Fleeting thoughts: Semantics as ordering: Semantics of scientific theories as well-ordering

Briefly: 

Semantics of scientific theories can be formally defined as that semantics developing in time to a logical well-ordering. That development can be qualified as Sisyphus’s for any discovery can return that semantics to a less ordered state. The quantity of information as a measure for ordering can feature the change of the semantics of scientific theories: The quantity of information increases gradually in “normal development”, but sharply decreases in “scientific revolutions”. The change of information in time, in a concrete scientific area, can serve for forecasting general trends, e.g. the probability of a new “scientific revolution” in it.

Preliminary statements:
Any semantics is underlain by a fundamental ontological and semiotic relation between semantic units (signs) whatever be and signified units, or roughly speaking, between “words and things”. If that relation is formalized as some kind of mapping in a rigorous mathematical sense, that mapping implies some corresponding ordering in both semantic and signified units. One can admit that ordering as the essence and objectivity of any semantics.

Furthermore, as the mapping is more unambiguous as semantics is more exact but less universal. One can offer a series of more and more unambiguous mappings converging to an ideal semantics, which will be ultimate and absolutely exact and unambiguous where the corresponding mapping between semantic units and signified is one-to-one. It will be referable only to those signified and to nothing else. Let us designate that ultimate and limit semantics as zero semantics. The one-to-one mapping corresponding to zero semantics implies a well-ordering both of semantic and designated units and therefore a series of consequent deductions starting from an initial element corresponding to the fundamental principles, postulates or axioms in the semantics of real scientific theories.

Thesis: Semantics of scientific theories can be formally defined as developing in time to zero semantics and therefore to a logical well-ordering. However, that development can be qualified as Sisyphus’s for any new discovery can return and sometimes really returns that semantics to a less ordered state. Thus, the quantity of information as a quantitative measure for ordering can feature the change of the semantics of scientific theories: The quantity of information, “I”, increases gradually in the so-called periods of normal development, but sharply decreases in “scientific revolutions”. The change of I in time, i.e. the function I(t) for a concrete scientific area can serve for forecasting the general trends of its future development, e.g. the probability of a new “scientific revolution” in it.

Arguments:

1 Any scientific theory shares the following formal set-theoretical structure: It can be considered as a hypothesis about the definitive property of a set containing potentially infinitely many members, from which are known a finite subset. Those potentially infinite members are the individuals referable to both theory and scientific area, to which the theory is relevant. The hypothetical property, which is properly the theory, is able to generate a theoretical counterpart of any real individual, i.e. member of the set. Thus a one-to-one mapping between the individuals of the “things” relevant to the theory and the “words” generated by it theoretically is necessary for any successful theory. On the contrary: any violation of that one-to-one mapping (for whether theoretical constructs without real counterparts or individuals without theoretical “twins”, or even individuals sharing one and the same theoretical “word” as well as one and the same individual admitting more than one theoretical correspondences) is an argument against the theory.

2 In fact, any argument against a given theory almost does not influence on the quantity of “objective” information for it almost does not change the objective probability for any theoretical image to have a single real counterpart. However, it is able to change crucially the subjective probability of that and thus the corresponding “subjective” information. Consequently, one needs that quantity, which is invariant to the transition from objective facts to subjective ones (expectations), for the estimation of

relevance of any scientific theory. Just probability and information (entropy) possess that property. Even more, they are the only widely enough utilized quantities sharing that invariance.

3 Given any anomaly (i.e. a violation of the above one-to-one mapping) in any theory. Its meaning for the relevance of the theory grounds on the subjective estimation (expectation) for the anomaly to turn out to be more or less a “rule” in future, i.e. as to the new members of the set of the objects of the theory, which are forthcoming to be investigated and become known. Thus a single anomaly is able to refute a theory while a series of anomalies cannot influence at all the confidence in it for the subjective probability and information of the former can be much higher than that of the latter.

4 Any “revolution” is conditioned much more by the expected ability of the corresponding theory to generate theoretical counterparts, i.e. relevant and unambiguous “words” for the investigated “things” (or “facts”) in future than to the demonstrated already ability for that theorization in the past. Nevertheless, the quantity of information is able to reflect that crisis in the confidence to any theory.

5 On the contrary: the periods of “normal” development almost do not change the subjective probability and information featuring the theory at issue. However, the objective probability and information increase gradually and smooth after extending more and more the set of known individuals relevant to the theory successfully isolating appearing anomalies (if any).

6 Thus the quantity of information is able uniformly to characterize both revolution and normal period correspondingly by its subjective and objective component. However, the involvement of both components needs information to be interpreted as a complex function, the mathematical formalism of which is already developed and interpreted in the theory of quantum information by dint of the complex Hilbert space. Transferring those interpretations mutually, one can speak equally well of the “wave function” of a theory and of the state of a quantum system as a “theory”.

7 Unitary operators can represent “revolutionary” change as a change of the “phase”, i.e. the ratio of the subjective and objective component of information. Self-adjoint operators make sense to normal change as a change of the module, i.e. increasing the range of the theory.

8 Any theory can be considered as a very extended notion. Then the explicitly expressed theory would correspond to the definition of that “notion”, i.e. to its intension, and the set of actual and potential individual relevant to it, to its extension. The “wave function” of a theory is interpretable as both finitely dimensional vector of its intension and subspace of its extension.