Limitations and the World Beyond

Abstract

The present topic is that of limits—the metamathematical limits of axiomatic systems, epistemic limits of explanation, linguistic limitations of expression, conceptual limits of conceivability, and, ultimately, questions of ontological and metaphysical limits as well. The limitations of axiomatic demonstration and of mechanical computation are clear from the Turing and Gödelian traditions. Section 16.2 pursues extensions and analogies to limitations intrinsic in the structure of explanation, restrictive on a Principle of Sufficient Reason (PSR). In Sect. 16.3 we consider the limitations on expression entailed by recursive linguistic structure, extending the argument from single languages to sets of possible languages and showing that even the properties of languages inevitably outstrip the properties expressible within those languages. Section 16.4 considers epistemic implications, extending the discussion beyond language to incompleteness of any body of conceivable truths in the face of a demonstrably larger realm of fact. Notwithstanding this amorphousness, something can nevertheless be shown regarding its general character that merits further consideration.