Field of Rationality and Category Theory, Michał Heller

The field of rationality idea was put forward by Joseph Życiński as a context in which the questions: "How do mathematical objects exist?" and "Why is mathematics so effective in the physical sciences?" could be better understood. The idea never went beyond its seminal stage. In the present study I try to make it less fuzzy by relating it to the ontologically interpreted category theory, where "ontological interpretation" should be understood "in the sense of Quine". Roughly speaking, the ontology in the sense of Quine does not aspire to establish what does exist, but rather what a given theory or doctrine assumes there exists. To construct such an ontology, one should paraphrase a given doctrine into the first order logical calculus and look for those variables that are bound by the existential quantifiers. Only those entities that correspond to such variables are postulated to exist. However, in principle each topos has its own "internal logic". Consequently, we should apply Quine's program to each such category individually (by employing its own internal logic), and speak about the ontology in the sense of Quine characteristic for a given category in terms of a given logical calculus (if Quine's program is realisable in this logic). However, the Quine ontological program does not directly refer to the internal logic of a given theory, but rather to the logic with the help of which we are interpreting this theory (the external or meta-logic), and this is expected to be the standard classical logic. This is also the case with respect to category theory. When developing this theory, for instance by proving theorems, we are using standard logical laws of inference. This is why we seem entitled to ontologically interpret category theory strictly following Quine's recipe (i.e. with the help of the first order logical calculus), but we should be aware that this could be conditioned by the fact that our brain is a macroscopic object embedded into the world having the ontology characteristic for the category of sets. Having these caveats in mind I sketch "ontological commitments" of category theory and briefly signal some underlying philosophical problems. *** This talk was delivered during the conference ���The Limits of Physics and Cosmology” organized by Copernicus Center for Interdisciplinary Studies. There exist various kinds of limits which are inherent in cosmological research and physics. The most conspicuous one is connected with scientific method itself. It manifests itself during attempts to construct (or discover) suitable mathematical structures to model the physical world. However, this limit may be even more profound. Over the past 300 years we have become accustomed to the mathematical and empirical method which has proven to be uncannily successful in describing the enclosing universe. But is there a fundamental limit to the method itself which is intrinsic to the structure of the world? Another group of limits which are more connected to ourselves – as human beings – are conceptual ones. The two monumental theories of physics of the 20th century – General Relativity and Quantum Mechanics - are incompatible and this inconsistency is not only on the mathematical level, but also on the conceptual. Usually scientists believe that the reconciliation between the relativist and quantum realms is possible. But is there a conceptual limit within ourselves as elements of the world?

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