Learning from Repetition

Gerald Sussman teaches a class at MIT called The Structure and Interpretation of Classical Mechanics, where students study classical mechanics with the aid of powerful Scheme notation.

Using Scheme has the benefit of exposing just how impressionistic popular calculus notation is. For example, that partial derivatives operate on function positions is something that emerges straight from the notation – making it much easier to understand the distinct between taking a derivative and evaluating it at a point or on a plane, and clarifying the distinction between a variable and a number.

The crucial ingredient is the compiler which processes the custom notation, an evaluator which is uniform across space and time. Even though the compiler is effectively a black box, the extent to which we are certain of its spatial and temporal invariance and its simplicity (i.e. it is incapable of willfully conspiring against you, or of distinguishing between non-special variable names), it provides for an effective learning environment.

In a way, this is a specific demonstration of a more general principle, that it is only possible to learn about the world to the extent that it repeats. The usual assumptions are those of uniformity over space and time. Such assumptions can turn out to be wrong, of course – the Lucas Critique warns against the use of historical data without properly conditioning the correlations against hidden contexts / structural variables.

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