There is a unique line through 2 points in the plane, and a unique conic through 5. These counts generalize to a count of degree d rational curves in the plane passing through 3d-1 points. Surprisingly, the problem of determining these numbers is connected to mathematical physics, and it was not until the 1990's that it was completely solved. For example, Kontsevich determined them with a celebrated recursive formula. Such formulas are valid when you allow your curves to be defined with complex coefficients.  Some of the solutions may be real, or integral, or defined over Q[i], but the fixed count does not see the difference. Homotopy theory on the other hand, studies continuous deformations of maps. In its modern form, it provides a framework to study shape in many contexts, including the motivic homotopy of algebraic varieties. This talk will introduce some interactions of homotopy theory with the arithmetic of solutions to enumerative problems in geometry. We will use this to completely determine an enriched "count" of degree d rational curves passing through 3d-1 points over an arbitrary field of characteristic not 2 or 3. This enumeration is joint work with Erwan Brugallé and Johannes Rau.