Categorical Ontology I - Existence

The present paper approaches ontology and meta-ontology through Mathematics, and more precisely through category theory. We exploit the theory of elementary toposes to claim that a satisfying ``theory of existence'', and more at large ontology itself, can both be obtained by means of category theory. For us, an ontology is a mathematical object: it is a category $E$, the universe of discourse in which our Mathematics (intended at large, as a theory of knowledge) can be deployed. The internal language that all categories possess, in the particular case of an elementary topos, is induced by the presence of an object $\Omega_E$ parametrizing the truth values of the internal propositional calculus; such pair $(E,\Omega_E)$ prescribes the modes of existence for the objects of a fixed ontology/category. This approach resembles, but is more general than, the one leading to fuzzy logics, as most choices of $E$ and thus of $\Omega_E$ yield nonclassical, many-valued logics. Framed this way, ontology suddenly becomes more mathematical: a solid corpus of techniques can be used to backup philosophical intuition with a useful, modular language, suitable for a practical foundation. As both a test-bench for our theory, and a literary divertissement, we propose a possible category-theoretic solution of the famous Tlön's ``nine copper coins'' paradox, and of other seemingly paradoxical construction in Jorge Luis Borges' literary work. We conclude with some vistas on the most promising applications of our future work

Categorical Ontology I - Existence

The present paper is the first piece of a series whose aim is to develop an approach to ontology and metaontology through category theory. We exploit the theory of elementary toposes to claim that a satisfying ``theory of existence'', and more at large ontology itself, can both be obtained through category theory. In this perspective, an ontology is a mathematical object: it is a category, the universe of discourse in which our mathematics (intended at large, as a theory of knowledge) can be deployed. The internal language that all categories possess prescribes the modes of existence for the objects of a fixed ontology/category. This approach resembles, but is more general than, fuzzy logics, as most choices of \clE and thus of \Omega_\clE yield nonclassical, many-valued logics. Framed this way, ontology suddenly becomes more mathematical: a solid corpus of techniques can be used to backup philosophical intuition with a useful, modular language, suitable for a practical foundation. As both a test-bench for our theory, and a literary divertissement, we propose a possible category-theoretic solution of Borges' famous paradoxes of Tlön's ``nine copper coins'', and of other seemingly paradoxical construction in his literary work. We then delve into the topic with some vistas on our future works

Categorical Ontology I - Existence

The present paper approaches ontology and metaontology through mathematics, and more precisely through category theory. We exploit the theory of elementary toposes to claim that a satisfying “theory of existence”, and more at large ontology itself, can both be obtained through category theory. In this perspective, an ontology is a mathematical object: it is a category, the universe of discourse in which our mathematics (intended at large, as a theory of knowledge) can be deployed. The internal language that all categories possess prescribes the modes of existence for the objects of a fixed ontology/category. This approach resembles, but is more general than, fuzzy logics, as most choices of E and thus of Ω E yield nonclassical, many-valued logics. Framed this way, ontology suddenly becomes more mathematical: a solid corpus of tech- niques can be used to backup philosophical intuition with a useful, modular language, suitable for a practical foundation. As both a test-bench for our theory, and a literary divertissement, we propose a possible category-theoretic solution of Borges’ famous paradoxes of Tlön’s “nine copper coins”, and of other seemingly paradoxical construction in his literary work. We then delve into the topic with some vistas on our future works

Categorical Ontology I - Existence

The present paper is the first piece of a series whose aim is to develop an approach to ontology and metaontology through category theory. We exploit the theory of elementary toposes to claim that a satisfying ``theory of existence'', and more at large ontology itself, can both be obtained through category theory. In this perspective, an ontology is a mathematical object: it is a category, the universe of discourse in which our mathematics (intended at large, as a theory of knowledge) can be deployed. The internal language that all categories possess prescribes the modes of existence for the objects of a fixed ontology/category. This approach resembles, but is more general than, fuzzy logics, as most choices of \clE and thus of \Omega_\clE yield nonclassical, many-valued logics. Framed this way, ontology suddenly becomes more mathematical: a solid corpus of techniques can be used to backup philosophical intuition with a useful, modular language, suitable for a practical foundation. As both a test-bench for our theory, and a literary divertissement, we propose a possible category-theoretic solution of Borges' famous paradoxes of Tlön's ``nine copper coins'', and of other seemingly paradoxical construction in his literary work. We then delve into the topic with some vistas on our future works

Categorical Ontology I - Existence

The present paper approaches ontology and metaontology through mathematics, and more precisely through category theory. We exploit the theory of elementary toposes to claim that a satisfying “theory of existence”, and more at large ontology itself, can both be obtained through category theory. In this perspective, an ontology is a mathematical object: it is a category, the universe of discourse in which our mathematics (intended at large, as a theory of knowledge) can be deployed. The internal language that all categories possess prescribes the modes of existence for the objects of a fixed ontology/category. This approach resembles, but is more general than, fuzzy logics, as most choices of E and thus of Ω E yield nonclassical, many-valued logics. Framed this way, ontology suddenly becomes more mathematical: a solid corpus of tech- niques can be used to backup philosophical intuition with a useful, modular language, suitable for a practical foundation. As both a test-bench for our theory, and a literary divertissement, we propose a possible category-theoretic solution of Borges’ famous paradoxes of Tlön’s “nine copper coins”, and of other seemingly paradoxical construction in his literary work. We then delve into the topic with some vistas on our future works

Categorical Ontology I - Existence

The present paper is the first piece of a series whose aim is to develop an approach to ontology and metaontology through category theory. We exploit the theory of elementary toposes to claim that a satisfying ``theory of existence'', and more at large ontology itself, can both be obtained through category theory. In this perspective, an ontology is a mathematical object: it is a category, the universe of discourse in which our mathematics (intended at large, as a theory of knowledge) can be deployed. The internal language that all categories possess prescribes the modes of existence for the objects of a fixed ontology/category. This approach resembles, but is more general than, fuzzy logics, as most choices of \clE and thus of \Omega_\clE yield nonclassical, many-valued logics. Framed this way, ontology suddenly becomes more mathematical: a solid corpus of techniques can be used to backup philosophical intuition with a useful, modular language, suitable for a practical foundation. As both a test-bench for our theory, and a literary divertissement, we propose a possible category-theoretic solution of Borges' famous paradoxes of Tlön's ``nine copper coins'', and of other seemingly paradoxical construction in his literary work. We then delve into the topic with some vistas on our future works

Categorical Ontology I - Existence

The present paper is the first piece of a series whose aim is to develop an approach to ontology and metaontology through category theory. We exploit the theory of elementary toposes to claim that a satisfying ``theory of existence'', and more at large ontology itself, can both be obtained through category theory. In this perspective, an ontology is a mathematical object: it is a category, the universe of discourse in which our mathematics (intended at large, as a theory of knowledge) can be deployed. The internal language that all categories possess prescribes the modes of existence for the objects of a fixed ontology/category. This approach resembles, but is more general than, fuzzy logics, as most choices of \clE and thus of \Omega_\clE yield nonclassical, many-valued logics. Framed this way, ontology suddenly becomes more mathematical: a solid corpus of techniques can be used to backup philosophical intuition with a useful, modular language, suitable for a practical foundation. As both a test-bench for our theory, and a literary divertissement, we propose a possible category-theoretic solution of Borges' famous paradoxes of Tlön's ``nine copper coins'', and of other seemingly paradoxical construction in his literary work. We then delve into the topic with some vistas on our future works

Functorial Erkennen

We outline a ‘formal theory of scientific theories’ rooted in the theory of profunctors; the category-theoretic asset stresses the fact that the scope of scientific knowledge is to build ‘meaningful connections’ (i.e. well-behaved adjunctions) between a linguistic object (a ‘theoretical category’ T ) and the world W said language ought to describe. Such a world is often unfathomable, and thus we can only resort to a smaller fragment of it in our analysis: this is the ‘observational category’ O ⊆ W. From this we build the category [O op , Set] of all possible displacements of observational terms O. The self-duality of the bicategory of profunctors accounts for the fact that theoretical and observational terms can exchange their rôle without substantial changes in the resulting predictive-descriptive theory; this provides evidence for the idea that their separation is a mere linguistic convention; to every profunctor R linking T and O one can associate an object O ] R T obtained glueing together the two categories and accounting for the mutual relations subsumed by R. Under mild assumptions, such an arrangement of functors, profunctors, and gluings provides a categorical interpretation for the ‘Ramseyfication’ operation, in a very explicit sense: in a scientific theory, if a computation entails a certain behaviour for the system the theory describes, then saturating its theoretical variables with actual observed terms, we obtain the entailment in the world