Leszek Nowak 2

(4) The neo-Millian paradigm. No mathematical structure fits any piece of realitywith full precision, there is always discrepancy between a mathematicalformalism and reality we want to describe with the theory. Idealization is a meansto fill a gap, i.e. to create a construction that would fall exactly under themathematical formalism serving thus as a m odel for the imprecise world we livein. Ideas of the kind may be found in Mill 2 and they are developed in the so-called Ludwig-approach (Ludwig 1981, Hartkaemper/Schmidt 1981).

An approach to idealization presented below could be termed neo-Hegelian asit refers to Hegel's idea that idealization ("abstraction") consists in focussing onwhat is essential in a phenomenon and in separating the essence from theappearance of the phenomenon. Y et, not all idealizations may be interpretedrealistically in this sense. Since I am interested here more in the methodologicalcontents of the conception in question than in the philosophical presuppositionsof it (cf. Part V below), I shall label that conception the idealizational approach to

science.In this paper, I would like to summarize the main results of this approach.

That seems to be worthwhile because of the fact that the significant majority of writings I shall refer to is in Polish. I shall also answer the main criticisms whichhave been recently put forward against the idealizational conception of science(cf. also my replies to older criticisms1974c, 1975c, 1976b). In particular, what Ikeep to be the main deficiencies of the idealizational approach to sciencerevealed in some critical papers (e.g., Kuokkanen and Tuomivaara 1992,Paprzycka and Paprzycki 1992, Balzer and Snoubek 1994, Hoover 1994) will becorrected which results in some significant amendments of this paper incomparison with its previous version (Nowak 1992).

(II )

The core of the idealizational approach to science

1. Idealization and the notion of significance

On the notion of essentiality. A scientific law is basically a deformation of phenomena being rather a caricature of facts than a generalization of them. Thedeformation of fact is, however, deliberately planned. The thing is to eliminateinessential components of it. It is taken for granted from the methodologicaltradition that not all the methodological notions need to be defined; some notionsmay be introduced as conceptual primitives, it is only the rest which is to bedefined with the aid of earlier terms, in the final instance with the aid of primitivenotions. As has been noted (cf. my 1980a, p. 97), the notion of influence is such a

2 Cf. Kotarbi ska (1974), Cartwright (1989), Hamminga and De Marchi (1994).


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The Idealizational Approach to Sceince 3

conceptual primitive whose legitimation is that it serves as a means to defineother notions of the conception under consideration. That notion was onlycharacterized formally in order to justify a construction of the notion of essentialstructure, and images of the essential structure, of a magnitude.

It has been presupposed that for every magnitude F there exists a set of all the parameters influencing it (the space of essential factors for F ). These parameterswere assumed to be differentiated as to the level of their significance for thedetermined magnitude. The relation: ... is more influential for .-.- than --- wassupposed to be antisymmetric and transitive. The most influential factors aretermed principal factors for F , the remaining ones are secondary factors for F . The (partially) ordered set of parameters of the kind is called the essentialstructure of F . The notion of significance (of one magnitude for another) isnormally adopted as a primitive one.

There have been several attempts to define that notion in more elementaryterms (Nowakowa et al. 1977, Nowak 1989, Machowski 1990) but thesedefinitions suffer from some more or less severe drawbacks (cf. Pogonowski1978 for criticism of Nowakowas proposal, and Kuokkanen and Tuomivaara1992, K. Paprzycka and M. Paprzycki 1992 for criticism of my own definition).Making use of these criticisms, I have corrected my (1989) formulations asfollows (cf.1997, 1998). A value b of factor B influences factor F iff given that,for some x, B( x) = b, it is, for some m, neither F ( x) = a1, nor F ( x) = a2,..., nor F ( x)= a m. The set { a 1,..., a m} is termed the exclusion range of F relative to B,b andsymbolized e B( F )b. The more essential a factor for F is, the greater is the range of values of F excluded by the fact that this factor adopts a given value. Accordingto this intuition, the notion of the essentiality level of B for F , e B( F ), is introducedas a ratio of the sum of all the ranges of exclusion of F relative to values bi of B to the cardinality of the set of all the values of F, Val ( F ). that is e B( F ) = ! # E B( F )b

i /# Val ( F ). If e B( F ) = 0, B is inessential for F , otherwise it is essential for F . Now, B is more essential for F than A iff B has greater essentiality levelrelative to F than A has, i.e. e B( F ) > e A( F ). B and A are equi-essential for F iff e B( F ) = e A( F ). Let us make the division of the set of factors into sets of equi-essential factors relative to F . The essential structure S F of the parameter F istermed the sequence of sets E 1 , ..., E k such that (a) B, A " E i iff e B( F ) = e A( F ), (b)for each A " E i , for each B " E i+1 , e B( F ) > e A( F ). Factors of the highestsignificance for F , i.e. those from the set E k , are termed principal factors for F ,

whereas all the remaining ones are secondary for F 3

.3 Kuokkanen and Tuomivaara (1992, note 5) found some real deficiences of my (1989)

definition of significance and their criticism inspired apart from that of Paprzycka andPaprzycki (1992) the change in my formulations. I do not agree, however, with their suggestion to eliminate the over-determination case. The point of the criticism is that definitionof e B( A)b admits the limiting case when the range of exclusion is identical with the set Val ( A) of all values of the parameter A, and hence Val ( A) determined by B relative to b may be empty.


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Assume that the sets of equi-essential factors for F are of the form: E k ={ H 1,..., H n},

E k-1 = { p1k-1 , ..., puk-1 k-1 }, ...., E 1 = { p11, ..., pu11}. The essential structure of F will then be of the form:

SF: H 1,..., H n H 1,..., H n; p1k-1 , ..., p

uk-1k-1

................................. H 1,..., H n; p1k-1 , ..., puk-1 k-1 ; .......... ; p11, ..., pu11

Below, we will usually adopt some simplification assuming that for a parameter F given is its essential structure S F whose secondary part is composed of singletons and the principal part counts many elements: E k ={ H

1,..., H n}, E k-1 ={ pk-1},..., E 1 = { p1}, or even reducing the set of principal factors E k to one factor only: E k ={ H }, as well.

The requirement of self-effective definitions. Some people ascribe the methodof idealization an intention to identify the "hidden essence of phenomena", for example "(According to the idealizational approach to science) in order todiscover the essence, the investigator would have to know it beforehand. What isthus the purpose of the method (of idealization L.N.]?" (Jorland 1995, p.277).This is understandable only on the supposition that, according to the quotedauthor, the notion of influence (and hence of the essential structure of amagnitude, etc) is to be self-effective. A term is self-effective iff its intension

provides a procedure to determine whether an arbitrary object belongs to theextension of that term or not. Now, it becomes clear why Jorland demands:

Nowak's "main task" should be "to give a criterion of 'influence', in order to tellwhether a magnitude belongs to the (mentioned) set (the space of essentialmagnitudes for a given magnitude L.N.] or not" ( ibid. , p.276). Yet, the notionof influence, even if defined in the above way, is, obviously, not self-effective.The problem is whether this is a deficiency of the proposed approach or not. I donot have any elaborated meta-methodology at my disposal, I must admit. Let usthen try to see what the history of methodology teaches us.

However, that was a deliberate move to admit this limiting case (the so-called total essentiality -cf. Nowak 1995). The importance of it perhaps is not too visible in philosophy of science butmetaphysically it is even crucial, at least for what I call the negativist unitarian metaphysics (cf.

Nowak 1997, 1998, 1998a).Moreover, I do not agree with the claim of Kuokkanen and Tuomivaara (1992, p.93) that the

over-determination case is at variance with my thesis that the maximum of the range of exclusionis the set of values of a given parameter minus singleton { w}, w being the value the parameter infact assumes. For the thesis does not follow from the definition of influence (quite to the contrary,the definition admits that the range of exclusion may be empty). It is, instead, a reconstruction of the thesis of (strict) determinism in the proposed framework and hence a philosophically orientedrestriction imposed upon the conceptual possibilities admitted by that framework.


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Let us note that there was a methodological orientation which put forward therequirement that all concepts be self-effective. That was Bridgman'soperationalism. However, the idea of reducing all concepts to mere bulks of operations failed entirely. It turned out, for instance, that this idea forces us toadmit that there is not one notion of length measured in various way but as manyas the (historically available) methods of measurement of distance.

Moreover, there are epistemological notions of undeniable significance whichare overtly not-self-effective. Take, for instance, the semantic notion of truth: " p"is true iff p. This Tarski-condition does not give any imaginable procedure todecide whether a given statement is true or not. Is this bad? Imagine that it is

possible to define truth as a self-effective concept. A general procedure enablingus to decide whether an arbitrary statement is true or false would follow fromsuch a definition. What would be the use for human creative thinking then?Creativity would be utterly superfluous in such a world: the question of truth

would be in this world simply decided by philosophers applying such a definitionto all possible statements. There, philosophy instead of understanding humancreativity, would just eliminate it with the aid of its alleged "algorithm".

On a similar basis a self-effective definition of influence equipping us not onlywith the notion of essentiality but also with a general criterion to decide whichfactors are essential for which ones would make useless the building of empiricaltheories. For in order to know whether, say, the velocity of a body is essential for its length not the testing of the theory of relativity would be necessary but simplya verdict of a methodologist applying his general criterion.

Indeed, such a definition-and-criterion of influence would make a serious task

of empirical sciences putting forward what may be termed essentialisthypotheses (cf. my 1980a, pp.111 ff ) entirely superfluous. To my understanding,it is not a deficiency but actually a merit of the idealizational methodology that itdoes not mix the notion and the criteria of essentiality. It is the task of methodology to explain the notion and the cognitive role of "essentiality" as itfunctions in science. But it is the task of empirical sciences to offer possiblymany different criteria of influence of which they may make use building their theories.

The purpose of the idealizational methodology. If not to offer any workablecriteria of essentiality, what is the purpose of the method of idealization? Simply:to reconstruct the way science works. According to the idealizationalmethodology, there are three main stages of scientific conduct:

I. pre-theoretical stage: postulation of essentialist hypotheses putting forward possible images of the essential structures of considered magnitudes;

II. theoretical stage: postulation of a body of idealizational hypotheses whichsubsequently undergo the process of concretization;


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III. empirical testing of the theory.

Another claim of the conception is that the three stages are mutually tied in thesense that what is decided in one may be questioned on another which forces thetheoretician to come back to the "earlier" stage (Brzezi ski 1977, 1985, my1980a, p.33). In other words, the "order of reconstruction" in methodology is notthe "order of justification" in science. The latter is not, strictly speaking, order atall but a network of mutual connections. In particular, it is not so that "after"gaining the "knowledge" of what is essential for what (stage I) the theoretician

builds an idealizational theory (stage II) and tests it (stage III). It is rather so thatit is only the test of a hypothetical theory which confirms the essentialisthypotheses adopted at the very beginning.

Knowledge, suppositions, hypotheses. If the above point of view requires a philosophical legitimation, it is this. There have appeared three basic notions of cognition in Western epistemology (Marciszewski 1972). First one was that of Plato: to know something meant to be able to recognize for certain the hiddenessence of things. Philosophy kept this notion of knowledge, and the belief that itis available for us, for more than one and a half thousand of years. This notion of knowledge is still present in Descartes. It was only Hume who called it intoquestion and opened a new tradition . There is no knowledge in the Platonicsense, what is available for us is nothing more than a (far from being certain)supposition based on experience. The only thing we are able to realisticallydemand of ourselves is to increase the probability of our convictions. What isavailable to us are more or less probable suppositions, and that is all. Popper opened the third epistemological tradition: nothing is certain, that is correct, butwe are not interested in increasing probability of our convictions at all. Were we,the most interesting for us would be tautologies which have the highest possible

probability. Or stereotypes whose probability is, in our assessment, very high.Instead, claims Popper, we are interested in new, risky hypotheses whose initialsubjective probability is always very low. It is novelty, or originality, of our hypotheses that matters to us.

The three ideas of human cognition as knowledge, supposition, or hypothesis are based on some metaphysical assumptions. The idea of knowledge presupposes essentialism, the ideas of supposition and hypotheticalcognition deny it. Now, the idealizational methodology attempts to combinemetaphysical essentialism rather in the style of Hegel than of Plato (cf. my

1977a-b, 1978) with the Popperian idea of hypothetical cognition. For it is nottrue that we cannot put forward hypotheses concerning the hidden essences of

phenomena. We can, and science is the best example of that. This becomes evenquite obvious, if we realize that science uses idealization and that the testing of a(n idealizational) theory is at the same time the main practical means to assessthe reliability of the essentialist hypotheses underlying it. A final refutation of


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such a theory implies that the initial view of what is essential for the explained phenomena was doomed to failure from the very beginning. Therefore, newessentialist hypotheses and a new project of a theory must be tentatively

proposed. Etc. The hidden essence is not something which can only bedogmatically believed in. Everything may be grasped hypothetically, for thehypotheticity lies in our attitude towards something, not in the nature of thatsomething; empirical facts or God may be grasped either dogmatically or hypothetically. The hidden essences of phenomena may be treated as subject of hypotheses as everything else.

A remark on methodology and philosophy. Let me add that the definition of essentiality plays an important role in a metaphysical conception for which it has

been elaborated (cf. my 1991a, 1992a), but for the idealizational methodology,even if correct, it means much less than one could expect. It turned out that the

primitive notions for the idealizational approach begin a bit earlier than expected,

and that is all. Obviously, technically this means a lot the conceptual structureof the theory becomes more clear and comprehensible but its explanatory power does not profit very much from such an innovation. Such is the dialecticsof definitions of the crucial notions of a theory: when they are lacking, everythingin the theory appears to be unclear and dependent on the sense which is attachedto the primitive notion. When such a definition is already given, it appears thatthe theory does not explain much more, if anything, than before.

In other words, the idealizational approach to science is practicallyindependent of the above definition. And that is what should be expected. Oneshould not mix the explanatory tasks of methodology of science with the

problems of philosophical understanding of science. Who accepts the body of concepts and hypotheses termed the idealizational approach to science may beequally well a follower of the instrumentalist (or relativist) vision of science or hemay support the realistic (in the aristotellian or platonist sense) vision of science.The present writer is inclined to believe in an interpretation of science of the

platonic origin but some people working on the same idealizational approach toscience are of other philosophical inclinations. How science is understooddepends on our reconstruction of the scientific practice given by methodology of science but also on our metaphysical views on the nature of reality and/or our epistemological understanding of the position of the cognitive subject. That iswhy, our explanations of the research practice do not prejudge our understanding

of science, and vice versa . That is why, any definition of the notion of essentialitydoes not matter too much for the methodology of science.

2. Basic ideas and notions

Idealization is not abstraction. The crucial point which follows from the proper understanding of the notion of idealization is that idealization is not abstraction


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(cf. my 1971c, 1975d, 1980a, pp.31ff) 4. Roughly, abstraction consists in a passage from properties AB to A, idealization consists in a passage from AB to A- B. For instance, the move from the notion of an open capitalist economy ( CEO )to the notion of a capitalist economy ( CE ) is an act of abstraction, whereas an actof idealization would consist, for instance, in a passage from CEO to the notionof a closed capitalist economy ( CE-O ).

Sometimes it is claimed that the method of idealization "is akin to the classicalabstraction and determination operations" known from the textbooks of logic(Jorland 1994, p.277). Not so. An abstract (general) statement:

(2.1)students are laborious

applies to our world directly, whereas, say, the statement::

(2.2) if ( ff ( x) & R( x) = 0, then s( x) = 1/2 gt 2( x)

does not. Instead, (2.2) applies to the ideal world in which freely falling ( ff ) bodies do not meet any resistance ( R) on their path(s) depending in the wayshown in formula (2.2) on the gravitational constant ( g ) and the time ( t ) of freefall.

Moreover, it is actually the "operation of abstraction and determination" whichsuffices to show why the analyzed claim fails. According to the famous classicalformula, as the intension of a series of terms increases (they become more andmore abstract), their extension decreases (they become more and more

4 This distinction (cf., e.g., Harre 1970 or my 1970c, 1971a,c) is perhaps quite obvious but far from being methodologically exploited. Quite the reverse so, the two procedures are often mixed.In part this is due to the prevalence of the empiricist tradition of "abstraction" in the philosophy of science which differs greatly from the Hegelian tradition (cf. Coniglione 1986, 1990). But in partit is also a matter of the terminology. Let us then compare the terminology applied here with thoseof other authors. For instance, Hempel (1952) and Cohen (1970) apply the term "idealization" inthe meaning similar to what is termed here so. Rudner (1966) and Barr (1971) apply the term"idealization" in the meaning close to what is termed here as ideation. Suppe (1972) terms"abstraction" roughly which is labelled here idealization but in reference to the data, not to thegeneral statements; Wojcicki (1974) employs in this context again the term "idealization".Zielinska (1981) labels "abstraction" which is termed below reduction and "idealization" which istermed below ideation. The intuitions of Dilworth (1990) are similar and covered by the sameterminology. Cartwright (1989) applies the term "abstraction" in the sense close to that which istermed below idealization, whereas "idealization" is used in the sense similar to what will betermed below "ideation". Etc. I must admit that the terminological confusion may be found in myearly writings as well. In (1968, pp.82ff, 1970b) to distinguish the idealizational procedure I hadused the term modelling (modelling conditions, modelling statements, etc.), then revealing theaffinity of the underlying ideas with Hegelian/Marxian intuitions I passed to the term"abstraction" (1970a), and finally recognizing an enormous confusion resulting from the latter decision (1971c, in English 1975d) I have finally decided to use the term "idealization"(1970c, 1971a, b and further on). But in all these cases there were merely terms being changed,the notion was the same all that time.


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determinate) and, vice versa , as the intension decreases, their extension increases.We have already considered an example of such a series of terms: "opencapitalist economy", "capitalist economy", "economy". Such a rule does notwork, however, for the series of idealizational terms, that is ones whose meaningcharacteristics (a set of meaning postulates, a partial definition, a definition, etc.)embraces at least one idealizing condition. The extension of the term "closedcapitalist economy" is not narrower than the extension of the term "capitalisteconomy"; the two have no elements in common, that is the intersection of theseextensions is empty. The "law of decreasing extension and increasing intension"from the textbooks of elementary logic does not hold for idealizational terms.

Idealization is counter-actual. By introducing idealizing conditions of the form p( x) = 0 the researcher eliminates factors thought to be secondary. What remainsis the factor considered principal for the determined magnitude. A hierarchy of factors considered to be the essential structure for F is termed the researcher's

image of the essential structure for that magnitude.An idealizational statement is a conditional possessing an idealizing condition

in its antecedent. Having established such a statement the researcher must takeinto account the neglected factor. He removes the condition replacing it by itsrealistic negation and introduces a correction in the formula (consequent) of thestatement. This procedure of concretization leads to a more realistic statementreferring to the less abstract conditions than the initial idealizational statement.

Thus, the idealizational structure of the F -phenomena is of the form:

(T ) T k , T k-1 , ..., T 1, T 0,

where Tk

is an idealizational law:Tk : if ( G( x) & p1( x) = 0 & p 2( x) = 0 & ... & pk-1( x) = 0 & pk ( x) = 0),

then F ( x) = f k ( H 1( x),..., H n( x)]

and T k-1 , ..., T 1, T 0 are its concretizations:

Tk-1 : if ( G( x) & p1( x) = 0 & p2( x) = 0 & ... & pk-1( x) = 0 & pk ( x) # 0),then F ( x) = f k-1( H 1( x),..., H n( x), pk ( x)]

..............................................................................................................T i: if ( G( x) & p1( x) = 0 & ...& p i( x) = 0 & pi+1 ( x) # 0 & ...

& pk-1( x) # 0 & pk ( x) # 0)then F ( x) = f i( H 1( x),..., H n( x), pk ( x),..., p i+1 ( x)].

..............................................................................................................T1: if ( G( x) & p1( x) = 0 & p2( x) # 0 & ... & pk-1( x) # 0 & pk ( x) # 0)

then F ( x) = f 1( H 1( x),..., H n( x), pk ( x),..., p2( x)]

T0: if ( G( x) & p1( x) # 0 & p2( x) # 0 & ... & pk-1( x) # 0 & pk ( x) # 0)then F ( x) = f 0( H 1( x),..., H n( x), pk ( x),..., p2( x), p 1( x)]


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with its subsequent concretizations 5, including T 0 lacking any idealizingconditions and being a factual statement 6.

5 Niiniluoto (1989, pp.34ff) proposes that instead of material implication, the formulation of the idealizational statement is to employ the counterfactual reading of if ... then as if it werethe case that ..., then it would be the case that... (briefly, $ ). As a result, the sequence (T k ), (T k-1), ..., (T 0) is proposed to be rewritten as:

Tk : (G( x) & p1( x) = 0 & p 2( x) = 0 & ... & pk-1( x) = 0 & pk ( x) = 0) $ F ( x) = f k ( H ( x)]Tk -1: (G( x) & p1( x) = 0 & p2( x) = 0 & ... & pk-1( x) = 0) $ F ( x) = f k-1( H ( x), pk ( x)]..............................................................................................................

T1: (G( x) & p1( x) = 0) $ F ( x) = f 1( H ( x), pk ( x),..., p2( x)]T0: G( x) $ F ( x) = f 0( H ( x), pk ( x),..., p2( x), p 1( x)]

where T 0 is a factual statement, whereas T k ,...,T 1 are counterfactuals claimed to be consequencesof T 0 (on the basis of the correspondence principle). Although I. Niiniluoto derives from thatsupposition important conclusions (1992, pp. 34ff, 42ff), the initial assumption seems to fail, both

because of the semantic and epistemological reasons.Semantically, (T 0) is a factual statement, indeed, as all the empirical objects G 0 of the universe

G satisfy the conditions p i ( x) # 0 ( i = 1 ,..., k ), but T 0 is not. Let us define (Nowak 1971a, pp.,1980a, pp. 99ff),

x " G 0 iff x " G & p1( x) # 0 & p2( x) # 0 & ... & pk-1( x) # 0 & pk ( x) # 0) x " G 1 iff x " G & p1( x) = 0 & p2( x) # 0 & ... & pk-1( x) # 0 & pk ( x) # 0)..................................................................................................

x " G k -1 iff x " G & p1( x) = 0 & p2( x) = 0 & ... & pk-1( x) = 0 & pk ( x) # 0) x " G k iff x " G & p1( x) = 0 & p2( x) = 0 & ... & pk-1( x) = 0 & pk ( x) = 0).

G 0 is a set of empirical (real) objects, G 1 is the set of p1-ideal types (of the first degree) of empirical objects, G 2 is the set of p1,2-ideal types (of the second degree) of empirical objects, etc.It is now visible that T 0 refers both to the empirical domain G

0 to which (T 0) applies and to theidealized domains G 1,...,G k-1 , G k that the idealizational statements (T 1), ... , (T k-1), (T k ) refer to,correspondingly. T 0 is then a kind of both supra-factual and supra-idealizational statement.The semantic status of conditionals of the kind deserves a careful analysis , to be sure, but (T 0)cannot be simply replaced by T 0, and the same applies to T i and (T

i), for 0 < i < k . (Notice thatalso Kuokkanen and Tuomivaara 1992, p.92 express some doubt as for the applicability of I.

Niiniluotos reconstruction in this respect).Epistemologically, adopting I. Niiniluotos stand the epistemological sense of the method of

idealization becomes dubious. Assume T 0 is somehow justified, say inductively. Once we knowthat, there is no need to visit ideal worlds (constructed nicely by Niiniluoto himself, 1992,

pp.43ff) with universes G i (0 < i % k ). All we need for explaining, predicting and programming theempirical world is T 0, allegedly factual a statement. Idealizations reduce to the role of counterfactual special cases of it and concretization becomes cognitively superfluous at all.

Facing the so-destructive implications of the otherwise largely and precisely elaboratedapproach, one should, I believe, rather to take a risk and remain with the not so precise conceptwhich at least allows us somehow realize the cognitive importance of the method which for thefirst glance is something really common in science.

6 Balzer and Zoubek (1994) pose two objections against the form of concretization employedin the text. First, this scheme does not cover all forms of scientific laws; for instance it does notcover purely qualitative laws (p.65). Second, while during the [concretization] transition the. ..connections [dependencies] may change, and will change, the value of property F remainsidentical (p.64). They add to the second point in the footnote This identity. . .is somewhat

puzzling and seems to make sense only for a metaphysical realist (p.64, n. 21).


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As to the first point, the criticism addressed to the above formulations is correct. The aboveschemes are far from being general enough to cover all forms of scientific laws. However,making a general scheme need be not the starting point but rather an (ideal) outcome of thedeveloping chain of explications. This procedure seems to be admissible on the following twoconditions: (a) the initial explication is natural, i.e. it covers the classical representatives of agiven kind, and (b) each of the elements of the chain of succesive explication is more general thantheir predecessors. The classical laws of physics seem to be classsical cases of laws of science[a] And the kind [b] of development of the idealizational approach to science was openly

postulated (cf. Nowak 1974a, pp.21, 273ff, 1980a, part III) and actualized in numerous writings(cf. summary below par. III and par. V) in order to cope with the complexity of types of idealization and concretization in science; in particular, also idealization and concretization for the purely qualitative laws can be somehow conceptualized (Nowakowa 1996).

Let us add that similar objection is put forward by Sintonen and Kiikeri (1995) complainingthat although evolutionary theory is an idealizing theory in some sense, the Pozna typeidealization schemata is not the best way to describe it (p.207/208). In fact, as it is visible fromthe form of Darwins laws (cf. above Chap 2), they do not fall under our standard formulae T k , T k-1,... etc. In the light of the above explanations, Darwinian laws presumable belong not to theclasssical centre of the colloquial class idealizations, but rather to its peripherry and somederivative schem of idealization is to be expected here. Indeed, Klawiter (1978, summary inEnglish 1989) has elaborated a scheme of adaptive idealizational statement, a scheme of theadaptive concretization etc. legitimizing thus the intuitive work on the idealization in the theory of natural selection ( astowski 1977, 1987, English summary 1994, astowski and Nowak 1982, cf.also above Chap. 2). However, there are difficulties in linking these schemata with the standardformulae T i and Klawiters construction remains an interesting but separate conceptualization of some kind of idealizations. In this sense to a certain extent, although not straightforwardedly,Sintonen and Kiikeris objections still presents a real problem.

Let us come back to the criticism of Balzer and Zoubek (1995). As to their second point, in F ( x) = n one should distinguish three things and the postulate of their identity through theconcretizational transition seems to have quite different methodological sense. For the sake of simplicity, let us consider the conditionals:

(T 1) if G( x) & p( x) = 0, then F ( x) = f 1( H ( x))(T 0) if G( x) & p( x) # 0, then F ( x) = f 0( H ( x), p( x)).

Now, let us distinguish the three cases.a. One is the identity of object x. That is something which is not postulated at all. It couldnt

be, as the object satisfying the conditions: G( x) & p( x) = 0 & p( x) # 0, and on the strength of (T 1) and (T 0) the conditions: F ( x) = f 1( H ( x)) & F ( x) = f 0( H ( x), p( x)) would be simply self-contradictory. Actually those satisfying (T 1) and (T 0) are different. They are, however, tied in aspecial manner: the former G 1 are ideal types of the first degree (cf. an explication in note 3aabove) of the latter G 0, i.e. empirical objects of the type G.

b. The second possibility is the identity of magnitude F in (T 1) and (T 0). that is in fact presupposed which simply means that the range of property F includes both empirical objectsfrom G 0 and their p-ideal types (of the first degree) from G 1. But this is, how it should be. Not

only the sun but also the mass point possess a mass. If the researcher wants to model the sun, thenhe/she simply ascribes the suns mass to it making the theoretical sun. All these so commonlyemployed in science operations are possible on the condition that scientific magnitudes (mass,velocity, etc.) are defined on a universe transcending that of the empirical discourse. That isadopted in our construction and very well. How to understand this metaphysically is a separatematter. My present views on the subject are given in (1995, 1998, cf. also Chap. 31 of this book).

c. The third possibility of understanding of the objection is to comprehend it as the identity of the value F ( x) of the magnitude F . This is admissible but only on the extreme case of what is


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Leszek Nowak 12

An idealizational law on F is thus identified with that of idealizationalstatements concerning F which is most abstract, i.e. neglecting all the factorsclaimed to be secondary for F , and takes into account what is considered to be

principal for F . This notion of law allows then to preserve the traditionalconnections between the concepts of law, regularity, the essence, etc. For simple idealizational theory ( T ) presupposes a certain view on what ifluences theconsidered magnitude F. All the factors taken to influence upon F form an imageof the space of essential factors for F. The factors thought to be secondary areomitted on the strength of the idealizing conditions whereas those considered to

be principal are taken as "independent variables" from the very beginning. In our simplified scheme of the idealizational theory of property F the image of thespace of essential factors of F , I( P F), is the set { H 1,..., H n, pk ,..., p2, p1}, H 1,..., H n

being the principal factors, whereas the remaining ones are secondary. Then onemay say that what is presupposed by the idealizational structure ( T ) is the imageof the essential structure:

I(SF): H 1,..., H n H 1,..., H n, pk ,

.................. H 1,..., H n, pk ,..., p2, H 1,..., H n, pk ,..., p2, p1.

The levels of this hierarchy of factors correspond to subsequent elements of thesimple idealizational structure. Obviously, that there is a linear order of thestrength of influence among the secondary factors upon F in the set I(P F), and thatthis order corresponds to the sequence of idealizing conditions, holds only in theextremely idealized picture of science. Below, a certain path from such asimplified scheme to more realistic, and thus more complicated, schemes of thescientific theory is outlined 7.

termed the degenerate concretization (Nowak 1974a, p.92, 1980a, p.192), when p proves to be,against the researchers expectation, inessential for F and hence for all elements of G (objectssatisfying G( x)), f 1( H ( x)) = f 0( H ( x), p( x)). As a result, the idealizing condition p( x) = 0 turns out to

be superfluous. Apart from the case of the (mistaken or instrumental) degenerate concretization,for genuine concretization when p is in fact influential for F , f 0( H ( x), p( x)) # f 1( H ( x)).

It follows from the above that the postulate of the identity that is presupposed in the presented approach is one of identity of the magnitude (cf. b) during concretization which, as far as I see it, gives no reason to a methodological objection.

7 An important simplification silently adopted here is that only properties are consideredwhereas relations are neglected. This results in the following: (i) predicates in the schemes of idealizational statement, its concretization etc. are monadic, (ii) they are first order-predicates,(iii) interactions between determinants of a given magnitude are neglected. Since all these effectsmight make, not without justification, an impression of assuming the "purely Aristotelianontology", I would like to comment on them in short.

Re: (i). This is easy to be removed, by introducing relations and relational predicates, on the price of a complication of the conceptual apparatus from the very beginning (cf. 1977).


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Anyway, the crucial idea is that the theory begins with reconstruction, in theform of the initial law, of the dependence holding on the first level, and thefurther concretizations of the law reconstruct more realistic dependencies holdingon subsequent levels of the structure. To put this in Hegelian terms: the (simpleidealizational) theory can be thus claimed to be a discovery of (what isconsidered to be) the hidden essence of F-facts and a reconstructi on of itsmanifestation through (what is considered to be) secondary influences 8.

Re: (ii). Indeed, for some types of statements this limitation ceases to be trivial. Namely, thisconcerns adaptive statements employed above in Chap. 2 (an initial analysis of these cf. my 1975)which appear in different branches of science (Klawiter 1977, Lastowski 1982, Kosmicki 1985)and which involve, in an irreducible manner, predicates of higher orders (Kmita 1976, p. 4lff).The question of applicability of the operations of idealization and concretization in the realm of these statements is analyzed by Klawiter (1977,1982) and Patryas (1979).

Re: (iii). It is presupposed here that no interactions between the elements of the

space of essential factors occur. This is silently assumed already in the adopteddefinition of the notion of an essential property. Namely, it does not embrace the case inwhich two properties G and G' are non-essential for F , if taken separately, but areessential for F , if taken together. This might be expressed by saying that although G andG' are insignificant for F , their interaction int (G, G') is essential for F . Therefore, theessential structure for F is, in Brzezi ski's (1975) terms, of the second kind (if there is,additionally, a single factor H essential for F ) or of the third kind (if the interaction isthe only element influencing F ). I have not included these problematics because itwould complicate our schemes very much. It will be mentioned below, section ( V, 4).

8 A doubt may raise as to the methodological status of the thesis stating that science appliesidealization (Batg 1974, Kirschenmann 1985). E.g., P. Kirschenmann claims that "it is not clear what kind of scientific practice Nowak would possibly count as an instance telling against hismethodology" (1985, p. 15). For instance, if somebody proven that reconstructing physical laws,or economic ones, etc. as idealizational statements leads to the distortion of their contents andwhat is considered to testify to their idealizational character (e.g. comments referring them to"inertial systems", "closed economies" and so on) in fact supports, for example, the view of their inductive nature, then the idealizational conception of science would be falsified. This would also

be a serious argument against the hypothesis of essentialism as it would then be difficult tomaintain both that the world is essentially differentiated and that our best form of knowledge of itdoes not reveal this ontological property it has.

Of course, the way in which methodologists `test' their conceptions is different from the way physicists take account of observation and the difference, to be sure, justifies enclosing the termin quotation marks. Yet, natural sciences are not the only form of science. What about the way thetheoreticians of literature test (or `test') their proposals? What I claim is that the level of thedevelopment of methodology is akin to that of the traditional humanistic disciplines (Nowak 1974a, p. 277ff).

When a methodological conception is unable to conceptualize in its own terms anyconcrete example of a piece of scientific practice, it would be recognized to be, for example , an interesting logical innovation, but its attachment to the methodology of science would be, I conjecture, denied. On the contrary, the fact that it is possible withinSneed-Moulines-Balzer's paradigm to reconstruct the whole of physical or economictheories is a significant argument supporting the structuralist theory of science. To besure, the method for testing methodological conceptions is on the level of the science of


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Leszek Nowak 14

Naturally, the image I(S F) of the essential structure S F may differ from thestructure itself, either basically or derivatively. The image is basically different(resp. similar) from the appropriate essential structure iff it wrongly ( resp. correctly) identifies the principal factor(s) for the investigated magnitude. I(S F)differs derivatively from the structure S F (resp. is similar to it derivatively) iff itdoes identify ( resp. does not) the principal factor(s) for the magnitude F but ismistaken ( resp. is correct) in enlisting the secondary factors for F. The analysis of

possible relationships between the essential structure S F and its image I (S F) andtheir epistemological meaning leads to the problematic of trut hfulne ss (Nowak 1977e, Nowakowa 1977, 1992) and will be discussed in Part VI 9 10.

literature; that is it reduces itself to quoting more or less accidental exaxnples. However,the more methodology comes to understand of science, the more rigorous, I conjecture,ways of testing its theories will be applied.

9 It is obvious that the role of abstraction in science may be conceptualized in a variety of ways. One may distinguish at least three of them: (a) treating idealizing conditions as antecedentsof some (idealizational) statements (cf. my 1970, 1971a, 1972), (b) considering the as axioms of atheory (Barr 1971), (c) identifying "counter-factual elements" as rules of the interpretation of atheory to some purified data (Suppe1972). It is also obvious that nobody knows a priori which of these conceptualizations, if any, is a proper one and at least because of this it is worth developingall of them. For me possibility (a) is the most intriguing because of its philosophical

presuppositions that allow, for example for the reconstruction of a large part of the Hegelian-Marxian heritage (Nowak 1980a, an alternative approach 1991, 1995, 1998,cf. also below Part V). But this is not a (methodological) argument since one could be found onlyin the explanatory power of a given conception as to what is taking place in science; only scientifc

practice may give, then, arguments supporting one of them and discriminating against the other.I suspect that the greatest problem the conceptions (b) and (c) meet when facing scientific

practice is the lack of the analogue of concretization of conception (a). But it seems that this may be better or worse met also in terms of these approaches.

Let us take for example conception (b) and consider the set (I k ) p1( x) = 0, . . . , pk ( x) = 0 of idealizing conditions. An idealizational theory will be termed a deductive system S k = (A k , C k ) inwhich set of axioms A k includes set of idealizing conditions I k , whereas C k are derivativeconsequences of A k . If S k was applied to the world, the results would be clearly false. However,we are not obliged to do so. We can say after all that some of the axioms, namely those of set Ik,have been accepted because of the requirements of simplicity and are to be removed. And so,condition pk ( x) = 0 is being replaced with its negation and a new deductive system S

k-1 = (A k-1,Ck-1) is put forward, in which A k-1 differs from A k only because of the said replacement of pk ( x) =0 by the realistic condition p k ( x) # 0 and set C k-1 differs from C k as much as the replacementchanges deductive sequences in the new system. The latter can be said to be a concretization of Sk. And so on. The full theory would then be composed of the sequence of systems S k , Sk-1, . . ,So.

This approach seems to be convenient to explain the use scientists sometimes makeof idealizing conditions; sometimes they use them as premisees of reasonings, indeed.Similarly, the (c)-approach allows for explanation of some intriguing aspects of thestructure of scientific theories (Kupracz 1991, cf. also below par. IV 2). Let us allow

people supporting different approaches to develop them, as this is the only availablemeans to state which of them has the largest explanatory power.


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10 Let us comment somewhat upon the famous thesis of N. Cartwright (1983) that the fundamental laws of physics do not satisfy the facticity requirement (p.60). She argues that

One of the chief jobs of the law of gravity is to help explain the forces that objectsexperience in various complex circumstances. This law can explain in only very simple,or ideal, circumstances. It can account for why the force is as it is when just gravity is atwork; but it is of no help for cases in which both gravity and electricity matter. Once theceteris paribus modifier has been attached, the law of gravity is irrelevant to the morecomplex and interesting situations (Cartwright 1983, p.58).

Then she refers to the rule of addition of vectors stating that it does not help too much as [n]aturedoes not add forces. For the component forces are not there, in any but a metaphorical sense,to be added; and the laws that say they are there must also be given a metaphorical reading(ibid. , p.59). All this, however, does not undermine the validity of fundamental laws.

Let us consider the scheme of the image I(S F) with n principal factors and k sets of (equi-essential) secondary factors. Assume that k = 0. that is all the sets of secondary factors for F aresupposed to be empty, and the only determinants of F are its principal factors H 1,..., H n. In such a

situation, introducing the idealizing conditions eliminating particular H i-s is possible only withoutany essentialist justification. In the extreme case, only H 1 of the determinant is accounted for andall the remaining H i-s are omitted via idealization, then only H 2 is accounted for and all theremaining H i-s (including H 1) are abstracted from, etc. In other words, the following series of idealizing/realistic conditions are postulated (Nowak 1971a, pp.184ff):

(i1) H 1( x) # 0 & H 2( x) = 0 & H 3( x) = 0 & .... & H n( x) = 0 (i2) H 1( x) = 0 & H 2( x) # 0 & H 3( x) = 0 & .... & H n( x) = 0.......................................................................................(in-1) H 1( x) = 0 & H 2( x) = 0 & ....& H n-1( x) = 0 & H n( x) # 0.

Under these, the appropriate idealizational statements are put forward:Tk 1: if G( x) & H 1( x) # 0 & H 2( x) = 0 & H 3( x) = 0 & .... & H n( x) = 0,

then F ( x) = f 1k ( H 1( x))Tk 2: if G( x) & H 1( x) = 0 & H 2( x) # 0 & H 3( x) = 0 & .... & H n( x) = 0,

then F ( x) = f 2k ( H 2( x)).....................................................................................................Tk n-1: if G( x) & H 1( x) = 0 & H 2( x) = 0 & ....& H n-1( x) = 0 & H n( x) # 0,

then F ( x) = f nk ( H n( x)).In this case, the concretization is possible only on the assumption of superposition (e.g., adding)of component influences into the global one: f k = &( f 1k , f 2k ,..., f n-1k ). Having assumed such a

principle of superposition, the idealizational statements T k 1, Tk

2,...., Tk

n-1 lead to:Tk : if G( x) & H 1( x) # 0 & H 2( x) # 0 & ....& H n-1( x) # 0 & H n( x) # 0,

then F ( x) = f k ( H 1( x), H 2( x),..., H n( x)).If k = 0 in fact, then T k is a factual statement T 0.

Thus, superposition is a concretization of a special sort. And it is as legitimate as everyconcretization is. Similarly, fundamental laws in the sense of Cartwright (of the form T k 1,..., T

k n-

1) form a special case of idealizational statements and are as legitimate as all the idealizations are.In (1989) Cartwright is closer to such an understanding of the matter Here is a nice formulation of the idea underlying the notion of concretization: in case of an ideal situation all other disturbing factors are missing. . . .When all other disturbances are absent, the factor manifests its

power explicitly in its behavior. When nothing else is going on, you can see what tendencies afactor reveals by looking at what it does. This tells you something about what will happen in verydifferent, mixed circumstances but only if you assume that the factor has a fixed capacity thatit carries with it from situation to situation. (Cartwright 1989, pp. 190-91).


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Leszek Nowak 16

The form of a law of science. The adopted form of a scientific law may appear today out-of-dated (indeed, Diederich 1994 calls it "anachronic"). Well, that isreally a rather traditional point of view. What matters, however, are arguments.My arguments would run as follows.

It is notoriously true that scientists use not the conditionals but the formulae,for example equations, calling them scientific laws. Prima facie then, thefollowers of the idea law is a predicate (e.g., the structuralists, Wjcicki 1974,1979 and others) are in a better position as what they claim does accord with thelinguistic custom in science. However, the way in which an expression is usedoften does not matter too much: sometimes it is of significance, sometimes it isconfused. Think, for instance, of a biologist who would be inclined to classify

plants according to the natural language a crucial category for him would beone of "vegetables" which is an absurd notion from a theoretical standpoint (cf.

below Chap. 20 ).

A proper criterion for such a discussion would be, I think, the following:assuming a given linguistic stipulation, try to explain from your standpoint thereasons underlying the contrary one adopted by your protagonist. The problemfrom my point of view is, then, to understand why the scientists call the formulaethe scientific laws. The answer is that they spontaneously look for the factorsconsidered to be principal for the investigated magnitudes. The way such a factor influences a given magnitude is grasped in the formula of the appropriateidealizational law; the antecedent of it abstracts instead from the working of factors treated to be secondary for this magnitude. The list of those factors alwayschanges and is never considered to be complete. In case of findingcounterexamples for the formula, the scientist normally assumes that the formulais correct, it is only the list of secondary factors which is incomplete and attemptsto find a source of the discrepancy, i.e. an hitherto unknown secondary factor causing the deviations. As a result, the antecedent of the conditional changes withthe formula which is kept in force (cf. 1980A, pp.201ff, also below section 6 andChap 5). Therefore, the conditional T i, quite "symmetric" from themethodologist's point of view is for the scientist evidently "non-symmetric" theconsequent of it is for him much more important than its antecedent. It is notsurprising that in his linguistic custom he focuses on what is crucial for him.

If I am not mistaken, given my assumptions it is possible to explain whyscience applies the terminology contrary to mine. I cannot, however, imagine

how the approach identifying a law with the appropriate predicate could explainthe fact that sometimes scientists explicitly formulate idealizational conditionalscalling them the laws of nature. For instance: Kittel et al . (1969, p.77) expressesthe law of inertia as follows: " a = 0, when F = 0". Marx formulated the law of value in the following manner: " if demand and supply balance each other, thenthe market prices of commodities correspond to their natural prices, that is, their


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values defined by the corresponding amounts of labor indispensable for the production of these commodities" (Marx 1845, p.141; italics of the original); etc.Is it really permissible for a methodologist to make the law of inertia "shorter"distorting the original formulation?

When a theory is realistic? Obviously, I do not take arguments of this kind as"decisive" in any sense. What can be seriously said is simply this: in the respectanalyzed above the conception of law as an idealizational conditional agrees withscientific facts better than one identifying the law with a formula. I am perfectlyaware that there are also respects under which the latter prevails the former. If thehistory of philosophy of science teaches us anything at all, it teaches us that no"decisive arguments" in our domain exist. Everything is instead a matter of

balancing arguments and counterarguments in order to keep the basic idea thatunderlies the whole conception working. The idea in question is that of realisticinterpretation of a scientific theory. For the structuralists, I guess, "realistic" in

science means "sufficiently close to the empirical facts"; this is an empiricisttradition or, rather, what remained out of it in the so sophisticated approach asstructuralism W. Diederich adheres to. For me, "real" means in science"essential", that is "not disturbed by the accidentalities"; this is the Hegeliantradition in which "real" stands in an intimate relation with "essential" (and "true"

cf. Part IV of this book). How to decide between the two without engaging inan open metaphysical discussion which I want to avoid here? As philosophers of science we should not develop an overt metaphysics, we are instead obliged torespect the metaphysical assumptions accepted in science. My conjecture is thatscience respects an ontology which is closer to Hegel rather than to Bacon.

The most straightforward reasons are these.

a. Consider the current terminology in science. It is Newton's most idealizedlaws for the mass points, inertial systems etc. that are termed "principles" nottheir numerous concretizations much closer to the empirical world. The closer isa statement to the empirical facts the lesser chance it has to gain the dignity of a"principle", "basic law", etc. Also proper names ("Ohm's law", "Lorentz'stransformations", "Domar-Harrod's model" etc.) are attached most often toidealizational laws (i.e., the most abstract idealizational statements), not to their concretizations closer to the actual facts.

b. The quite spontaneous criteria of evaluation in science incline us to name acrucial discovery (an interesting or innovative idea etc.) a new proposal of theidealizational law in the given domain, and not any concretization of the alreadyestablished law; the terminological custom noted above is only a manifestation of this. Were the attitude of scientists close to the one adopted by my critic, onecould rightly expect the reverse to be true. For the new idealizational law as suchdoes not contribute to the diminishing of the discrepancies with facts (sometimes


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Leszek Nowak 18

the opposite holds true), it is the concretizations of the old idealizational law thatdo.

c. Most important changes in science, sometimes referred to as revolutionary,consist actually in replacing one idealizational law by another (e.g., theAristotelian principle of inertia by the Galilean-Newtonian one), not in their concretizations although it is actually the former, not the latter, which make thewhole machinery closer to empirical facts.

And so on, and so forth. All this testifies not to any negligence of theempirical testing in science but, I would risk saying, to a better understanding of its role. First and foremost, nobody knows in advance what is essential for what.That is a matter of theoretical hypotheses which undergo constant testseliminating less adequate images of the hidden, essential sides of reality.

Some errors of idealization . Given the idealizational statement:

(2.3) if ( G( x)& p ( x)= 0 & q( x)= 0 & r ( x)= 0, then F ( x)= f ( H ( x))one thing is forbidden: to relate it directly (i.e., without concretization or

approximation) to reality using the following reasoning pattern:

(2.4) if ( G( x)& p ( x)= 0 & q( x)= 0 & r ( x)= 0, then F ( x)= f ( H ( x))G(a)' F (a)= f ( H (a).

This is the fallacy of reification of idealization. The reason for considering thisreasoning pattern fallacious is obvious. In inference (2.4) the conclusion does notfollow from the premisees. On the other hand, if the enthymematic premisees,

p(a) = 0, etc. will be added to the scheme (2.4), the body of premisees will proveto be contradictory. For, according to the knowledge on which our idealizationalstatement t is based, p(a) # 0, q(a) # 0 and r (a) # 0.

Let us add for the sake of symmetry that, given the factual statement:

(2.5) if ( G( x), then F ( x) = f ( H ( x)),

which is falsified by finding such a that G(a) & F (a) # f ( H (a), it is forbidden toadd an idealizing condition ad hoc , that is, without making an effort to remove itand to correspondingly correct the formula of the statement. In other words,

passage from (2.5) to (2.2) is forbidden as long as it is merely a means of savinga threatened theorem.

3. Approximation

Normally, however, final concretization is not met in science. Normally, after introducing some corrections the procedure of approximation is being applied.that is all the idealizing conditions are removed at once and their joint influenceis assessed as responsible for the deviations up to a certain threshold (. Therefore,


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explaining the F -phenomena the researcher will be referring to the simpleapproximative structure of the type:

(AT ) T k , T k-1 , ..., T i, AT i,

where Tk , T

k-1, ..., T

iare of the above form whereas AT

iis an approximation of

T i, i.e. a factual statement of the form:

AT i: if ( G( x) & p1( x) # 0 & ... & p i ( x) # 0 & pi+1 ( x) # 0 & ... & pk-1( x) # 0 & pk ( x) # 0),

then F ( x) ) ( f i ( H ( x), pk ( x),..., p i+1( x)].Obviously, given the threshold (, it is both possible that already the

approximation of the idealizational law AT k is true and that it is actually falseand some concretization steps are necessary to obtain a formula complicatedenough to deviate from the empirical F- facts by (. In the second case

idealizational statements Tk

, Tk-1

, ..., Ti+1

are approximatively false (that is,their approximations AT k , AT k-1 , ..., AT i+1 are true) and it is only theidealizational statement T i (i < k ) that is approximatively true (i.e., itsapproximation At i is true). In science one may find examples of both cases whichis to say that pure idealizations, i.e. idealizational laws that do not apply to theempirical fa cts even approximatively are fully legitimate (Nowak 1973, 1974a,

pp. 158-60) 11

Approximation proves then to be a subsidiary means in relation to theconcretization procedure. When a researcher is not in a position (or, there is nocognitive need) to apply the latter, he refers to approximating his idealizationalstatements.

4. Idealizational theory and explanation

The sequence ( AT ) is somewhat better approximation to theories that are built inthe actual scientific practice than the simple idealizational structure ( T ). Still, it isfar removed from the scientific practice. When building an idealizational theoryin science, more statements are equipped in one and the same list of idealizingconditions. Then they are concretized by gradually admitting the previouslyneglected secondary properties and modifying the formulas of these statements.The laws become more and more complicated and therefore ever closer to theempirical reality. And also the body of them in subsequent models of theincreasing realism becomes larger and larger. This procedure continues until themost realistic model becomes a sufficient approximation of the given system.

11 Compare Cartwrights thesis that there are fundamental laws which are not evenapproached in reality. They are pure fictions (1983, p.153). Let us note that in the Polishmethodological literature the idea of admissibility of idealizational laws whose approximationsare empirically false mentioned in the text has been vividly discussed, usually criticically(Wjcicki 1974, Krajewski 1974b, Siemianowski 1976, and others).


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Leszek Nowak 20

The structure of a scientific theory is thus given by a sequence of models M k , M k-1, ..., M i, AM i, where M k is the most abstract model equipped with k idealizingconditions, M k-1 ...M i being its subsequent concretizations, finally, AM i is anapproximation of the least abstract of these models M i to the empirical reality (cf.

Nowak 1971a, 1980a). There are two ideas of explanation. One is that to explain that F (a ) means that

it is always the case that F ( x), where x ranges over the class G whose member isa. This suffers from the famous objection of F eyerabend: it is strange to answer to the question why F (a) by recourse to the facts that F (b) , F (c) ,... (a, b, c... aremembers of G), that is to the facts which are ununderstandable well as the initialfact to be explained. The other tradition is that to explain means to find theessence of what is to be explained. And this is actually the idea which is adoptedin the idealizational approach to science.

The idealizational law, given a definite image of the essential structure of thedetermined magnitude, is an idealizational statement which neglects all thefactors claimed, truly or not, to be secondary. Such a statement refers then to(what is considered to be) the way in which the principal factors influence thegiven magnitude, i.e. to (what is considered to be) the regularity. Theconcretizations of the law reveal instead how the regularity manifests itself in theconditions closer and closer to reality . All of them reconstruct how the essence isdeformed by all the actual disturbances, that is how the phenomenon deviatesfrom its essence.

The model of perfect explanation is thus the following: to explain perfectly acertain F- fact means (1) to identify and select in an accepted idealizational theory

a sequence of statementsTk , T k-1 , ..., T 1, T 0

where

(i) its first member is the idealizational law of the magnitude F and theremaining ones are the subsequent, and all, concretizations of the law provided

by that theory,

(ii) to deduce from the last member of the sequence, i.e. the factual statementT0, and appropriate init ial conditions C , the statement E (explanandum )describing the given F -fact 12.

12 N. Cartwright puts in question the deductive-nomological (D-N) model of explanation emphasizing that [i]t is never strict deduction that takes you from thefundamental equations at the beginning to the phenomenological laws at the end (1983,

p.104). Identifying the fundamental laws with idealizational ones andphenomenological laws with their (final) concretizations or approximations of their (far enough) concretizations one may state that her criticism is both too strong and too


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In the limiting case, i.e. when n = 0, we obtain the usual D-N model of explanation:

T0 C ' E .The model of approximate explanation differs from the above only by

reference to a sequence:

Tk , T k-1 , ..., T i, AT i.

That is why not the given F- fact but a class of F- fact s de fined by the threshold of approximation ( can be derived from such a structure 13.

weak. Too weak because the relation of concretization is not a weakened deduction,

but a relation sui generis . Too strong because the D-N model is not simply wrong asCartwright calls it (1983, p.107). In the light of the idealization-concretization (I-C)model of explanation, the D-N model holds good in the extreme, factualist case. The I-Cmodel is but a generalization of the D-N model: it says that the D-N model works for thefactual statements but it is wrong for the idealizational laws. Cf. Chap. 16 .

13 Diederich (1994) claims that in the model of explanation (let us take for the sake of simplicity its non-approximativist version) which I defend the whole story with concretization isredundant because what actually explains the fact F under the initial conditions E is the factuallaw T 0. Two arguments are given for this supposition. First, we use in explanation only the finalexpression T 0 & E which is to perfectly agree with Hempel's model - because we cannot apply theidealizational laws to the real conditions. Second, all the idealizational statements areconsequences (the limiting case) of the factual presumption T 0.

I do not agree with the first argument. The legitimation for the use of idealizational statementsin explanation is their role in deriving the factual statement T 0 via concretization. They legitimateT0, not E directly. It will be useful to employ the well-known distinction between "explaininglaws" and "explaining the facts". One could say that the idealizational premisees, including thefirst of them, i.e. the law T k , explain the (general) factual statement T 0, whereas the latter explainsthe (singular) statement about the fact F .

The second argument is much more subtle for it refers to a controversial problem of thelogical relationship between an idealizational statement and its concretizations. That is really aserious problem and it may be solved in various ways. One is them is what Diederich claims:concretization is a special case of the relation of entailment. But that position is not mine. It isvisible that T i-1 is not more general than T i : their ranges of application do not intersect. That is soobvious that it should be explained how it is possible that the prominent methodologist does notacknowledge this very fact. The reason, I conjecture, is that concretization is (mis)conceived as anoperation of deleting an idealizing condition (cf. also Krajewski 1977), not as one of replacing itwith an appropriate realistic condition. For instance, if concretization is understood as a passagefrom:

(t) if ( G( x)& p( x) = 0 & q( x) = 0, & r ( x) then F ( x)= f ( H ( x))to ,

(t) if ( G( x)& p( x) = 0 & q( x) = 0, then F ( x)= f ( H ( x, r ( x)) ,further to:

(t) if ( G( x)& p( x) = 0, then F ( x)= f ( H ( x), r ( x), q( x)) ,and finally to:


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Leszek Nowak 22

5. Testing idealizational laws

The rule of delayed falsification. The procedure of testing the idealizationalstatements is not easy to imagine from the standpoint of the idealizationalapproach to science. In part, the matter is easy to comprehend. Given theidealizational law T k , one may identify a special sort of conditions C , call themclassical ones, and approximate that law for the classical conditions, that is toform the approximation of that law limited to C :

AT k/c : if ( C ( x) & p1( x) # 0 & ... & p i( x) # 0 & pi+1( x) # 0 & ... & & p i-1( x) # 0 & pk ( x) # 0)then F ( x) ) ( f k ( H ( x)].

This statement can be tested directly. If the outcome is positive, it confirms theidealizational standard from which it is a deviation. If not, the standard T k isdisconfirmed.

If the classical conditions cannot be found, the researcher may create them.That is how the role of experiment is explained the sense of it is to secureapproximation for the idealizational laws (Nowak, 1971a, p. 215, for a moresophisticated account cf. Patryas 1976, summary in 1982).

If, however, no classical cases can either be found or created there, i.e. if theapproximation of a given idealizational law is false for all the subsets of its actualrange, then the problem is how an idealizational statement could be tested againstthe empirical data:

(5.1) F (a) = k

(t) if ( G( x), then F ( x)= f ( H ( x, r ( x), q( x), p( x))then it is obvious to claim that concretization consists in generalization. However, (t) differs fromct, and so do statements (t) and cct, and ( t ) and cct. The statement (t) is neither factual, as ccctis, nor idealizational as t is. (t) is a more general statement applying both to the ideal worldsdeprived of factors p and/or q and/or r and the actual world in which all these factors operate.

How to interpret this metaphysically is another matter, and my present views on that subject, Iwould like to add, differ from those of the seventies, i.e. ones underlying the writings W.Diederich deals with. At present, I am rather inclined to think that D. Lewis's doctrine of modal

possibilism requires a significant strengthening and to admit the existence of the ideal worldsincluding the empty world ("nothingness") in which all the magnitudes are idealized (reduced - cf.1991, also Chap. 31). But the outlining of this metaphysical proposal (cf. 1998) would lead us toofar here. For the task of this discussion it suffices to say that what I have not changed are theabove schemes of an idealizational statement and its concretization. And those schemes imply thatthe former is not a special case of the latter. It is only the formula (consequent) of theidealizational statement which is - if taken in itself - a special case of the formula of itsconcretization. But the unit of science are, I am still inclined to think, not the formulae themselves- let alone the formulae treated formally, without a substantive (e.g. economical) interpretation -

but the conditionals.


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For the sake of clarity, let us consider simplified schemes. The answer is easy incase of a factual statement:

(5.2) if G( x) , then F ( x) = f ( H ( x))

Indeed, under the asssumptions(5.3) G(a) & H (a) = l ,

if it is the case that

(5.4) k = f (l ),

then the factual statement (f) is confirmed, otherwise it is not. Y et, how thestatement with idealizing conditions:

(5.5) if G( x) & p( x) = 0 & q( x) = 0 , then F ( x) = f ( H ( x))

can be tested against data concerning the empirical object a i.e. one which does

not meet any of these conditions is not clear at all.One thing seems to be certain: Poppers paradigm of falsification (Popper

1959) according to which science tests its theories following the modus tollendotollens (MTT), does not work in the case of idealizational statements at all. For if (5.5) is an idealizational statement, then neither (5,1) or its negation cannot bederived from (5.5). In both cases the error of reification (cf. above 2) iscommitted. What may follow from (i) are merely positive idealizationalobservational statements:

(5.6) if p(a) = 0& q(a) = 0, then F (a ) = k

or negative ones:

(5.6)* if p(a) = 0 & q(a) = 0, then F (a) # k which dismisses Popperian rule of falsification, because statements of the formcannot be found with the aid of observation.

A tentative solution to this problem was the following (cf. my 1971a, 1980a).The idealizational statement is concretized step by step by admitting the

previously neglected secondary properties and modifying its formula. The lastidealizational statement is approximated to reality and AT i is obtained. Whether AT i is true, or not, only experience will decide. If

* F (a) f i( H (a) , pk (a) ,..., p i+1 (a))* % ( then AT i is confirmed (directly, and indirectly so is also T k ). If not, then T k isdisconfirmed. In case of our simplified example, what is necessary is aconcretization of (i) of the form:

c(5.5) if G( x) & p( x) # 0 & q(a) = 0 , then F ( x) = f ( H ( x), p( x)).Under assumptions:


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Leszek Nowak 24

(5.7) G(a) & H (a) = l & p(a) = & & q(a) # 0, if it is the case that

(5.8) *k f (l, )* % (,then the concretization c(5.5) is confirmed, otherwise it is not.

Such a grasp makes it necessary to get rid of the famous claim that a universalfor which a counterexample has been found is to be rejected. Not so for theidealizational universals. One should distinguish between real counterexamplesand prima facie counterexamples. A fact is a prima facie counterexample of T k if it negates the approximation AT k of it but there is such a concretization T i of T k

that the same fact does not negate the approximation AT i any more; that meansthat the deviation inherited in this fact is smaller than the threshold of deviationsfrom the corrected formula in the consequent of T i. On the other hand, a realcounterexample is one which negates all approximations AT k,... AT 1, and also thefinal concretization T 0.

Now, in actual scientific practice the prima facie counterexamples are taken asconfirmations, and not disconfirmations, of the idealizational law. And when afact negating such a statement is being found, then the main effort of theoreticians is to prove that it is merely a prima facie counterexample, i.e. that itsuffices to concretize that law in order to explain the discrepancy and take whatseems to negate the law as a confirming case (Nowak 1971a, 1980a, pp. 163-64)14. It has also been argued that the outlined grasp of testing allows for includingthe well-known idea that a theory provides empirical facts with a definiteinterpretation (Klawiter 1975a).

How to empirically discriminate among idealizations? Sometimes thecognitive usefulness of the rule of delayed falsification is put in doubt. Hoover (1994) asks the following question. Assume that there are two competing

14 This is the rule applied by Marx in Capital (cf. my 1971a, 1980a). Hence, Hamminga andDe Marchi (1994) correctly observe that Marx would manage with counterexamples in that style(p. 38). It is not clear, however, on what basis they distinguish McCulloch's rule of dealing withcounterexamples ("Whereas you - erroneously - 'see' a counterexample, I teach you to recognizethat it is actually an example" - ibid. ) and contrast it with the Marxian one. I would say that theMarxian rule actually consists in revealing to the critic that where s/he sees a counterexample (toa too early approximation of the idealizational law) it proves to appear (after further steps of concretization) actually anexample confirming the law.

Hoover (1994) claims that the rule of delayed falsification outlined in the text makes of theinitial, most idealized model something similar to "a Lakatosian hard core" ( p.49). In a sense -yes. But still there is a significant difference. The hard core is immune from the negativeoutcomes of experience due to adoption of the additional hypotheses making possible areinterpretation of those outcomes. Not so in case of the rule in question. For it results from whatis inherent in a model without any additional statements.


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idealizational theories each developing according to the rule of delayedfalsification. There will then take place progress in explaining empirical facts

better and better within each theory taken separately. But what about their commensurability? How could we decide which of them is better not internally

but externally, that is by reference to empirical facts? The author provides us witha decisional procedure in econometric terms. That is really an important problemfor the idealizational methodology and I would like to reconstruct such a

procedure in general terms.

Let us consider a simple conceptual structure called, in the idealizationalapproach to science, a linear approximative explanatory chain:

T: T k , ..., T i, AT i.

Let also the competing approximative explanation of the same facts be given:

S: S n , ..., S i , AS j .

It invokes another image of the essential structure of the magnitude F , composedof K, qn ,...,q j+1 ,...,q i. Assume also that both T and S develop according to therule of delayed falsification and that they have reached their maximallyconcretized statements, T i and S j correspondingly. How can we decide which iscognitively better?

The answer may be given by reference to the following criterion:

(a) Take an arbitrary object a from the range of the investigated magnitude F and measure the intensity F (a). That explanation which gives smaller discrepancywith F (a) is cognitively better for the object a; for example if ( = * F (a ) f ( H (a),

pk (a),..., p i+1(a )]*

.The empirical material is left uncorrected and lacking therefore any control.Example: Marxian theory of reproduction (for a reconstruction cf. Nowak 1980a,

pp. 25-28).

(iv) An intuitive-undeveloped-operationalized-purified-theory : . Example: the theory of cognitive dissonance (for a reconstructioncf. Nowak 1971a, pp.212-15).

(v) An intuitive-developed-operationalized-naive-theory: .

Example: the Darwinian variation of the theory of evolution (for a reconstructioncf. Lastowski and Nowak 1982).

(vi) A formal-developed-speculative-naive-theory: Example:Marxian theory of value (for a reconstruction cf. Nowak 1971a; moresophisticated accounts cf. Balicki 1978 and Hamminga 1990).


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Leszek Nowak 68

(vii) An in tuitive-developed-speculative-naive-theory : .Example: thetheory of human rationality (for a reconstruction cf. Patryas 1979, pp.8-46) or Marxian theory of classes (for a reconstruction cf. Jasinska and Nowak 1976; analternative interpretation Tuchanska 1980, pp.18-21).

(viii) An intuitive-undeveloped-speculative-naive-theory: .Example: Weberian theory of bureaucracy (for a reconstruction cf Nowak 1971a,

pp. 99-100) or Engels' theory of primitive societies (for a reconstruction cf.Burbelka 1980, pp. 29-40).

4. Some other expansions

Let us briefly refer to some other expansions of the idealizational approach toscience.

Interactions. Brzezi ski (1976, summary in 1975) notes that the basic model of the idealizational approach to science works only under the tacitly adopted

assumption that all the factors essential for a given magnitude are inessential for one another. The assumption is, however, evidently false. The author introducedthe notion of interaction, distinguished three types of essential structures (isolated

with no interactions, purely interactive, mixed), generalized the form of idealizational statement, its concretization etc. An alternative approach tointeractions is proposed by Gaul (1985).

Paprzycki and Paprzycka (1992) notice that the initial approach to idealizationis based on the intuition that the more a law is concretized, the better its accuracy,i.e. the level of aggreement of the theoretical data (calculated by the law) withthe empirical values of the given magnitude. Explicating both the notion they

prove that it is in fact the case i.e. the level of accuracy of an idealizationalstatement and of essentiality of a given factor for the corresponding dependentmagnitude from that statement order the set of determinants paralelly only onthe assumption that the considered determinants are strictly independent, andhence there are no interactions among them. This outcome testifies to thesignificance of the problematics of interaction and poses the problem of what, if anything, remains from the initial intuition in case of the pure (or mixed)interactive idealization.

Heterogeneity of factors. Brzezi ski, Burbelka et al. (1976) find another simplification on which the core ideas of the idealizational methodology are

based, viz. that all the factors essential for a given magnitude exert upon the latter a homogeneous influence, i.e. an influence that could be expressed in one and thesame dependency. If the principal factor is heterogeneous in relation to thedetermined magnitude, then instead of one idealizational law the set of k 0 idealizational laws is to be reconstructed. If, additionally, the first of thesecondary factors is heterogeneous as well, then instead of one concretization of each of these laws the set of k 1 concretizations of each of k 0 laws is to be


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established, etc. As a result, instead of simple idealizational theory a tree-idealizational theory is to be built.

Mutual significance . Another simplification tacitly adopted in the initialapproach is, according to Brzezi ski, Burbelka et al . (1976), one postulating thatfactors are one-sidedly essential one for another. It is, however, well-known thatfactors sometimes mutually influence one another. Taking this into account theauthors transform simple linear theories into structures of more or lesscomplicated form depending on whether the principal and/or secondary factorsare mutually essential or not. The general schemes proposed by the authors allowus to explain why empirical theories contain many axioms in their initial models.The answer is: because the set of axioms reveal mutual influences of a given setof factors.

Enlargement of a factor. Zielinska (1976) observes that the standard form of concretization consisting in a passage from the formula f(H) = H 0 to the formula

g ( f(H), h(p) ] = H 1 presupposes that the secondary factor p is an enlargement of H 0. This, in general, need not be the case. F or instance, it is not so in the passagefrom Clapeyron's law to that of van der Waals (Batog 1976, Kuipers 1985,

p.198). The author generalizes the notion of concretization in order to cover allthe possible cases of the kind, including the one created by van der Waals'sconcretization.

5. Idealizational approach is self-applicable

As has been observed (Nowak 1976d), all the extensions of the initialidealizational approach fall under the following model:

a. the inadequacy of the initial approach in the light of facts from a science, or the history of it, is stated;

b. a simplifying assumption which must be adopted, if the initial approach isto hold, is put forward;

c. the initial approach is modified; it is argued that the new version of theidealizational approach meets two conditions: first, it passes into the initial one if the simplifying assumption ( ad b ) is adopted anew, and, second, it allows toexplain what rejects the initial approach ( ad a) .

It is not difficult to see that the procedure consists in application of the rule of dialectical correspondence in methodology. The initial approach is formulated ata certain level of methodological abstraction (Nowak 1980a, p.189). If it turnsout to be inadequate, then the additional simplification is revealed, under whichthe initial approach is still acceptable. But this simplification must be removedand this conception modified. Thus, the new conception dialectically correspondsto the earlier version of the idealizational theory of science. The idealizational


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Leszek Nowak 70

theory of science thus develops in the same manner as according to it sciencedoes. It is self-referential also in its dynamics 27.

(VI)Truth and the Cognitive Progress

1. The Essentialist Concept of Truth

Usually it is shown that the classical definition of truth creates a dichotomousnotion of truth whereas the theory of science, particularly the theory of scientific

progress, requires a comparative one. But this seems to be misleading since itforces us to construct notions of truth starting from the idea of the classicalconception and differing from the latter only in being comparative notions.However, it appears that the classical definition of truth is an entirely poor point

of departure for the theory of cognitive progres

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