Mathematical Structuralism from the Standpoint of (Modal) Set Theory Khamdamov T.V. Automation Science. A Conceptual View

Mathematical structuralism is characterized by the acceptance of the thesis that the subject matter of mathematics is structures. These structures can be understood in various ways. Structures, whatever they are, are “implemented” by specific collections of objects. One of the ways to classify different variants of mathematical structuralism relates to the possibility of constructing a theory of structures and how such a theory should be constructed. In other words, this method concerns whether it is possible to describe any mathematical structures using some unified mathematical theory, and what kind of mathematical theory it is. From the point of view of the set-theoretic version of mathematical structuralism, mathematics is the science of structures, which can be described using the apparatus of set theory. Specific mathematical theories “grasp” various kinds of set-theoretic structures, and the relation of realization that takes place between these structures and mathematical theories is supposed to be described by model-theoretic means. This version of structuralism presupposes a non-structural understanding of set theory, and the axioms of structure theory have a fixed interpretation. This version of set theory assumes that the entire universe of sets has already been built, that all stages of the cumulative hierarchy have already been realized, which conflicts with the very idea that every stage in the cumulative hierarchy is followed by the next stage. Another consequence of understanding the axioms of set theory as truths with respect to a fixed structure, as well as the fact that set theory as a basis receives a special status and is not interpreted in a structural way, is that such an approach does not imply any kind of set-theoretic pluralism. That is, only one set-theoretic structure is considered the only correct description of the universe of sets, while any alternative options proceeding from other axioms are discarded as false. In order to get rid of the conflict between the fact that the axioms of the theory of structures assume the universe of sets is already built, and all stages of the cumulative hierarchy of sets are already realized, and the principle of indefinite extendability, one can turn to modalities. Due to the fact that the axioms of set theory are understood in a modal way, it is possible to explain in a potentialist way the totality of all sets without the assumption that there is some universe of sets fixed once and for all. Accordingly, this is consistent with the principle of indefinite extendability. Set-theoretic pluralism still turns out to be significantly limited. That is, modal set-theoretic structuralism still does not provide a structuralist explanation for set theory.

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