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Math's Bill Dunham Questions the Origins of the Möbius Function

August 16, 2019
Books on the Mobius Function

Visiting Professor and Research Associate Bill Dunham has received the Carl B. Allendoerfer Award from the Mathematical Association of America (MAA) for his article "The Early (and Peculiar) History of the Möbius Function."

While the math concepts discussed in the paper may be beyond the understanding of most lay readers, Dunham's article is a highly accessible piece of historical sleuthing. In the article, Dunham argues that the Möbius function, which is usually traced back to an 1832 paper by its namesake August Ferdinand Möbius, can be found in Leonhard Euler's classic 1748 text, "Introductio in analysin infinitorum."

Dunham credits Collier Library as the source of inspiration for the article. He explained the inspiration in his response to the MAA on receiving the award:

"It is a thrill to receive the Allendoerfer Award for my article on the origins of the Möbius function. Many thanks to the MAA and to those committee members who directed this honor my way.

"Let me share a little story. When Penny [Dunham's spouse] and I retired from Muhlenberg College in 2014, we moved to Bryn Mawr on Philadelphia’s Main Line. This put us near Bryn Mawr College, and we were pleased when their mathematics department gave us an affiliation that let us enjoy a new academic home a few blocks from our real one.

"As a historian, I especially appreciated the College’s excellent science library. There, one might find an old book that once belonged to Charlotte Angas Scott, Bryn Mawr’s first math professor, or a volume bearing the signature of Emmy Noether, the illustrious mathematician who was welcomed by Bryn Mawr after fleeing Nazi Germany in 1933.


Dr. Dunham with Mobius Book

"One day my browsing led me to the collected works of August Ferdinand Möbius. I figured I’d thumb through it to find the famous Möbius function from the theory of numbers. But nothing inside smacked of number theory. It took some time before I spotted a version of the function buried within an 1832 paper on analysis. This suggested that there was more to the topic than meets the eye, a thought reinforced when I found that Leonhard Euler had stumbled upon the same function in his famous "Introductio in analysin infinitorum" from 1748. My attempts to unravel the history of this idea became an article for Mathematics Magazine. And here we are.

"The take-away from my little tale: grazing through a great library can have unexpected rewards."

Dunham is a historian of mathematics who has written/edited six books on the subject, including Euler: The Master of Us All (MAA, 1999) and The Calculus Gallery (Princeton, 2005). Since retiring from Muhlenberg College in 2014, he has held visiting positions at Princeton, Penn, Cornell, Harvard, and at Bryn Mawr College.

Established in 1976, the Carl B. Allendoerfer award is made to authors of expository articles published in Mathematics Magazine. Carl B. Allendoerfer was a 1932 graduate of Haverford College known for his work in topology. Allendoerfer began his teaching career at Haverford and then went on to teach at the University of Washington. He was president of the Mathematical Association of America from 1959-60.

Department of Mathematics