Frege and his groups

Frege's decent's dissertation Rechnungsmethoden, die sich auf eine Erweiter ung des Grossenbegriffes grunden (1874) contains indications of a bold atte mpt to extend arithmetic. According to it, arithmetic means the science of magnitude, and magnitude must be understood structurally without intuitive support. The main thing is insight into the formal structure of the operati on of 'addition'. It turns out that a general 'magnitude domain' coincides with a (commutative) group. This is an interesting connection with simultan eous developments in abstract algebra. As his main application, Frege studi es iterations of functions. He does not yet pose the question of existence proofs. Measurement of magnitudes is also connected to numbers, but the dis cussion is here ambiguous in a way which calls for the systematic account o f numbers in Grundgesetze.