In the period very roughly from the beginnings of modern physics (1905) up to Alan Turing's description of the Turing machine in 1938, one of the focal points of dispute in the theory of knowledge was the foundations of mathematics.

The main players in this struggle are:

Gottlob Frege: the founder of Logicism, the position that the whole of mathematics can be reduced to a set of relations derived one from the other solely by means of logic, without reference to specifically mathematical concepts such as number. Wittgenstein attempted to carry Frege's concepts of mathematics over to the natural language, with predictably inane results. Frege was also the inspiration for Rudolph Carnap and the various schools of Logical Positivism which continued to wrestle with the problems generated by the new physics. Frege took no part in the struggle after 1903, and died in bitterness and isolation in 1925 having failed to complete a system based on his concept without the appearance of contradictions or logical flaws. His project was later continued by Bertrand Russell and Alan Whitehead.