Carl Gottfried Neumann was the son of the physicist and mineralogist Franz Ernst Neumann (1798-1895) and visited the famous Mathematics and Physical seminar, which was founded in 1834 by his father and Carl Gustav Jacob Jacobi in Königsberg. He studied in Königsberg among others, Otto Hesse, where he received his doctorate in 1856. After his habilitation in 1858 in Halle, he worked first as a lecturer and from 1863 as associate professor. He moved in 1863 to Basel in 1865 to Tübingen in 1868 and received a professorship at the University of Leipzig, where he worked 42 years.
Carl Neumann married on 7 April 1864 in the Berlin St. Matthew Church Mathilde Hermine Elise Klose, daughter of the Privy registrar Heinrich Theodor Klose. The marriage remained childless.
Neumann died on 27 March 1925 and was buried in the New St. John's Cemetery in Leipzig. Neumann's nephew, the mathematician Ernst Richard Neumann was able to improve the results of his uncle to potential theory.
1895 Neumann was appointed a corresponding member of the Bavarian Academy of Sciences. In 1897 he received the Order Pour le Mérite for Sciences and Arts and the Bavarian Maximilian Order for Science and Art. The Faculty of Arts Königsberg honored him in the years 1906, 1916 and 1922. When the University of Leipzig in 1909 celebrated its 500th anniversary, Neumann was appointed as her then- senior to the Privy Council.
Neumann worked on the Dirichlet problem and the Neumann boundary conditions are named after him. It can be considered as one of the founders of the theory of integral equations, the Neumann series is named after him.
His name is connected with the "method of arithmetic means " and the "Theory of Bessel and spherical functions ". In the field of mathematical physics, he wrote articles especially for analytical mechanics and potential theory. He introduced the logarithmic potential. Particularly striking is the very special way of working Neumann, which impresses with clarity and simplicity in the thought processes and ideas. Mathematics and physically he also explained the deviation of the plane of polarized light by electric and magnetic forces. In physics, he kept all his life firmly to the Newtonian action at a distance theory. For electricity and heat distribution in a ring he solved the problems by introducing new coordinates in difficult equations. Furthermore, he explained at the time unknown questions about the steady state temperature in a homogeneous body.
Together with Alfred Clebsch he founded the journal Mathematische Annalen.