The assessment of a model depends not only on whether it fulfills certain theoretical and empirical requirements but also on how these requirements are fulfilled. That is to say, models are also assessed on the basis of how they have been constructed, whether it is done in a rigorous way or not. What is taken to be rigorous depends on the underlying assumption of what a model actually is: whether it is seen as an instrument or as a formal concept. Although one usually thinks of an instrument as being a physical device such as, for example, a thermometer or a ruler, in economics models are non-material. However, despite their non-materiality they function as empirical objects, see Chapter 1. This ‘instrumental’ view contrasts with the more received formalistic account of mathematical objects, which take an ‘axiomatic’ view in that they consider models to be formal axiomatic abstractions.1 Both the ‘instrumental’ and the ‘axiomatic’ views will be explored on the basis of specific developments in the history of index number theory. The two quotations above, by Hilbert and Bracey, reflect the difference between the Axiomatic Index Theory and Irving Fisher’s work on index numbers and represent the two respective ‘axiomatic’ and ‘instrumental’ approaches. While in one approach the existence of inconsistencies is a capital sin, in the other approach it is accepted that the best instrument sometimes has to be a compromise between incompatible requirements.