[2][3] His mother Luise Florentine Hagen (born 1800) was the sister-in-law of mathematician Friedrich Wilhelm Bessel.

[2] His doctoral thesis of 1856 was supervised by mathematician Friedrich Julius Richelot and focused on the application of the theory of hyperelliptic integrals to classical mechanics.

[2] Two years later, he wrote his habilitation in the University Halle on the mathematical treatment of the Faraday effect, supervised by mathematician Eduard Heine.

[2] This work earned him the position of lecturer (Privatdozent) and in 1863 was appointed as extraordinary (ausserordentlicher) professor at the University of Halle.

[2] The same year, together with Alfred Clebsch, Neumann founded the mathematical research journal Mathematische Annalen.

[2][3] Wilhelm Eduard Weber described Neumann's professorship at Leipzig as for "higher mechanics, which essentially encompasses mathematical physics," and his lectures did so.

[5] Hermann von Helmholtz criticized Weber electrodynamics, including Neumann's work, for violating of the conservation of energy in the presence of velocity-dependent forces.

[2] Neumann remained critical of the works of Helmholtz and Heinrich Hertz on Maxwell's electrodynamics, but appreciated their action principles.

[2] He also argued that for Newtonian mechanics to make sense there should exist an imovable object in the universe called the body Alpha, from which all speeds can be measure relative to it.

[8] Several objects developed later in mathematics are named after his Neumann problem including the Neumann–Neumann methods and the Neumann–Poincaré operator by Henri Poincaré.