Материал: искусственный интеллект

what counts is the local pattern of di erences. Hjelmslev called “relation” the conjunction typical of the textual process, and “correlation” the disjunction typical of the whole system of language. The relations between the signs of a text acquire meaning according to their positions in the system of correlations which constitutes language.

At this point, the di erence in the modes of observational access becomes quite obvious. The observational access to a text is immediate, since it just consists in reading a sequence of signs. But access to the system of language is quite complex: it arises from the analysis of an immense (a priori unlimited) corpus of speeches and texts. Yet both types of access have a predictive value. Understanding of a text makes it possible to foresee to a certain extent what follows it, by constraining the field of future possibilities. As for knowledge of the system of language, it constrains any text and speech to fit with “grammatical” rules.

The reason for the “complementary” nature of synchronic and diachronic approaches to language is easily identifiable from there. The observation of a speech or a text forces us to accept a certain creative freedom, and consequently opens the way to a future destruction of the system that presently constrains it. The observation of the language system, on the other hand, makes it necessary to declare outlaw any deviation from it, and to set strict boundaries of what can be said without a time limit.

The most tempting analogy, although arguably a partial one, is with the third variety of Bohrian complementarity: the complementarity between actuality and potentiality. Here, the actuality is that of the text, while the potentiality is that of the system of the language, capable of generating all the texts that follow its rules and also capable of carrying a sentence of banishment against the texts which deviate from such rules.

5Early Quantitative Applications of Quantum Theory to the Human Sciences

We must now examine the quantitative aspects of the epistemological analogy between quantum theory and the human sciences. Giving this analogy a formal translation is a decisive test for its relevance. The most delicate question for researchers in this field was how to collect in a formalism the constraints of the common epistemological configuration, while leaving aside the peculiarities of the various domains to which it extends. Let us admit, as we have said before, that the quantum formalism translates above all the limits of the activity of objectification. Does this mean that every symbol of the quantum formalism can be related to this epistemological constraint? And should we infer that the quantum theory formulated by physicists from 1925 can be transposed immediately to a number of problems of psychology, sociology, or economics? The answer to these questions is “no”. Indeed, several features of the quantum formalism are derived from specific domains of the physical science, from mechanics to electrodynamics. An exemple is the structure of the Hamiltonian operator in the Schr¨odinger

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equation: its form is identical to that of the Hamiltonian function of classical mechanics, and it unambiguously expresses the connection of quantum formalism with this domain of physics. One must therefore go further up the scale of generality, and identify the epistemological core of quantum formalism after having put aside its “physical” envelope. Where do we find this nucleus: in the probabilistic algorithm of quantum theory, in its Hilbert space structure, or in the underlying structure of “orthocomplemented lattices”, which was extracted by Birkho and Von Neumann and formulated as “quantum logic”? Each of these options has been explored (albeit sporadically) during the twentieth century. I will mention three of them.

According to Jean-Louis Destouches (1909–1980), it is the structure of the probabilistic algorithm that expresses what quantum formalism owes to the epistemological situation confronting microscopic physics [8]. Destouches thus tried to build what he called a “general theory of predictions” capable of providing probabilistic evaluations ; and he identified in it the particular features that make it possible to arrive at quantum or classical versions of this kind of theory. During his research, worked out jointly with Paulette Destouches-F´evrier, he obtained an important result that retrospectively justified his initial program. This result is stated in the following theorem: “If a theory (of predictions) is objectivist, it is in principle deterministic and one can thus define an intrinsic state of the observed system ... Conversely, if a theory (of of predictions) cannot be considered as objectivist, that is to say if it is irreducibly subjectivist, then it is not deterministic in principle; as a consequence, such theory is essentially indeterministic” [9]. In the latter case, the theory is bound to be probabilistic, whereas in the former case the use of probabilities is only due to the ignorance of the intrinsic state of the system. The result is remarkable, but the sentences by which it is expressed involve a vocabulary that may trigger misunderstandings. To begin with, the couple of terms “objectivist-subjectivist” expresses the opposition between an epistemological situation where the work of objectification can be carried out, that is to say, where it leads to objectspecific autonomous determinations, and another situation (typical of quantum physics) where the phenomena are inseparable from the instrumental context that allows them to manifest. This translation of the term “subjectivist”, with its inappropriate connotations, by a more neutral term as “contextualist”, is justified by the definition given by Paulette Destouches-Fevrier herself: “(We call) ‘objectivist theory’ a theory in which measurement results can be considered as intrinsic properties of observed systems, and ‘subjectivist theory’ a theory in which the measurement results cannot be ascribed to the observed system as intrinsic properties, but only to the complex apparatus-system, with no possible analysis that would ascribe part of the result to each one (...)”. Taking into account these definitions, the heart of the theorem can be stated as follows: a theory allowing to predict phenomena indissociable from their mode of access is “essentially indeterministic”. The ineliminable use of probabilities is therefore, according to Jean-Louis Destouches and Paulette Destouches-F´evrier, the generic mark of the epistemological situation of microscopic physics.

That being granted, is there a way to identify, among the features of the probabilistic formalism of quantum theory, something that is specifically expressive of this epistemological situation, after having set aside what belongs to the physical domain to which it applies? Doing that is precisely one of the goals that Destouches set for himself when he developed his general theory of previsions in the late 1930s. This general theory of previsions, he writes, makes it possible to “... separate hypotheses about predictions from the strictly physical assumptions” [7].

As a first step, Destouches defines the initial “prediction element” that is characteristic of a given experiment. A “prediction element” is a mathematical entity that can be used to associate a probability distribution to each measurement that can be made after some given experimental preparation.

The second step consists in calculating the evolution of the prediction element in time. This is done by using a unitary operator, which has the property of ensuring that the sum of probabilities evaluated from the prediction element remains equal to 1 at any time.

The third step consists in making a list of “eigen (or proper)” prediction elements, which provide a probability 1 for one of the values that the selected variable can take, and 0 for all other values of this variable.

At the fourth step, one determines the set of coe cients such that the final prediction element can be written as a linear combination of the proper prediction elements, weighted by these coe cients (we thus generate a vector space of prediction elements that may, if certain additional conditions are met, acquire the structure of a Hilbert space).

At the fifth step, finally, the probability of each value of the measured variable is calculated. This last stage is especially interesting because from it, one may bring out a characteristic imprint, on the form of the probabilistic evaluations, of the epistemological situation of inseparability of the phenomena vis- `a-vis their experimental modes of access. When the predictions concern such inseparable phenomena, one can prove a theorem stating that the probabilities are the square modulus of the previous coe cients. This is the “Born rule”, which generates probability distributions that are isomorphic to the intensities of a wave. Through this theorem demonstrated by Paulette Destouches-F´evrier, Born’s rule and the wave-like e ects typical of quantum mechanics have both been shown to be direct consequences of the limit to objectification that characterizes microscopic physics.

Now, we are certain that there is indeed a feature of the quantum predictive formalism (the Born rule) which directly expresses the epistemological situation of indissociability of phenomena with respect to their modes of access. But what about other features ? What in the structure of the general theory of predictions is still connected to physics? Two things, essentially: (a) the definition of each variable, because it depends on the procedure used for its measurement; and

(b) the structure of the unitary operator that is used to calculate the evolution of the prediction elements, because it expresses the dynamics of the process under consideration. In standard quantum mechanics, this evolution operator

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is inserted in the Schr¨odinger equation ; it involves a Hamiltonian operator derived from classical mechanics or electrodynamics. All the rest of the predictive formalism (including the vector space structure) is much more general than any physical theory. A momentous consequence of this generality was drawn by Destouches in the 1950s: the quantum-like theory of predictions applies to “many other domains” than physics. In particular, it was applied by Destouches to biology and to “questions of econometrics” [7], thus showing its relevance for some human sciences.

Two other authors (Satosi Watanabe and Patrick Heelan) sought the similarity between quantum physics and the human sciences in an even deeper structure, underlying the probabilistic formalism. They found it in the “orthocomplemented lattice algebra”, which replaces in quantum theory the ordinary Boolean algebra of the empirical propositions of classical science. This structure is at once looser and more general than that of Boolean algebras; it can be considered as a non-Boolean network of Boolean subalgebras. As Watanabe pointed out [28], the use of an orthocomplemented lattice algebra instead of a Boolean algebra is a mark of a deep alteration of the epistemological situation. Indeed, the Boolean algebra of empirical propositions is underpinned by a postulate according to which “each predicate corresponds bi-univocally to a defined set of objects that satisfy the predicate”. In other words, Boolean algebras apply to a corpus of propositions which define subsets of objects characterized by the intrinsic possession of a predicate. Things become very di erent when a measurement result can no longer be assigned to an object as its intrinsic attribute. If this is the case, if we must suspend the attribution of predicates to objects, if we cannot even set apart “primary qualities” belonging to objects from “secondary qualities (or predicates)” relating to experimental methods, then the former postulate is no longer valid, and Boolean algebra no longer governs all empirical propositions. What comes in the place of Boolean algebras is a non-distributive orthocomplemented lattice algebra which articulates Boolean subalgebras within a structure that is more universal than the latter.

Watanabe ascribes these results a generality that far exceeds physics alone. In order to test their generality, he applies them to the composite structure of everyday language. This language, he points out, combines e ortlessly mentalist and physicalist elements in the same propositions. The option one adopts regarding the legitimacy or illegitimacy of such a combination partly determines the position one occupies in the debate on the mind-body problem. Considering that the mentalist predicates (about ‘inner’ states) should not be combined in the same sentence with physicalistic predicates (on the states of the body), but that both are legitimate, is to engage on a path that leads to dualism. Giving priority to physicalistic predicates (considering mentalist predicates as redundant) is to engage in the path of reductionism or even eliminativism. It remains to be seen what are the conditions of possibility of the curious association of the two kinds of predicates which is so common in ordinary language. Watanabe begins with emphasizing that this association is by no means obvious. The famous remark made by Gilbert Ryle, according to which the mentalist predi-

cates are dispositional in nature, whereas the standard physicalist predicates are of a categorical nature, makes their juxtaposition in the same sentence almost baroque. This di erence in nature also renders a strict correlation between mentalistic and physicalist predicates implausible: (a) a pure disposition may admit to being conditioned by a fairly wide range of physical states, and (b) information derived from complex mental states are probabilistic, whereas the physicalist propositions are of the assertoric type. In addition, the modes of access to the two types of predicates are profoundly di erent, not to say incompatible. Access to macroscopic physicalist predicates takes place through a single observation or measurement, whereas access to the dispositional traits of mentalist predicates can only be achieved through the study of an open-ended sequence of behaviors. Besides (as Bohr pointed out), access to the entirety of the alleged physical substratum of a dispositional mental predicate would destroy this substratum, and would thus be eminently disturbing for the mental state. The two languages, mentalist and physicalist, thus prove to be mutually exclusive in a sense very similar to that of the conjugated variables of quantum mechanics.

Therefore, if we want to understand how ordinary language is able to combine mentalist and physicalist propositions in one and the same discourse, it is necessary to aknowledge that this language can be underpinned by a nonclassical logic. The latter logic is shown to be isomorphic to quantum logic, i.e. to a non-distributive orthocomplemented lattice. This is enough to see the mind-body problem under a new light. The mind-body problem stems from the wrong attempt to project on a single Boolean logic two classes of propositions and predicates that are mutually exclusive due to the incompatibility of the corresponding modes of access. The mind-body problem is then dissolved when one has accepted to take this duality of modes of access into account (yet without hypostatizing them into property dualism).

Patrick Heelan later extended Watanabe’s cogent analysis to any contextdependent language that one purports to unify through a common metaor trans-contextual language [16]. To illustrate this extension, Heelan applied the former analysis in ethno-linguistics. According to him, a meta-contextual language (namely a language that can be used by speakers of two linguistic subgroups who attempt to communicate with each other) is necessarily underpinned by a logic that is isomorphic to the quantum logic of Birkho and Von Neumann.

6 Epilogue

Perhaps, however, the most interesting lesson that can be drawn from these applications of quantum theory to the human sciences does not concern the latter sciences, but physics itself. In view of the successes obtained in perceptual psychology or in decision theory by applying a protoor quasi-quantum formalism, it becomes di cult to deny the epistemological-reflective meaning of quantum mechanics. For the only plausible common feature of psychology, sociology, economics, decision-making, and quantum mechanics is the type of act of knowledge that these disciplines bring into play. Bohr’s provocative statement