The terms covariant and contravariant arose first in mathematics, or perhaps mathematical physics. Fairly recently they have been adapted to the theory of types for computer languages. I thought I would explain the connection that I see between these two worlds.

### In Math and Physics

The simplest use of these terms in math is to distinguish two ways of defining a vector in terms of coordinates. If I speak of the slope of a plot of land I could as well speak of the gradient of the height of that plot from point to point. A point here belongs to a 2D space with latitude and longitude as coordinates. I could speak of the gradient of any field that ascribes a number to each point in the field. If I deal with weather patterns I will likely think about temperature and pressure gradients. These gradients are vectors, they have a direction and magnitude.

If we compute these gradients in miles then the numbers we get will be in degrees per mile. If we choose kilometers, then the degrees per kilometer will be less than the degrees per mile. The mile is larger than the kilometer and thus leads to larger vector components. There are more degrees per mile than degrees per kilometer. For the same gradient the components are smaller when the units are smaller. As a vector, the gradient is thus deemed to be covariant.

By contrast if we speak of velocity then a vector component is smaller when the unit is bigger. There are fewer miles per hour than kilometers per hour. Velocity is thus deemed contravariant.

Golf scores are contravariant—small wins. High jumping scores are covariant—large wins.

### In Programming Languages

Some modern languages involve a type theory where a particular value may be of several types at once. Usually the types are organized in a tree where one type is above another if all values of the first type are also of the second type. In any case a type may be said to be larger than another when every value of the second is also of the first. As temperature depends on location, so may one type depend on another.

In C I may write typedef struct{zot q; int p;} tx;. If I change zot to a larger type (more values) then the type tx will also be larger with more values. This kind of production of new types from old is covariant.

If I write in C typedef int fun(zot); this means that fun is the type of a function that takes a zot and returns an int. If I modify the definition of zot to include more values, then there will be fewer functions that can cope with the expanded set of possible arguments—i.e fewer functions that conform to the modified type zot. This form of type construction is called contravariant thereby.

Penrose’s “The Road to Reality” says that Lagrangians and Hamiltonians have this duality. I wish I understood this.
Google knows these terms in many more math contexts than in language contexts. One reference to language concepts may be found at OO-Semantics.

I have resurrected a sequence of very interesting email that was once found here, and which was captured thus by the Internet Archive Wayback Machine. It speculates on history of such terms in the mathematical context.