He entered military service at the beginning of World War I with the rank of Hauptmann in the 68th (6th Royal Saxon) Field Artillery Regiment.
In November 1923, he and his wife joined the Nazi Party (NSDAP) just before it was outlawed as a result of the Beer Hall Putsch.
He then joined the National Socialist Freedom Party, a Nazi front organization, becoming its Gauleiter in Pomerania on 4 April 1924.
[5] Consequently, Vahlen was placed on indefinite leave on 1 May 1927 and his newly appointed Deputy, Walther von Corswant, was effectively put in charge.
[4] Also in May 1927, Vahlen faced disciplinary actions stemming from an incident a few years earlier when he was Rector at the University of Greifswald.
[6] Upon his dismissal, Friedrich Schmidt-Ott increased the funding Vahlen had been receiving for his work for the German Navy since 1922.
In that year, he became an ordinarius professor of mathematics at the Humboldt University of Berlin, as successor to Richard Edler von Mises,[9] who emigrated from Germany as a result of the Law for the Restoration of the Professional Civil Service, which was in part directed against professors with Jewish ancestry, which von Mises had.
Through a manipulation of the election process by Vahlen and his supporters, he was selected as president of the Prussian Academy of Sciences effective 1 January 1939 and remained in this post until 1 April 1943.
[6][7][13] In April 1944, Vahlen moved to Vienna after his Berlin apartment was destroyed in an air raid and again taught at the Technische Hochschule Wien.
[14] Vahlen gained his doctorate with Beiträge zu einer additiven Zahlentheorie, and continued to specialise in number theory, but later turned to applied mathematics.
His 1902 paper in Mathematische Annalen recounts William Kingdon Clifford's construction of his 2n dimensional algebra with n − 1 anti-commuting square roots of −1.
His motivation was to unify the theory of motions in Euclidean, hyperbolic, and elliptic space, which is obviously in the spirit of Clifford.