| I propose a novel way of viewing the connection between mathematical discourse and the mathematical logician's formalizations of it. We should abandon the idea that formalizations are accurate descriptions of mathematical activity. Instead, logicians are in the business of supplying models in much the same way that a mathematical physicist formulates models of physical phenomena or the hobbyist constructs models of ships.;I first examine problems with the traditional view, and I survey some prior work in the spirit of the logic-as-model, including that of Gerhard Gentzen, Georg Kreisel, and Imre Lakatos. The utility of the present framework is then demonstrated in three case studies.;First, this approach solves a purported problem with precise semantics for vagueness. Precise semantics supposedly err by replacing the defining characteristic of vagueness (imprecision) with precise cut-offs, but on the logic-as-model approach this precision can be seen as an artifact serving to simplify the model but corresponding to nothing actually found in vague languages.;Second, on the logic-as-model view the advantages claimed for branching quantifiers, i.e. providing the expressiveness of second-order languages while avoiding their semantic intractability, are illusory. On the one hand, second-order formulations of mathematical concepts are usually far more natural than branching quantifier formulations; on the other hand, branching quantifiers are no more tractable than second-order languages.;Finally, I examine the claim that Gottfried Leibniz's (seemingly incoherent) infinitesimal calculus is nothing other than Abraham Robinson's non-standard analysis. Although non-standard analysis is a useful model, there are other formalizations of Leibniz's mathematics as fruitful as, yet incompatible with, the non-standard approach, a fact hard to reconcile with the idea that logic provides accurate descriptions.;After the case studies, I examine the objectivity of logic from the logic-as-model framework. This issue is difficult enough on traditional accounts of logic, and would seem to become even more difficult from the present point of view. Nevertheless, a rich picture of the objectivity of logic is worked out with interesting new sorts of indeterminacy arising from the idea that logic provides models and not descriptions. |
