How did Milton Babitt teach math at Princeton and research math, without an undergraduate degree in math? Even if Babbitt taught merely elementary calculus, don't universities hire someone with at least an undergraduate degree in math to teach it? 
I quote Schoenblog.com:



  Babbitt came to teach at Princeton in 1938—before there was even a music department. We did not have offices. We lived under very Spartan conditions.” When the U.S. entered World War II in 1941, Babbitt was working toward one of the first graduate degrees offered in music at Princeton. During the second semester of that year, he taught music and math (mostly elementary calculus [I bolded.]). “Everybody suddenly began taking math courses that semester after Pearl Harbor. I was teaching on Saturdays.” The music section disappeared as everyone was either drafted or preparing for technical jobs in the army. Babbitt was sent by the Army to Washington, but soon was sent back to teach, because mathematicians were considered top priority. Babbitt stayed with the math department until the end of the war, when he made the not-so-difficult decision to return to music.



In avouching his 'mathematical research in Washington, D.C', Wikipedia hints that Babbitt was skilled at more than elementary calculus. 



  Babbitt's father was a mathematician, and it was mathematics that Babbitt intended to study when he entered the University of Pennsylvania in 1931. However, he soon left and went to New York University instead, where he studied music with Philip James and Marion Bauer. There he became interested in the music of the composers of the Second Viennese School and went on to write a number of articles on twelve tone music, including the first description of combinatoriality and a serial "time-point" technique. After receiving his bachelor of arts degree from New York University College of Arts &amp; Science in 1935 with Phi Beta Kappa honors, he studied under Roger Sessions, first privately and then later at Princeton University. At the university, he joined the music faculty in 1938 and received one of Princeton's first Master of Fine Arts degrees in 1942 (Barkin &amp; Brody 2001). During the Second World War, Babbitt divided his time between mathematical research in Washington, D.C., and Princeton, where he became a member of the mathematics faculty from 1943 to 1945 (Barkin &amp; Brody 2001).
The first part of the answer is that in the 1940s, requirements for hiring at American universities were clearly different than they are now. American universities at the time had not developed the world-leading position they have now, and hiring decisions may have had something to do with it -- though it also goes the other way around: If you're not at the level of Harvard today, you need not shoot so high in your hiring; you may also not be able to attract the same level of talent.

Secondly, the early 1940s were particularly difficult, with a large number of talented people needed elsewhere in the nation and not available to hire at universities. In addition, many others needed to get an education and so the demand for teachers was particularly high -- formal qualifications be damned.

Thirdly, in reality, universities have always (and continue) hired people who are qualified, but degrees do not equal qualification. It is generally true that qualified math professor applicants will have a PhD today, but that is not universally true and universities can and do hire people who don't have one if a candidate has demonstrated exceptional qualification in other ways. Babbitt may have been able to show that he knows what he was talking about in ways that convinced those who hired him that insisting on some kind of degree or title was pointless.