Interpretability

We can often simulate one model inside another, providing a translation of meaning from the first realm to the second.

In mathematics we can all be polyglots, expressing the same meaning in multiple languages, interpreting one mathematical structure within another. We find a simulation of one mathematical realm inside another, and having done so we are able to express all the mathematical facts about it entirely in terms of the language and structural features of this new host structure. In many cases, we can furthermore translate the new structure back into the original structure. Often these iterated translations proceed with perfect accuracy and no loss of meaning, although in other circumstances the meaning may begin to drift with repeated translations—a mathematical version of the telephone party game.

To interpret one model in another, we should define a simulated domain of suitable objects in the host structure to serve as the individuals of the proxy interpreted model—we allow ourselves to use finite tuples, not just points, and also we allow the convenience of a definable equivalence relation, simulating the interpreted model as a quotient by this congruence. On this new domain, we define analogues of the atomic structure of the interpreted model, always using only the language and structure of the host model. In this way, we produce a translation of the underlying language of the interpreted model into the language of the host.

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This is the first of several posts on interpretability.

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To get clear on the notion, let us explore a series of specific examples of interpretability. Eventually, we shall get clear on the distinctions between mutual interpretability, bi-interpretability, direct interpretations, faithful interpretations, synonymy, definitional equivalence, and more.