  Pasch's Lectures on Modern Geometry dealt with projective geometry. The first axiomatization of Euclidean geometry that was up to Pasch's standards—Foundations of Geometry by David Hilbert (b. 1862, d. 1943)—appeared in 1899 and exercised enormous influence on twentieth century mathematics and philosophy. Hilbert invites the reader to consider three arbitrary collections of objects, which he calls ‘points’, ‘straight lines’ and ‘planes’, and five undefined relations between (i) a point and a straight line, (ii) a straight line and a plane, (iii) three points, (iv) two pairs of points (‘segments’) and (v) two equivalence classes of point triples (‘angles’). The conditions prescribed in Hilbert's 20 axioms—including the Axiom of Completeness added in the second edition—are sufficient to characterize the said objects and relations up to isomorphism. Isomorphism—i.e., structural equivalence—can hold, however, between different, intuitively disparate, systems of objects. Hilbert availed himself of this feature of axiomatic theories for studying the independence of some axioms from the rest. To prove it he proposed actual instances (models) of the structure determined by all axioms but one, plus the negation of the omitted one. Frege complained that the geometric axioms retained in these exercises could be applied to Hilbert's far-fetched models only by tampering with the natural meaning of words (cf. Alice's conversation with Humpty Dumpty). Hilbert replied on 29 December 1899: 