Periodic Reporting for period 1 - C-FORS (Construction in the Formal Sciences)
Berichtszeitraum: 2023-01-01 bis 2025-06-30
Around 1900, set theory—the foundation of mathematics—was threatened by paradox. Russell discovered a set that gives rise to a contradiction: the set of all sets that are not a member of themselves. The solution turned out to be to regard sets as successively formed (or “constructed”) by gathering available objects into a single set. This “process” is iterated a great (in fact, infinite) number of times. By contrast, analogous paradoxes still pose a threat to intensional entities, such as propositions, properties, and various kinds of groups, which are needed in formal semantics and formal ontology, as well as philosophy. Here there is still no agreed-upon solution, a century after set theory was placed on a secure footing.
To remedy this, we extend the “constructional” approach to provide a secure foundation for these intensional entities as well. We use philosophy to make sense of the highly idealized form of “construction” that is invoked, which far outstrips what we humans (or even our computers) can in fact construct. Additionally, we seek inspiration from ideas in constructive mathematics to develop and apply some logical-mathematical tools appropriate for the study of intensional entities. One such idea is so-called predicativity: very roughly, we ban even some mildly circular definitions, which generalize over a domain to which the defined entity would belong. Another such idea is non-instantial generality: having a single generic explanation of why some generalization holds, as opposed to a highly conjunctive explanation that passes through each instance of the generalization. For example, we can explain why every object a has a singleton set {a}, not by considering each and every object, but in terms of a general recipe for constructing singleton sets.
Our aim is to provide consistent foundations for the mentioned intensional entities, user-friendly enough to be helpful in formal semantics, formal ontology, and mathematics. If successful, our work has the potential to do to these disciplines what the constructional approach to sets did to set theory. We aim to provide a foundation for groups (e.g. teams, committees), nominalization (properties derived from adjectives or verbs, such as wisdom or running), and structured propositions.
To remedy this, we extend the “constructional” approach to provide a secure foundation for these intensional entities as well. We use philosophy to make sense of the highly idealized form of “construction” that is invoked, which far outstrips what we humans (or even our computers) can in fact construct. Additionally, we seek inspiration from ideas in constructive mathematics to develop and apply some logical-mathematical tools appropriate for the study of intensional entities. One such idea is so-called predicativity: very roughly, we ban even some mildly circular definitions, which generalize over a domain to which the defined entity would belong. Another such idea is non-instantial generality: having a single generic explanation of why some generalization holds, as opposed to a highly conjunctive explanation that passes through each instance of the generalization. For example, we can explain why every object a has a singleton set {a}, not by considering each and every object, but in terms of a general recipe for constructing singleton sets.
Our aim is to provide consistent foundations for the mentioned intensional entities, user-friendly enough to be helpful in formal semantics, formal ontology, and mathematics. If successful, our work has the potential to do to these disciplines what the constructional approach to sets did to set theory. We aim to provide a foundation for groups (e.g. teams, committees), nominalization (properties derived from adjectives or verbs, such as wisdom or running), and structured propositions.
We have a weekly project research seminar where all central project collaborators gather to present and discuss work in progress. Each semester the seminar has a well-defined theme, to ensure focus and a shared agenda.
In our first 2.5 years, the group members have published 14 scientific articles, organized 8 workshops, and given more than a hundred talks at various academic events.
Our main achievements so far are the following.
1. Clarifying the constructional approach. We explain the idea of construction as a form of definition. Available entities and facts are successively used to provide an exhaustive description of certain new entities, which are thereby “constructed”. The idea, in short, is to fully describe the “new” entities in a way that relies solely on entities already “constructed”. Additionally, we have developed a logical-mathematical framework for constructionalism based on three modules: certain constructors (e.g. set formation or number abstraction), input to the constructors (e.g. some objects or properties), and the global structure of the constructional process (e.g. which constructional possibilities are compatible). Each module can be adapted to suit one’s needs, which makes the framework highly user-friendly.
2. Constructionalist approaches to intensional entities. We develop an idea due to the great French mathematician Poincaré’s: the guardrail against paradox, when constructing intensional collections, is a requirement of definitional stability. We must ensure that, as more and more entities are constructed, a classification that is used in a definition never “changes its mind”. We show this idea of definitional invariance to subsume and generalize the more traditional approach of banning “vicious circles”.
3. Absolute generality. How can we generalize across a totality that is forever “under construction”, such that not all its members are available? We develop an answer based on the mentioned idea of non-instantial generality. We show that this answer validates a logic intermediate between constructive and intuitionistic.
4. Desiderata for foundational theories of three key applications of intensional entities. Intensional entities are applied in formal semantics and formal ontology in connection with nominalization, group formation, and highly structured propositions. Each application faces paradoxes. The project will develop a secure foundation for each application. To ensure that these foundational theories will be useful to practitioners of the relevant fields, we have developed desiderata or design specifications for them, in a highly interdisciplinary manner, working alongside experts in semantics and formal ontology.
In our first 2.5 years, the group members have published 14 scientific articles, organized 8 workshops, and given more than a hundred talks at various academic events.
Our main achievements so far are the following.
1. Clarifying the constructional approach. We explain the idea of construction as a form of definition. Available entities and facts are successively used to provide an exhaustive description of certain new entities, which are thereby “constructed”. The idea, in short, is to fully describe the “new” entities in a way that relies solely on entities already “constructed”. Additionally, we have developed a logical-mathematical framework for constructionalism based on three modules: certain constructors (e.g. set formation or number abstraction), input to the constructors (e.g. some objects or properties), and the global structure of the constructional process (e.g. which constructional possibilities are compatible). Each module can be adapted to suit one’s needs, which makes the framework highly user-friendly.
2. Constructionalist approaches to intensional entities. We develop an idea due to the great French mathematician Poincaré’s: the guardrail against paradox, when constructing intensional collections, is a requirement of definitional stability. We must ensure that, as more and more entities are constructed, a classification that is used in a definition never “changes its mind”. We show this idea of definitional invariance to subsume and generalize the more traditional approach of banning “vicious circles”.
3. Absolute generality. How can we generalize across a totality that is forever “under construction”, such that not all its members are available? We develop an answer based on the mentioned idea of non-instantial generality. We show that this answer validates a logic intermediate between constructive and intuitionistic.
4. Desiderata for foundational theories of three key applications of intensional entities. Intensional entities are applied in formal semantics and formal ontology in connection with nominalization, group formation, and highly structured propositions. Each application faces paradoxes. The project will develop a secure foundation for each application. To ensure that these foundational theories will be useful to practitioners of the relevant fields, we have developed desiderata or design specifications for them, in a highly interdisciplinary manner, working alongside experts in semantics and formal ontology.
The PI has developed a Critical Plural Logic, which describes which objects can coexist at a single stage of a constructional process. We pioneer the use of Critical Plural Logic as a framework for constructionalism. This gives constructionalism a simple modular structure, where the mentioned logic codifies both the many objects that are simultaneously available to serve as input to the constructors and the global structure of the constructional process. This framework is versatile and highly user-friendly; in particular, it avoids the unnecessary clutter of extant approaches.
A second breakthrough is our approach, philosophically as well as mathematically, to abstraction, in the sense of the great German mathematician and logician Frege. For example, by examining what all pairs or all triples have in common, abstraction yields the numbers two or three, respectively. Our approach is based on the simple and natural idea of abstracting on available objects to characterize yet further objects. The approach is technically simple and elegant. Furthermore, it solves the thorny problem of distinguishing consistent from inconsistent forms of abstraction by identifying a vast and natural class of consistent forms, namely, those where the abstraction defines “new” objects entirely in terms of objects that have already been defined.
A second breakthrough is our approach, philosophically as well as mathematically, to abstraction, in the sense of the great German mathematician and logician Frege. For example, by examining what all pairs or all triples have in common, abstraction yields the numbers two or three, respectively. Our approach is based on the simple and natural idea of abstracting on available objects to characterize yet further objects. The approach is technically simple and elegant. Furthermore, it solves the thorny problem of distinguishing consistent from inconsistent forms of abstraction by identifying a vast and natural class of consistent forms, namely, those where the abstraction defines “new” objects entirely in terms of objects that have already been defined.