Frameworks, Models, and Math

One could go mad seeking a vocabulary to speak of vocabulary, a language to speak of language.

A framework is a shared semantic web, a range of possible pragmatic moves to make in language-games played with fellow adherents as well as with proponents of alternatives. In short, it is a language.

A model is a much more regimented language; its moves are fewer but more penetrating, the domain of its meanings narrower but, one hopes, more illuminating if kept within those confines. In economics, all they taught us were models. Even at the not-particularly-mathematical George Mason University Department of Economics, there was no substitute for the value of a formal model. Simplicity in the model was pared with sophistication in collecting the larger menu of models; we were taught to be undoctrinaire about models that might add up to an inconsistent whole, so long as the application brought us closer to truth for the matter at hand. Somehow, though, these models all fit comfortably inside of a utilitarian framework, albeit the more qualitatively sensitive and uncertainty-emphasizing Austrian variety.

Models turn out to have an unusual portability. The economic models, as I mentioned, were clearly utilitarian in design. They existed in a hermeneutic circle with utilitarian frameworks; the model as the part and the framework as the whole. The law of supply and the law of demand, perfect competition or price discrimination, are all models that are seemingly incomprehensible without a foundation of utilitarian assumptions.

And yet, these assumptions can be relaxed. Perhaps not entirely eradicated. But a humanist like Deirdre McCloskey can comfortably turn these models into metaphors and integrate the “P(rudence)” values into a framework which includes “S(acred)” ones.

But such an integration has its costs, or at least its impact. McCloskey is no mystic. Integrating the economists’ prudent models transforms the framework they are integrated into, just as the models themselves must be transformed as they enter into a new hermeneutic circle with a different whole.

Of course, Gadamer emphasized that all understanding is a creative act, and transformative. All fusions of horizons leave both horizons forever changed. The utilitarian is transformed merely in the act of applying his model to a specific case, just as certainly as the humanist is transformed by integrating models of utilitarian origin.

As I thought about these things, my mind wandered to the question of math. Math, like models, is quite regimented. So much so that – again, like models – it requires a less regimented language in order to provide the resources to explain it, discuss what is going on in a given example, and so on. Nevertheless, there are few things more portable than math. Mathematicians do their work in a staggering variety of vernaculars. And their work can be understood without too much explanation by mathematicians who do not share a common tongue.

It is no wonder that math so dazzled the Pythagoreans and Platonists, seeming to transcend the contingencies of language as it does. But it does not truly transcend those contingencies. The positivist dream of a perfectly rational language is long dead; mathematics requires the resources of unregimented, highly contingent language in order to be understood and to be maintained (never mind further developed). Math itself is, as I said, a regimented language. But it is not just another language. There truly is something miraculous about it, and about the portability of models and of meanings, across the creative and transformative gulf of fused horizons.

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