Vagueness: A Mathematician’s Perspective

I comment on the question of how to deal appropriately with vague information by formal means. I am critical of attempts to endow vague concepts with semantics in analogy to crisp concepts that refer to a mathematical structure. I argue, to the contrary, that vagueness pertains to the variability of the models associated with a particular mode of perception; the crucial point from my perspective is that we may choose between structures of differing granularity. I see the primary challenge as finding a solution to the problem of reasoning simultaneously at different levels of granularity—a problem to which a canonical solution is unlikely to exist. The idea of modelling vague concepts by fuzzy sets is defended, and additionally, two logical calculi of possible practical value are suggested.