Intensionality: From Philosophical Logic to Metamathematics
The article is devoted to the study of the status of intensionality in the exact contexts of logical and mathematical theories. The emergence of intensionality in logical and mathematical discourse leads to significant obstacles in its formalization due to the appearance of indirect contexts, the uncertainty of its indication in the theoretical apparatus, as well as the presence of various kinds of difficult-to-account semantic distinctions. The refusal to consider intensionality in logic is connected with Bertrand Russell's criticism of Alexius Meinong's intensionality ontology, and with Willard Van Orman Quine's criticism of the concept of meaning and quantification of modalities. It is shown that this criticism is based on a preference for the theory of indication over the theory of meaning, in terms of the distinction "Bedeutung" and "Sinn" introduced by Gottlob Frege. The extensionality thesis is explicated; by analogy with it the intensionality thesis is constructed. It is shown that complete parallelism is not possible here, and therefore we should proceed from finding cases of extensionality violation. Since the construction of formal logical systems is to a certain extent connected with the programs of the foundations of mathematics, the complex interweaving of philosophical and purely technical questions makes the question of the role of intensionality in mathematics quite confusing. However, there is one clue here: programs in the foundations of mathematics have given rise to metamathematics, which, although it stands alone, is considered a branch of mathematics. It is not by chance that, judging by the problems arising in connection with intensionality, there is a growing suspicion that intensionality can play a significant role in metamathematics. As for the question of the sense in which metamathematics results can be considered mathematical, in terms of the presence of intensional contexts in both disciplines, it is a matter of taste: for example, the autonomy of mathematical knowledge as a result of the desire of mathematicians to eliminate the influence of philosophy that took place in the case of David Hilbert may be worth considering in the context of mathematics. Thus, the rather vague concept of intensionality receives various explications in different contexts, whether it is philosophical logic or metamathematics. In any case, the detection of context intensionality is always associated with a clear narrowing of the research area. It is obvious that the creation of a more general theory of intensionality is possible within a more general framework, in which logic and mathematics must be combined. In this respect, we can hope for the resumption of a logical project, which would be a purely logical consideration made of the natural and the mathematical.