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AHRC Research Grant awarded

AHRC Research Grant: Inexpressibility and reflection in the formal sciences

We are delighted to report that Dr Volker Halbach (CUF Lecturer in Philosophy and Fellow of New College) and Dr Gabriel Uzquiano (CUF Lecturer in Philosophy and Fellow of Pembroke College) have been awarded a research grant by the Arts and Humanities Research Council of more than £730,000 for a major project on Inexpressibility and reflection in the formal sciences.

The three-year project will be devoted to a study of the adequacy of formal systems for describing the subject matters of the formal sciences and for expressing what is implicit in the acceptance of theories in the abstract sciences.

Formal systems designed to capture theories in the abstract sciences such as mathematics, linguistics, computer science and philosophy express only partially what we implicitly endorse when we accept the informal theories that are supposed to be captured by those systems. These formal systems are incomplete with respect to their expressive resources and to their consequences. For instance, it has been argued that under certain general conditions absolute general quantification cannot be expressed in such formal frameworks. Similarly it has been claimed that the open-endedness of certain domains of discourse is not expressible or describable in standard formal systems. Furthermore, the soundness of a theory, which is implicit in the acceptance of a theory, cannot be adequately expressed or proved within the formal systems aimed to capture pre-formal theories of certain abstract subject matters. In particular, in the formal frameworks one cannot even express the claim that all consequences of the theory are true and even approximations to this general claim are not provable in the system.

The project led by Dr Halbach and Dr Uzquiano will investigate how these deficiencies in the expressive and reflective power of formal systems for theories in the abstract sciences can be overcome, and thereby will make explicit what is implicit in the acceptance of the theories. This work will yield new insights into the adequacy of formal systems for capturing theories of abstract objects and therefore also in the applicability and significance of formal deductive systems and in their scope and significance. The results of the project will have ramifications for mathematicians, computer scientists, formal linguists and nearly anyone who employs formal systems in order to express their theoretical commitments.

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