Types in programming languages are commonly thought of as a way of preventing certain bad things from happening, such as multiplying a number by a string. But this is only half of the benefit of types: it is what types are against. Types in programming languages are also what enable some good things to happen, such as selecting the right implementation of a heterogeneous operation like comparison or printing based on type information; this is what are types for. This ability is surprisingly powerful, and gives rise to a variety of highly expressive generic programming techniques. Jeremy illustrates with some examples based on the rank-polymorphic array operations introduced in Iverson’s APL: not only does the type information prevent array shape errors, it is what directs the lifting of operations across array dimensions.

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