The second move of the mathematical fictionalist strategy is to provide such nominalistic premises. Field has done that in one important case: Newtonian gravitational theory. He elaborates on a work that has a respectable tradition: Hilbert's axiomatization of geometry (Hilbert 1971). What Hilbert provided was a synthetic formulation of geometry, which dispenses with metric concepts, and therefore does not include any quantification over real numbers. His axiomatization was based on concepts such as point, betweenness, and congruence. Intuitively speaking, we say that a point y is between the points x and z if y is a point in the line-segment whose endpoints are x and z. Also intuitively, we say that the line-segment xy is congruent to the line-segment zw if the distance from the point x to the point y is the same as that from the point z to w. After studying the formal properties of the resulting system, Hilbert proved a representation theorem. He showed, in a stronger mathematical theory, that given a model of the axiom system for space he had put forward, there is a function d from pairs of points onto non-negative real numbers such that the following ‘homomorphism conditions’ are met: