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One of the issues at stake in the discussion about the origin and meaning of geometrical axioms was to establish the preconditions for the possibility of spatial measurement. A related issue was to analyze the concept of number to gain insights into its relation to that of magnitude. Despite the traditional definition of arithmetic as the theory of quantities, numbers cannot be identified as magnitudes. Numbers can only represent magnitudes in measurement situations. In order to justify the use of numbers in modeling measurement situations, some conditions are required. The study of these conditions is now known as measurement theory. Helmholtz has been acknowledged as one of the forefathers of measurement theory. However, the connection between Helmholtz's analysis of measurement and his inquiry into the foundations of geometry has not received much attention.This chapter deals with the philosophical aspect of Helmholtz's theory and with the psychological interpretation of Kant's forms of intuition proposed by Helmholtz. The psychological part of Helmholtz's theory of measurement may not overcome compelling objections. Nevertheless, I rely on the reception of Helmholtz's views about measurement in neo-Kantianism to reconsider the transcendental structure of Helmholtz's argument for the applicability of mathematics: additive principles can be established independently of the entities to be measured, although they are necessary for judgments about quantities to be valid.

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