He earned a Bachelor of Science degree in physics from Rensselaer Polytechnic Institute and a PhD in mathematics from Syracuse University.
Studying under Abe Gelbart, Finn completed a thesis titled On some properties of the solution of a class of non-linear partial differential equations.
[2] At the beginning of his career, Finn did research on minimal surfaces and quasiconformal mappings and later in his career on mathematical problems of hydrodynamics, such as mathematically rigorous treatments of capillary action.
For the academic years 1958–1959 and 1965–1966, he held Guggenheim Fellowships.
He died eight days later, in Palo Alto, California, on August 16, 2022.