Kolmogorov’s Theorem: From Algebraic Equations and Nomography to Neural Networks

We trace the developments around Hilbert’s thirteenth problem back to questions concerning algebraic equations.Its solution, namely Kolmogorov’s superposition theorem of 1956, is stated in an elaborate form and its relation with neural nets is explained. A detailed proof allows to initiate discussions concerning implementability.We address individuals interested to form an opinion about the hotly debated applicability of the superposition theorem but also the philosophically inclined readers that want to learn the background of a mathematical problem with an eventful history, and who, by studying its proof will get a sense of the difference between construction and existence in mathematics.