Born in New York City, Finkelstein obtained his Ph.D. in physics at the Massachusetts Institute of Technology in 1953 and taught at Stevens Institute of Technology through 1960, while he also held a Ford Foundation Fellowship at the European Organization for Nuclear Research from 1959 to 1960. From 1964 to 1976, he was professor of physics at Yeshiva University. He became a member of the faculty at Georgia Tech in 1980.