 One wonders how it could happen that for fifty years nobody challenged this nonsense. In fact some people did challenge it, notably the geometer Moritz Pasch who in section 12 of his Vorlesungen über Neuere Geometrie (1882) insisted that geometric axioms tell us nothing whatever about the meanings of ‘point’, ‘line’ etc. Instead, he said, the axioms give us relations between the concepts. If one thinks of a structure as a kind of ordered \(n\)-tuple of sets etc., then a class \(\Mod(T)\) becomes an \(n\)-ary relation, and Pasch’s account agrees with ours. But he was unable to spell out the details, and there is some evidence that his contemporaries (and some more recent commentators) thought he was saying that the axioms may not determine the meanings of ‘point’ and ‘line’, but they do determine those of relational terms such as ‘between’ and ‘incident with’! Frege’s demolition of the implicit definition doctrine was masterly, but it came too late to save Hilbert from saying, at the beginning of his Grundlagen der Geometrie, that his axioms give ‘the exact and mathematically adequate description’ of the relations ‘lie’, ‘between’ and ‘congruent’. Fortunately Hilbert’s mathematics speaks for itself, and one can simply bypass these philosophical faux pas. The model-theoretic account that we now take as a correct description of this line of work seems to have surfaced first in the group around Giuseppe Peano in the 1890s, and it reached the English-speaking world through Bertrand Russell’s Principles of Mathematics in 1903.