BOOK VI - Page 5

Nagel and Feyerabend on Meaning Variance

Semantic incommensurability is a special case of the more general semantic phenomenon that Feyerabend calls “meaning variance”, the phrase that he uses to refer to semantic change.  Accordingly it is instructive to consider firstly Feyerabend’s thesis of meaning variance.  This thesis is argued in his “Explanation, Reduction, and Empiricism” in Minnesota Studies in the Philosophy of Science (1962), where he opposes it to the contrary thesis that he calls meaning invariance, which he finds characteristic of the neopositivist philosophy and specifically of the positivist views of Carl Hempel and Ernest Nagel.  Together with Paul Oppenheim, Carl Hempel set forth the “nomological-deductive” thesis of scientific explanation in “Logic of Explanation” in Philosophy of Science (April, 1948), and a later statement by Hempel is given in chapters five and six of his Philosophy of Natural Science (1966).  Nagel set forth his thesis of reduction of theories in chapter eleven of his Structure of Science (1961).  Hempel and Oppenheim emphasize the logical-deductive nature of scientific explanation of individual events, while Nagel addresses more explicitly the semantical aspect of theoretical explanation and reduction.  Since the semantical aspect is at the center of Feyerabend’s thesis of meaning variance, a brief consideration of Nagel’s discussion of the reduction of theories is in order, to understand what Feyerabend is opposing.  As it happens, Nagel might also be said to have a thesis of meaning variance, but his positivist view of semantical change is not the same as Feyerabend’s.

Initially the logical positivist interest in reduction was part of the Vienna Circle’s Unity of Science program.  When it became evident that this program is unmanageably ambitious, the reductionist program was limited to the characteristically logical positivist problem of relating theoretical terms in theories to an observation-language reduction basis.  This type of reduction is accomplished by what Carnap called “reduction sentences” and by what Hempel called “bridge principles”.  Nagel is in the logical positivist tradition, but his treatment of logical reduction is somewhat less programmatic and more closely related to episodic developments in the history of science.   And he is more interested in those cases in the history of science, in which a relatively autonomous theory is absorbed by or “logically reduced to” some other more inclusive theory, a type of development that he believes is a recurrent feature of the history of modern science.  In this type of episode the set of theoretical statements or experimental laws, as the case may be, that is reduced to another theory is called the “secondary science”, while the theory to which the reduction is effected is called the “primary science”. 

Reductionism is a type of explanation in science, and Nagel explicitly defines it as the explanation of a theory or of a set of experimental laws established in one area of inquiry to a theory formulated in some other domain.  He is principally interested in those types of reduction in which concepts are required for describing phenomena in one area that were not formerly employed in the other area, even when the two areas were described with the same vocabulary.  He refers to this type of reduction as a “heterogeneous” reduction, because it describes a qualitative dissimilarity between the phenomena in the domains of the two theories involved in the reduction.  On the other hand a reduction without different vocabulary and describing a qualitative similarity is what he calls a “homogeneous” reduction.  Nagel finds only the heterogeneous type to be problematic.

Nagel employs a theory of meaning in which a descriptive term may have as many meanings as there are explications, which proliferates equivocations.  He illustrates his thesis in his examination of the heterogeneous reduction of thermodynamics to statistical mechanics and of the semantics of the term “temperature”, as that term’s meaning is affected by the successful reduction.  Even before the reduction is made, there is much to be said about the semantics of the terms involved, because a term such as “temperature” has several meanings resulting from overtly performed instrumental operations.  Nagel exemplifies the multiple meanings of the term “temperature” by noting that a person who understands temperature in terms of an ordinary mercury thermometer would have difficulty understanding what is meant by a temperature of fifteen thousand degrees, if he also knew that no mercury thermometer could be used to measure such an extreme temperature.  But if the person had studied physics, he would discover that the term “temperature” in physics has a broader application from a more embracing set of rules of usage describing other measurement procedures.

Nagel invokes Paul W. Bridgman’s idea of “operational definitions” set forth in the latter’s Logic of Modern Physics (1927), and states that such rules of usage are explications aimed at specifying the meanings of descriptive expressions such as “temperature” in terms of other observable ones, which in any given context must be traced to certain descriptive expressions that are selected to be observable primitive expressions.   It is noteworthy that in Nagel’s theory of semantical specification as in Bridgman’s, each such specification describing an alternative measurement procedure constitutes a cognitively distinct meaning of the observation term.  Yet these multiple meanings are not unrelated, since the diverse measurement procedures will yield the same measurement values where more than one is deemed applicable.  Thus the term is empirically unambiguous while at the same time it is cognitively (i.e., semantically) equivocal. Nagel extends Bridgman’s semantical thesis for observation terms to theoretical terms.  He gives as examples of theoretical explications of “temperature”, the explication in the science of heat with the help of statements describing the Cournot cycle of heat transformation, and therefore in terms of such theoretical primitives as “perfect nonconductors”, “infinite heat reservoirs” and “infinitely slow volume expansions”. 

Nagel emphasizes that while the term “temperature” is explicated in the science of heat in terms of both theoretical and observational primitives, it is not the case that the term understood in the sense of the first explication is cognitively synonymous with “temperature” construed in the sense of the second.   This is one way in which the thesis of multiple meanings serves the logical positivist well: the positivist does not want the meanings of observation terms to be contaminated with the meanings of theoretical terms.  It is therefore important to him that the set of meanings supplied by the various theoretical explications and the set supplied by the observational explications be separate and distinct.  The thesis that multiple explications do not result in cognitive synonymy but rather in empirically unambiguous cognitive equivocation, enables him to say that even when a revolutionary new theory is developed, it will produce a new set of theoretical explications but will not revise the set of observational explications.  In this way there is meaning variance in the theoretical meanings, and yet there is also meaning invariance in the observational meanings.  It is interesting that Nagel’s approach is different from Carnap’s, because the latter distinguishes theoretical terms as having “incomplete” semantics, such that theoretical terms could change their meanings by becoming more complete even in a heterogeneous reduction.  Carnap did not employ any thesis of empirically unambiguous equivocation like Nagel; Nagel is more faithful to Bridgman.

Nagel next considers the formal conditions for a heterogeneous reduction.  In the reduction of thermodynamics to statistical mechanics, the Boyle-Charles law is made a logical consequence of the principles of mechanics, when these principles are supplemented by a hypothesis about the molecular constitution of a gas, a statistical assumption about the motions of molecules, and a postulate connecting the experimental notion of temperature with the mean kinetic energy of the molecules.  Nagel sets forth two formal conditions for the reduction: the condition of connectability and the condition of derivability.

The first condition, connectability, requires that assumptions be introduced which postulate suitable relations between what is signified by a descriptive term (e.g., “temperature”) in the secondary science, and traits represented by theoretical terms already present in the primary science (e.g., the kinetic energy of molecules). This is done by “coordinating definitions” or “correspondence rules”, as Nagel also calls them, which have the same functions as what Carnap called “reduction sentences”, and what Hempel calls “bridge principles”.  By whatever name, these are the sentences that connect theoretical terms occurring in a theory with the observation terms in the empirical statements the theory explains.  Both the primary and secondary theories involved in a reduction are presumed to have whatever coordinating definitions they need before the reduction is effected.

The second condition, derivability, requires that together with the above mentioned assumptions all the laws of the secondary science including those containing the connected terms, must be logically derivable from the theoretical premises in the primary science and their associated “coordinating definitions”.  When both of these conditions are satisfied, the reduction can be effected, and the experimental and theoretical laws of the secondary science are made logical consequences of the theoretical assumptions including the coordinating definitions of the primary science.

After his discussion of the formal conditions, Nagel extends his semantical thesis of multiple meanings to reduction.  After the reduction of thermodynamics to statistical mechanics is accomplished, the term “temperature” can be explicated in terms of the mean kinetic energy of molecules, and it thereby acquires still another meaning.  This is the outcome of satisfying the condition of connectability.  He explicitly denies that the connection made by the assumptions employed in the reduction are “logical” connections between established meanings of expressions, because the assumptions would then assert that there is either a synonymy or a one-way entailment in the relation to a theoretical expression in the primary science.  Nagel maintains that the connecting assumptions are initially conventions that merely assign the additional meaning, and which later become empirical statements, because further development of the theory makes it possible to calculate the temperature of the gas in some indirect fashion from experimental data other than the temperature value obtained by actually measuring the temperature of the gas.  He rejects as “unwitting double talk” the objection to his thesis that the reduction occurs due to a redefinition of the term “temperature”.  He maintains that the term “temperature” cannot be cognitively synonymous with the phrase “mean kinetic energy of molecules”.  He says that the terms in each of the two sciences have meanings unambiguously fixed by codified rules of usage or by established procedures appropriate to each discipline, and that these established meanings are not lost or changed as a result of the reduction.

Feyerabend is critical of the views of Hempel and Nagel, and he takes a fundamentally different view, fundamental because Feyerabend advances his “pragmatic theory of observation” in opposition to the positivist naturalistic view of observation.   This point of departure places Feyerabend in the same company as Einstein, Heisenberg, Popper and Hanson, all of whom reject the positivist separation of theory and observation.  On the positivist view observation statements are the products of natural processes that supply the observation language with its distinctive semantics.  Feyerabend on the other hand affirms an artifactual theory of meaning, when in “Explanation, Reduction, and Empiricism” he bases his pragmatic theory of observation on the distinction between nature and convention.  In his view this distinction implies, contrary to the positivist view, that the observational status of a statement must be separated from its meaning.  Thus Feyerabend says that an observation sentence is distinguished from other sentences of a theory not by its meaning content but by the “cause of its production”, by which he means that its production conforms to certain behavioral patterns. This distinction begs elaboration that Feyerabend does not provide, but his pragmatic theory of observation gives Feyerabend an alternative to any reductionist thesis such as Nagel’s.

Feyerabend maintains that when a transition is made from one theory to another theory of wider scope, which Nagel calls the secondary and primary sciences respectively, what actually happens is semantically much more radical than the incorporation of an unchanged theory into the context of the primary theory, unchanged, that is, with respect to the meanings of the secondary theory’s main descriptive terms as well as to the meanings of the terms of its observation language.  What happens is not a reduction, but the complete replacement of the ontology and perhaps the formalism of the secondary science by the ontology and the formalism of the primary science, and a corresponding change in the meanings of the descriptive elements of the formalism of the secondary theory, providing that these elements of the formalism of the secondary theory are still used.

Feyerabend states that contrary to the positivist reductionist thesis, the replacement affects not only the theoretical terms of the secondary science, but also at least some of the observational terms occurring in its test statements.  He opposes the positivist thesis that a comprehensive theory merely orders facts, and maintains that a general theory has a deeper influence on thinking.   This deeper influence is the semantical influence of the context of the primary theory on the empirical statements and vocabulary of the secondary theory.   The consequence of the distinction between nature and convention, which separates observability and meaning, is what Feyerabend calls the “contextual theory of meaning”.  Other philosophers refer to this idea as relativized semantics.  His theory of meaning description implies a wholistic approach, because he says that the contextual determination of meaning is not confined to a single scientific theory or even to a single language.  Thus the unit of language involved in the test of a specific theory is not just the theory taken together with its own consequences, but rather is a whole class of mutually incompatible and factually adequate theories.  This class is the context by which meanings are to be made clear.

Feyerabend’s rejection of the positivist naturalistic causal theory of meaning and his proposal of his conventionalist contextual theory of meaning, lead him to attack two basic assumptions that he finds in Nagel’s theory of reduction and explanation.  These assumptions are (1) deducibility and (2) meaning invariance.  Meaning variance is one of the reasons that deducibility is impossible, but in addition to meaning variance, there are purely quantitative reasons why deducibility is impossible.  In his treatment Nagel gave the reduction of Galileo’s physics to Newton’s physics as an example of a homogeneous reduction, one in which there is no meaning change resulting from the reduction.  But Feyerabend says that there is a quantitative deviation between the Galilean and the Newtonian physics, an inconsistency due to the fact that one and the same set of observational data is compatible with very different and mutually inconsistent theories. 

This inconsistency that makes deduction logically impossible has two sources.  Firstly universal theories always make claims about phenomena that are beyond those that have actually been observed or that might be available at any particular time; it is this characteristic that makes them universal.  Secondly the truth of any observation statement, such as a statement reporting a measurement reading, can be asserted only within a certain margin of error.  The first reason allows for theories that differ in domains where experimental results are not yet available.  The second reason allows for such dif­ferences even in those domains where observations have been made, provided that the differences are restricted to the margin of error in the observations.

The principal reason that deducibility is impossible in explanation and reduction of general theories is the inconsistency produced by the meaning variance, the semantical change resulting from the change of context.  To illustrate this Feyerabend considers the purported reduction of the Aristotelian theory of motion to Newton’s theory.  He says that in this case Newton’s theory offers the same quantitative measurements as Aristotle’s, so there is no quantitative inconsistency.  The reduction is achieved in the apparently simple manner of equating the concept of impetus in the Aristotelian theory with the concept of momentum in Newton’s theory.  On this approach the procedures and assumptions of Newton’s theory supposedly fix the meanings of the descriptive terms in the impetus theory.  But Feyerabend maintains that the concept of impetus as fixed by the usage established in the Aristotelian theory of motion cannot be defined contextually in a reasonable way in the Newtonian theory, because the Aristotelian usage involves laws that are inconsistent with Newtonian physics.  Thus contrary to Nagel, the concept of impetus is not logically explicable in terms of the theoretical primitives of the primary science in a reduction, even if equating impetus with momentum is proposed as a physical hypothesis instead of an analytical one.  Such a physical hypothesis merely says that wherever momentum is present, then impetus will also be present, and the measurements will be the same in both cases.

Feyerabend also finds meaning variance in the purported reduction of phenomenological thermodynamics to the kinematic theory of gases, the heterogeneous reduction case considered in detail by Nagel. He describes Nagel’s view as a claim that the terms in the statements that have been derived from the kinetic theory with the help of correlating hypotheses will retain the same meanings that they originally had within the phenomenological theory.  And he states that Nagel repeatedly emphasizes that these meanings are each fixed by its own procedures that is by the procedures of the phenomenological theory, whether or not the theory has been or will be reduced to some other discipline.  Thus the term “temperature” as fixed by the established usages of phenomenological thermodynamics, as Nagel says, is such that its application to concrete situations entails the strict nonstatistical law.  Feyerabend states that the kinematic theory does not offer such a concept.  There does not exist any dynamical concept in the phenomenological law, while on the statistical account fluctuations between two levels of temperature is allowed.  He therefore says that the thermodynamic concept and the kinetic statistical concept of temperature are “incommensurable”, and that replacement rather than incorporation or derivation characterizes the transition from a less general theory to a more general one. 

Feyerabend notes that both he and Nagel say that incorporation into the context of the statistical theory changes the meanings of the main descriptive terms of the phenomenological theory, but he adds that this is “double talk” by Nagel, because the law that has been reduced is no longer the same law.  He says Nagel’s view of change of meanings is somehow supposed to leave untouched the meanings of the main descriptive terms of the discipline to be reduced.

There is a sense in which Nagel’s view indeed involves double talk.  This double talk is not an inconsistency in Nagel’s thesis, but rather is a logical consequence of his semantical thesis, the view that the terms in science are equivocal and have multiple meanings.  But Feyerabend prefers to reject any such equivocation that would permit semantical continuity through the reduction.  Instead he prefers to retain the univocity in the terms at any point in time, and to affirm a change from one meaning of a univocal term to another new one, even at the expense of a semantical continuity in the empirical explications.   Consideration of the nature of this semantical discontinuity introduces the roles of inconsistency and especially “incommensurability”.

In his “Explanation, Reduction, and Empiricism” Feyerabend describes two ways in which theories can be related to each other such that meaning variance may occur.  Those two ways are inconsistency and incommensurability.  Given two historically successive theories denoted T and T' respectively, the theory T will differ from the theory T', either (1) if T is inconsistent with T' in the domain of deduced empirical laws where T and T' overlap, or (2) if the set of empirical laws that follow from theory T' are incommensurable with those following from T.  When the relation is that of inconsistency, the two theories are commensurable, which is to say semantically comparable.  Feyerabend references Popper saying that the new and superior theory T' implies laws that are different from and superior to those implied by theory T.  In this case the laws deduced from theory T' correct and replace those deduced from T, just as occurred in the case of Newton’s theory correcting and replacing Kepler’s and Galileo’s laws.

When theories T and T' are incommensurable, however, they do not have any comparable observational consequences.  It is not even possible to say that the empirical laws that are deduced from one are superior or inferior to those that are deduced from the other.  This semantic incommensurability is admitted by Feyerabend’s wholistic pragmatic theory of observation.  On this theory of meaning nature does not determine the content of thought and therefore does not guarantee consistency or even comparability of meaning.  Instead the content of thought is a human artifact not unlike any work of art, and there may result differences between people’s thinking that are so fundamentally different, that they may admit no basis for comparison or common denominator; they may be incommensurable.

In his “On the ‘Meaning’ of Scientific Terms” reprinted in Realism, Rationalism, and Scientific Method, Feyerabend describes a theory and its predecessor to be incommensurable, if prior to the time the theory is proposed, there exists no more general concept having an extension that includes the extensions of the concepts of the two theories.  He considers Einstein’s relativity theory to be incommensurable with Newtonian celestial mechanics, because prior to Einstein the Reimann metric did not include time, and he says that this change in the transition to Einstein’s theory was drastic enough to exclude common elements between the two theories.  He also considers quantum theory to be incommensurable with classical physics, because prior to its advent the conservation laws were not applied to virtual states. 

Later Feyerabend further elaborated on his concept of semantic incommensurability by drawing upon the Sapir-Whorf hypothesis and specifically upon Whorf’s thesis of linguistic relativity.  Both Kuhn and Feyerabend briefly reference Whorf in their works published in the 1960’s, and Feyerabend’s elaboration of his thesis of semantic incommensurability is to be found in his Against Method published in 1975.  But before turning to this work, a summary of the Sapir-Whorf hypothesis is in order.

The Sapir-Whorf Hypothesis

Benjamin Lee Whorf (1897-1941) was a cultural anthropologist and linguist by avocation, who received a BA degree in chemical engineering in 1918, and spent his career with an insurance company eventually becoming Assistant Secretary, an officer of the corporation.  He became interested in linguistics in 1924 and was almost completely self-educated in linguistics except for some nondegree courses that he took from Edward Sapir, a cultural anthropologist and linguist at Yale University.  Sapir encouraged Whorf to study the language of the Hopi American Indians, and he financed Whorf’s field studies.  These studies occasioned Whorf’s formulation of the Sapir-Whorf hypothesis, the thesis of linguistic relativity for which Whorf is now best known.  This thesis is still controversial, and is in conflict with such absolutist views as Chomsky’s thesis of innate linguistic universals.  Whorf wrote many articles, but few of those that he submitted to academic journals were accepted and published in his lifetime in spite of the intrinsic merit of the papers.  A posthumous anthology of his writings, Language, Thought and Reality, was published in 1956 (ed. Carroll, MIT Press).

It may be said that there is an earlier and a later, expression of Whorf’s thesis.  The earlier statement made in the 1930’s is his thesis of “cryptotypes” or “covert categories”, while the more mature statement is the explicit statement of linguistic relativity made in “Science and Linguistics” in 1940.  Whorf exemplifies the idea of the cryptotype with grammatical categories for gender.  Gender may be manifested either by overt or by covert indicators.  They are overtly manifested by morphemes, which are formal markers that occur in such languages as Latin or German.  They are covertly manifested in English by what Whorf calls their “reactance”, their association with definite linguistic configurations such as lexical selection, word order that is also class order, or in general by some kind of patterning.  More precisely: overt categories are those having a formal mark that is present in every sentence containing a member of the category, while covert categories are all others, even those that are marked nonphonetically but occur only in certain types of sentences.  And he defines his idea of reactance as a special type of “rapport”, an idea that is roughly equivalent to the general idea of structure in language. 

Rapport is the linkage between the elements of language that enables these elements to have semantical effect.  It is governed by what Whorf calls “an invisible central exchange”.  This invisible central exchange of linkage bonds is what gives rise to the covert categories, since they are submerged, subtle and elusive meanings corresponding to no actual word, but having a functionally important rôle in the grammar of a language.   Words of a covert category are not distinguished by a formal mark but rather by a semantical class, by an idea that gives the grammatical class its unity, which is manifested by “common reactance”.   Semantically the covert category is what Whorf calls a deep persuasion of a principle behind some phenomenon, like the ideas of inanimation, substance, force, or causation.

The later and more relevant expression is the thesis of linguistic relativity, the thesis that language structure controls thought.  Whorf locates his development of linguistic relativity in the history of cultural anthropology in the lineage of Franz Boas and Edward Sapir.  Boas had shown that a language could be analyzed sui generis, that is without forcing upon the language the categories of the “classical” tradition.  Then in 1921 in his book Language Sapir inaugurated the linguistic approach to thinking, demonstrating the importance of linguistics to cultural anthropology.  According to Whorf comparative linguistics now reveals that the background linguistic system, the grammar of each language, is not merely a sentence-producing instrument for voicing ideas but rather is the shaper of ideas.  And this is the essence of his thesis of linguistic relativity.  The human mind cuts up nature, organizes it into concepts, and ascribes significance, because men are parties to an agreement that holds throughout the speech community, and that is codified in their language.  Not all observers are led by the same physical evidence to the same picture of the universe, unless their linguistic backgrounds are similar or in some way can be “calibrated”.  For Whorf’s term “calibrated” one is tempted to substitute Feyerabend’s term “commensurated”, except that Feyerabend does not believe that semantically incommensurable theories can ever be commensurated.

Whorf further elaborates on his linguistic relativity thesis in his “Language, Mind and Reality” (1942).  In the context of a discussion of the Mantric Art of India he distinguishes two great levels: the realm or level of meaning or lexication, and the higher and controlling level of patterning of sentence structure that guides words which occur at the lexical level and that is more important than words.  Lexication, the partitioning of the whole manifold of experience and the assigning of the parts to words, makes the parts stand out in artificial and semifictitious isolation.  This process of lexication is controlled by the patterning function of sentence structure and thus by the organizing at a higher level, where the combinatory scheme occurs.   These patterns are not individual sentences, but rather are schemes of sentences and designs of sentence structure.  The patterns are manifested by using the mathematical or grammatical formulas into which words, values or quantities may be substituted.  Each language does this partitioning and pat­terning in its own way, and each has its own characteristic form principles, that make consciousness a mere puppet, whose linguistic maneuverings are held in unsensed and unbreakable bonds of pattern. 

These passages suggest similarities between Whorf’s view and Feyerabend’s contextual theory of meaning, save for the fact that Feyerabend does not restrict the term “meaning” to a lexical function.  As it happens, Whorf explicitly states in several of his later articles that his thesis of linguistic relativity applies to empirical science.  He views it as applicable not only because science including mathematics consists of language, but also because an awareness of the effect of language on the foundations of thought will facilitate what he describes as “science’s next great march into the unknown”.  He expresses regret that philosophers and mathematicians do not even have apprenticeship training in linguistics, and he states the opinion that further development in logic will proceed with the investigation of the structures of diverse languages. 

Like later philosophers, Whorf views the various specialized sciences as different languages, because he finds that there exist communication problems among the researchers in the different specialties, just as there are such problems among the speakers of different natural languages.  He maintains that these communication problems do not simply breed confusion about details that the expert translator could resolve.  The problems are much more perplexing, since the language of science is a “sublanguage”, which incorporates certain points of view and certain patterned resistances to widely divergent points of view. 

These resistances not only isolate artificially the particular sciences from one another, but they also operate to restrain the scientific spirit from taking the next great step in its development, a step which entails viewpoints unprecedented in science and involving a complete severance from tradition.  This great episode will unify the diverse sciences, and will be based on the discovery of the aspect of language consisting of patterned relations.  The approach to reality through mathematics as used in science today is merely one special case of this. 

Whorf proposed that there is a premonition in language of an unknown and vaster world, which is quite different from the world as it is currently understood through the structure of the Indo-European languages, which insist on substantives.  The apparent necessity of substances is purely a result of the “Ayrian grammar”.  The logic of Aristotle is provincial, because it is based on the ideology of substantives, while modern physics with its emphasis on fields casts doubt on this ideology. Whorf prognosticates the emergence of a new type of language for science that is even more universal than that presently used, because it will be a transcendental logic of relations of pure patternment.

Whorf’s premonition of an unknown and vaster world was more prescient than he probably knew.  Today he might have referenced the phenomenon of nonlocality, had he known of J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt’s experiment, implementing John Bell’s inequality for the famous EPR thought experiment, to say nothing of string theory.  If there is a language of pure patternment, it is the mathematical statement of the modern quantum theory, which does not translate unambiguously into the substantive language of ordinary discourse.  Even the practice of scientific realism does not conclusively resolve the issue of whether the electron’s wave and particle aspects are instantiated as two aspects of one and the same entity, as Heisenberg maintains, or whether they are instantiated as two separate entities, as Bohm maintains, because mathematics does not contain substantive syntactical categories.  The “individual” in mathematics is the measurement instance and not the substantive entity, and Heisenberg had to conjure a peculiar substantive “entity” he called a “potentia”

If Bohm is correct that the duality issue occurs in what he calls the “informal” language and not in the mathematical formalism, then Hanson’s observation that the mathematical expressions of the wave mechanics and the matrix mechanics can be transformed into one another does not necessarily support his thesis that independence through such transformation implies any semantics or ontology for the Copenhagen duality interpretation, unless perhaps one redefines “entity” in terms of Max Born’s ontological criterion of invariance, i.e., quantities having the same value for any system of reference independently of transforma­tions.  This might be construed as ontological relativity, if one excludes Quine’s requirement that any theory subordinate to our initial “home language” must be interpreted by reference to this home language, which Whorf views as our Indo-European language of substantives.


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